Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UFE0047089/00001
## Material Information- Title:
- Environmentally and Socially Responsible Operations
- Creator:
- Turken, Nazli Z
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2014
- Language:
- english
- Physical Description:
- 1 online resource (159 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Business Administration
Information Systems and Operations Management - Committee Chair:
- CARRILLO,JANICE ELLEN
- Committee Co-Chair:
- PAUL,ANAND ABRAHAM
- Committee Members:
- RAJAPAKSHE,THARANGA
GEUNES,JOSEPH PATRICK - Graduation Date:
- 8/9/2014
## Subjects- Subjects / Keywords:
- Adoption rates ( jstor )
Advertising campaigns ( jstor ) Environmental transport ( jstor ) Fees ( jstor ) Fundraising ( jstor ) Nonprofit organizations ( jstor ) Objective functions ( jstor ) Pollutant emissions ( jstor ) Shelters ( jstor ) Transportation ( jstor ) Information Systems and Operations Management -- Dissertations, Academic -- UF operations - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Business Administration thesis, Ph.D.
## Notes- Abstract:
- This dissertation focuses on environmentally and socially responsible operations and is divided into two research substreams: (i) environmentally responsible operations; (ii) socially responsible operations. In the second chapter, we study the effects of environmental regulations on the facility location and capacity acquisition decisions of a company. We extend the traditional facility location and capacity acquisition problems to include regional production and global transportation emissions regulations, and carbon tax. We incorporate civil,criminal penalties as well as clean-up costs and injuctive relief as a consequence for violating the regulations. We utilize a realistic data set from the auto industry gleaned from publicly available resources to estimate the parameters of our model. We perform regression analysis to describe the relationship between production size and emissions. The model we present in chapter two is a nonlinear, nonsmooth maximization problem that is known to be NP-complete. We propose an algorithm to solve this problem by taking advantage of the known discontinuity points. We then explain the strategic supply chain decisions of companies under environmental regulations. In the third chapter, we utilize metrics from queuing theory to explain five possible performance measures we have identified for animal shelters. We use M/G/k/k queues with reneging, and without reneging to represent traditional and adoption guarantee shelters. We, then compare the adoption guarantee and traditional animal shelters in these performance measures. As a final part of our analysis, we provide optimal resource allocation policies that minimize these performance measures along with the costs. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2014.
- Local:
- Adviser: CARRILLO,JANICE ELLEN.
- Local:
- Co-adviser: PAUL,ANAND ABRAHAM.
- Statement of Responsibility:
- by Nazli Z Turken.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Turken, Nazli Z. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 968131638 ( OCLC )
- Classification:
- LD1780 2014 ( lcc )
## UFDC Membership |

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PAGE 1 ENVIRONMENTALLY AND SOCIALLY RESPONSIBLE OPERATIONS By NAZLI TURKEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014 PAGE 2 Â© 2014 Nazli Turken PAGE 3 To my family PAGE 4 4 ACKNOWLEDGMENTS I first would like to thank my supervisor committee chair Dr. Janice Carrillo for her continuous patience, encouragement and support. I would also like to extend my gratitude to my committee member s Dr. Anand Paul, Dr. Tharanga Rajapakshe, and Dr. Joseph G eunes. Special thanks to my family and friends for their continuous love and support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 1.1 Environmental Implications of Strategic Supply Chain Decisions: The Role of Location and Scale ................................ ................................ .......................... 12 1.2 Resource Allocation for Non Profits: A Case of Animal Shelters ...................... 13 1.3 Resource Allocation of Animal Shelters with Capacity Expansion .................... 14 1.4 Overview of the Dissertation ................................ ................................ ............. 15 2 ENVIRONMENTAL IMPLICATIONS OF STRATEGIC SUPPLY CHAINS: THE ROLE OF LOCATION AND SCALE ................................ ................................ ....... 16 2.1 Motivation ................................ ................................ ................................ ......... 16 2.2 Literature Review ................................ ................................ .............................. 20 2.2.1 Facility Location and Capacity Acquisition Literature .............................. 21 2.2.2 Contemporary Environmental Models in Operations Research ............... 21 2.2.3 Economics and Regulatory Limitations ................................ .................... 22 2.2.4 Contribution to the Literature ................................ ................................ ... 23 2.3 The Model ................................ ................................ ................................ ......... 24 2.3.1 EUFLP1: Emissions Tax Regulation ................................ ........................ 25 2.3.2 EUFLP2: Regional Production Emissions Regulation ............................. 27 2.3.3 EUFLP3: Transportation Emissions Regulation ................................ ...... 28 2.4 Analysis ................................ ................................ ................................ ............ 29 2.5 Solution Methodo logy for EUFLP3 ................................ ................................ .... 35 2.6 Realistic Data Set ................................ ................................ ............................. 39 2.7 Results of the Computational Experiments ................................ ....................... 41 2.7.1 The Base Case ................................ ................................ ........................ 41 2.7.2 Fi xed Costs of Capacity Acquisition and Demand ................................ ... 42 ................................ ....... 42 2.7.4 The Effect of Regional Environmental Production Constant ( ) ... 43 2.7.5 The Effect of Regional Production Environmental Penalty ...................... 44 2.7.6 The Effect of Transportation Emissions Constant, Transportation Emissions Limit and Penalty ( , ) ................................ . 45 2.8 Concluding Remarks and Future Directions ................................ ..................... 46 PAGE 6 6 3 RESOURCE ALLOCATION OF NONPROFITS: A CASE OF ANIMAL SHELTERS ................................ ................................ ................................ ............. 55 3.1 Motivation ................................ ................................ ................................ ......... 55 3.2 Literature Review ................................ ................................ .............................. 60 3.3 The Model ................................ ................................ ................................ ......... 62 3.3.1 Adoption Guarantee Animal Shelter: M/G/k/k (No Bulk Arrivals, No Priorities) ................................ ................................ ................................ ....... 64 3.3.2 Traditional Animal Shelter: M/G/k/k No Bulk Arrivals, No Priorities) ....... 64 3.4 Performance Comparison ................................ ................................ ................. 65 3.5 Resource Allocation for Adoption Guarantee Shelters ................................ ...... 70 3.5.1 Mean Demand Rate ................................ ................................ ................ 73 3.5.2 Mean Waiting Time ................................ ................................ .................. 78 3.5.3 Mean Rejection Rate and Mean Adoption Rate ................................ ...... 80 3.5.4 Traffic Intensity ................................ ................................ ........................ 83 3.6 Resource Allocation for Traditional Shelters ................................ ..................... 85 3.7 Concluding Remarks ................................ ................................ ......................... 87 4 RESOURCE ALLOCATION OF ANIMAL SHELTERS WITH CAPACITY EXPANSION ................................ ................................ ................................ ........... 98 4.1 Motivation ................................ ................................ ................................ ......... 98 4.2 Realistic Data ................................ ................................ ................................ .... 99 4.3 Model Description ................................ ................................ ........................... 102 4.3.1 Model for Adoption Guarantee Shelters ................................ ................ 102 4.3.1.1 Numerical Experiments for Adoption Guarantee Shelters ............ 103 ................................ ................................ ...... 104 4.3.1.3 The Effect of Adoption Fees, Capacity, and Advertiseme nts on the Reputation ( , ) ................................ ................................ 104 4.3.1.4 The Effect of Reputation and Fundraising on Donations ( ) 105 4.3.1.5 Objective Function Weights ( ) ................................ ..... 106 4.3.1.6 The Effect of Advertisements or Fundraising on the Mean Demand Rate (G, L) ................................ ................................ ............. 107 4.3.1.7 Per Unit Cost of Operating a Primary Care Area (v) .................... 108 4.3.1.8 Summary of the Results for A doption Guarantee Shelter ............ 108 4.3.2 Model for Traditional Shelters ................................ ................................ 110 ................................ ............ 111 4.3.2.2 The Me an Euthanization Rate Plus Mean Rejection Rate Problem ................................ ................................ ................................ 111 4.4.Concluding Remarks ................................ ................................ ....................... 112 5 CONCLUSIONS ................................ ................................ ................................ ... 120 APPENDIX A EUF LP DATA ................................ ................................ ................................ ........ 123 PAGE 7 7 B RESOURCE ALLOCATION CALCULATIONS ................................ ..................... 125 Proof of Proposition 3: ................................ ................................ .......................... 125 Adoption Guarantee Shelter Solutions ................................ ................................ .. 126 Mean Demand Rate Problem ................................ ................................ ......... 126 Mean Waiting Time Problem ................................ ................................ .......... 128 Mean Rejection Rate Problem ................................ ................................ ........ 130 Traffic Intensity Problem ................................ ................................ ................. 131 Mean Adoption Rate Problem ................................ ................................ ........ 133 Traditional Shelter Solutions ................................ ................................ ................. 133 Mean Demand Rate Problem ................................ ................................ ......... 133 Mean Waiting Time Problem ................................ ................................ .......... 135 Mean Effective Euthanization Rate Plus Mean Rejection Rate ...................... 137 Traffic Intensity ................................ ................................ ............................... 139 Mean Adoption Rate ................................ ................................ ....................... 142 C CHARACTERISTICS OF THE DERIV ATIVES OF THE BLOCKING PROBABILITY ................................ ................................ ................................ ...... 144 D COMPLETE DATA AND RESULTS ................................ ................................ ...... 145 LIST OF REFERENCES ................................ ................................ ............................. 153 BIOGRAPH ICAL SKETCH ................................ ................................ .......................... 159 PAGE 8 8 LIST OF TABLES Table page 2 1 Summary of model parameters ................................ ................................ .......... 50 2 2 Summary of decision variables ................................ ................................ ........... 50 2 3 Model parameter estimates and sources ................................ ............................ 51 2 4 Numerical experiment ranges ................................ ................................ ............ 52 2 5 Performance of the algorithm ................................ ................................ ............. 52 3 1 Notation ................................ ................................ ................................ .............. 91 3 2 Performance measures for adoption guarantee shelters ................................ .... 91 3 3 Performance measures for traditional shelter ................................ .................... 92 3 4 Comparison of Impact Metrics ................................ ................................ ............ 92 3 5 Summary of decision variables ................................ ................................ ........... 92 3 6 Summary of parameters ................................ ................................ ..................... 92 3 7 Summary of adoption guarantee result ................................ ............................... 94 3 7 Continued ................................ ................................ ................................ ........... 95 4 1 Summary of data ................................ ................................ .............................. 116 A 1 Summary of parameter estimates ................................ ................................ ..... 123 D 1 Form 990 data ................................ ................................ ................................ .. 145 D 2 Mean Demand Rate Regression Data ................................ .............................. 145 PAGE 9 9 LIST OF FIGURES Figure page 2 1 Non smooth plant co sts ................................ ................................ ..................... 50 2 2 The effect of gamma on network dispersion ................................ ....................... 53 2 3 The effect of on network dispersion ................................ ...................... 5 3 2 4 The effect of on transportation emissions ................................ ............. 54 3 1 Rehoming process at adoption guarantee and traditional shelters ..................... 91 3 2 The mean demand rate problem solution ................................ ........................... 96 3 3 The mean adoption rate problem solution ( is dominant) ............................... 96 3 4 The mean adoption rate problem solution ( is dominant) ............................... 97 3 5 The optimal with approximation objective function comparison .......................... 97 4 1 The effect of advertisements on reputation vs optimal results .......................... 117 4 3 The effect of the we ight of donations ................................ ................................ 118 4 4 Changes in G and the optimal adoption fees ................................ .................... 118 4 5 The weight of donations vs optimal solutions ................................ ................... 119 A 1 Plant size versus emissions ................................ ................................ .............. 123 PAGE 10 10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ENVIRONMENTALLY AND SOCIALLY RESPONSIBLE OPERATIONS By Nazli Turken August 2014 Chair: Janice Carrillo Major: Business Administration This dissertation focuses on environmentally and socially responsible operations and is divided into t wo research substreams: (i) environmentally responsible operations; (ii) socially responsible operations. In Chapter 2 , we study the effects of environmen tal regulations on the facility location and capacity acquisition decisions of a company. We extend the traditional facility location and capacity acquisition problems to include regional production and global transportation emissions regulations, and carb on tax. We incorporate civil,criminal penalties as well as clean up costs and injuctive relief as a consequence for violating the regulations. We utilize a realistic data set from the auto industry gleaned from publicly available resources to estimate the parameters of our model. We perform regression analysis to describe the relationship between production size and emissions. The model we present in Chapter 2 is a nonlinear, nonsmooth maximization problem that is k nown to be NP complete. We propose an algorithm to solve this problem by taking advantage of the known discontinuity points. We then explain the strategic supply chain decisions of companies under environmental regulations. In Chapter 3 , we utilize metrics from queuing theory to explain five possible performance PAGE 11 11 measures we have identified for animal shelters. We use M/G/k/k queues with reneging, and without reneging to represent traditional and adoption guarantee shelters. We, then comp are the adoption guarantee and traditional animal shelters in these performance measures. We, then provide optimal resource allocation policies that maximize these performance measures along with the costs. In Chapter 4, we include the capacity expansion as a decision variable for our model and perform numerical experiments using realistic data gleaned from publicly available sources. PAGE 12 12 CHAPTER 1 INTRODUCTION 1.1 Environmental Implications of Strategic Supply Chain Decisions: The Role of Location and Scal e In different regions of the world, several environmental regulations are enforced to push companies into compliance, albeit the cost of monitoring does not allow every company to be monitored. Consequently, it is essential for the environmental regulatio ns to be designed in a way to persuade companies to comply willingly. The first step in designing effective environmental regulations requires understanding the behaviors of companies under different environmental regulations. Most of the current literatur e on emissions reduction has been focused on carbon emissions. In Compensation and Liability Act (CERC LA) and the Superfund Amendments and Reauthorization Act for hazardous substances on the facility location and capacity acquisition decisions of a company. We extend the traditional facility location and capacity acquisition problems to include regional pr oduction and global transportation emissions regulations as well as carbon tax. We incorporate civil and federal penalties as well as clean up costs as a consequence for violating the regulations. We utilize a realistic data set from the auto industry g leaned from publicly available resources to estimate the parameters of our model. We perform regression analysis to describe the relationship between production size and emissions. The model we present in Chapter 2 is a nonlinear and nonsmooth maximizatio n problem that is known to be NP complete. We first show results for special cases and then propose an algorithm to efficiently solve this problem by taking advantage of the PAGE 13 13 known discontinuity points. The Pseudo Facility Linear Estimation Algorithm can be used to solve any facility location where more than one fixed cost occurs as long as the user is aware of the discontinuity points. The results of the realistic computational experiments illustrate that with regards to the regional production environmenta l penalty, an increase in the lump sum dollar amount associated with the penalty is much more effective than a decrease in the actual limit of damage tolerated. These results have implications for the policy maker as well. Constantly reducing the environme ntal limits without increasing the penalties does not force the companies to comply with the regulations. On the contrary, choosing intermediate levels of environmental limits allow the companies to take advantage of economies of scale and avoid the risk o f incurring penalties. The companies can avoid the penalties by dispersing their network and creating small to medium sized plants with small regional but large global environmental impact. As an extension to this paper, we try to identify incentives drivi ng small and medium plants to willful improvement. We study the effects of different environmental technologies on the facility location and capacity acquisition decisions. The preliminary results show that a company will invest in a new technology in all locations or none. 1.2 Resource Allocation for Non Profits: A Case of Animal Shelters Weisbrod (1975) defines non profits as extra governmental providers of collective consumption goods. He explains the existence of non profit organizations as the res p onse to the excess de m and from g ov ernme n tal organizations that can only ser v e to the le v el of the median voter . Non profits differ from for profit organizations in numerous ways: the distribution of wealth, measurement for services provided, and proof of completion of services. For profit organizations distribute wealth among shareholders whereas non profits can u se wealth only to invest PAGE 14 14 in resources for the organization. In addition, non profits receive limited funding making resource allocation a key factor in determining the future of the organization. The services a donor expects from a non profit organization include distribution of goods or collective consumption of goods (i.e. education, food distribution or humane housing of animals) albeit there is not a tangible way to provide proof of completed services for some of these measures. These uncertainties and intangibilities make it challenging to analyze the operations of non profits. metrics from queuing theory to explain five possible performance measures we have identified for a nimal shelters. We use M/G /k/k queues with reneging, and without reneging to represent traditional and adoption guarantee shelters. We, then compare the adoption guarantee and traditional animal shelters in these performance measures using real data. Our r esults show that the traditional shelters perform better in mean rejection rate, mean waiting time, and traffic intensity and perform very poorly in effective mean euthanization rates. We perform numerical analysis to observe how sensitive these conclusion s are to the changes in the parameters. We find that mean adoption rate is the parameter with the most impact on the performance measures. Specifically, when the mean adoption rate is above a threshold, the performance of the adoption guarantee shelter app roaches the performance of the traditional shelter. As a final part of our an alysis, we formulate a mathematical model to find the optimal budget allocation policy for adoption guarantee shelters. 1.3 Resource Allocation of Animal Shelters with Capacity E xpansion In addition to allocating resources to improve performance, non profit organizations also aim to extend their service. To achieve this, they must allocate a PAGE 15 15 portion of their resources to capacity expansion. In the case of animal shelters, the orga nization can improve its capacity by purchasing more primary care areas and care givers. However, it is important to identify the right time and amount of capacity to be added to ensure the continuation of services. In Chapter 4, we incorporate capacity a s a decision variable to the model we presented in Chapter 3. The model we introduce is mixed integer, non linear and known to be NP complete. We perform numerical experiments using realistic data to give recommendations on capacity expansion. We collect d ata from Asilomar Accords, Form 990s, Charity Navigator and shelter websites to estimate the parameters. For the parameter values that are not readily available, we perform regression analyses to estimate their values. Our results show a trade off between advertisements and fundraising activities. When the marginal benefit of advertisements is larger than the marginal benefit of fundraising, the animal shelter should invest all resources to advertisements and vice versa. We also identify scenarios where th e organization should consider investing in capacity. 1.4 Overview of the Dissertation In Chapter environme ntally responsible substream of dissertation. Chapter Allocation of Non Chapter 3 focuses on the socially responsible/continuous help component of my research interests. Chapter 4 incorporates capa city decis ions to the non profit resource allocation problem to study the growth of organizations . PAGE 16 16 CHAPTER 2 ENVIRONMENTAL IMPLICATIONS OF STRATEGIC SUPPLY CHAINS: THE ROLE OF LOCATION AND SCALE 2.1 Motivation The manufacturing strategy area traditionally considers choices concerning a their classic article, Wheelright and Hayes (1985) list capacity decisions (such as the amount, timing and type) as well as facilities deci sions (such as size, location and specialization) as two key structural manufacturing decisions which will ultimately drive lant expansion in relation to the scale of the manufacturing plants under consideration. The benefits of economies of scale in creating larger plants are well documented in terms of increased productivity and lower costs. However, reliance on fewer large p lants increases the costs of transportation, and consequently, emissions. To illustrate, General Motors (GM) recently announced expansion and investment in 17 of its manufacturing plants throughout the United States in response to increased demand, (Terlep 2011). Through a joint partnership in China, GM has also invested in plan t expansion and is planning on building more plants due to the automobile industry are typically based on criteria such as increased demand, labor costs, and exchange ra tes, other environmental concerns are becoming increasingly important in these expansion decisions. According to a GM report on sustainability, GM sell and buy where we build. This practice makes commercial sense, not only for our PAGE 17 17 minimize handling dama ge, preserve natural resources, minimize shipping and use less fossil fuel Sustainability Report (2012)). Another key issue concerns the impact of both formal and informal environmental regulati ons on the plant expansion decision. The regulations on greenhouse gases can be divided into two types: carbon tax and cap and trade. In this paper we focus on the carbon tax policy, which involves a penalty for every ton of carbon emitted by the company. These regulations are mostly implemented in the 34 countries that are in the Organization for Economic Cooperation and Development (OECD), which includes some states in the U.S. as well as many European countries. The cap and trade (also known as emission s trading) program was introduced in the U.S by the Acid Rain Program associated with the 1990 Clean Air Act. The cap and trade program is essentially a market based tool to reduce emissions. The firms in this program receive emissions allowances, with a The firms then sell or purchase allowances to meet the overall limit. Several other regions use this policy for emissions reduction, especially the European Union. In addition to greenhouse gas regu lations, a hazardous substance regulation (CERCLA) is implemented in the U.S. The Environmental Protection Agency (EPA) releases a list which includes reportable quantities for several hazardous substances. If a company exceeds this reported quantity, the n the company has to report their emissions annually, and if found to be non compliant the company must recompense for PAGE 18 18 the violation. There are four kinds of penalties in the U.S. including civil, criminal, cleanup and federal facilities penalties. The civ il and criminal penalties create an upper bound on the amount of penalty charged to a company per day per item. The cleanup penalty depends on the environmental damage induced by the emissions. In addition, injunctive relief, a part of the civil penalties, requires any non compliant company to bring their facilities into compliance. In the years from 2007 to 2011, over 170 cases of criminal enforcement activities for the environmental regulations were reported. Among these cases, several companies were su bjected to fines and restitution up to $370M, (EPA NETs 2012). Once a company violates an environmental regulation, there are several factors driving the magnitude of the penalties that it must pay, including the following: degree of willfulness or neglige nce, history of noncompliance, ability to pay, degree of cooperation and other factors that are specific to the case. These fines could have been avoided if the company had considered environmental regulations in advance within their manufacturing network . To illustrate, the company can consider investing in environmental damage abatement technologies or redesigning their network to bring their facilities into compliance. More recently, GM has issued statements confirming an updated policy towards environ mental regulations. with all governmental entities for the development of technically sound and financially ort, 2012). In addition to the production emissions, some companies are reconsidering the environmental effects of their facility network design due to transportation considerations. Ocean Spray recently redesigned their distribution network and opened PAGE 19 19 a n ew plant in Florida reducing their emissions from transportation by 20% (Cheeseman, 2013). In 2008, Unilever created its own internal transport management organization called Ultralogistik with a goal of carbon emissions reductions through network redesign . Specifically, Unilever constructed regional distribution hubs to reduce the total distance travelled in Europe by 175 million kilometers (Unilever.com, 2013). In this paper, we analyze the supply chain design decisions for a multi plant manufacturing network taking into account the environmental impact as well as the transportation and production costs. We study the location and capacity decisions under the lens of contemporary environmental considerations. We explicitly model the trade off between ec onomies of scale with regards to manufacturing in large plants along with the environmental impact due to emissions and regulations. In addition, we identify conditions under which a dispersed manufacturing network is appropriate as opposed to a manufactur ing network with larger centralized plants. In particular, we address the following research questions: What are the trade offs with regards to plant size between economies of scale and environmental implications? How do national and regional environmenta l regulations impact on plant size and location decisions? How should a firm configure its manufacturing network in response to changes in transportation costs, plant size and environmental concerns? Under what circumstances is a dispersed manufacturing plant strategy appropriate? Under what circumstances is a single large centralized plant appropriate? PAGE 20 20 Can solutions for the environmental facility location problem be identified in an economical manner? This paper is organized as follows: Section 2 .2 highl ights the contemporary literature on capacity expansion and facility location problems along with the environmental considerations. In Section 2. 3, we develop a detailed model, which guides a firm with its capacity expansion and scale choices, given carbon emissions taxes, regional regulations on production emissions and transportation emissions regulations. We also incorporate transportation decisions into the model to explicitly capture the trade off between plant size and the dispersion of the supply net work. In Section 2. 4, we provide some analytical results for some special cases. In Section 2. 5, we analyze the model which is, nonlinear integer and discontinuous with concave production costs and introduce an algorithm that solves the problem by taking a dvantage of known discontinuity points. In Section 2. 6, we describe the realistic data set that constitutes the basis of our computational experiments that we report on in Section 2. 7. In Section 2. 8, we offer managerial insights into the results obtained from the computational experiments, and provide future directions for research. 2.2 Literature Review During the course of conducting the study presented in this paper, we utilize three streams of re search, which we briefly discuss . In particular, the lite rature on (i) facility location and capacity acquisition decisions, (ii) the incorporation of environmental issues in mathematical models, and (iii) environmental regulations are all quite relevant to this work. In closing, we position the paper in the con text of these three streams of literature and reiterate our contribution to this literature PAGE 21 21 2.2.1 Facility Location and Capacity Acquisition Literature Strategic supply chain decisions including capacity expansion and facility location problems have been s tudied since the 1960s. Manne (1961) was amongst the first to study capacity expansion with probabilistic growth. One of the most basic representations of the facility location problem is the uncapacitated facility location (UFLP). Efroymson and Ray (1966) analyze the single period plant location problem where the plant costs are piecewise linear concave. They introduce a Branch and Bound method to solve their version of the UFLP. Verter and Dincer (1992) note that single period UFL and capacity acquisitio n decisions (CAP) had typically been handled separately in the literature, and they provide a model integrating the UFL and CAP decisions. Verter and Dincer (1995) base their solution on the Dualoc algorithm by Erlenkotter (1978) along with a progressive l inear approximation of the continuous concave plant costs. In addition, Verter and Dincer (1995) identify the conditional dominance property, whereby each market will be fully served by a dominant facility (full server). For a summary of the related lite rature on facility location problems, see Melo et al. (2009). 2.2.2 Contemporary Environmental Models in Operations Research More recent operations literature calls for research which incorporates (Angell and Klassen, 1999, Dangayach and Deshmukh 2001, and Corbett and Klassen, 2006). Angell and Klassen (1999) identify two different perspectives on how the environment influences operations management via component resources or operating constraints. These authors also PAGE 22 22 (2001) offer a thorough review of the manufacturing strategy lit erature categorizing each paper by content, methodology and outlet. They also identify papers concerning both environmental issues and manufacturing strategy and note that more research is needed which connects these two important areas. Corbett and Klasse n (2006) also link environmental and operations strategy via a resource based view of the firm by Corbett et al. (1995) use mathematical programming to optimally allocate resources for the decontamination of polluted sites by utilizing a quantitative environmental measure previously developed by Jacobse and Wolbert (1988). Kraft et al. (2013) analyze the impact of regulatory uncertainty concerning certain potentially (1992) develop a model which is equivalent to a nonlinear program in which they represent the environmental damage with carbon emissions. Klassen and Vachon (2003) utilize a survey data from several Canadian manufacturing plants and find that supply chain collaboration significantly affects the level and form of investment in environmental technologies. Diabat and Sim chi Levi (2009) utilize mixed integer programming (MIP) to study the supply chain network problem with a carbon emissions tax. 2.2.3 Economics and Regulatory Limitations While contemporary considerations of carbon emissions seem to dominate the popular p resources have existed for decades (Pashigian, 1984). A body of literature within the PAGE 23 23 manuf acturing plants. To illustrate, Snir (2001) focuses on product stewardship, which used to ensure that the company is in compliance with the regulations and to identify the liabilities the company will face in case of noncompliance. A body of prior literature points to the limitations of such regulatory environments on total manufacturing c apacity at the aggregate level (Gray and Shadbegian, 1993), while some of the literature finds little or no significant negative impact on these capacity investments (Shadbegian and Gray, 2005). Furthermore, such regulations can occur at the federal, stat e or community level. More recently, Chen and Monahan (2010) highlight the role of informal or voluntary regulations within certain communities and firms in moderating environmental impact associated with classic operations decisions. 2.2.4 Contribution t o the Literature totaled over $125 billion per year, a level that represented more than 2% gross national hat the effects of pollution control were in both political and economic in nature. Thus, it is vital for a firm to incorporate environmental issues into its strategic decision making concerning its supply chain. We extend the traditional capacity acquisi tion literature by incorporating the impact of both production emissions limitations and transportation emission constraints imposed by regional governments. The aforementioned model is a discontinuous, non linear integer minimization problem which is kn own to be NP complete. The Pseudo Facility Linear Approximation Algorithm can be applied to any PAGE 24 24 facility location problem with numerous fixed costs as long as the user is aware of the disco ntinuity points. We propose an algorithm that takes advantage of th e known discontinuity points. In addition, we solve limited versions of the model to gain deeper theoretical insights concerning the optimal solutions to the problem. Finally, we utilize information from several auto companies within the U.S. to illustrat e numerical examples of the model distinguishing the circumstances under which a dispersed or centralized plant network is optimal. Our results show the impact of various regulatory schemes (such as target emissions levels and penalties) on the optimal pl ant network. By modeling the effects of regulations on the facility location decisions, we can identify the environmental limits and penalties that will drive the company to compliance. We specifically find that creating stricter regulations without high penalties will not assure compliance. Furthermore, we find scenarios where high polluting companies will be non compliant even if the aforementioned levels of limits and penalties exist. In addition, we identify the magnitude of environmental damage a comp any causes by choosing a dispersed or centralized facility location layout. 2.3 The Model In a traditional facility location and capacity acquisition problem there are fixed and variable costs of building at each location. The fixed location cost allows us to capture the differences in construction costs between countries and regions. The varia ble location cost represents the equipment and other costs related to building a new facility. We model the demand as deterministic assuming that the short term variability in demand is insignificant. The demand at each location differs and can be satisfi ed through production from any of the regions, and a cost is incurred for transporting products between two regions. In addition to the fixed cost of establishing a PAGE 25 25 new facility in any region, we incorporate the capacity acquisition and production cost wit h a general function including concave elements. Specifically, economies of scale in . The notations used are shown in Table 2 1 and Table 2 2 . : In this paper, we consider emissions (carbon) tax, regional production regulations and transportation regulations. We introduce our model in three steps: the first model (EUFLP1) is a UFLP model with emissions tax considered. The second model (EUFLP2) is a nonsmooth, nonlinear integer problem with concave production costs incorporating emissions tax, and regional production emissions limit and penalties. In EUFLP3, we finally incorporate these two elements, and the transportation emissions and penalties int o the model. In Section 2.4 , we show analytical results for some special cases for EUFLP1 and EUFLP2. Our complete model (EUFLP3) is known to be NP complete, thus, in Section 2. 5 we propose an algorithm to solve it. 2.3.1 EUFLP1: Emissions Tax Regulation In our model instead of limiting the tax policy to only carbon emissions we use the term emissions tax to include all hazardous substances from production. We incorporate emissions tax into our model as a part of the variable cost, assuming a linear relationship between production size and emissions ( . To motivate our modeling choice with regards to the relationship between production size and emissions, we utilize linear and exponential regression analysis to support our linearity assu mption. We gathered the production/sales rates from various resources and total hazardous chemical releases from EPA Toxic Release Inventory for 5 different auto manufacturing companies between the years of 2005 2010, and found an R squared PAGE 26 26 value of .763 3 for linear versus an R squared value of 0.5722 for exponential. There are several different hazardous substances that are released to the environment but to evaluate the overall effect we summed up the releases for all substances for each company from al l plant locations in U.S. The R squared values shows that linear is a better fit for the relationship between production and emissions. (Refer to Table A 1 and Figure A 1 in the Appendix for the data and regression graphs.) The objective function of EUFLP1 can be written as follows: Subject to the following constraints: 1. The binary variable indicating an expansion must be equal to 1 if there is any produ ction at location i. 2. The demand at each region is satisfied . 3. The constraints defining the ranges of the variables The first and second terms are the fixed and variable costs of capacity acquisition and emissions tax. The third term is the cost of production with economies of scale. Finally, the fourth term is the total cost of transporting between locations. Constrain t sets (1) and (2) are similar to those typically associated with UFLP problem. Specifically, the demand at each location can be fulfilled from local production or with PAGE 27 27 the excess supply transferred from other regions. Constraint set (3) delineates the fe asible range for the descriptive and binary variables. EUFLP1 is a classical UFLP with emissions tax included in the variable capacity acquisition cost. In the next step, in EUFLP2, we introduce the regional production environmental regulations and penalti es. 2.3.2 EUFLP2: Regional Production Emissions Regulation We introduce environmental constants for production emissions ( ) to reflect the percentage of waste from production activities. Similar to EUFLP1, we assume a linear relationship betwee n production and emissions. These environmental constants will change depending on the industry and the environmental technology adopted by the company. In addition, environmental limits/regulations are typically established at a local level and can vary widely between different regions of the world. These different types of environmental requirements are captured in our model by an environmental constraint. If the production emissions in a region exceed , then the binary indicator for nonco i, equals 1 and the company pays environmental penalties. We allow for both a fixed penalty associated with production emissions as well as a variable penalty based on the magnitude of the violation. The objective function for E UFLP2 with regional production emissions becomes: Subject to constraints (1), (2), (3) and (4). PAGE 28 28 In constraint (4), we introduce the combined term as the pollution residuals: 4. The binary variable indicating a violation of a regional environmental regulation must be equal to 1 when the emissions from production in region i exceeds the limit. The penalties issued also vary between different regions of the world. In our model we use the penalty example utilized by the EPA which includes several facto rs such as civil and criminal monetary penalties as well as a Superfund penalty. The fourth and fifth terms in the previous objective function are the costs of violating the environmental limits and are paid by the company as a lump sum of , and a vari able cost per unit for each region the environmental regulations that is violated. The fixed penalty can be interpreted as the civil or criminal penalty enforced. The variable penalty reflects the clean up costs associated with emissions as well as th e necessary investments to bring the operations to a level that complies with environmental regulations. Production regulations are not the only concern, as 28% of the greenhouse gas emissions in the U.S. result from transportation. As a third step, in EUF LP3, we introduce the transportation emissions regulations and penalty. (EPA.GOV) 2.3.3 EUFLP3 : Transportation Emissions Regulation We capture the trans portation emissions through the parameters . These parameters change depending on the distances between the regions and the available technology in the producing region. We assume that there is a one time penalty for PAGE 29 29 violating a transportation emissions regulation. The objective function and the constraints for EUFLP3 is shown below: Subject to constraints (1), (2), (3), (4) and (5). 5. T he binary variable indicating a violation of a transportation emissions regulation must be equal to 1 when the total emissions from trucks exceed the specified limit. ; 2.4 Analys is In Section 2.4 , we present our analytical findings for the special cases of EUFLP1 and EUFLP2 to develop key concepts associated with the problem. We first highlight the similarities between UFLP and EUFLP1, and utilize an important property of UFLP to gain further insights. Then we show the scenarios where a single noncompliant facility dominates solutions with multiple compliant facilities for EUFLP2. We also analyze the impact of the environmental constraints on the optimal solutions, for the special case where all other cost parameters are similar. Finally, we highlight the situation where an increase in the environmental constant reduces the plant size and increases the number of plants. EUFLP1, the environmental facility location problem with emis sions tax, is a classical UFLP model and the conditional dominance property defined in (Verter and Dincer, 1995) still holds. According to this property, each market will be fully served by a dominant facility (full server) that varies with demand given th at all other parameters PAGE 30 30 remain the same. From this result, the demand at market j will be fully satisfied by the plant location that can provide the lowest cost. The effect of emissions or carbon tax on s demand is similar to the effect of any other linear variable costs on the decision. A region with lower emissions tax is not necessarily dominant over a region with higher emissions tax, as the optimal decision is dependent on the total cost including fi xed and variable costs of capacity, production and transportation costs and the emissions tax. In the case of symmetric plant locations (i.e. all of the costs and distances to market j are the same for each supply location) with location A having a lower e missions tax, market j will be fully served from location A. However, the symmetric locations scenario does not exist in reality and the reason companies move their production to lower emissions (carbon) tax regions is not solely due to low emissions taxes but also the benefits of lower corporate taxes and wages and relaxed regulations emerging in new markets. The lower overall cost is the factor which drives the companies to less regulated regions which is also supported by the conditional dominance proper ty. In addition to the emissions tax, many regions in the world employ command and control type regulations. EUFLP2 incorporates the regional production emissions regulations which create a soft capacitated facility location problem and the conditional dom inance property no holder holds. In a soft capacitated facility location problem, an extra cost is incurred for opening a facility larger than the allowed capacity limit. Similar to the capacitated facility location problem, the optimal solution follows th at of an extreme flow pattern. In the capacitated facility location problem, this extreme flow solution contains at most one flow amount between the upper and lower bounds. Thus, PAGE 31 31 the demand at each market will be served by either: (i) 1 facility that fully satisfies the demand (full server), (ii) Q facilities at their capacity limit sizes that partially satisfy the demand (partial servers), or (iii) Q 1 partial servers and 1 remainder server, where the remainder server is not at its capacity. In the remaind er of Section 2.4 we will abbreviate the compliant and non compliant facilities as CF and NCF and the remainder compliant server as RCF. Furthermore, we define symmetric locations for market j as locations with the same costs, scale parameters, emissions l imits and penalties as well as the same distances to market j. Before we complete our analysis we introduce the following mathematical notations: , = the size of the facility at location i. We choose symmetric location s to illustrate how the results of EUFLP2 and EUFLP3 differ from the traditional UFLP. Proposition 1: For the EUFLP2 problem, in the case of symmetric locations, it is less expensive to open a single non compliant facility than q non compliant and Q q comp liant facilities to serve demand at location j if . compliant full server, , and q noncompliant out of Q total servers, , are given below: We omit the subscripts i,j when unnecessary. PAGE 32 32 In the case of symmetric locations, if the difference in environmental penalties are higher compared to the fixed costs of capacity acquisition and the benefit from economies of scale, a single noncompliant facility is better than q noncompliant facilities . Proposition 2: (a) For the EUFLP2 problem when : In the case of symmetric locations, market j will be served by a non compliant full server instead of Q compliant partial servers if , and by Q compliant partial servers, otherwise. (b) For the EUFLP2 problem when : In the case of symmetric locations, market j will be served by a non compliant full server instead of Q 1 compliant partial servers plus 1 c ompliant remainder server if ) and by Q 1 compliant partial servers plus 1 compliant remainder servers, otherwise. Proof: The cost functions with symmetric locations for Q compliant partial servers are show n below: The cost function for Q 1 compliant partial servers and 1 remainder server is: Let Q* + R= PAGE 33 33 When the fixed costs of building Q 1 extra facilities minus the loss from economies of scale for opening smaller facilities exceed the total environmental penalty paid by the company, it is optimal to have a single full noncompliant server. The company will pay civil/criminal and cleanup penalties and will bring their operations up to compliance. Otherwise, the company will open Q compliant fac ilities and not incur the penalties. A similar scenario holds for RCF. Proposition 3: In the case of symmetric locations with location B having less strict environmental regulations, the optimal solution will include a facility in location B. Proof: We kn ow from proposition 2 that the demand at market j will be satisfied in one of the three forms. Given that the optimal solution is a single facility, the demand location j will be served from the facility with the lowest cost. Let where is the objective function value if the noncompliant facility is opened in location i. All costs other than ar e not affected by , for all i not B. Given the optimal solution is Q partial servers, the optimal solution will be the combination of Q locations that give the minimum cost. Given that all costs are the PAGE 34 34 same we can write for all t not B where i . is the set of Q facility combinations and is a combination containing location B. We can show similarly that Proposition 4: In the case of symmetric locations, decreases with an increase in if . Proof: Let , and Q* = . We can write if , is decreasing in Q. As the company becomes more polluting, it is more likely to open a single noncompliant facility. We proved the conditions under which a single noncompliant facility becomes optimal in Proposition 2. This condition strengthens when is increasi ng. Note that this result mirrors empirical evidence in the industrial ecology literature. In his empirical paper in 1984, Pashigian showed that the number of facilities decreased and facility sizes increased with environmental regulations. PAGE 35 35 2.5 Solution Methodology for EUFLP3 The EUFLP3 model presented in Section 3 constitutes an extension of the Capacitated Facility Location and Capacity Acquisition Problem (CFL&CAP) studied in Verter and Dincer (1995). From an analytical perspective, the primary differe nce is the penalties to be paid when the company exceeds the regional emission limits and aggregate transportation emission limits. These penalties result in discontinuities in the total costs of capacity acquired at eac h plant, as depicted in Figure 2 1 . Furthermore, the environmental regulations represented in constraints (3) and (4) act as capacity limitations on the total amount of goods produced by a particular facility. Consequently, the dominant facility property presented in Verter and Dincer (199 5) no longer holds. The problem is clearly NP complete. Holmberg (1994) studied a similar problem with stepwise linear facility costs. He linearization technique, with or w ithout improvements, is a very interesting approach for according to the plant locations. The unit price, the fixed cost, the environmental penalty, variable expansion and production costs and transportation costs are facility separable. Similar to Verter and Dincer (1995) and Holmberg (1994), the above plant cost function can be approximated by two linear segments. Each linear segment represents a pseudo facility th at is related to the plant and is defined by the regional production emissions limit. The first pseudo facility represents the case where the company complies with the limits, and the second pseudo facility is when the company decides to incur the penaltie s. The upper bound, on the first pseudo facility is the production PAGE 36 36 size that the environmental limit allows and the lower bound, is zero. The lower bound of the second pseudo facility, is the upper bound of the first pseudo f acility plus one. The upper bound for pseudo facility two, is the total demand. Once the pseudo facilities are created, the updated model reduces to the IP below. We can reformulate the problem utilizing the piecewise linearization method. Le t represent the number of pseudo facilities or linear segments, and be the end point of linear segment k, where . The cost at linear segment k is and TD= . Initially all variables are set to two and the penalties for first pseudo facilities will be set to zero. Min Z 4 = Subject to the following constraints: 1. The binary variable indicating an expansion should be 1 if there is an expansion at location i with pseudo facility k. 2. The pseudo facilities are capacitated. 3. There can only be one pseudo facility opened at region i. 4. The binary variable indicating a violation of a transportation emissions regulation must be equal to 1 when the total emissions exceed the specifie d limit. PAGE 37 37 5. The demand at each region is satisfied. 6. The constraints defining the ranges of the variables. Using the aforementioned properties of EUFLP3, we propose the following algorithm: Pseudo Facility Linear Estimation Algorithm Step 1. Initialize Input: For i , get data Set : TD= , , , and , , , , , , , Set , number of pseudo facilities, to 2 for each i. Step 2. Solve the Updated IP using branch and bound For i obtain the optimal values for and PAGE 38 38 Set , , , Step 3. Improve the approximation For For , If then separate this range into two new pseudo facilities. , and , , and , and , Update the costs and bounds accordingly. Step 4. Terminate If no new pseudo facility is generated. Set , , , Report and as optimal. Else Go to step 2. This algorithm can be used to solve any facility location problem where more than one fixed cost occurs as long as the user is aware of the discontinuity points. In addition, if no global constraints are present in the UFLP with discontinuities, a CFLP solver can be utilized instead of an IP solver which will reduce the computation time. PAGE 39 39 2.6 Realistic Data Set Section 2.6 , we outline the realistic data set we assembled as a basis of the computational ex periments to be discussed in Section 2.7 . To motivate our examples, we collected data from several resources for the automotive industry. Table 2 3 shows the different parameters and data sources that were utilized to estimate param eter values for the model. We gathered financial information from 10 K and 20 Fs of 5 major car manufacturers, including GM, Ford, Toyota, Honda and Hyundai. Please refer to Table A 1 for the complete data. While most of the estimate calculations were str aightforward, the estimates for the transportation and production emissions constraints warrant further explanation. The transportation emissions are obtained from the U.S. Energy Information Administration (EID). The data given for carbon dioxide, methan e or nitrous oxide is given in kilograms per gallon of fuel used or per miles. In order to complete the calculations concerning transportation emissions, we also utilize a simple average miles per gallon estimate. According to the data given on city data. com and eia.gov, the car carrier trucks travel an average 8 miles per gallon and release 10.15 kilograms of carbon dioxide per gallon. Thus, on average 1.27 kilograms of carbon dioxide per mile is released into the atmosphere per truck. Note that a full c ar carrier typically holds 10 cars. Consequently, we estimate the carbon dioxide emitted per car as .278 pounds/car mile. The emissions data for nitrous oxide and methane are similarly pounds/car mile and pounds/mile. There are several standards for transportation emissions such as the ones established by the EPA and National Highway Traffic Safety Administration (NHFSA). PAGE 40 40 However, these emission standards are set to regulate the per vehicle emissions rather than the lon g term effects of the gases that are released to the environment. As the focus of our model is on the total environmental impact of a company, therefore an environmental limit that incorporates the frequency of the shipments and the distance traveled need s to be incorporated. Thus, we utilize the limits provided in the National Enforcement Trends and Case Statutes published by the EPA. These values are used as an upper bound for the penalty the company would have to pay for transportation emissions violati ons. In order to find the cumulative transportation emissions in each region, we multiply the distances by the number of cars that are shipped from the production region to the receiving region and the pounds of greenhouse gases emitted per car mile which results in the total pounds of emissions. Next, we estimate the parameters associated with plant level emissions. The environmental penalty data, the environmental constant and the limit are estimated using www.transportreviews.com, Environmental Protec Settlements and Toxic Release Inventory (TRI) reports from the EPA, respectively. The total release data which is the sum of all the chemica ls released. In estimating the regional production limits, we base the calculations on the amount of pollutants enforcement trends. (See Table A 1 in Appendix for the complete dataset.) In order to estima te the distances, we divided the Contiguous U nited States into twenty regions , and calculated the distances to and from each of the region center. PAGE 41 41 For the numerical experiments, we identified a base case from the range of values that were established for th e parameters as shown in Table 2 4 . These values were chosen so that the base case results in a centralized solution where the effect of economies of scale is dominant. This allows us to pinpoint the scenarios that make the effect of environmental regul ations and penalties dominant. Specifically, the environmental limits have no effect on the optimal network structure and all demand is satisfied from a single facility in the base case. 2.7 Results of the Computational Experiments The instances of the linearized model are then solved using Matlab 2010a/Matlab2012b. We first outline the results for the Ba se Case. The remainder of Section 2.7 focuses on sensitivity analysis of the optimal solution to changes in certain model parameters. Table 2 5 shows t he performance of the algorithm compared to Lingo 13. 2.7.1 The Base Case In the base case with 20 symmetric regions (i.e. where the costs, regulations and penalties are the same in every region), the company opens one plant that satisfies the demand in every region. This plant exceeds both the regional production environmental limits and the transportation emissions limit. The economies of scale effect is high enough that the company realizes value from opening one plant and then pays the penalties. This is the case of a single noncompliant facility location solution similar to that found in Proposition 2. Moreover, these results are robust when we vary both the variable capacity expansion costs and also the variable manufacturing costs within a wide ran ge. Note that an emissions tax policy is essentially an increase in variable capacity acquisition costs. If a symmetric carbon tax exists in all regions, then this PAGE 42 42 policy does not have an impact on the network dispersion. This result is consistent with th e conditional dominance property stating that the overall cost is more effective in facilities decisions than the emissions tax. 2.7.2 Fixed Costs of Capacity Acquisition and Demand In the classical UFLP case, an increase in the fixed costs translates into a more centralized network if the transportation costs are negligible. The EUFLP3 behaves similar to the UFLP case, however the size of the plants and the fixed cost threshold that drives the company to centralization are influenced by the environmental r egulations. Specifically, as the fixed costs increase, there are thresholds whereby the limits associated with transportation and production emissions drive the solution. The values for demand at all plants were varied between 50,000 and 1,000,000 units per region. In EUFLP3, high demand translates to a higher possibility of environmental penalties. Thus, when the demand and fixed costs are both high, it is optimal for the company to have a centralized network. In general, when fixed costs are low, the c ompany disperses its network as demand increases until the demand reaches a threshold. Above this threshold, the company can never comply with the regulations, thus it decides to incur the penalties with less plants. To summarize, high demand translates t o a centralized network regardless of whether the fixed costs are low or high. 2.7.3 0.9 which provided enough economies of sca le to drive the company to violate both environmental limits and incur penalties with a single plant. Note that while a small value is associated with lower economies of scale, it also indicates reduced unit PAGE 43 43 production costs. Thus, even with small size d plants the company can benefit from economies of scale allowing compliance by dispersion to be feasible. In order to fully between 100,000 lbs and 600 ,000 lbs. From Figure 2 3, when low or high, is not affected by economies of scale. The variable only for the intermediate values of The threshold is reached. From Figure 2 2 approximately .6. As a benchmark, Manne (1967) comments that a 0.65 value is appropriate for many manufacturing firm s, thereby placing most companies above the capacity acquisition plan particularly if the environmental limits are kept at an intermediate level . 2.7.4 The Effect of Re gional Environmental Production Constant ( ) For this experiment, we vary the regional environmental production constant between 0.0 and 0.5, and the results are shown in Figure 2 3 . Recall that reflects the percentage of production which contributes to environmental waste such that a high value represents a high polluting company. For both low and high transportation costs, a high value (specifically higher than 0.25) causes the company to open one single plant and incur both penalties. This result is consistent with Proposition 4 and Pashigian (1984), as with an increase in , the single facility location solution becomes more likely as it is harder for the company to comply with the regulations in every region. However, our results emphasize the importance of PAGE 44 44 considering transportation costs concurrently with production pollution when determining an appropriate plant network strategy. When transportation costs and regional production environmental constants are low, the company should centralize its network to avoid the costs associated with excessive production penalties. This result is important because it indicates that low polluting companies with low er transportation costs have more flexibility in establishing their plant network to minimize total costs. Conversely, companies with large transportation costs and low production waste disperse their network when production emissions are low. 2.7.5 The Effect of Regional Production Environmental Penalty The combination of pollution contribution ( ), the fixed penalty charge ( ), the variable penalty ( ), and the pollution threshold ( ) together determine the gravity of the environmental penalty. We varied between $250M and $5000M for alternate values of the pollution contribution . Note that the upper bound for the environmental penalty according to the data we have collected is $2500M. However, we extend ed our analysis to pinpoint the environmental penalty that fully disperses the network. In particular, if the fixed penalty is below a threshold, then a single plant solution is dominant for all values of . As this penalty increases, then a more dispersed network of plants is optimal. Moreover, higher values of are associated with a greater number of plants when the company disperses the network. Recall that the variable regional environmental penalty ( ) is an additiona l cost incurred for the number of units produced over the regional limitation. As the variable regional penalty increases, the plant network becomes more dispersed depending on the provided pollution threshold . When the pollution thresho ld is high, PAGE 45 45 then the firm always chooses a centralized solution regardless of the variable environmental penalty. Therefore, the effect of the variable regional environmental penalty is moderated through the pollution threshold . In summary, it appears that than the environmental limit itself. The key managerial insight is that as long as the environmental penalties are greater than the fixed costs of building Q 1 extra facilities minus the loss from economies of scale for opening smaller facilities, then the penalties firms. 2.7.6 The Effect of Transportat ion Emissions Constant, Transportation Emissions Limit and Penalty ( , ) Similar to the regional production constant and limit, the transportation emissions and the transportation emissions limit determine when the company will incur the transportation emissions penalty . When is lower than a threshold, the transportation emissions penalty has a significant effect on the network dispersion. However, above this threshold, the company but more by and . Figure 2 4 illustrates the impact of increases in the transportation emissions limit on the transportation emissions released. From these numerical results, an increase in the transportation emission penalty is associated with a decrease in emissions, particularly w hen the transportation emissions limit is at an intermediate level. Note that intermediate values for the transportation limit drive the company to minimize its transportation emissions. In this range, it is feasible for the company to comply with the transportation regulations by designing a dispersed network. For lower values, the firm PAGE 46 46 incurs the penalty and centralizes its network. For higher values, the transportation limit There is also an interaction between the regional pollution thresholds and the global transportation emissions threshold In general, when is low for all of the regions, it is optimal to open a single plant and incur all of the penalties under low transportation costs, regardless of the transportation limitations. When and are both at intermediate levels, the company disperses the network. Therefore, the transportation emissions regulation s are most effective when there are intermediate values for the transportation emission limits accompanied by a high tran sportation emissions penalty. 2.8 Conclu ding Remarks and Future Directions While much of the current literature has focused on emissi ons from a single plant, we analyze the impact that production and emissions regulations have on a the environmental regulations are not taken into consideration th ere is a trade off between transportation costs and fixed costs. For these problems, a demand region is the impact of emissions taxes, regional production level en vironmental limits and global transportation emissions regulations. Essentially, we find that when the global limit on transportation emissions is relatively low, then a more dispersed production network is optimal. As for the variable capacity expansion or variable manufacturing costs, we find that they do not have a significant effect on the network dispersion for the realistic PAGE 47 47 instances. An immediate consequence of this result is that having a symmetric carbon tax policy (i.e. in all regions simultane ously) will not reduce regional production environmental emissions. In this situation, the company can simply increase prices to compensate for the carbon (emissions) tax. If a company has very high demand in each region such that the dispersion efforts w ill not make the company compliant, the company is better off centralizing its network. We find a demand threshold for each company where the company conforms by considering new technologies (greener technologies during the construction phase rather than l ater) rather than dispersing its production network. This option has not been incorporated into our model but can be considered as an extension in the future. In general, high total demand translates to a centralized network. Intuitively, a low regional production environmental limit should force the company into compliance. However, numerical results illustrate that with regards to the regional production environmental penalty, an increase in the lump sum dollar amount associated with the penalty is muc h more effective than a decrease in the actual limit of damage tolerated. When non compliance becomes costly with a large fixed penalty, both the regional production environmental damage and the global transportation emissions are reduced. When the enviro nmental penalties are high and the environmental limits are at intermediate values, the company will disperse its network even if the benefit from economies of scale is great. If the civil/criminal penalties are high, the companies lose profits even if the y are not paying the penalties. In order to comply with the regulations the company disperses its network which reduces the benefits of economies of scale. PAGE 48 48 Economies of scale, the primary driver of network centralization in UFLP, is not as dominant for man y scenarios in the EUFLP3 case. An industry or a company with high pollution production is better off having a single large plant and trying to invest in environmental abatement policies or incurring the civil penalties and injunctive relief. However, a company or an industry with low pollution production is better off dispersing the network so it is not incurring penalties but also opening the largest possible plants that comply with the environmental limits. The same scenario holds for the transportati on emissions penalty. If the company has a highly polluting fleet, the optimal decision is to have a centralized network. The company is incurring the penalty even if the network is dispersed. In the intermediate cases, the company should consider disper sing the network to reduce the transportation emissions. These results have implications for policy makers as well. Constantly reducing the environmental limits without increasing the penalties does not force the companies to comply with the regulations. O n the contrary, choosing intermediate levels of environmental limits allow the companies to take advantage of economies of scale and avoid the risk of incurring penalties. If the companies can still benefit from economies of scale without having to invest in costly abatement policies, they will try to disperse their network and comply with the regulations. This type of compliance by dispersion reduces the regional production environmental damage and the transportation emissions. By setting the regional prod uction limit and penalties at optimal values, a national standard for environmental damage could be achieved. Dispersion will create small and medium sized plants that do not have high individual environmental impact. PAGE 49 49 To summarize, in order to reduce the regional production environmental damage and transportation emissions the policy makers should choose intermediate limits but high penalties. Furthermore, the companies with low and medium pollution should consider dispersing their network to avoid penalti es and reduce their costs. The companies with high pollution should resort to other resources for compliance or take the risk of being penalized. As future research in this area, the total effect of small and medium sized plants can be considered. The sma ll and medium sized plants have small individual environmental impact but in total their impact can be large. In addition, the effect of different environmental abatement technology options on facility loca tion problems can be explored. PAGE 50 50 Figure 2 1. Non smooth plant costs Table 2 1 . Summary of model parameters Model P arameters Number of possible plant locations Number of markets Fixed cost of location i Unit costs at location i Unit costs of expansion at location i Unit emissions tax Unit cost of production at location i Demand at location i En vironmental limit at location i One time penalty of going over the regional environmental limit The unit cost of transportation between lo cations i and j (includes trade tariffs) Regional environmental constant between 0 and 1 Transportation emissions constant for transferring items between locat ions i and j The constant for economies of scale at location i The distance between location i and j The transportation emissions penalty The transportation emissions limit The unit penalty for going over the regional environmental limit Total Demand Table 2 2 . Summary of decision variables Decision V ariables PAGE 51 51 1 if any new facility is established at location i, otherwise 0 1 if the regional environmental limit is violated, otherwise 0 The number of units produced in location i for location j The total number of units produced in location i 1 if transportation emission limit is violated, otherwise 0 Table 2 3 . Model parameter estimates and s ources P arameter Description of Data Source Capital Expenditure (NET PP&E Capital Expenditure)/Output Output or Sales EPA threshold for the given toxic materials Civil penalty paid by the company/Estimated Value of Complying Actions Shipping estimates Total Releases/ Output Emissions in g/mile Expenses/Output Revenue/Output EPA Penalties/Output Yahoo! Finance 10K or 20F Yahoo! Finance 10K or 20F Market Share Report er, EPA Toxic Release Inventory National Enforcement Trends (EPA) Transport Reviews website Toxic Release Inventory Eia.gov or city data.com Yahoo! Finance 10K or 20F Yahoo! Finance 10K or 20F National Enforcement Trends(EPA) PAGE 52 52 Table 2 4 . Numerical experiment ranges Parameter Range Base Case Fixed Costs(mil$) 595 14,325 1,000 Variable Manufacturing Costs($1000s) 12.600 61.695 18 Variable Expansion Costs($1000s) 3.420 15.117 6 Transportation Costs($ per car per mile) .02 .1.5 0.06 Demand(units) 50,000 17,500,000 300,000 Regional Environmental Percent 0 0.95 0.1 Regional Production Limit(thousand pounds) .002 1,308,000 90 Regional Penalty for exceeding limit(mil$) 0 2500 1,000 Economies of Scale Constant 0 1 0.9 Selling Price(1000$) 14.5 61.3 20 Transportation Emissions Constant(pounds) 0.000139 .278 0.0001 Transportation Emissions Limit(thousand pounds) .002 1,308,000 30 Transportation Emissions Penalty(mil$) 0 4636 1,000 Variable Regional Environmental Penalty($mil/ unit) 0 0.268 0.1 Table 2 5 . Performance of the algorithm Number of Locations CPU Time (seconds) Algorithm Lingo 4 1.1807 52 5 1.2015 972 6 1.2126 15663 8 1.2329 >19200 16 1.4854 >19200 20 2.3705 >19200 45 3427.6 >19200 PAGE 53 53 Figure 2 2 . The effect of gamma on network dispersion Figure 2 3 . The effect of on network dispersion 0 5 10 15 20 25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of Plants Economies of Scale Constant( ) Number of Demand Locations=20 Elimit=100,000 lbs Elimit=300,000 lbs Elimit=400,000 lbs Elimit=600,000 lbs 0 5 10 15 20 25 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Number of Plants Regional Production Environmental Constant Number of Demand Locations=20 Transportation Cost:High Transportation Cost:Low PAGE 54 54 Figure 2 4 . The effect of on transportation e missions 525 530 535 540 545 550 555 560 565 570 575 500 510 520 530 540 550 560 570 580 590 600 Transportation Emissions Released(thousand pounds) Transportation Emissions Limit(thousand pounds) G=$2500M G=$1000M PAGE 55 55 CHAPTER 3 RESOURCE ALLOCATION OF NONPROFITS: A CASE OF ANIMAL SHELTERS 3.1 Motivation In 2011, the number of charities and private organizations that is registered with the Internal Revenue Service exceeded 1.6 million(IndependentSector.gov). Weisbrod(1974, 1979) describes non profits as providers of public goods and explains the existence of non profit organizations as the res p onse to the excess de m and from g ov ernme n tal organizations that can only ser v e to the le v el of the median voter . This excess demand is social responsibility that is an external cost to the government or the people. The services nonprofits provide range from protection of the environment to bringing performing arts to the community to supplying shelter, protection a nd resources to humans or animals. In our paper, we choose to focus on animal shelters as the stray animal population and euthanization is a significant problem. According to ASPCA, 5 7 million animals enter shelters each year and almost 50% of them are e uthanized. The trap and euthanize program intuitively may seem as an economically feasible solution to the problem but socially it is not acceptable. Moreover, Oxford Lafayette Humane Society approximates the total number of stray dogs and cats in the US t o be 70 million. To the being of the community may be seen as miniscule. In their report, Fitzgerald and Wilkinson mental health and wellbeing, as well as their negative impacts at the regional and national level such as tourism, distribution of costs and benefits, and damages to indigenous cultures. In several communities, outdoor cats are known to be the suspect PAGE 56 56 in many deaths of endangered birds (Cats vs Rare Bird, Wayne Parry). In Cape May, home of the Annual World Series of Birding and one of the prime bird watching spots in North America, where bird watching brings $2 billion to New Jersey economy every year, the cost of feral cats can be estimated to b e very high. One of the many challenges non profit organizations including animal shelters face today is the optimal allocation of resources and obtaining sufficient monetary do nations to continue their services. Non profit organizations, can use their efforts to receive funds from donations, grants and advertisements. They need to allocate their resources, like manpower and money, efficiently to organize fundraising activities, qualify for grants, set pricing policies and invest in advertisement to improve their services and funds. However, the return on these investments are highly stochastic causing an unstable source of funds. Among the nonprofit organizations, some are more overlooked than others. An example is the Best Friends Animal Society, the largest sanctuary for abused and abandoned animals in the US. with four stars or a score of 61.1.1 out of 70 at Charity Navigator, received 23.48% of the contributions the World W ildlife Fund, a leading wildlife conservation organization with three stars or 55.63 out of 70, received in 2012 (Charitynavigator.org). This difference in donations could be explained by the priority of the problems they address, however, a difference in funding between the same types of organizations also exist. For example: Greenville Humane Society, highest ranked animal shelter in 2012, received only 0.025% of the contributions Best Friends Animal Society has received. Sarsted and Schloderer(2010) bel ieve that the reputation of a nonprofit organization can be considered as an intangible asset, and they show that PAGE 57 57 universal ranking/rating system does not exist, many donors utilize the rankings of organizations such as Charity Navigator or Guidestar. Charity Navigator currently uses two types of metrics to evaluate non profits: financial health, and accountability and transparency. This ranking system quantifies certa in characteristics of the organization adequately, but it is not able to explain the differences in funding due to reasons related to alignment to mission. The significant effect of reputation on willingness to donate has been shown, alas a common reputati on metric capturing all aspects of an organization does not exist for non profits. In our paper, we try to capture the effect of reputation, fundraising activities and advertisements on the donations the animal shelter receives. We incorporate several fact ors known to be effective on increasing donations to appropriately represent the reputation of an animal shelter. In addition to donations from advertisement or fundraising activities, animal shelters also receive funds from adoption fees to cover their ex penses or to invest in growth. In addition to obtaining sufficient funding for the existence and growth of the organization, non profit organizations also need to stay true to their mission, and show the alignment between their program expenses and their goal. Sawhill and Williamson economically and that we need other metrics to quantify an organizations success in its mission. Unfortunately, unlike the financial measures, there are no common performance measures change depending on the organization, however Sawhill and Williamson(2001) were able to provide guidelines to find these performance metrics by PAGE 58 58 introduced by The Nature Conservancy contains three types of measures for non profit organizations: (i) impact measures: measure progress toward mission, (ii) act ivity measures: measure progress toward programs that help achieve the mission (iii) capacity measures: measure toward the necessary requirements for the non profit to exist. In the case of The Nature Conservancy, the impact measures are the biodiversity h ealth and threat abatement, the activity measures are the projects launched and sites protected, and the capacity measures are the public and private funding, total membership and market share. We identify impact measures related to animal shelters by anal yzing the mission statements and the goals of several animal shelters. Perhaps the most significant problem with animal shelters is euthanization. In an effort to reduce needless program in 2014 (Sorentrue, 2014). We propose the mean euthanization rate metric to quantitatively capture the performance of shelters in this impact measure. The euthanization problem would be eliminated if there was enough demand to adopt all homeless animals. At the Fort Worth Animal Shelter, in the last years, the goal has been increasing the live release rate or the number of animals not euthanized. We capture this measure through the mean demand rate and the mean adoption rate metrics. The animals who are neither adopted nor currently in shelter or euthanized are turned away from animal shelters creating a stray population. ACT clinic, an ur paper, we use the mean rejection rate metric to PAGE 59 59 measure the success of this mission. Another measure concerning any nonprofit ion statement (cityofboston.gov). We capture this goal through the traffic intensity metric. Finally, for any animal that is not adopted, rejected or euthanized, the amount of time spent in a shelter has a significant effect on its wellbeing. Clermont Cou get animals ... adopted out, as quickly as possible, so we have more space for the next In this paper, we identify two types of animal shelters: (i) traditional and (ii) adoption guarantee. Traditional shelters are shelters that accept and perform euthanization due to behavioral or health problems, space issues(i.e. Animal Control). Adoption guarantee shelters do not perform eu thanization and are also identified as no kill shelters. We first compare these shelters in the aforementioned impact measures. We find that traditional shelters perform better in most of these measures except for the mean euthanization metric. Moreover, w e provide optimal resource allocation and adoption fee policies that maximize the performance of animal shelters. We utilize the idea of social and economical performance measurements from Sawhill and Williamson(2001) to simultaneously maximize the impact measures and the funding of animal shelters. (For the rest of the paper we will refer to investments as activity measures and expected return on investments and the amount leftover from adoption fees minus the operational costs as capacity measures.) We f ormulate 6 different optimization problems to provide animal shelters with the most suitable objective for their organization. PAGE 60 60 In particular, we address the following research questions: 1. What are the impact, activity and capacity measures for animal shel ters? 2. How do different organizations/shelters compare in their impact measures? 3. How should these organizations allocate their resources and set their adoption fees to maximize their performance? 4. In what circumstances a negative adoption fee is optimal? 5. Is there an objective function that performs better than others? This paper is organized as follows: In Section 3.2 , we provide a brief literature review on research on non profit organizations. In Section 3. 3, we first propose quantitative metrics to represe nt the impact measures of animal shelters. We then compare two types of animal shelters for impact measures: traditional shelters and adoption guarantee shelters ( Section 3. 4). We show through comparison that traditional shelters perform better in most imp act metrics except for the mean euthanization rate. Another important subject concerning nonprofit organizations is the allocation of resources to achieve the organizations mission. In Section 3. 5, we provide optimal resource allocation and fee policies th at maximize the performance metrics of an organization. 3.2 Literature Review Most of the literature on non profit organizations has focused on comparison between non profit and for profit organizations. The profit maximization problem has been studied ex tensively for for profit firms/organizations. In contrast, profit maximization is not a suitable objective for non profit firms. The objective of the non profit organizations has been assumed to be the maximization of quality or quantity of services render ed in detail for organizations with tangible services such as hospitals, PAGE 61 61 universities and performing arts.(Newhouse 1970, Feldstein 1971, James and Neuberger 1981). Lee (1971) proposes a model that maximizes the organizations use of inputs. Another common objective is the maximization of the budget. (Tullock 1966, Niskanen 1971). Tullock and Niskanen also considered different types of nonprofits: purely donative and purely commercial. We assumed the revenue of animal shelters to be obtained through donation s as well as adoption fees. For a detailed literature review please refer to Hansmann (1980). In later studies, Verheyen (1998) explores the internal and external budget models of nonprofit organizations and the integration of managerial and professional decisions. On another topic, Lien et al.(2014), study the resource distribution operations of a nonprofit organization, and provide a heuristic for discrete resource demand distributions. In his paper, Kingma(1993) being aware of the similarities in uncer tainties between stock returns and returns on funds for non profit organizations, utilizes financial portfolio theory to minimize the financial risk of nonprofit organizations. He explains the portfolio managers seek to nonprofit sector seek to provide a certain level of services while minimizing nonprofit donations, and he tries to minimize the variance of net revenues given that there are no real changes in the expected funding from nongovernment sources, and the relat ive growth of any funding is the same from all sources. Similarly, Chabotar (1989), PAGE 62 62 Gronbjerg (1990, 1991a, 1991b), Chung and Tuckman (1991a, 1991b) study revenue predictability and revenue growth for nonprofit organizations. On another note, some sources believe that the objective of nonprofit defined as maximizing the net revenues vs keeping the expenses as low as possible (A WealthEngine White Paper). In this report, the authors also identify several factors that might affect the return on investment such as the type of organization, fundraising targets, size and wealth of the target donor audience, number and type of fundraising staff employed, and also the extent and foc us of the fundraising strategy. In Section 3.3 , we present our model: we utilize queuing theory to represent the impact metrics of animal shelters, financial portfolio theory to represent the expected return on donations, and general optimization to provi de the resource allocation policy that will maximize the impact metrics, the expected return from donations and fees collected from services less the operating costs of a shelter with k capacity. 3.3 The Model We identified the following impact metrics for any type of shelter: the mean demand rate, the mean waiting time, the mean rejection rate, traffic intensity and the mean adoption rate. These impact measures support the animal shelters in achieving their mission. In addition to these common impact me asures, we also propose a shelter specific impact metric for shelters with euthanizations (i.e. traditional shelters); the mean effective queuing theory to estimate these impac t measures and compare them for two different type of shelters: traditional and adoption guarantee shelters. (See Fomundan and Herman (2007)). The adoption guarantee shelters are euthanization free, thus are PAGE 63 63 represented by an M/G/k/k queue without reneging while the traditional shelters are represented by an M/G/k/k queue with reneging due to the euthanization policies. In an M/G/k/k queuing system, there is a single class of animal arrivals that have an exponential interarrival time distribution with a mea n arrival rate of and a generally distributed departure rate with mean rate of Âµ, where and Âµ represent the mean arrival rate for the population and the mean demand rates, respectively. There are servers in the system which represent the maximum c apacity for care at a shelter. In an M/G/k/k queue, there is no queue space, in the event of a new arrival, when all the primary care areas are full, the animals are turned away from the shelter (i.e. rejections). In traditional shelters, in addition to the rejections, an animal that has been in the shelter for more than a certain amount of time units is euthanized with probability p and kept at the shelter with probability 1 p. This waiting time is known to be rene ging time and it is generally distributed with mean . Figure 3 1 illustrates the rehoming process of adoption guarantee and traditional shelters. Let j represent the shelter type, where j is 1 for adoption guarantee shelters and 2 for traditional shelt ers. P k,j is defined as the probability that a shelter of type j is full (i.e. has k animals), and is dependent on arrival, demand and euthanization rates. If a shelter j is full (has k animals), the animals will be turned away from the shelter at a mean k,j ] and accepted to the shelter at a P k,j ]. The impact measures we identified can be represented by the following metrics: (i) mean demand rate (ii) mean waiting time, (iii) mean rejection rate, (iv) mean effective euthanizat ion rate, (v) traffic intensity/utilization, and (vi) mean adoption rate. The mean demand rate is the demand for animal adoptions. If the animal shelters did not have PAGE 64 64 capacity constraints or an abundant amount of animals were available, this measure will e qual to the mean adoption rate. However, the mean adoption rate is limited by the animals that are available at the shelter an d the capacity of the shelter. Table 3 1 generalizes notation we will be using in this paper. 3.3.1 Adoption Guarantee Animal She lter: M/G/k/k (No Bulk Arrivals, No Priorities) Type 1 or adoption guarantee shelters, are euthanization free shelters which are represented by an M/G/k/k queue. To facilitate the comparison between the two types of shelters, we assume that , and . M/G/k/k queues are a special case of M/G/c/k truncated queues and the performance measure calculations are well known (Gross 1985): ( 1 ) ( 2 ) The performance meas ures for M/ G/k/k queues are summarized in T able 3 2 . (For tractability reasons we represent functions with f (.) and separate collection of terms using [ ].) 3.3.2 Traditional Animal Shelter: M/G/k/k No Bulk Arrivals, No Priorities) Traditional animal shelters are represented by an M/G/k/k queue with reneging and retainment. In this case , the animals depart from a traditional shelter in two ways: (i) adoptions, (ii) euthanizations. The animals depart by an adoption with a rate of ,this corresponds to the animals that reneged but has been retained (not euthanized), and the animals that never reneged. For the rest, in p of the time, the animals that reneged are euthanized. The steady state equations can be written in a PAGE 65 65 similar way as type 1 shelters. The steady state equations for the M/G/k/k queuing system with reneging and retainment are shown in Table 3 3 (Pallabi, 2013) 3.4 Performance Comparison In Section 3.4 , we compare the traditional and adoption guarantee shelters in the aforemen tioned impact metrics. The first metric, the mean demand rate, represents the populations demand for adopting animals in a shelter. We assume this metric to be the same for any type of shelter. This means that each shelter have the same opportunity to per form well in terms of adopting out animals. We first present some observations and propositions to aid in comparing the different impact metrics. According to HSUS, 6 8 million animals enter shelters each year and about 50% of them are euthanized (humanesociety.org). The euthanizations are performed due to health, behavior problems and sometimes space issues. Despite the high number of euthanizations, it is easy to find an empty animal shelter at any time. In observation 1, we can show that the tra ditional animal shelters are more likely to be empty than the adoption guarantee shelters: Observation 1. The probability of a traditional shelter being empty is larger than or equal the probability of an adoption guarantee shelter being empty given that t he arrival and departure rates are the same. That is, . Proof: We know that . We can easily show that Where, PAGE 66 66 and From observation 1, we see that traditional shelters may be performing unnecessary space related euthanizations. On the contrary, adoption guarantee or no kill shelters are less likely to be empty and utilize their capacity more efficiently. Any animal tha t is not admitted to the shelter, not euthanized or placed in a home will be turned away from the shelter becoming strays and trying to survive on their own. It is essential for animal shelters to admit and adopt out as many animals as possible. To compare the mean rejection rate measure of adoption guarantee and traditional shelters, we use the property that the blocking probability of any type of shelter is decreasing and strictly convex as a function of the departure rate. Proposition 1. The blocking pr obability is strictly convex decreasing in departure rate , and capacity, k. Proof: This result is evident as Harel (1990) showed that is convex and decreasing in . Since , we can say . is also kno wn as the blocking probability, when multiplied with represents the rejection rate in our model. We can say that the mean rejection rate is at least as much as or higher in adoption guarantee shelters than in traditional shelters. It is easy to see tha t also holds. In proposition 1, we show that the probability of a shelter being full/ the probability of an animal being turned away from a shelter is smaller for traditional shelters than adoption guarantee shelters given that the arrival and mean demand rates PAGE 67 67 are the same. A smaller blocking probability and a smaller stray population is achieved at traditional shelters at the expense of euthanizations. We also see that increasing the departure rate or the capacity of a shelter will decrease the mean rejection rate at a shelter. From convexity, we can also say that the blocking probability decreases at a slower rate with increasing capacity or the mean demand rate. Any shelter with a goal to reduce the stray population, can achieve their miss ion by spay/neuter programs, increasing their capacity, and increasing the mean departure rate. However, the more the shelter increases the capacity or the mean demand rate or the mean euthanization rate, the smaller the changes in the mean rejection rate will be. Another concern related to the wellbeing of animals is identified as the behavioral of 10 cats who stay in shelters more than 90 days develop behavioral pro blems such as overstimulation/aggression or fearful/shyness (Parry, 2000). Next, we show that the waiting times at the two types of shelter are stochastically ordered. Proposition 2. Waiting time in a traditional shelter is stochastically smaller than or e qual to the waiting time in an adoption guarantee shelter ( ). Proof: Let denote the waiting time random variable in the adoption guarantee shelter. Let the distribution function of be . Then the waiting time random variable i n the traditional shelter random variable, say , is a mixture of the random variables Min( ,T) and , where T is the threshold random variable at which the animal may be euthanized with some probability, p. The distribution of , is PAGE 68 68 It is immediate that F( ) < F( ), hence This proves that, an animal is more likely to spend longer time in an adoption guarantee shelter than a traditional shelter. We also know that the mean waiting time at a traditional shelter is smaller than the mean waiting time at an adoption guarantee shelter. A shelter trying to improve its waiting time should try to increase the departures from the animal shelter. There are two ways of achieving this: by increasing ad options or by accepting euthanization although this is not socially preferable. Nonprofit organizations, specifically animal shelters must use their resources efficiently. One way to measure efficiency in queuing systems is traffic intensity/utilization w hich represents the percentage of the time the resources are busy. When it comes to efficiency, the utilization/traffic intensity of a traditional shelter is less than the traffic intensity/utilization of an adoption guarantee shelter if the capacity is be low a threshold. To show this, in Proposition 3, we first prove certain characteristics of the traffic intensity for an M/G/k/k queue. Proposition 3: Traffic intensity for any type of shelter, , for an M/G/k/k queue is semistrictly quasiconcave, fu rthermore it is decreasing as a function of for any Proof: See Appendix From Proposition 3, we know that the traffic intensity is decreasing as a function of for both type of shelters if th e capacity is below a threshold. Under this condition, the traffic intensity of the traditional shelters is smaller than the traffic intensity of PAGE 69 69 adoption guarantee shelters. The adoption guarantee shelters are better at utilizing their capacity when compa red with traditional shelters. The comparison of the adoption rate impact measure depends on propositions 1 3. The mean adoption rate of a shelter differs from the mean demand rate of a shelter. The mean demand rate represents how fast an animal is ado pted once it becomes available for adoption, whereas the mean adoption rate depends on the mean demand rate, the capacity of the shelter and the animal admittance rate. The larger the mean demand rate, the faster the animals can be rehomed. Similarly, the larger the animal shelter the more animals rehomed. The mean adoption rate of traditional shelters is less than the mean adoption rate of adoption guarantee shelters. While adoption guarantee shelters adopt out every animal they admit, traditional shelters adopt out only a percentage. The rest of the admitted animals at traditional shelters are euthanized. In T able 3 4 , we show a summary of the comparison of the common impact metrics. From Table 3 4 out guarantee shelter. In proposition 4, we show some characteristics of the mean effective euthanization rate. Please note that the mean euthanization rate represents the frequency that an individual animal is euthanized, whereas the mean effective euthaniz ation rate refers to the whole shelter. PAGE 70 70 furthermore it decreases with if . Proof: Mean effective euthanization rate, , is increasing with k and is decreasing with k and result follows from Proposition 2. Intuitively, the mean effective euthanization rate of an animal shelter should decrease with an increase in the capacity, k. However, an increase in capacity also decreases the mean rejection rate, and an animal shelter with a low mean rejection rate is more likely to accept animals with behavioral or health problems. This inc rease in admittances will also increase the mean effective euthanization rate. The same result holds for the effect of arrival rate on the mean effective euthanization rate. In order to reduce the mean effective euthanization rate, the animal shelters can decrease capacity, mean euthanization rate, arrival rate or increase the mean demand rate. Animal shelters employ several methods to reduce euthanization rates, increase adoptions or simply improve the efficiency of their operations. In addition to achievi ng these goals they need to manage their resources efficiently to continue providing services. 3.5 Resource Allocation for Adoption Guarantee Shelters In Section 3.5 , we introduce the activity and capacity measures, and we provide the optimal resource allocation policies for adoption guarantee animal shelters to invest in programs that maximize their performance metrics. Activity metrics measure the progress toward pr ograms that help achieve the mission. Animal shelters can invest in programs to assist the organization reach their mission and long term goals. They can PAGE 71 71 invest in programs that benefit the welfare of the animals by increasing the adoption rate or by inves ting in a wide range of fundraising activities that increase the monetary donations. The amount of resources an animal shelter dedicates to these programs represent the activity metrics. These activity metrics are limited by the resource capacity of the or ganization, thus a+d<= m . For simplicity, we will refer to adoption increasing programs as advertisements and monetary donation increasing programs as fundraising activit ies for the rest of the paper. The third set of metrics we introduce are the capacity metrics which measure toward the necessary requirements for the non profits to exist. In the case of the Nature Conservancy, the capacity measures were the public and private funding, total membership and market share. We identify the capacity measures for animal shelters to be the donations the organization receives. In our model, we use a similar approach to Kingma (1993) to represent the return on investment from different fundraising programs. Table 3 5. Table 3 6 list the terminology used in Section 3 .5 : The expected return from fundraising activities take the form: . is the risk free donations any shelter would receive. According to the 2010 Nonprofit Fundraising Survey, organizations with larger investments in fundraising activities saw higher increases in their donations in 2010. Hence, we assume that the expected return from fundraising activities is directly affected by the fundraising investment amount d. represents the difference between the mea n animal shelter reputation and the reputation of one specific animal shelter, and is how sensitive the expected return is to a 1% change in reputation. The donations available to animal shelters are limited and fairly competitive PAGE 72 72 Thus, an animal she , is decreasing with adoption fee rates, increasing with investments in advertisements and the size of the organization due to the high visibility effect (Weiss et al.(1999), Kotha et al.( 2001)). represent how much the reputation factor is affected by a change in adoption increasing investments, the size of the animal shelter and the adoption fee rates, respectively. In addition to t he return on investments, advertisements increase the mean demand rate and take the form In our resource allocation model, we maximize the performance metrics we defined. Given these measures, the optimization problem for the adoption gu arantee shelter can then be written as: s.t Where the first term is an impact measure. and represents the objective function including performance measure l for shelter type j . The second term is the expected return from advertisements or fundraising activities as monetary donations. The third term is the donations received through an imal adoption fees minus the cost of operating an animal shelter of capacity size k. before investments and average reputation of all shelters as . The activity metr ics are bounded by the resource limits of the organizations, and are incorporated into our model as a constraint. These activities increasing the mean demand rate or the PAGE 73 73 monetary donations, impact the return on investments and the impact metrics of the she lter. The monetary component of the objective function represents the capacity measures that ensure the organization continue to exist. 3.5.1 Mean Demand Rate In the mean demand rate problem, we simultaneously maximize the mean demand rate, the expected r eturn on investments and the donations from adoption minus the operating costs. We first introduce an important property of the blocking probability that aid in our analysis. We show that the blocking probability is jointly convex in a and f. Theorem 1: Th e blocking probability B or is jointly convex in a and f. B is strictly increasing convex in f, strictly decreasing convex in a and has decreasing differences in a and f . Proof: B is convex and decreasing in . We need to show that: B( [1 t]x+ty)) [1 t]B( (x))+tb(g(y)) Since is linear, we can write: B( (1 t)x+ty))= B((1 t) (x)+t y)) (1 t)B( (x))+tB( (y)) (1 t)B( (x))+tB( (y)) Please see Appendix B for the second result. In Table 3 7 we show the optimal solution to the adoption guarantee problem for different impact measures: To better explain the aforementioned results we perform further analysis: Observation 2 (Mean Demand Rate): The exact optimal adoption fee that maximizes t he mean demand rate problem is decreasing as a function of a, if the PAGE 74 74 marginal effect of a on the expected returns on investment (ROI) is larger than the marginal effect of d on ROI. Conversely, is increasing as a function of a if the marginal effect of a on both the mean demand rate and ROI is less than the marginal effect of d on ROI. Proof: In Appendix B, we see that . Thus, if , and if . In Observation 2, we see that if the marginal effect of a on ROI is larger than the marginal effect of d on ROI, the adoption fee is decreasing. An investment in adoption increasing programs increase both the repu tation of the nonprofit organization and the mean demand rate. Generally, an increase in demand will translate to an increase in prices as well. In this case, as the mean demand rate increases, the adoption fees decrease. This result can be explained by th e fact that three types of benefit is gained from an investment in a : (i) increase in mean demand rate, (ii) increase in expected return due to improvement in the reputation of the organization, and (iii) increase in the mean adoption rate. This can also b e explained with the non profit nature of the organization. If the organization can receive enough funds through donations, they do not need to charge a fee for their services. Conversely, if the marginal effect of a on the objective function from increas es in mean demand rate and ROI, is smaller than the marginal effect of d on the objective function, then adoption fee is increasing with a . In this scenario, as the organization PAGE 75 75 decides to invest more in a , the benefits on ROI decrease . In order to make up for these losses, the organization must increase the adoption fees. In summary, if the marginal benefit of investing in adoption increasing programs is higher than the marginal benefit of investing in fundraising, the organization will decrease their adoption fees as the objective of a nonprofit organization is not to make profit but to provide a service. Otherwise, the organization is already losing on ROI by investing in a and must increase adoption fees to cover up their expenses. Ob servation 3(Mean Demand Rate): The optimal adoption fee with approximation, that maximizes the mean demand rate problem is increasing as a function of k, if the marginal effect of a on the expected returns on investment (ROI) and the mean demand ra te is larger than the marginal effect of d on ROI. Proof: We know from Cardoso(2009) that , and it is easy to see that . Thus, if , and if . If the marginal benefit of advertisements is larger than the marginal benefit of fundraising, the optimal adoption fee increases as a function of the capacity, k. An animal shelter will increase adoption fees as a response to cover up expenses, and it will decrease fees as donations become sufficient enough to support the expenses. The monetary gain from a is not sufficient to cover up the expenses of a large shelter as the effect of a on donations is indirect (through increasing reputation). Hence, the PAGE 76 76 org anization must increase adoption fees. If the marginal benefit of d is larger than the marginal benefit of a , the optimal adoption fee decreases as a function of capacity. The fundraising activities have a more direct effect on the organizations donations but have no effect on the impact metrics. In this scenario, the organization has sufficient funds to support a large shelter, but it still has to lower adoption fees to increase the mean demand rate. This result is the same for the heavy traffic approximat ion. The capacity, k, has the opposite effect on the optimal resource allocation to advertisements with approximation , . is decreasing as a function of k, if the marginal effect of a on the objective function is larger than the marginal ef fect of fundraising and the adoption fee on the objective function( Advertisements and capacity both increase the reputation of the organization and the mean demand rate, if the marginal benefit of a is large, o ne unit increase in a will have a significant impact on the objective function especially with a large capacity. Hence, the organization can spare more resource to invest in fundraising. We also observe that is decreasing as a function of the init ial mean demand rate, enough that the effect of advertisements is insignificant. Observation 4(Mean Demand Rate): The capacity measures component of the optimal objective function valu e with approximation , , is larger than zero when . In this scenario, the optimal adoption fee with approximation , , can be set to a negative value. PAGE 77 77 Proof: when , where We see that when the risk free return on investments in resources is larger than a threshold, the animal shelters can provide incentives for adopters, ie veterinary or supply discounts given that an adoption screening process is in place. To evaluate the s cenarios when the adoption can be negative we perform some numerical experiments. We choose as a base case that satisfies the above conditions and we vary between 0 and 10. This graph represents the scenario where , hence the optimal adoption fee, , is decreasing as a function of a. In Figure 3 2, the shaded area represents the risk free return rate that allows the organization to provide incentives for adopters. We see that as is increasing, the optimal adoption fee and the o ptimal allocation to advertisements is decreasing. In this scenario, a small increase in advertisements will lead to higher benefit on ROI. The organization can invest less and still receive high return on advertisements. Observation 5 (Mean Demand Rate): The objective function for the mean demand rate problem, , is increasing as a function of k when , and decreasing as a function of k, otherwise. Proof: If we remove the integrality condition of k, we can write: PAGE 78 78 if where since (Cardoso 2009). If the cost of maintaining a unit of shelter capacity, is smaller than a threshold, ,an increase in the capacity of the animal shelter will improve the objective function. the operating cost. 3.5.2 Mean Waiting Time F or the mean waiting time problem, we cannot obtain exact solutions, thus we explore the heavy traffic approximation solutions. The possible program investments, a and d are competing for resources, consequently the adoption fee is decreasing as a function of d while it is increasing as a function of a. As the organization allocates more resources to advertisements, it does not need to decrease the adoption fee to improve reputation, the waiting time or the mean adoption rate. Conversely, if the organizatio n allocates more resources to fundraising, it needs to decrease the adoption fees to reduce the waiting time and increase the mean adoption rate. The discrepancy between the optimal adoption fees for the mean demand rate problem and the mean waiting time p roblem is caused by the rate of change differences between the two impact metrics. We see that, the rate of change in the mean demand rate is larger than the rate of change in the mean waiting time with respect to a, assuming While the optimal adoption fee for the mean demand rate problem is not affected by , the optimal solution for the mean waiting time problem is affected by the initial mean demand rate. , is increasing as a function of the initia l mean demand rate, . PAGE 79 79 This difference between the mean demand rate problem solution and the mean waiting time problem solution is explained by the fact that the rate of change for the mean waiting time is affected by the initial mean demand rate while t he rate of change for the mean demand rate is not. By reducing the adoption fee, an animal shelter can improve the demographic it reaches regardless of its initial mean demand rate, causing an increase in the mean adoption rate. This increase in the mean adoption rate will decrease the mean waiting time, however the change in the mean waiting time will still depend on the initial demand rate. Observation 6 (Mean Waiting Time):The optimal adoption fee with approximation for the mean waiting time problem, is increasing as a function of k, if and decreasing if . Proof: The optimal adoption fee with approximation that maximizes the mean demand rate problem is increasing as a function of k, if the mar ginal effect of a on the expected returns on investment (ROI) is smaller than the marginal effect of d on ROI. Proof: We know from Cardoso(2009) that , and it is easy to see that . Thus, if , and if . For the mean waiting time problem the optimal adoption fee with approximation is increasing as a function of k when the marginal benefit of f and d exceed the marginal benefit of a. PAGE 80 80 Similar to observation 5, the objective function, , is increasing as a function of k when , and decreasing as a function of k, otherwise. This threshold for the mean demand rate problem and the mean waiting time problem are the same since neither of the impact measures are a function of k. 3.5.3 Mean Rejection Rate and Mean Adoption Rate We see that the optimal adoption fee solution to mean rejection or mean adoption rate problem shows similar characteristics to the mean demand rate case. The optimal adoption fee is decreasing as a function of a when the marginal benefit of a is larger tha n the marginal benefit of d. Observation 7 (Mean Rejection and Mean Adoption Rate): The exact optimal adoption fee that maximizes the mean adoption (rejection) rate problem is decreasing as a function of a, if the marginal effect of a on the expecte d returns on investment (ROI) and the mean adoption (rejection) rate is larger than the marginal effect of d on ROI. Conversely, is increasing as a function of a if the marginal effect of a ROI is less than the marginal effect of d on ROI. Proof: In Appendix B, we see that . Thus, if , and if . Observation 8(Mean Rejection and Mean Adoption Rate): The exact optimal adoption fee that maximizes the mean adoption (rejection) rate problem is PAGE 81 81 increasing as a function of k, if the marginal effect of a on the expected returns on investment (ROI) is larger than the marginal effect of d on ROI. Proof: We know from Cardoso(2009) that , and it is easy to see that . Thus, if , and if . Observations 7 and 8 are similar to Observations 2 and 3 since the mean adoption rate impact measure and the mean demand rate impact measure are closely related. An improvement in the mean demand rate also improv es the mean adoption rate and the mean rejection rate. The difference between the optimal adoption fees with approximation for the mean adoption rate problem and the mean demand rate problem is . This difference is due to the fact that the mean demand rate problem represents the demand rate for any animal, while the mean adoption rate depends on the capacity of the shelter, k. The optimal resource allocation to advertisements with approximation that maximize the mean adoption rate or the mean rejection rate problem shows similar characteristics to the mean demand rate problem. is decreasing as a function of k when the marginal benefit of a exceed the marginal benefit of d. We see that . If the organization invests one unit in advertisement or decreases the adoption fee by one unit, the mean demand rate will improve by one unit, whereas the mean adoption rate will improve by k units. Therefore, the optimal ado ption fee with PAGE 82 82 approximation and the optimal resource allocation to advertisement with approximation are larger for the mean demand rate problem. Observation 9 (Mean Adoption (Rejection) Rate): The capacity measures component of the optimal objective fun ction value with approximation , , is larger than zero when . In this scenario, the optimal adoption fee with approximation , , can be set to a negative value. Proof: The capacity measures component of the objective function value is when , where In observation 9, we demonstrate the risk free return threshold that will allow the animal shelter to provide incentives for adopters and still receive sufficient funding. Figure 3 3 illustrates the behavior of the optimal adoption fee with approximation , , and the optimal resource allocation to advertisements with approximation , when the marginal benefit of fundraising on ROI is dominant. We see that, , is decreasing as a function of k, and is increasing as a function of k. In order for the animal shelter to provide incentives for the adopters and still have sufficient monetary donations, the risk free return rate should in the shaded range or larger. Observation 10 (Mean Adoption/Rejection Rate): The objective function for the mean adoption/rejection rate problem, , is increasing as a function of k when , and decreasing as a function of k, otherwise. PAGE 83 83 Proof: If we remove the integrality condition of k, we can write: if where since (Cardoso 2009). Observation 10 shows the threshold per unit capacity operating cost, . The difference between the threshold per unit capacity cost for the mean demand rate problem and the mean adoption (rejection) rate problem is: and it is the res ult of the effect of k on the mean adoption (rejection) rate impact measure. A unit increase in k will increase the mean adoption rate, and consequently the objective function value, while the mean demand rate measure is not affected by k. This extra incre ase in the objective function from the mean adoption rate translates to a lower per unit operating capacity threshold. 3.5.4 Traffic Intensity The optimal adoption fee for the traffic intensity/utilization problem behaves similar to the previous solutions . The difference between the optimal adoption fees of the mean demand rate problem and the traffic intensity problem is: . The traffic intensity represents the average occupancy of an animal shelter. The optimal adoption fee with approximation for the traffic intensity problem is larger and the difference is increasing with k. PAGE 84 84 Observation 11 (Traffic Intensity): The capacity measures component of the optimal objective function value, , is larger than zero when . In this scenario, the adoption fee, , can be set to a negative value. Proof: The capacity measures component of the objective function value is when , where In Figure 3 4, we see the behavior of the optimal solution with approximation to the traffic intensity problem for the dominant case. The optimal adoption fee with approximation is negative and increasing as a function of k, and the optimal resource allocation to advertisements with approximation is decreasing as a function of k. The risk free return rate threshold is decreasing as a functio n of k. In this specific scenario, as the capacity becomes larger, the objective function increases. In Observation 10, we explore the scenarios where the objective function increases as a function of capacity. Observation 12 (Traffic Intensity): The obj ective function for the traffic intensity problem, , is increasing as a function of k when , and decreasing as a function of k, otherwise. Proof: If we remove the integrality condition of k, we can write: PAGE 85 85 if where since (Cardoso 2009). The difference between the threshold per unit capacity cost for the mean demand rate problem and the mean adoption (rejection) rate problem is: and it is the result of the effect of k on the traffic intensity impact measure. A unit increase in k will decrease the traffic intensity if , and consequently the objective function value, while the mean demand ra te measure is not affected by k. This extra decrease in the objective function from the mean adoption rate translates to a larger per unit operating capacity threshold for traffic intensity. On the contrary, when , . 3.6 Resource Allocation for Traditional Shelters The resource allocation problem for the traditional shelter case is similar to the adoption guarantee shelter case: s.t In the traditional shelter case, the mean demand rate also includes the mean euthanization rate. The performance measures are obtained from Section 3.4. Due to the complexities of the impact measures the results are implicit (Table 3 8). PAGE 86 86 We see from Table 3 8 that the optimal adoption fee formula is identical for all of the impact measures. However, this term also includes the optimal resource alloc ation to advertisements. The optimal resource allocation to advertisements and donations can be calculated using the expressions. The optimal adoption with heavy traffic approximation for traditional animal shelters is inversely influenced by th e effect of the adoption fee on the mean demand rate. As the effect of the adoption fee on the demand rate increases the organization can reduce its adoption fees. also decreases as function of the third objective function weight, , and the capa city, k. As the animal shelter puts less importance on covering its expenses, the organization can reduce its fees and improve its impact measures. The effect of the adoption fee, and the ratio of the effect of fundraising on the donations to marginal effe ct of advertisements on the mean demand rate increase the adoption fee. On the contrary, the ratio of the effect of advertisements on the donations to the effect of advertisements on the mean demand rate decreases the adoption fee. Observation 13 (Tradit ional Shelter): The objective function for the mean euthanization and mean rejection rate problem, , is increasing as a function of k when , and decreasing as a function of k, otherwise. Proof: If we remove the integrality condition of k, we can write: PAGE 87 87 if where since (Cardoso 2009). The difference between the per unit capacity cost threshold for the mean demand rate problem and the mean effective euthanization problem is: if since . This difference is due to the reduced mean adoption rate from euthanizations and the effect of k on the mean effective euthanization rate. The per unit capacity cost threshold is larger for the mean demand rate problem if the effect of an increase in k o n the mean demand rate is larger than the effect of an increase in k on the objective function of the mean effective euthanization problem. 3.7 Conclu ding Remarks In this paper we identify the optimal adoption fees, and the optimal resource allocation to advertisements and monetary donation increasing programs that maximize the impact metrics, the expected return on investment and the donations leftover after operating expenses are paid. We find that, in general, the optimal adoption fee is decreasing as a function of a , if the marginal benefit of advertisements is larger than the marginal benefit of funding. In this scenario, the organization is receiving enough funding as monetary donations by improving its reputation that it can now decrease the adoptio n fees. The optimal adoption fee is increasing as a function of a, if the marginal benefit of advertisements is high. An animal shelter will increase adoption fees as a response to cover up expenses, and it will decrease fees as donations become PAGE 88 88 sufficient enough to support the expenses. The monetary gain from a is not sufficient to cover up the expenses of a large shelter as the effect of a on donations is indirect (through increasing reputation). Hence, the organization must increase adoption fees. In add ition to the dynamics of the optimal adoption fee as a function of a and k, we also identify scenarios where the animal shelter may provide incentives for adopters. The optimal resource allocation to advertisement with approximation is decreasing as a fun ction of k. Advertisements and capacity both increase the reputation of the organization and the mean demand rate, if the marginal benefit of a is large, one unit increase in a will have a significant impact on the objective function especially with a larg e capacity. Hence, the organization can spare more resources to invest in fundraising. We also observe that is decreasing as a function of the initial mean demand rate, enough that the effect of advertisements is insignificant. The dynamics of the optimal adoption fees are the same for each impact metric, however the actual optimal adoption fee expressions are different for each. The optimal adoption fee for the mean demand rate pro blem differs from the optimal adoption fee for the mean waiting time problem due to the rates of changes in the optimal values with an increase in a . The optimal adoption fee solutions for the mean demand rate problem and the mean adoption (rejection) rate problem differ because the mean demand rate measures the departure rate of a single animal from a shelter, while the mean adoption rate considers the capacity of the shelter as well. We see that . If the organization invests one unit in advertisement or decreases the adoption fee by one unit, the mean demand rate will improve by one unit, whereas the PAGE 89 89 mean adoption rate will improve by k units. Therefore, the optimal adoption fee with approximation and the optimal resource all ocation to advertisement with approximation is larger for the mean demand rate problem. The difference between the threshold per unit capacity cost for the mean demand rate problem and the mean adoption (rejection) rate problem is: an d it is the result of the effect of k on the mean adoption (rejection) rate impact measure. A unit increase in k will increase the mean adoption rate, and consequently the objective function value, while the mean demand rate measure is not affected by k . T his extra increase in the objective function from the mean adoption rate translates to a lower per unit operating capacity threshold. The difference between the optimal adoption fees of the mean demand rate problem and the traffic intensity problem is: . The traffic intensity represents the average occupancy of an animal shelter. The optimal adoption fee with approximation for the traffic intensity problem is larger and the difference is increasing with k.The difference between t he per unit capacity cost threshold for the mean demand rate problem and the mean adoption (rejection) rate problem is: and it is the result of the effect of k on the traffic intensity impact measure. A unit increase in k will decrease the traffic intensity if , and consequently the objective function value, while the mean demand rate measure is not affected by k . This extra decrease in the objective function from the m ean adoption rate translates to a larger per unit operating capacity threshold for traffic intensity. On the contrary, when , . PAGE 90 90 We also evaluated the objective function at their optimal values with approxima tion to identify the best performing objective. We find that when the marginal benefit of a exceeds the marginal benefit of f and d together, the traffic intensity problem gives the highest optimal objective function value. Otherwise, the order of the optimal objective function values with approximation effect on the mean demand rate and the effect of all variables on ROI. PAGE 91 91 Table 3 1. Notation Symbol Meaning Mean arrival rate of animals at a shelter Mean adoption rate of animals at shelter type j Capacity of shelter type j Mean reneging(euthanization) rate of animals at traditional shelters Probability that there are n animals in shelter type j Mean waiting time of an animal in shelter type j Mean rejection rate of an animals in shelter type j Mean number of euthanizations in traditional shelters The utilization of a shelter type j Mean number of adoptions at shelter type j Maximum allowed number for mean number of euthanizations Figure 3 1. Rehoming process at adoption guarantee and traditional shelters Table 3 2. Performance measures for adoption guarantee shelters Impact Measure Formula Mean Demand Rate Mean Waiting Time Mean Rejection Rate: Traffic Intensity: Mean Adoption Rate: Primary Care Area Primary Care Area Primary Care Area Queue Size=0 Primary Care Area Primary Care Area Primary Care Area Queue Size=0 PAGE 92 92 Table 3 3. Performance measures for traditional shelter Table 3 4 . Comparison of Impact Metrics Mean Demand Rate Mean Waiting Time Mean Rejection Rate Traffic Intensity 1 Mean Adoption Rate Table 3 5 . Summary of decision variables Decision Variables a The percentage of the resources allocated for programs increasing mean demand rate d The percentage of the resources allocated for fundraising programs f Adoption fee per animal per time Table 3 6 . Summary of parameters Parameters Fundraising reputation factor for the animal shelter Mean fundraising reputation factor for all animal shelters 1 This inequality holds when the condition satisfying Proposition 3 holds. Impact Measure Formula Mean Demand Rate: Mean Waiting time: Mean Rejection Rate: Mean Effective Euthanization Rate: Traffic Intensity: Mean Adoption Rate: PAGE 93 93 Sensitivity of the returns to the reputation factor Risk free return on investment for any shelter The effect of programs increasing adoptions(a), adoption fees(f), and shelter size(k) on reputation v The effect of percentage of resources allocated to programs increasing monetary donations on expected return on investments Per unit cost of keeping up a shelter The weight of the performance metric t, t PAGE 94 94 Table 3 7. Summary of adoption guarantee result Range Advertisements (a) Mean Demand Rate 0 m Mean Waiting Time 0 m Mean Rejection/Adoption Rate 0 m Traffic Intensity 0 m PAGE 95 95 Table 3 7. Continued Table 3 8. Traditional animal shelter r esults Impact Measure f Mean Demand Rate Mean Waiting Time Mean Rejection Plus Euthanization Rate Traffic Intensity Mean Adoption Rate Fundraising (d) Adoption Fee Rate(f) Mean Demand Rate m 0 Mean Waiting Time m 0 Mean Rejection/Adoption Rate m 0 Traffic Intensity m 0 N/A PAGE 96 96 Figure 3 2. The mean demand rate problem s olution Figure 3 3. The mean adoption rate problem s olution ( is dominant) PAGE 97 97 Figure 3 4. The mean adoption rate problem solution ( is dominant) Figure 3 5. The optimal with approximation objective function comparis on PAGE 98 98 CHAPTER 4 RESOURCE ALLOCATION OF ANIMAL SHELTERS WITH CAPACITY EXPANSION 4.1 Motivation The biggest difference between non profit organizations and for profit organizations is ownership. For profit organizations can distribute their wealth among the shareholders, whereas nonprofit organizations can only use the surplus from their activities to provide services, self preserve and grow. In the case of non profit organizations that provide education, healthcare or care, growth usually requires investing in more equipment, workforce or both. Animal shelters can provide care for more animals by ad ding primary care areas and care givers. Capacity expansion in an animal shelter is also a viable solution for overcrowded shelters in many situations. In order to increase capacity, some of the limited resources must be allocated to adding more capacity. In Chapter 4, we incorporate capacity expansion decisions to the resource allocation problem. In this scenario, the resources of the organization must be divided between advertisements, fundraising activities and the capacity expansion. The model we intr oduce is a non linear mixed integer problem and known to be NP complete. We perform numerical experiments to understand the dynamics between changes in alternate resource allocation plans. We collect data from several resources to estimate the parameters a llowing us to perform numerical experiments with realistic data. We explain the estimations of the parameters in detail in Section 4.2 . The key research questions that we address in Chapter 4 are: 1. What are realistic data for the animal shelter setting? 2. Giv en the realistic data, what are appropriate resource allocation schemes for alternate shelters? PAGE 99 99 3. When should an animal shelter consider adding more capacity? 4. Under what circumstances should the animal shelter undertake large fundraising efforts? 5. When is adv ertising the most appropriate mechanism to aid in the adoption process? 6. Should adoption guarantee shelters utilize a different resource allocation strategy than the traditional shelters? 4.2 Realistic Data We utilize realistic data obtained from Asilomar A ccords, Form 990s , shelter websites and Charity Navigator to estimate the parameters of our model. Asilomar Accords include information about yearly intake, adoptions, euthanizations as well as the number of healthy and unhealthy animals and transfer to an d from other organizations. The information available provides a way to track shelter statistics and live release rates. Specifically, we use the beginning and ending shelter counts , total intake, adoptions and total euthanasia. The beginning count is the number of animals that are in the shelter or in foster care at the beginning of a reporting period. Similarly, the ending shelter count is the number of animals at the shelter or at foster care at the end of a reporting period. Intake is the number of live animals received at the shelter from the public, incoming transfers from within and outside the coalition, and from guardians requesting euthanasia. Adoptions are the number of animals that the organization rehomed within the public . Total euthanasia fiel d shows the number of animals that are euthanized regardless of their health or behavioral status. In addition to the information from Asilomar Accords, we collect the financial information from Form 990s. Form 990s are annual reports that non profit organ izations must file with the IRS that include information about the mission of the organization, its finances and programs. We specifically collected the contributions, gifts and grants, PAGE 100 100 program service revenue, advertising and promotion expenses, fundraisi ng income and expenses and operating expenses. Contributions, gifts and grants are the donations the organization receives. This number is separate from the fundraising income. Program service revenue is only applicable to organizations that charge a fee f or their services (i.e hospitals, schools, animal shelters). Advertising and promotion expenses include all printed and electronic media advertisements, internet site link costs and independent contractor fees for advertisements. We also collected the sco res and rankings from Charity Navigator to represent reputation in our model. The adoption fees were collected from individual shelter websites and the maximum adoption fee was used in our calculations . In Table 4 1 , we see the base case and the ranges o f the parameters we identified. To estimate the arrival, adoption and euthanization rates, and the capacity of the shelters we use Asilomar Accord records for 2011. We collected the information from 159 and 149 animal shelters for dogs and cats, respective ly. The arrival, departure and euthanization rates are taken from Total Intake, Adoptions and Total Euthanasia fields on the Annual Animal Statistics reports . The capacity represents the maximum amount of animals that can be kept at the shelter at any time and it is calculated as the maximum of the beginning and ending shelter count in 2011. The reputation of any organization is intangible. Charity Navigator, a non profit organization that provides aid to donors, has a scoring and rating system that allows the donors to compare different charities. Charity Navigator uses three dimensions to ra te non profit organizations: (i) financial measure, (ii) accountability and transparency and (iii) results reporting. We use these ratings/scores to estimate the reputation of the PAGE 101 101 organization. Among the 1.6 million charities, Charity Navigator rates appro ximately 7500 charities, and 24 charities that are listed in Table D 1 represent the animal shelters that participate in Asilomar Accords program and also rated by Charity Navigator. We use the Charity Navigator score of an animal shelter to represent the reputation of a firm, and the average reputation is calculated by taking the mean of the scores of the all the animal shelters in our data. We collected financial information from Form 990s of the 24 aforementioned charities. To calculate , t he effect of programs increasing monetary donations on expected return on invest fundraising expenses. The budget parameter, m, is directly taken from the contributions , gifts and grants field. While mo st of the estimate s were readily available , the estimates for the effect of advertisements, adoption fees , and capacity on the expected donations and the mean demand rate warrant further analysis. We performed two different multiple regression analyses to estimate these parameters. In the first regression analysis, we tried to find the effect of advertisements, adoption fees, capacity and reputation difference on the expected donations ( ) . is the reputation difference and it is the . The R square and the adjusted R square values for this analysis were 0.72 and .63. (Refer to Appen dix D for complete analysis.) In the second regression analysis, we estimate the effect of advertisements and adoption fees on the mean adoption rate (G,L) . To complete this analysis, we utilize the adoption rate and financial information for 2010 and 201 1. We also collected the PAGE 102 102 adoption fee information from the shelter websites and used the maximum adoption fee reported. The number of animal shelters that have both Asilomar Accords and Form 990s for both 2010 and 2011 were low, hence the R squared and adj usted R squared values were low. We still use the coefficients as a starting point for G and L in our numerical analysis. 4.3 Model Description 4.3.1 Model for Adoption Guarantee Shelters The resource allocation problem including the capacity as a variable for adoption guarantee shelter i s: s.t The objective function is similar to the resource allocation problem descri bed in Section 3.5. In all k parameters are replaced with , where is the capacity expansion size variable and is the initial capacity parameter. The first constraint is the resource constraint indicating that the organization can invest its resources on either advertisements, , fundraising activities, and capacity expansion, at a per unit cost of . The second constraint limits the capacity expansion at an animal shelter due to space limitations. The third constraint limits the adoption fees. The organization can set negative adoption fees, providing incentives for adoptees up to . PAGE 103 103 The first term of the objective function represents the impact measures that were defined in Section 3.5. The mean demand rate, , and the mean waiting time, 1/( are not affected by the capacity measure . The mean rejec tion and mean adoptio n rates, , , are increasing con cave with respect to , as is known to be convex with (Cardoso 2009). Traffic intensity , is also decreasing and convex The second term of the objective fun ction represents the donations the organizations receives. The only term involving in this component is and is linear. The third term of the objective function is the revenue from adoption fees less the operating cost. We know that is concave in and v is linear with respect to 4.3.1.1 Numerical Experiments for Adoption Guarantee Shelters In the base case scenario, the optimal solution is to invest all the resources to advertisements and set the adoption fee to $600 for all impact measure problems . During our experiments we observed similar trends for different impact measures. In general, we see that changing the risk free donation rate, , the weight of the impact measure, , per unit cost of capacity e xpansion, c, the initial reputation or the average reputation, or the initial mean demand rate , , have no effect on the optimal solution (i.e. the resource allocation plan), but they directly affect the objective function value. The risk free donation rate is the donations any shelter will receive regardless of its reputation. This result is consistent with the results in Section 3.5. The risk free donation rate only increases the total funding available to the organization but it has no e ffect on how the resources are allocated. The costs associated with capacity are the PAGE 104 104 per unit expansion cost and the per unit operating cost. For the base case scenario, per unit operating cost associated with maintaining the shelters is much larger than the per unit expansion cost, making changes in the per unit expansion cost insignificant to the optimal solutions as the objective of the organization is to improve the mea n demand rate as much as possible with the available resources. 4.3.1.2 We varied the values of the mean arrival rate between 1000 and 3000. The results show that the changes in the mean arrival rate does not change the optimal resour ce allocation structure. However, we see a threshold where the optimal adoption fee shifts from the lower bound ( $200) to the upper bound ($600). When arrival rate is high, so is the adoption rate for adoption guarantee shelters. In the low arrival rate c ase, the donations are sufficient enough to cover the expenses and provide incentives for adoptees, whereas for the high arrival rate donations are not enough. 4.3.1.3 The Effect of Adoption Fees, Capacity, and Advertisements on the Reputation ( , ) We range the values of between 0 and 12, and see that when the marginal benefit of advertisements is lower than the marginal benefit of fundraising, the animal shelter allocates all resources to fundraising activities (Figure 4 1). Once investing in advertisements become more v aluable, i.e higher marginal benefit, the organization allocates all resources to advertisements. The changes in has no effect on the capacity or the adoption fees as an increase in donations has no effect on the impact or off between the resource allocation in advertising or fund raising. This may have future PAGE 105 105 implications for the organization, as these two alternate activities may require different sets of organizational competenci es to implement effectively. For the range of values we utilized in the numerical experiment, the eff ect of adoption fees on the reputation, had no influence on the optimal solutions. This can be explained by the difference in magnitudes of the possible values of a and f . We changed the values of between 0 and 10, and we observed a threshold w here the animal shelter expands the capacity of the shelter to the upper bound. As a result of the amount spent on capacity expansion, the resource allocation to advertisements decreases. We also observe that the adoption fee is not affected by . There fore, it appears that when the influence of capacity on reputation increases, then capacity expansion becomes a more viable option. In this situation, not only are the operational benefits of increasing capacity important, but also the reputational effect s as well. 4.3.1.4 The Effect of Reputation and Fundraising on Donations ( ) In Figure 4 2 , we see how the optimal solution and the optimal objective function values change with the effect of fundraising on donations for the mean demand rate impa ct measure ( . We observe a similar trend with all impact measures with the exact solutions slightly different from each other (Please refer to Appendix D for complete numerical results.). As the effect of fundraising on donations increase s , the optimal resource allocation to advertisements decrease s and the optimal allocation to fundraising increases. Specifically, the organization invests all resources either in advertisements or fundraising activities. For the ranges we have studied, the optimal adop tion fee is first decreasing until a threshold and increasing thereafter for the mean PAGE 106 106 waiting time and mean rejection rate impact measures. The threshold is the point where the marginal benefit of advertisements and marginal benefit of fundraising equal ea ch other . For the mean demand rate and the traffic intensity , the adoption fee slowly decreases with . As the effect of fundraising on donations, , increase, the animal shelter gains more from a single unit of resource investment in fundraisi ng , all owing the organization to obtain most of its funds from donations and reduce its adoption fees. We ranged the values for between 0 and 60,000. An increase in translates to an increase in the effects of advertisements, adoption fee, capacity and the relative reputation on the donations. As we increase the effect of reputation from the donations, the animal shelter first allocates all resources to fundraising activities up to a threshold. This threshold is determined by the point where the effects of reputation and fundraising activities on the donations equal each other. Beyond this threshold, the reputation is dominant over fundraising activities for raising donations, thus the animal shelter will invest all resources to advertisements. 4.3.1.5 O bjective Function Weights ( ) The weight of the donations, , has no effect on the optimal resource allocation. The organization allocates all resources to advertisements due to the higher marginal benefit. In contrast, the optimal adoption fee is set to the lower bound when is zero, and to the uppe r bound when is larger than zero. This can be explained by the fact that when the organization is not receiving any funding through donations, it has to cover its expenses solely through adoption fees. The effect of the weight of donations, , for the mean dema nd rate problem is shown i n Figure 4 3. When the weight of the expected donations, , is zero, the PAGE 107 107 optimal solution is to expand the capacity by 100 units at a per unit cost of $1000 and allocate the rest of the resources to advertisements. The objective of the animal shelter in this case is to improve the impact measure and donations. As a result, the organization lowers the adoption fees to increase the mean demand rate and increases the capacity to improve its reputation. 4.3.1.6 The Effect of Adverti sements or Fundraising on the Mean Demand Rate (G, L) In Figure 4 4, we illustrate the changes in the optimal adoption fees for each impact measure with respect to increases in the effect of advertisements on the mean demand rate. For the mean waiting time and the mean adoption/rejection rate problems, the optimal adoption fee is $600 for any values of G while the optimal adoption fee is decreasing for the mean demand rate problem and increasing for the traffic intensity problem. In the mean demand rate pro blem, when G is zero, the animal shelter needs to increase adoption fees to cover expenses as the mean adoption rate is not as high. In the traffic intensity problem, when G is zero, the impact measure can only be improved by the adoption fee, hence the an imal shelter reduces the adoption fees. The difference between the mean demand rate and the traffic intensity problem is caused by the fact that an increase in advertisements or a decrease in adoption fees influence the mean demand rate impact measure by a single factor and the traffic intensity impact measure by factors. We observe similar results to the optimal solutions with changes to the effect of adoption fee on the mean demand rate. PAGE 108 108 4.3.1.7 Per Unit Cost of Operating a Primary Care Area (v ) In general, we observe that the animal shelter cannot expand the capacity due to the high costs of operating and construction. In the base case scenario where per unit cost of expansion is $1000, the changes in per unit operating costs has no effect on t he capacity expansion decision. To fully understand the effect of per unit operating costs, we reduce the per unit capacity expansion cost to $100. The results show that when per unit cost of capacity expansion is low, the capacity expansion size decreases with per unit operating costs up to a threshold. Beyond this threshold, the costs associated with expansion are too high and the organization invests all resources to advertisements. 4.3.1.8 Summary of the Results for Adoption Guarantee Shelter In genera l, capacity is introduced as a variable to the resource allocation problem for the adoption guarantee animal shelters, due to the high cost of construction and operating, the organization is better off allocating all of its resources to either advertisemen ts or fundraising activities. If the marginal benefit of advertisements is larger than the marginal benefit or fundraising activities then all of the resources will be allocated to advertisements and vice versa. The organization will invest in capacity exp ansion only under some circumstances when: (i) the effect of capacity on donations is larger than the effect of fundraising activities on donations, (ii) the weight of the third term in the objective function are zero, and (iii) both per unit cost of opera ting and per unit cost of capacity expansion are low. In the first scenario, the animal shelter receives more donations by expanding its capacity and consequently increasing visibility and reputation than through fundraising activities. This applies to org anizations with low fundraising efficiency. In the second case, when the weight of the second term in the PAGE 109 109 adoption fees. Thus, the animal shelter will set the adoption fees to the upper bound and expand capacity. An increase in the capacity is accompanied by an increase in the mean admittance rate, and consequently the mean adoption rate. In the case where the weight of the third objective is zero, the animal shelter wil l expand capacity as the cost of operating is not a factor in the decision and the marginal benefit of capacity is high. For the third case, the organization will expand capacity as much as possible up to threshold. Above this threshold, the costs associat ed with a single primary care area are too high. In general we see that risk free donations, the weight of the impact measures, per unit cost of capacity expansion, the initial reputation or the average reputation of all shelters does not influence optima l solutions. In the base case, per unit cost of capacity expansion is insignificant compared with the per unit operating cost of capacity. In order for the capacity expansion cost to be a significant factor, per unit operating cost should be low. For the o ther parameters, the organization will try to improve its performance regardless of the current values. The effect of advertisements, adoption fees and capacity on reputation influence the structure of the optimal solution differently. There is an explici t trade off between advertisements, fundraising and capacity expansion. When the marginal benefit of any of these is the highest, the organization will invest in that specific activity. In the case where the dominant effect is that of capacity on donations , the organization will invest in capacity up to the limit and allocate the rest of the resources to advertisements. The PAGE 110 110 magnitude of the adoption fees is insignificant compared to the advertisements and donations, as a result the effect of adoption fees on reputation are also insignificant. The effect of fundraising on donations follows a similar trend, when fundraising is dominant the animal shelter should invest all r esources to donations and vice versa. The weight of the donations only affects the monetary component of the objective function, therefore the base case resource allocation solution is still optimal. However, we find that the optimal adoption fee is at the upper bound when the weight of the through adoptions. Or if the organization does not put any importance on operating revenue, t hen they should expand the capacity to the limit and allocate the rest of the resources to advertisements. The results of the numerical experiments for each impact measure show similar trends for most of the cases. When we focus on the effect that the adv ertisements or adoption fees have on the mean demand rate, the results are different for each impact measure. We find that, when the effect of advertisements on the mean demand rate is zero, the animal shelter reduces adoption fees or increases adoption f ees for the traffic intensity and mean demand rate problems, respectively. These differences stem from the fact that for the traffic intensity problem, any increase in the mean demand rate has an effect multiplied by the capacity of the shelter. Whereas, an increase in the mean demand rate for the mean demand rate problem increases the impact measure linearly. 4.3.2 Model for Traditional Shelters In Section 4.3.2 we perform numerical experiments using real data to solve the resource allocation problem inc luding capacity as a decision variable. The resource PAGE 111 111 allocation problem including the capacity as a variable for the traditional shelter becomes: s.t Most of the results for the resource allocation problem are similar to the adoption guarantee case. Thus, we only discuss the cases specific to the traditional shelter case. 4.3.2.1 The Mean Similar to the mean demand rate of the adoption guarantee shelter, a change in the mean euthanization rate has no effect on the opti goal is to improve its performance regardless of the current depa rture rate. 4.3.2.2 The Mean Euthanization Rate Plus Mean Rejection Rate Problem In the adoption guarantee shelter problem, any animal that is accepted to the shelter is adopted. In the traditional shelter problem, the animals that are not adopted include euthanizations and rejections. In order to capture the animal population that is not rehomed we consider these together. The results for this impact measure followed a similar trend as the adoption guarantee shelter except for the weight of the donations t erm. In Figure 4 5, we see the effect of changes in the weight of the second term on the optimal solutions for the traditional shelter case. As increases, the adoption fees decrease and capacity increases. When donations are not dominant (i.e low weigh t), PAGE 112 112 the adoption fee is at the upper bound. In this case, the animal shelter is improving impact measures by allocating all resources to advertisements. When donations become dominant, the negative effect of a positive adoption fee becomes significant. As a result, the animal shelter reduces adoption fees to the lower bound. We find that the animal shelter increases capacity by 100 primary care areas beyond a certain threshold. Similar to the adoption fee case, as the donations become dominant, all of the factors that improve it also increase. The difference in the threshold values for the switches in adoption fees and capacity expansion is due to the difference between the and coefficients. Also, the animal shelter allocates all of its resources to advertisements up to a threshold. Beyond this threshold, the animal shelter allocates some resources to capacity expansion and the remainder to advertisements. 4.4.Conclu ding Remarks As a final part of our analysis, we reformulate the resource allocat ion problem for both types of shelters to include capacity as a decision variable. To gain further insights to this mixed integer problem , we perform numerous numerical experiments utilizing public data . In general, the results show that when the marginal benefit of advertisements is high, it is better to allocate all resources to advertisements and nothing to fundraising activities or capacity expansion. When the marginal benefit of fundraising is high, the animal shelter should allocate all resources to f undraising activities. Utilizing the realistic data, t here are several factors that have no effect on the capacity expansion or resource allocation decision of the animal shelter: risk free PAGE 113 113 donations, the importance of the impact measure to the organizatio n, per unit capacity expansion cost, initial reputation or the average reputation of all animal shelters. For all of the parameters except per unit capacity expansion cost and the importance of the impact measure, this result is expected. The organization will try to improve its performance regardless of these initial factors. When we consider the per unit capacity expansion cost, true capacity expansion cost includes the per unit operating cost as well. In general, per unit operating cost is much larger co mpared with the per unit capacity expansion cost. An animal shelter needs to consider both per unit capacity expansion cost and per unit operating costs before making the decision to add more primary care areas , and these costs appear to be prohibitive . In the case of the importance of the impact measure, the impact measure is not enough to change the resource allocation decision of an organization. The main goal of a non profit is t o provide continuous service, hence in certain cases covering expenses beco mes more important over impact measures. . We also note that the arrival rate of animals to a shelter does not affect the resource allocation structure of the organization for investments in advertisements, fundraising or capacity expansion . The arrival r ate, however affects the optimal adoption fees, because when the arrival rate is less than the departure rate, the organization can provide incentives for adoptees rather than charging a fee . In this scenario, the organization can cover its expenses from d onations when the arrival rate is high, donations are not sufficient enough to cover the expenses. The effect of advertisements, adoption fees and capacity influence the optimal solutions differen tly. In general, we see a trade of f between advertisements, fundraising PAGE 114 114 and capacity. If the marginal benefit of any of these is dominant, the organization should invest all of its resources on this specific activity. The effect of adoption fees on the reputation is not large enough compare d with advertisements, fundraising or capacity to influence the optimal resource allocation strategy. The effect of fundraising on donations follows a similar trend, when fundraising is dominant the animal shelter should invest all resources to donations a nd vice versa. The importance an animal shelter puts on a specific performance measure may differ depending on their particular mission . If an animal shelter believes that the donations are not as important to the organization as providing service, they w ill invest all their resources to advertisements and set the adoption fees to the maximum possible expenses. An other animal shelter may put importance only on the impa ct measures and donations. Since this type of donations, it will increase capacity, invest in advertisements and reduce adoption fees to improve reputation to receive the maximum amount of donations as possible. The influence of advertisements or adoption fees on the first term is different for each impact measure. We find that, when the effect of advertisements on the mean demand rate is zero, the animal shelter reduces adoption fees for the traffic i ntensity problem and increases adoption fees for the mean demand rate problem. These differences stem from the fact that for the traffic intensity problem, any increase in the mean demand rate has an effect multiplied by the capacity of the shelter. Wher eas, an increase in the mean demand rate for the mean demand rate problem increases the PAGE 115 115 impact measure linearly. Therefore, if a shelter is more concerned about utilization or efficiency measures (i.e. traffic intensity), then it should reduce the adoption fees. The animal shelter will invest in capacity expansion only when per unit operating and per unit capacity expansion costs are low or zero, or the effect of a large shelter on reputation is extremely high. The results to the traditional shelter probl em show similar trends except for one special case. If receiving donations is a dominant objective for the animal shelter, then it is optimal to allocate resources to capacity expansion and to advertisements. The organization can also provide incentives fo r adopters to improve impact measures. PAGE 116 116 Table 4 1. Summary of data Parameters Meaning Source Base Case Initial reputation of the organization Score from Charity Navigator (CN) 56.60 Average reputation of all shelters Mean score for all available shelters (CN) 56.60 The effect of reputation on donations Estimated using regression(990) 54288.9 8 Risk free donations Estimated using regression(990) 0 The effect of advertisement on reputation The effect of adoption fee on reputation The effect of capacity on reputation Estimated using regression(990) Estimated using regression(990) Estimated using regression(990) 9.76E 05 0.038 0.072 The effect of fundraising on reputation Fundraising revenue/fundraising expenses(line 8c/line 8b 990) 1.82 v Per unit cost of operating (maintaining) a primary care area Values from Part IX, Line24 / capacity(990, AA) 17854.98 m Budget Contributions (990) 3876470 G The effect of advertisements on mean demand rate Estimated using regression (AA) 0.015670769 L The effect of adoption fee on the mean demand rate Estimated using regression(AA) 4.242619088 The weight of the performance metric t, t 1 Arrival rate of animals Intake (AA) 3380 Departure rate of animals including euthanizations Live Release(AA) 1624 Current capacity of the shelter Max(Beginning Count, Ending Count) 114 Euthanization rate Euthanizations(AA) 651.3851 p The euthanization probability Euthanizations/Intake(AA) .12 PAGE 117 117 Figure 4 1. The effect of advertisements on reputation vs optimal results Figure 4 2 . The effect of fundraising on donations 0 500000 1000000 1500000 2000000 2500000 3000000 3500000 4000000 4500000 0 4 8 12 The Effect of Advertisement on Reputation Advertisement Fundraising Adoption Fee 0 500000 1000000 1500000 2000000 2500000 3000000 3500000 4000000 4500000 2 6 10 14 Fundraising Donation Coefficient Advertisement Fundraising Adoption Fee PAGE 118 118 Figure 4 3 . The e ffect of the weight of donations Figure 4 4. Changes in G and the optimal adoption fees -300 -200 -100 0 100 200 300 400 500 600 700 0 1 2 3 W3 The Weight of Operating Revenue Fundraising Adoption Fee Capacity -300 -200 -100 0 100 200 300 400 500 600 700 0 1 2 3 ADOPTION FEE G The Effect of Advertisement on Mean Demand Rate Mean Demand Rate Mean Waiting Time Mean Adoption/Rejection Rate Traffic Intensity PAGE 119 119 Figure 4 5. The weight of donations vs optimal solutions -300 -200 -100 0 100 200 300 400 500 600 700 0 1 2 3 4 5 6 7 W2 The Weight of Donations Adoption Fee Capacity PAGE 120 120 CHAPTER 5 CONCLUSIONS In Chapter 5, we summarize the concluding remarks of our analyses from Chapters 2 4. In Chapter 2, we analyze the impact that production and emissions location decisions where the environmental regulations are not taken into consideration there is a trade off betwee n transportation costs and fixed costs. For these problems, a EUFLP3, we capture the impact of emissions taxes, regional production level environmental limits and gl obal transportation emissions regulations. Essentially, we find that when the global limit on transportation emissions is relatively low, then a more dispersed production network is optimal. A low regional production environmental limit should force the company into compliance. However, numerical results illustrate that with regards to the regional production environmental penalty, an increase in the lump sum dollar amount associated with the penalty is much more effective than a decrease in the actual li mit of damage tolerated. When non compliance becomes costly with a large fixed penalty, both the regional production environmental damage and the global transportation emissions are reduced. I n order to reduce the regional production environmental damage a nd transportation emissions the policy makers should choose intermediate limits but high penalties. Furthermore, the companies with low and medium pollution should consider dispersing their network to avoid penalties and reduce their costs. The companies w ith PAGE 121 121 high pollution should resort to other resources for compliance or take the risk of being penalized. In Chapter 3, we identify the optimal adoption fees, and the optimal resource allocation to advertisements and monetary donation increasing programs tha t maximize the impact metrics, the expected return on investment and the donations leftover after operating expenses are paid. We find that, in general, the optimal adoption fee is decreasing as a function of a , if the marginal benefit of advertisements is larger than the marginal benefit of funding. In this scenario, the organization is receiving enough funding as monetary donations by improving its reputation that it can now decrease the adoption fees. The optimal adoption fee is increasing as a function of a, if the marginal benefit of advertisements is high. An animal shelter will increase adoption fees as a response to cover up expenses, and it will decrease fees as donations become sufficient enough to support the expenses. The monetary gain from a is not sufficient to cover up the expenses of a large shelter as the effect of a on donations is indirect (through increasing reputation). Hence, the organization must increase adoption fees. In addition to the dynamics of the optimal adoption fee as a functi on of a and k, we also identify scenarios where the animal shelter may provide incentives for adopters. The optimal resource allocation to advertisement with approximation is decreasing as a function of k. Advertisements and capacity both increase the rep utation of the organization and the mean demand rate, if the marginal benefit of a is large, one unit increase in a will have a significant impact on the objective function especially with a large capacity. Hence, the organization can spare more resources to invest in fundraising. We also observe that is decreasing as a function of the initial mean PAGE 122 12 2 demand rate, enough that the effect of advertisements is insignificant. In Chapter 4, we reformulate the resource allocation problem for both types of shelters to include capacity as a decision variable. To gain further insights to this mixed integer problem, we perform numerous numerical experiments utilizing public data. In general, the results show that when the marginal benefit of advertisements is high, it is better to allocate all resources to advertisements and nothing to fundraising activities or capacity expansion. When the marginal benefit of fundraising is high, the animal shelte r should allocate all resources to fundraising activities. The effect of advertisements, adoption fees and capacity influence the optimal solutions differently. In general, we see a trade off between advertisements, fundraising and capacity. If the marginal benefit of any of these is dominant, the organization should invest all of its resources on this specific activity. The animal shelter will invest in capacity expansion only when per unit operating and per unit capacity expansion costs are low or zero, or the effect of a large shelter on reputation is extremely high. The results to the traditional shelter problem show similar trends except for one special case. If receiving donations is a dominant objective for the animal shelter, then it is optima l to allocate resources to capacity expansion and to advertisements. The organization can also provide incentives for adopt ers to improve impact measur e. PAGE 123 123 APPENDIX A EUFLP DATA Figure A 1. Plant size versus emissions Table A 1. Summary of parameter e stimates Exchange rate Unit sales (mils units) North America Sales (mils units) Capital Exp CAPEX (mils $) Expenses /Output ($/unit) (Net PPE Capex) /Output($/unit) Total Release /output Revenue /output ($/unit) 2010 11.8 General Motors N/A 2.21 N/A 4012.00 58634.10 6887.38 0.25 61334.24 Ford Motor N/A 5.52 N/A 4092.25 55312.53 3455.24 0.51 23346.13 Toyota Motor (87.78 â€¢/$) 7.24 2.10 7169.36 29598.37 8940.58 0.48 30790.89 Honda Motor (87.78 â€¢/$) 3.51 1.46 3547.05 27140.91 5280.84 0.55 28989.13 Hyundai Motor 5.79 1.07 3545.13 15693.97 4380.08 0.08 16822.83 2009 10.60 General Motors N/A 2.04 N/A 5431.00 61685.79 15117.96 0.22 51361.77 Ford Motor N/A 4.87 N/A 4059.00 24602.34 3817.92 0.51 23897.04 Toyota Motor (93.57 â€¢/$) 7.57 2.21 6460.79 29644.47 8623.87 0.35 30065.75 Honda Motor (93.57 â€¢/$) 3.39 1.30 3523.94 25884.29 5535.56 0.52 27030.43 Hyundai Motor 4.95 0.89 3303.68 15220.74 6172.25 0.06 14464.60 2008 13.50 General Motors N/A 2.98 N/A 7530.00 57097.95 10776.92 0.22 49976.18 Ford Motor N/A 5.53 N/A 6492.00 28795.37 3990.06 0.49 25955.17 Toyota Motor ( 103.36 â€¢/$) 8.91 2.96 14324.40 26069.36 8478.91 0.30 26241.32 Honda Motor ( 103.36 â€¢/$) 3.52 1.50 5796.55 27018.25 4260.02 0.62 27539.94 Hyundai Motor 4.21 0.80 3949.80 14464.76 4950.33 0.12 17167.26 y = 0.4323x 63660 RÂ² = 0.7633 y = 169650e 6E 07x RÂ² = 0.5722 0 5 10 15 20 25 30 35 40 0.00 10.00 20.00 30.00 40.00 50.00 60.00 Total Release(Pounds) x 100000 Production (Sales) (units) x 100000 Linear and Exponential PAGE 124 124 2007 16.50 General Motors N/A 3.87 N/A 7542.00 47657.87 9173.78 0.27 46543.57 Ford Motor N/A 6.56 N/A 6022.00 27116.86 4609.76 0.61 23551.33 Toyota Motor (117 â€¢/$) 8.52 2.94 1429.55 21766.32 6354.84 0.40 24010.88 Honda Motor (117 â€¢/$) 3.93 1.85 5590.00 24061.68 4793.51 0.52 26137.15 Hyundai Motor 2.33 0.47 3483.97 12594.82 9403.05 0.21 32116.57 2006 17.10 General Motors N/A 4.13 N/A 7902.00 50979.39 4361.21 0.28 49567.76 Ford Motor N/A 6.70 N/A 6848.00 26435.42 9939.60 0.69 23901.00 Toyota Motor (116.30 â€¢/$) 7.97 2.56 13099.39 20657.44 5976.84 0.44 22682.73 Honda Motor (116.30 â€¢/$) 3.65 1.79 5391.80 24098.41 3417.87 0.55 26104.12 Hyundai Motor 2.36 0.46 4669.68 30187.24 11590.18 0.20 31475.17 2005 17.50 General Motors 4.52 N/A 8179.00 46377.82 7090.53 0.34 42630.37 Ford Motor 6.77 N/A 7516.00 26364.42 4900.40 0.86 26131.96 Toyota Motor (110.22 â€¢/$) 7.41 2.27 594.59 20671.50 7017.39 0.51 22719.36 Honda Motor (110.22 â€¢/$) 3.39 1.68 2610.61 24553.74 4086.96 0.85 26509.26 Hyundai Motor 2.40 0.46 4124.27 27506.05 10447.00 0.95 28782.89 PAGE 125 125 APPENDIX B RESOURCE ALLOCATION CALCULATIONS Proof of Proposition 3: We first prove that the numerator , is nonnegative concave, and the denominator , k is positive and linear. Therefore , is semistrictly quasi concave . We then show that decreasing as a function of for any = Where, = .(Jag erman 1974(23)) Without loss of generality, we remove to simplify expressions when unnecessary. PAGE 126 126 is strictly negative for k >0 and it vanishes for k =0. (See Cardoso (2009), Harel (1990)). Harel (1990) showed that is convex in , we can represent using the methodology from Jagerman (1974). from convexity of the blocking probability. (Harel, 1990). if This shows that traffic intensity is decreasing as a function of , when . We know that: is concave in , is positive linear, the function is semi strictly quasiconcave (Avriel et al., 1988). Adoption Guarantee Shelter Solutions Mean Demand Rate Problem We can write the Lagrange an for the mean demand rate as follows: We can write the KKT conditions and the second order conditions as below: PAGE 127 127 , ( 3 ) ( 4 ) , d ( 5 ) , [ ] = 0 ( 6 ) The bordered Hessian is: is quasi concave when the Bordered Hessian is negative semidefinite if: Insert ( 6 ) into ( 3 ) and get: ( 7 ) Insert ( 7 ) into ( 4 ) : PAGE 128 128 We utilize a heavy traffic approximation from Shakhov (2010): Mean Waiting Time Problem We can write the objective function: We can write the KKT conditions and the second order conditions as below: , ( 8 ) ( 9 ) , d ( 10 ) , [ ] = 0 ( 11 ) The Bordered Hessian is: is quasi concave when the Bordered Hessian is negative semidefinite if: PAGE 129 129 Insert ( 10 ) into ( 8 ) to get: ( 12 ) We know that . Thus, . Insert ( 12 ) into ( 9 ) : We complete our analysis using the heavy traffic approximation from Shakhov (2010): PAGE 130 130 Mean Rejection Rate Problem We can write the Lagrange an for the mean adoption rate as follows: We can write the KKT condit ions and the second order conditions as below: , ( 13 ) ( 14 ) , d ( 15 ) , ( 16 ) The Bordered Hessian is: is quasi concave when the Bordered Hessian is negative semidefinite if: PAGE 131 131 Insert ( 15 ) into ( 13 ) to get: ( 17 ) Insert ( 17 ) into ( 14 ) to get the following equation: Solving this equation we get the exact results Traffic Intensity Problem We can write the KKT conditions and the second order conditions as below: , ( 18 ) ( 19 ) , ( 20 ) , [ ] = 0 ( 21 ) if PAGE 132 132 The Bordered Hessian: is quasi concave when the Bordered Hessian is negative semidefinite if: Equations ( 18 ) and ( 19 ) are first order differential equation with quadratic denominators, thus the exact solution are not shown. We can use the heavy traffic approximation and find: ( 22 ) Insert ( 22 ) into ( 19 ) to find: PAGE 133 133 Mean Adoption Rate Problem We omit this calculation as minimizing is very similar to maximizing . Traditional Shelter Solutions Mean Demand Rate Problem We can write the Lagrange an for the mean demand rate as follows: We can write the KKT conditions and the second order conditi ons as below: , ( 23 ) , , [ ] = 0 The Bordered Hessian: PAGE 134 134 is quasi concave when the Hessian is negative semidefinite if: Insert Error! Reference source not found. into Error! Reference source not found. to get: Insert ( 28 ) into Error! Reference source not found. to get the following equation: We can then use heavy traffic approximation to obtain the results. PAGE 135 135 We utilize a heavy traffic approxima tion from Shakhov (2010): Mean Waiting Time Problem We can write the objective function: We can write the KKT conditions and the second order co nditions as below: , ( 24 ) ( 25 ) PAGE 136 136 , d ( 26 ) , [ ] = 0 ( 27 ) The Bordered Hessian is: is quasi concave when the Bordered Hessian is negative semidefinite if: , We know that . PAGE 137 137 We complete our analysis using the heavy traffic app roximation from Shakhov (2010): Mean Effective Euthani zation Rate Plus Mean Rejection Rate We can write the KKT conditions and the second order conditions as below: , , , [ ] = 0 PAGE 138 138 if The Bordered Hessian: is concave when the Hessian is negative semidefinite if: PAGE 139 139 ( 28 ) Where We can then use heavy traffic approximation to obtain the results. Traffic Intensity We can write the KKT conditions and the second order conditions as below: , , , [ ] = 0 PAGE 140 140 + The Bordered Hessian: is concave when the Hessian is negative semidefinite if: PAGE 141 141 We can then use heavy traffic approximation to obtain the results. PAGE 142 142 Mean Adoption Rate We can write the KKT conditions and the second order conditions as below: , , , [ ] = 0 The Bordered Hessian: is concave when the Hessian is negative semidefinite if: PAGE 143 143 We can then use heavy traffic app roximation to obtain the results. PAGE 144 144 APPENDIX C CHARACTERISTICS OF THE DERIVATIVES OF THE BLOCKING PROBABILITY We can find the first and second order conditions of B: is strictly negative for x>0 and it vanishes for x=0. (The same as but multiplied by G) Harel (1990) showed that is convex in , we can represent using the methodology from Jagerman (1974). And for f we have: if since is strictly convex and decreasing as a function of k. PAGE 145 145 APPENDIX D COMPLETE DATA AND RESULTS Table D 1. Form 990 data Name Contributio ns Fundraising Income Program Service Expenses Highest Adoption Fees Capacit y F tilde Arizona Animal Welfare League 3232146 29670 84980 350 103 4.957916 7 Arizona Humane Society 10627266 796322 210783 200 1745 52.17 Berkeley East Bay Humane Society 886202 0 109381 175 13 34.97 East Bay SPCA 3317673 0 25250 150 89 64.16 Humane Society of Silicon Valley 6040482 413649 46325 350 81 56.1 Humane Society of Boulder Valley 2195486 17910 6800 5 99 102 49.84 Longmont Humane Society 1248359 97969 21475 295 99 37.65 Animal Rescue Foundation 4592358 697600 391908 175 9 60.34 Jacksonville Humane Society 2901660 78252 217976 125 93 51.31 Humane Society of Tampa Bay 2300281 129163 31346 150 9 67.2 Animal Welfare League 1472755 27287 12641 135 751 56.46 Anti Cruelty Society 3982051 10651 168985 170 106 58.68 Tree House Humane Society 4681229 52565 91157 85 0 65.86 Last Chance Animal Rescue 1095700 0 373953 100 25 36.99 Nevada Humane Society 1894871 164740 131348 150 193 51.45 Animal Humane New Mexico 2555674 0 46392 150 135 64.69 Erie County SPCA 5815485 116195 319670 350 170 65.89 Humane Society of Greater Dayton 1067935 163142 79958 300 46 53.44 Society for the Improvement of Conditions for Stray Cats 548737 114489 58172 216 33 65.04 Cat Adoption Team 702413 16973 70596 125 0 50.02 Oregon Humane Society 9027061 695759 338714 100 107 65.28 Dane County Humane Society 2133050 38076 4297 350 61 63.5 San Diego Humane Society 13725255 532326 49120 195 328 62.62 Animal Friends Rescue Project 6991153 422207 164174 150 69 63 Table D 2. Mean Demand Rate Regression Data Adoptions Advertisement Adoption Fee 2011 2010 Difference NAME OF ORGANIZATION: The Haven for Animals 3 158 155 4780 250 AME OF ORGANIZATION: Berkeley East Bay Humane Society 1446 256 1190 109381 175 NAME OF ORGANIZATION: Home At Last Animal Rescue 3378 1307 2071 90 200 NAME OF ORGANIZATION: Purrfect Cat Rescue 2909 1099 1810 1140 125 AME OF ORGANIZATION: Sunshine Rescue 0 179 179 0 100 AME OF ORGANIZATION: Tri Valley Animal Rescue 367 1405 1038 35698 200 AME OF ORGANIZATION: Valley Humane Society 1 5741 5740 0 150 NAME OF ORGANIZATION: Grateful Dogs Rescue 460 1555 1095 754 250 PAGE 146 146 NAME OF ORGANIZATION: Adams County Animal Control 17 535 518 280 100 Funding Regression Results SUMMARY OUTPUT Regression Statistics Multiple R 0.849765336 R Square 0.722101125 Adjusted R Square 0.625590777 Standard Error 2993605.49 Observations 23 ANOVA df SS MS F Regression 4 4.4244E+14 1.11E+14 12.34255 Residual 19 1.70272E+14 8.96E+12 Total 23 6.12711E+14 Coefficients Standard Error t Stat P value Intercept 0 #N/A #N/A #N/A 84980 5.297596507 5.142603223 1.030139 0.315881 350 2082.739242 5788.149713 0.35983 0.722944 103 3920.35509 1695.2334 2.312575 0.032113 4.957916667 54288.97787 27520.91611 1.972644 0.063267 Mean Demand Rate Regression Results SUMMARY OUTPUT Regression Statistics Multiple R 0.294851644 R Square 0.086937492 Adjusted R Square 0.231906259 Standard Error 2602.449591 Observations 8 PAGE 147 147 ANOVA df SS MS F Regression 2 3869211.77 1934606 0.285646 Residual 6 40636463.23 6772744 Total 8 44505675 Coefficients Standard Error t Stat P value Intercept 0 #N/A #N/A #N/A 4780 0.015670769 0.025809413 0.607173 0.565989 250 4.242619088 6.175747382 0.68698 0.517752 Complete Numerical Results Mean Demand Lambda 1000 1400 1800 2200 a 3874889.344 3874889.344 3874889.344 3874889.344 d 0 0 0 0 f 200 200 200 600 k1o 0 0 0 0 Z 19227168.34 19147169.78 19067169.78 19093387.89 Mean Waiting Lambda 1000 1400 1800 2200 a 3874889.344 3874889.344 3874889.344 3874889.344 d 0 0 0 0 f 200 200 200 600 k1o 0 0 0 0 Z 19227168.34 19082612.32 19002612.61 19032227.09 Mean Adoption Lambda 1000 1400 1800 2200 a 3874889.344 3874889.344 3874889.344 3874889.344 d 0 0 0 0 f 200 200 200 600 k1o 0 0 0 0 Z 19084012.32 19004412.61 19034427.09 Traffic Lambda 1000 1400 1800 2200 PAGE 148 148 a 5546194.859 3874889.344 3874889.344 3874889.344 d 0 0 0 0 f 300 200 200 600 k1o 0 0 0 0 Z 19082612.32 19002612.61 19032227.09 Mean Demand Rate Qa 0 4 8 12 a 61361.65546 3876454.007 3876465.436 3876461.5 d 3815070.682 0 4.33256366 0 f 600 600 600 600 k1o 0 0 0 0 Z11 6143896.112 8.41794E+11 1.68359E+12 2.52539E+12 Mean Waiting Time Qa 0 4 8 12 a 62968.92954 3876454.007 3876465.435 3876461.5 d 3813465.203 0 4.332450288 0 f 600 600 600 600 k1o 0 0 0 0 Z21 6140854.146 8.41794E+11 1.68359E+12 2.52539E+12 Mean Rejection Rate Qa 0 4 8 12 a 58602.03873 3876454.007 3876465.435 3876461.5 d 3817830.603 0 4.332450288 0 f 600 600 600 600 k1o 0 0 0 0 Z31 6152179.181 8.41794E+11 1.68359E+12 2.52539E+12 Traffic Intensity Qa 0 4 8 12 a 56427.49969 3876454.007 3876465.435 3876461.5 d 3818014.238 0 4.33254448 0 f 600 600 600 600 k1o 0 0 0 0 Z41 6121069.042 8.41794E+11 1.68359E+12 2.52539E+12 Mean Demand Rate Qf 2 6 10 14 a 3818745.237 3818745.237 3818745.237 3818745.237 PAGE 149 149 d 0.054532664 0.054532664 0.054532664 0.054532664 f 199.905357 199.905357 199.905357 199.905357 k1o 33 33 33 33 Z11 4.46361E+12 4.46364E+12 4.46368E+12 4.46371E+12 Mean Waiting Time Qf 2 6 10 14 a 3876464.103 3876464.103 3876464.103 3876464.103 d 3.156501299 3.156501299 3.156440845 3.156538937 f 600 600 600 600 k1o 0 0 0 0 Z21 4.53107E+12 4.5311E+12 4.53112E+12 4.53115E+12 Mean Rejection Rate Qf 2 6 10 14 a 3876464.103 3876464.103 3876464.103 3876464.103 d 3.156392195 3.156392195 3.156398846 3.156435503 f 600 600 600 600 k1o 0 0 0 0 Z31 4.53107E+12 4.5311E+12 4.53112E+12 4.53115E+12 Traffic Intensity Qf 2 6 10 14 a 3876464.103 3876464.103 3876464.103 3876464.103 d 3.156513028 3.156513028 3.156300363 3.156341207 f 600 600 600 600 k1o 0 0 0 0 Z41 4.53107E+12 4.5311E+12 4.53112E+12 4.53115E+12 Mean Demand Rate Qd 2 6 10 14 a 3.8749E+06 66698.45 502.4242 350.9301 d 0.0000E+00 3809735 3875961 3876117 f 6.0000E+02 600 389.3422 388.7282 k1o 0.0000E+00 0 0 0 Z11 19801390.39 22412371 37684767 53184405 Mean Waiting Time Qd 2 6 10 14 a 3.8749E+06 1145.403 6.235134 0.012138 PAGE 150 150 d 0.0000E+00 3875307 3876464 3876470 f 6.0000E+02 200 387.4443 387.4216 k1o 0.0000E+00 0 0 0 Z21 19740227.15 21408129 37684661 53190540 Mean Rejection Rate Qd 2 6 10 14 a 3.8749E+06 1145.403 115.4886 0 d 0.0000E+00 3875307 3876355 3876470 f 6.0000E+02 200 387.8547 387.4197 k1o 0.0000E+00 0 0 0 Z31 19743607.15 21408961 37688069 53193919 Traffic Intensity Qd 2 6 10 14 a 3.8749E+06 314103.6 2042.342 237.226 d 0.0000E+00 3562366 3874427 3876226 f 6.0000E+02 600 395.0923 161.0943 k1o 0.0000E+00 0 0 0 Z41 19740227.15 22238856 37685186 53180416 Mean Demand Rate w2 0 1 2 3 a 3876470 3876469.201 3876469.201 3876469.201 d 0 0 0 0 f 600 200 200 200 k1o 0 0 0 0 Z11 53694 9.06214E+12 9.06214E+12 9.06214E+12 Mean Waiting Time w2 0 1 2 3 a 3.7838E+06 3876469.201 3876469.201 3876469.201 d 0.0000E+00 0 0 0 f 6.0000E+02 200 200 200 k1o 0.0000E+00 0 0 0 Z21 7467.72002 9.06214E+12 9.06214E+12 9.06214E+12 Mean Rejection Rate w2 0 1 2 3 a 3.5247E+06 3876469.201 3876469.201 3876469.201 d 0.0000E+00 0 0 0 PAGE 151 151 f 6.0000E+02 200 200 200 k1o 0.0000E+00 0 0 0 Z31 4087.7 9.06214E+12 9.06214E+12 9.06214E+12 Traffic Intensity w2 0 1 2 3 a 5.7156E+04 3876469.201 3876469.201 3876469.201 d 1.6269E+06 0 0 0 f 6.0000E+02 200 200 200 k1o 0.0000E+00 0 0 0 Z41 7467.16882 9.06214E+12 9.06214E+12 9.06214E+12 Mean Demand Rate w3 0 1 2 3 4 a 3776466.843 3876464.103 3876464.103 3876464.103 3876464.103 d 3.156586616 0.108108201 0.108108201 3.156549778 3.156531635 f 200 600 600 600 600 k1o 100 0 0 0 0 Z11 4.53106E+12 4.53107E+12 4.53107E+12 4.53106E+12 4.53106E+12 Mean Waiting Time w3 0 1 2 3 4 a 3776466.843 3876469.533 3876469.533 3876464.103 3876464.103 d 3.156361622 0.108108201 0.108108201 3.156391181 3.156391181 f 200 600 600 600 600 k1o 100 0 0 0 0 Z21 4.53106E+12 4.53107E+12 4.53107E+12 4.53106E+12 4.53106E+12 Mean Rejection Rate w3 0 1 2 3 4 a 3776466.843 3876469.533 3876469.533 3876464.103 3876464.103 d 3.15643701 0.108108201 0.108108201 3.156571111 3.156489062 f 200 600 600 600 600 k1o 100 0 0 0 0 Z31 4.53106E+12 4.53107E+12 4.53107E+12 4.53106E+12 4.53106E+12 Traffic Intensity w3 0 1 2 3 4 a 3776466.843 3876464.103 3876464.103 3876464.723 3876466.547 d 3.156307084 3.156226038 3.156226038 0 0 f 200 600 600 600 600 PAGE 152 152 k1o 100 0 0 0 0 Z41 4.53106E+12 4.53106E+12 4.53106E+12 4.53106E+12 4.53107E+12 Mean Demand Rate G 0 1 2 3 a 3766026.959 3818915.856 3818915.856 3818915.856 d 73480.5 0.054532664 0.054532664 0.054532664 f 130.414 199.905357 199.905357 199.905357 k1o 28 33 33 33 Z11 4.40198E+12 4.4638E+12 4.46365E+12 4.46364E+12 Mean Waiting Time G 0 1 2 3 a 3132990.015 3132990.015 3132990.015 3132990.015 d 505594.9465 505594.9465 505594.9465 505594.9465 f 590.5457757 590.5457757 590.5457757 590.5457757 k1o 79 79 79 79 Z21 3.66204E+12 3.66204E+12 3.66205E+12 3.66205E+12 Mean Rejection Rate G 0 1 2 3 a 3132990.015 3132990.015 3132990.015 3132990.015 d 505594.9465 505594.9465 505594.9465 505594.9465 f 590.5457757 590.5457757 590.5457757 590.5457757 k1o 79 79 79 79 Z31 3.66204E+12 3.66204E+12 3.66204E+12 3.66204E+12 Traffic Intensity G 0 1 2 3 a 3132990 3133159.405 3133159.405 3133159.405 d 505594.9465 505594.9465 505594.9465 505594.9465 f 390.1829471 504.4265378 504.4265378 504.4265378 k1o 79 79 79 79 Z41 3.66204E+12 3.66224E+12 3.66214E+12 3.66211E+12 PAGE 153 153 LIST OF REFERENCES Abilock, H., L. 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Her recent research has been publi shed in Springers H andbook of Newsvendor Problems: Model s, Extensions and Applications. She is also a member of the Institute for Operations Research and the Management Sciences(INFORMS), Manufacturing & Service Operations Management Society (MSOM), and Production and Operations Management Society (POMS). |