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PAGE 1 1 OPTIMAL PERSONNEL SCHEDULING UNDE R UNCERTAINTY USING CONDITIONAL VALUE AT RISK WITH APPLICATION TO HOSPITAL PHARMACIST TIMETABLE ASSIGNMENT PROBLEMS By JOHN ROBERT STRIPLING IV A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 John Robert Stripling PAGE 3 3 I dedicate this thesis to the US Air Force whom has given me many great opportun ities and has pushed me to complete my m while working full time PAGE 4 4 ACKNOWLEDGMENTS I would like to first acknowledge Dr. Boginski for his steadfast encouragement and support throughout my thesis research and development Se cond, I would like to thank Judi Shivers. She has enabled active duty military students to pursue their higher education despite a very challenging environment of deployments, multiple out of area business trips, and occasionally confusing bureaucracy. I thank Dr. Pa rdalos for suppor ting my work on this thesis and becoming a co chair on my thesis defense board. Finally, I would like to acknowledge my family and friends. The unending support and encouragement I receive from all of you is one of the great blessings in my life. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ..................... 9 CHAPTER 1 OVERVIEW OF PERSONNEL SCHEDULING OPTIMIZATION ............................. 11 History ................................ ................................ ................................ ..................... 11 Personnel Scheduling Methods ................................ ................................ .............. 12 Optimization o f Conditional Value At Risk ................................ .............................. 14 2 OVERVIEW OF LINEAR AND INTEGER PROGRAMMING ................................ .. 16 Linear Programming Overview ................................ ................................ ............... 16 Integer Programming Overv iew ................................ ................................ .............. 17 3 CONDITIONAL VALUE AT RISK OPTIMIZATION ................................ ................. 20 CVaR ................................ ................................ ................................ ...................... 20 The A pproach ................................ ................................ ................................ ......... 22 Constraints ................................ ................................ ................................ .............. 22 4 HOSPITAL PHARMACY OVERVIEW ................................ ................................ ..... 24 What a Hospital Pharmacy Does ................................ ................................ ............ 24 Background Information o n the Marietta Hospital ................................ ................... 24 Pharmacy Organization ................................ ................................ .......................... 25 5 MODELS AND COMPUTATIONAL RESULTS ................................ ....................... 26 The Objective a nd Constraints ................................ ................................ ................ 26 The Model ................................ ................................ ................................ ............... 29 The Results of t he Case Study ................................ ................................ ............... 34 6 CONCLUSION ................................ ................................ ................................ ........ 38 7 RECOMMENDATIONS ................................ ................................ ........................... 39 PAGE 6 6 APPENDIX A RAW DATA ................................ ................................ ................................ ............. 40 B THE DATA WITH THE OPTIMIZATION PROBLEM ................................ ............... 44 LIST OF REFERENCES ................................ ................................ ............................... 52 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 54 PAGE 7 7 LIST OF TABLES Table page A 2 Raw pharmacy order requests for a second week ................................ .............. 41 A 3 The average pages of requested orders broken down into weekdays and weekends. ................................ ................................ ................................ .......... 42 A 4 Here is the variance of for samples of a Sunday ................................ ................ 43 B 1 The input setup for the weekend solution with the results of the optimizer as the answers. ................................ ................................ ................................ ....... 44 B 3 Weekend constraints ................................ ................................ .......................... 46 B 4 Weekday constraints ................................ ................................ .......................... 48 B 5 Weekday CVaR constraints ................................ ................................ ................ 50 B 6 Weekend CVaR constraints ................................ ................................ ................ 51 PAGE 8 8 LIST OF FIGURES Figure page 3 1 A graphic depictio n of VaR and CVaR [18] ................................ ......................... 21 5 1 The weekday day pharmacy schedule ................................ ............................... 26 5 2 The weekend day pharmacy schedule ................................ ............................... 26 5 3 ................................ .......................... 27 5 4 A table of required Pharmacist for any given hour ................................ .............. 28 5 5 The table of hourly wages based on a given hour of the day ............................. 30 5 6 The cost of starting a pharmacist shift on any given hour of the day .................. 30 5 7 The schedule results of the solver with CVaR Constraints for weekday schedule. ................................ ................................ ................................ ............ 36 PAGE 9 9 Abstract of the Thesis Presented to the Graduate School of the University of Florida in P artial Fulfillment of the Requirements for the Degree of Master of Science OPTIMAL PERSONNEL SCHEDULING UND ER UNCERTAINTY USING CONDITIONAL VALUE AT RISK WITH APPLICATION TO HOSPITAL PHARMACIST TIMETABLE ASSIGNMENT PROBLEMS By John Robert Stripling IV D ecember 2010 Chair: Vladimir Boginski Major: Industrial and Systems Engineering Mathematical optimization has become an indispensable tool in many organizations to drive improved efficiencies There are many techniques to opt imize processes mathematical ly in order to minimize or maximize a given objective function. C onditional V alu e at R isk methodology has become a popular new tool in the optimization community. Implementing a stochastic view of optimization, it permits the confidence level of a soluti on to be part of the optimization process. This theory is based around the C onditional Value at Risk measure which provides an upper bound on the allowed losses or excesses of a solution. We show that it can be applied to general personnel timetable assi gnment problems to provide a robust and useful optimization solution. This developed optimization techniques allow management level confidence in personnel scheduling solutions and accounts for real world uncertainties facing organization s throughout the world. One industry that must contend with variable demand and is being driven to cut personnel costs is hospitals. One aspect of hospital manning that makes a great candidate for personnel timetable optimization is the hospital pharmacy. Most hospital s PAGE 10 10 in the United States have their own pharmacy to handle the large and varied needs of their patient population. These pharmacies employ a variety of different workers. The most expensive employee s and ultimately the driving force in the pharmacy are the pharmacist s Their job is to process, monitor, give guidance, and distribute medicine for that we considered as a benchmark case study in this thesis is a h ospital located in Marietta, Ga. Work ing with the h ospital management, we have been able to get a statistically relevant amount of data on how the pharmacy operates which has opened the door to applying useful optimization techniques to this established way of doing business In this study, w e utilized C onditional Value at R isk constraints in the context of a personnel scheduling integer programming problem to d erive a robust solution The results of the study pro duced a schedule that saves a substantial amount of money per year while better covering high demand times of the pharmacy The result of this study is a new optimized schedule that shows the efficiency of using C onditional Value at Risk constraints in the considered settings PAGE 11 11 CHAPTER 1 OVERVIEW OF PERSONNEL SCHEDULING OPTIMIZATION 1.1 History There has been a great deal of work regarding personnel timetable assignments, or personnel scheduling for short, optimization often with a basic goal of simply automating the work that goes into scheduling. Many personnel s chedules must be created on a weekly basis in a very dyn amic environment where employee availability and business demands are an ever changing variable. Studies have shown that it takes a manager between 8 and 14 hours to schedule 70 100 employees for one week [7] Th ere is e ven data showing that hospitals in particular involved 10% of their personnel in the scheduling process [11] Advanced s tudies have shown that many medium to large sized organizations find that they have established local optimums in particular de partments a t the cost of developing larger scale, organization wide schedule optimum s [17] There is great need and demand for optimization. Reducing labor cost, particular unnecessary overtime charges, while still meeting the high demands of the busines s world are goals of any successful organization Linear and integer programming have often been the method of choice to optimize scheduling problems in th e last 20 years. Though linear and integer programming are powerful method s, they do have limitat ions. Many of the implemented integer programming models involved 100 to 500 variables. Typically these variables represent a reduced set of variables than are actually present in the scheduling equation, for example not putting in inputs such as 15 minu te breaks or employee preference s [7] This shows that often the optimized schedules are based on inc omplete models and fail to account for enough factors to produce the best schedule. PAGE 12 12 There are also examples of scheduling probl ems exceeding the capabili ties of popular software optimization tool s. One limitation of personnel schedule optimization work has shown that human schedulers typically want evaluation tools that show possibilities and can warn of m istakes made in the human schedule versus a mathematically optimized schedule This shows that in practice, human schedulers are usually unwilling to allow the process to be totally automated. Additionally h uman s chedulers almost always want the ability to ignore certain predefined constraints for different situation which form the basis of optimization solution [17] Though this is the reality of trying to solve a personnel scheduling problem there is a good bit that can be done to work with in this real world limitation 1.2 Personnel Scheduling Methods One approach that has been used to enhance mathematical programming as a scheduling optimization approach is to utilize complex dissatisfaction criteria in the formula to optimize Using mathematica l programming, one can adjust weights of certain rules, estimated employee dissatisfaction with particular schedules, estimated profits, and ability to match organizational demands to find an optimum schedule that is based on a big picture view of the oper ation Focusing the scheduling problem on dissatisfaction of the various parties involved versus a more simplified optimum schedule allows the solution to be more applicable in the real world [8] Management is then able to adjust the different weighting as various factors change over time to better represent the ever changing priorities of an organization and create a personnel schedule that meets these priorities. PAGE 13 13 W eighting of dissatisfaction is a management decision highlighting how important manageme n t involvement is in creating effective scheduling tool s Such an approach encompass es individual dissatisfactions such as higher weight for a more senior employee over a junior employee or a particular customer that demands a service or good on a more re strictive timeline The more specific criteria is added to the formula the more complex and large mathematical program gets [8] Issues with computational time as well as the amount of effort and investment required grows quickly as the scheduling prob lem includes more and more factors. A second approach to personnel scheduling optimization that has been utilized in recent years is implementing artificial intelligence into scheduling optimization. This is primarily accomplished by fusing management sci ence with modern artificial intelligence developments. This fusing permits a macro view of optimization, particularly large optimization problems, to be implemented during the optimization process [7] Multiple criteria can be optimized separately then v iewed as a collect ion to determine where local minimums have been created base d on par ticular criteria. Each criterion has a weight assigned to it that aids in the selection of the overall optimum solution. Implemented with a limited decision memory this appro ach creates a tolerance for bad optimization decisions within the optimization process The artificial intelligence approach also allows for new constraints to be added within the optimization problem leading to a more efficient optimization as ad ditional schedules are solved [7] This approach develops an improving process and can avoid some of the local minimum pitfalls of other scheduling optimization methods while PAGE 14 14 actually getting better over time. This method is very sophisticated and requir es the use of complex artificial intelligence to implement. A nother popular approach that is used in modern scheduling optimization is a Bayesian Stochastic model [9] The advantage to this method is that it takes variance from the schedule into accoun t in the optimum solution. The model can be seen as a standard mathematical programming model with two additions. It begins with Bayesian forecasting of the system s statistics to allow for an approximation of the system variance on minimal date. It the n adds a stochastic analysis of what data should be used to give reasonable confidence in the optimization through a normal linear program. The use of effective forecasting strengthens the sto s ke into account the variances of the system being optimized through personnel scheduling creates an important real life cons ideration in the optimization [9] The strength of this method is that it allows the overall method to change over time and still account accurately for the variance of the process. This is accomplished by constraining the data to recent data and using forecasting to enhance the results in order to determine the statistical characteristics of the process. In addition, the stochasti c optimization allows the scheduled solution to better cover the full range of demands and constraints and not just a point prediction. 1.3 Optimization of Conditional Value at Risk The approach s elected for this study is the conditional value at risk optimization of personnel timetable assignments This is a recent developme nt in optimization that allows risk levels to be set while using a stochastic solution to the random variables that are present in most optimization problems. This approach focuse s on the minimizing PAGE 15 15 losses, or when applied to personnel scheduling minimizing the understaffing, based on the variance of the overall system [14] Conditional Value at Risk optimization was primarily developed for financial and insurance industry purpose s but this study shows that the method expands to personnel scheduling optimization as well. This method was chosen to develop an optimized schedule that was not only efficient but would also prevent a major shortfall in critical health care that can occu r in the hospital environment. Further details of this method are describe d later in this document. PAGE 16 16 CHAPTER 2 OVERVIEW OF LINEAR AND INTEGER PROGRAMMING 2.1 Linear Programming Overview Linear Programming is a mathematical method of solving optimizatio n problem s of many different kinds. Linear programming was made possible in 1947 when George Dantzig developed an algorithm to solve linear programming problems called the simplex method [16] Put simpl y, linear p rogramming involves determining values fo r a set of decision variables which will minimize or maximize an object function subject to a set of constraints [12] Through a rather straightforward process, the requirements and goals of real world optimization problems can be put into mathematical fo rmat [6] There are limitations to this approach but overall linear optimization is a very popular method used across a broad spectrum of optimization problems. The simplex method is the method of choice for automating linear programming problems in a c omputer program. R ather elaborate and complex optimiz ation problem can be solved in minutes and often in mere seconds The algo rithm requires that t he objective function, which is maximization or minimization goal of the optimization problem, and the con straints are put into what is called standard form Standard form is an ordered series of equations translated by the system engineer from object to be optimized and the restraints of the problem [8] Once the optimization problem is put into standard fo rm the coefficients and products are converted into a matrix. The first step in the simplex method once this matrix has been formed from the standard form is to find what is known as an entry va riable of the matrix. The entry variable is selected based the objective function largest positive coefficient. A ratio test of each row following row 0 is performed. The row with the largest ratio is determined PAGE 17 17 based on the c oefficient of the row entry variable selected from row 0 divided into t he row solution. The row with the smallest positive ratio becomes the pivot row [16] The matrix is manipulated using elementary row operations focusing on the pivot to create a modified matrix where the entry variable is 1 in the pivot row and 0 in all o ther now 0 rows [16] This process is repeated until all basic variables have been discovered and optimized. 2.2 Integer Programming Overview The difference between integer programming and linear programming is that the input variables to the linear pr ogram must be integers At first glance this may seem somewhat trivial. In cases where the optimum solution inputs are very small such as 2.2 employees to work a shift, the usefulness of the linear programming optimization without integer restrictions c an become minimal Restricting some or all of the input variable s to integers transforms the program into an integer program ming problem Restricting some but not all of the input variables to integers is also known as a mixed integer programming [16] Integer programming is computed different ly than linear programming and has a number of different mathematical properties. Integer programming and mixed integer programming are generally categorized in a computational class called nondeterministic polyno mial time (NP) hard problem s NP hard problems cannot be easily solved computationally and often can become unsolvable due to the very large computational time to produce a solution. Integer programming requires heuristic search methods to determine an o ptimum versus a convergent search method seen in most linear programming algorithms [7] For optimization problems that have large number of variables, selecting an integer or PAGE 18 18 mixed integer programming approach may make the solution impossible to produce with current technology. There are a number of algorithms that have been developed over the years to solve integer problems. Three of the most popular algorithms that are currently used are the Branch and Bound method, Implicit Enumeration, and the Cutt ing Plane algorithm. In general the Branch and bound method is best used for standard integer programming, implicit enumeration is best used for Boolean integer programming, and cutting plane is best used for mixed integer programming [16] LINGO, the pr ogram used to solve the optimization problem set forth in this thesis, and in most integer program solving software uses the branch and bound method. The branch and bound method begins with the basic principle that if one solve s the integer or mixed int eger program as a linear program and the solution is all integers then the optimum integer program is the same linear program optimum. The integer program feasible solution region [16] This al lows the use of some conditional based linear programming to be applied to account for the integer requirement. The branch and bound method begins by taking the equations in standard form, removing the integer requirement, and producing a linear program s olution. It then establishes this output as an upper bound to the integer program since it is a subset of the linear program. If the solution is not an integer programming solution, meaning the solution contains integers for the variables requir ed to be integers then the algorithm moves on to the next step. The algorithm takes an arbitrary integer variable and selects the two integers nearest to the optimum value found in the linear program. From there it branches out two linear programs PAGE 19 19 based on either of the integer values. It will continue branching using the two surrounding integers to the optimal solution until the either the linear program solution is an integer only solution or all the integer variables have been branched off as integers. Work i s saved by pruning branch es and all their respective sub branches when the solution to that branch is either infeasible or yields an output less than an already established possible integer solution [16] Once all the branches have been pruned or solved w ith an integer solution the solution to the integer programming problem has been computed. PAGE 20 20 CHAPTER 3 CONDITIONAL VALUE AT RISK OPTIMIZATION The general premise of optimization using conditional Value at R isk constraints is its focus on minimizing the r isk of losses. This minimization is based a calculated conditional V alue at R isk (CVaR) rather than a more well known Value at R isk (VaR). When VaR is used as the upper bound to losses t here are a number of mathematical difficulties that arise such as la ck of sub additivity and convexity. U tilizing CVaR helps solve optimization problems by work ing with an upper bound that does not have these mathematical difficulties [14] This allows for condi tional CVaR to be used in mathematical programming to determi ne a more useful optimization. Because CVaR VaR this approach can be broadly accepted as a more conservative method that produces a much easier computationally optimization solution [14] 3.1 CVaR One of the most well known measures used in robust optimization under conditions of uncertainty is VaR. The VaR establishes an upper bound for a particular loss distribution [18]. A percentage is assigned, that is the chance expected losses would exceed a set maximum acceptable loss amount, (3.1) VaR is t he most widely applied risk measure for stochastic optimizations mainly because it is conceptually easy to grasp and apply [18]. An attractive alternative to using VaR as a measure of loss in optimization problems is to use CVaR. CVaR is derived by tak ing a weighted average between the value at risk and losses exceeding the value at risk. Mathematically CVaR is broken down to this equation PAGE 21 21 CVaR has the mathematical properties of being sub additive and convex. These properties allow it be used in optimization using mathematical programming [16] CVaR is more conservative than VaR and is a conditional ex pectation measure of potential loss Figure 3 1 A graphic depiction of VaR and CVaR [18] O ptimization solutions based on CVaR will always satisfy the VaR upper bound When the return loss function is normal, the VaR and CVaR solutions are identical [16] With these beneficial properties of CVaR in place a new approach to optimization is made possible PAGE 22 22 3.2 The Approach Let L (x, Y ) be a loss function for a given personnel schedule where the decision va riables for employees working are a vector x T he variable personnel demand s are the y vector standing for the uncertainty of employee requirements This Y vector can be assumed to have a probability density function p(y), but can be relaxed so as not to be strictly adhered to [14] This probabil ity density function can be approximated f or p(y). This can be accomplished through either a derived analytical expression or a Monte Carlo simulation for drawing samples from p(y). The optimization is now formulated to show a probability of not exceed ing an upper bound, with the following equation (3.3) This equation can be manipulated to take into account both VaR and CVaR in to one new function labeled This combined equation is defined as (3.4) Because the min CVaR is equal to min we are able to optimize CVaR and find the VaR in one equation [16] In addition to that quality the equation is convex and linear allowing linear programming to utilized. 3.3 Constraints The basis of optimization through CVaR is that each organization has a specific amount of risk they are willing to take in a given situation. In the case of hospital pharmacy scheduling, the risk they can accept is somewhat high T hat is if the manning happens to fall below demand they get behind and perhaps some problems arise but PAGE 23 23 can be caught up on later shifts In cases such as emergency room manning, the manning must meet demand or people can die. The ability to constrain the amo unt of risk associated with manning optimization is critical to producing a meaningful and effective optimization. Base d on the CVaR constraints explained earlier, a mathematical programming model can be developed that will optimize a personnel schedule w ith adjust able risk associated with that optimization The mathematical programming model will have the standard objective function to minimize and any other relevant constraints that were explaine d in detail in a later section. The key element of the m odel to implement CVaR based optimization is adding the risk constraints. To approximate the random distribution of scheduling demand, scenarios are developed based on observations of the variations and employing Monte Carlo simulations to those observati ons. These individual scenarios and the sum of the losses in these scenarios are factored into the optimiz ation in the following equation [16] (3.5) w probability that the max tolerable loss is not exceeded. S is the number of scenarios used to represent the random distribution and t s is an extr a set of variables. There will be 2S + 1 constraints added to linear ize the model These constraints can be expressed as follows: (3.6) PAGE 24 24 CHAPTER 4 HOSPITAL PHARMACY OV ERVIEW 4.1 What A Hospital Pharmacy Does To continue this thesis, some background information on the hospital pharmacy is in order. Pharmacies are often known from the commercial stores w here prescription medicine is normally given out after a visit to the doctor. In reality, pharmacies are in many different locations and have different roles. Most pharmacies can be easily identified by a few symbols that are used to mark a pharmac y or s omething pharmacy related. There are a number of different kinds of pharmacies. As stated earlier, the well known aspect of pharmacy, this thesis is focused on another kind o f pharmacy Pharmacies within hospitals may have more complex medication management issues whereas pharmacists in community pharmacies often have more complex busines s and customer issues [ 12 ] Another notable difference for the hospital pharmacies compared to other pharmacies is the medicine needed can be very critical and needed in a short period of [4] Due to the complexity, severity, and timeliness required out of the ho spital pharmacy, they must maintain a 24 hours manned schedule that is able to meet the needs of the hospital [4] 4.2 Background Information on the Marietta Hospital The Marietta hospital is run by a health care cooperation The health care cooperati on is a corporation that specializes in providing healthcare services. The corporation currently runs five hospitals in Georgia. In addition, the y provide a variety of other healthcare services and employ over 11,000 people [ 10 ]. The Marietta hospital i s PAGE 25 25 located in Marietta, GA and serves most of central and northern Cobb County. It is not a trauma center and is not a teaching hospital [ 12 ]. The importance of these facts is that trauma centers have fa r more requirements. The Marietta h cy requirements would be very different and more complex if it was a trauma center. There is no educational portion of the hospital pharmacy which would also affect staffing requirements of the hospital pharmacy. 4.3 Pharmacy Organization The Marietta h ospital pharmacy has a fairly flat organizational structure. The pharmacist all work as equals in each shift regardless of seniority. There are pharmacist technicians that take direction from the pharmacists as needed and have their own requirements and scheduling. The pharmacists all report to th e hospital pharmacy manager who typically works from 8 a m to 5 p m each weekday [1] The hospital pharmacy manager reports to the director of hospital pharmacy. He is in charge of overall daily operations o f the many aspects of a running a hospital pharmacy. Oversight of the director is provided by a Pharmacy and Therapeutics Committee selected by the health care cooperation [2] PAGE 26 26 CHAPTER 5 MODELS AND COMPUTATIONAL RESULT S 5.1 The Objective and Constraints The h ospital pharmacy has been operating for many years based on an established pharmacist rotation schedule. This schedule has the advantage s of being simple to administer, of being established, and meets the requirements of both the hospital and the ph armacy staff. The current shifts used are seen in the following tables. Name Shifts Weekday Day Shift 0700 1600 7 Pharmacist Evening Shift 1300 2400 6 Pharmacist Night Shift 2300 0900 3 Pharmacist Figure 5 1. The weekday day pharmacy schedule Name Sh ifts Weekend Day Shift 0700 1600 4 Pharmacist Evening Shift 1300 2400 4 Pharmacist Night Shift 2300 0900 3 Pharmacist Figure 5 2. The weekend day pharmacy schedule The shifts are eight hour shifts for the regular day, and two ten hours shifts for the evening and night shifts That general shift schedule applies for both weekdays and weekends The pharmacists have periods of overlap in their schedules to help high demand hours throughout the day. The weekend has fewer pharmacists because there are f ewer hospital requirements on the weekend. The number of pharmacist working at a given hour is less of an exact science than a schedule that has been effective over the years. The objective of this thesis is to utilized CVaR constraints with an integer p rogramming problem to create an optimum personnel schedule for the hospital PAGE 27 27 Information on the costs that go along with each schedule d pharm acist has been provided by the hospital pharmacy management different based on a number of factors but they are all paid hourly not salary The average hourly wage of $50 per hour is used for this thesis. It is a good average pay for the pharmacist based on an overall assessment of pharmacist pay Certain hours of the 24 hour d ay have extra hourly pay associated with it as compensation for working outside of normal hours. This incentive pay structure is seen in the following table. Hours Shift Incentive Pay 0700 1600 $0.00 per hour 1700 2200 $5.00 per hour 2300 0600 $8.00 pe r hour Figure 5 3. The pharmacist shift incentive pay table These amounts are added to the hourly wage of the pharmacists to give an incentive for working a less desirable schedule. This pay scale is included in the objective formula for minimizing c osts. The required number of pharmacist per hour is not an established value. It has been informally established by the standard set schedule. Thanks to the management s oftware used by the Marietta hospital p harmacy the hospital data with pages of req uests for pharmacy support broken down per hour was provided for this thesis The raw number of requests per hour of the pharmacy can be found in Appendix A. The break down according to pharmacist professionals at the hospital is that a pharmacist can ha ndle 27 pages of requests per hour. By taking multiple days of pharmacy requests data and determining an average, a pharmacy requirement can be established based on the hour of a given day Based on the collected data from the h ospital PAGE 28 28 p harmacy and the c onsensus that a pharmacist typically han dles 27 requests per hour a requirement for pharmacists per hour was created. This requirement is seen in the following table Figure 5 4. A table of required Pharmacist for any given hour As seen from the above table the required pharma cist broken into two schedules, weekday and weekend. This is based on the pharmacist staff requirements. The pharmacy staff has a few requirements that must be factored into the optimization problem. The pharmacists require that they have a steady schedule over Time Period Weekdays Pharmacists Weekend Pharmacists 0 3 2 1 2 2 2 2 2 3 2 2 4 2 1 5 2 1 6 2 2 7 2 2 8 5 4 9 6 4 10 7 6 11 8 7 12 8 7 13 7 6 14 7 6 15 7 6 16 8 7 17 7 5 18 7 4 19 5 3 20 4 3 21 4 3 22 3 3 23 3 3 PAGE 29 29 the week and a steady schedule over the weekend. This means that whatever optimized schedule is created must be the same for all the days of the week and a consistent schedule for the two days of the weekend. Another requireme nt of the pharmacists is that when they work a shift th ey must work either a full eight hour or ten hours shift. As seen from above the requirements are broken down into a weekend day and weekend requirement to match the scheduling requireme nts. The eight hour and ten hour shift requirement is factored into the objective function. 5.2 The Model The model was formulated based on the objective goal and constraints. The objective function is to minimize the cost of emplo ying pharmacists while still meeting the required pharmacists need ed by the hospital for any given hour in the day, 24 hours a day, and seven days a week. Microsoft Office E used as the tool to solve the optimization problem. Excel was c hosen because it is used by the Marietta hospital staff it is easy to use and its uses the branch and bound as its solver algorithm. The first step to applying integer programming on this problem is by putting the objective and constraints into stand ard form. The first equation of the standard form is the objective function. To develop the objective function the cost of starting an eight hour shift or a ten hour shift at any given hour must be calculated. The first step of this process is to establ ish the hourly wage of a pharmacist per hour of the day based on the incentive pay and the average base pay of $50 dollar an hour. The following table is a break down of the hourly wage of an average pharmacist working on a given hour. PAGE 30 30 Time Period Hourly Wage 0 000 $58.00 0 1 00 $58.00 0 2 00 $58.00 0 3 00 $58.00 0 4 00 $58.00 0 5 00 $58.00 0 6 00 $58.00 0 7 00 $50.00 0 8 00 $50.00 0 9 00 $50.00 10 00 $50.00 11 00 $50.00 12 00 $50.00 13 00 $50.00 14 00 $50.00 15 00 $50.00 16 00 $50.00 17 00 $55.00 18 00 $55.00 19 0 0 $55.00 20 00 $55.00 21 00 $55.00 22 00 $55.00 23 00 $58.00 Figure 5 5. The table of hourly wages based on a given hour of the day Once the hourly wage per hour has been calculated the cost of starting a pharmacist to work either an eight hour or ten h our shift on any given hour of the day must be calculated. These calculations are in the following table. Time Period 8 Hour Shift 10 Hour Shift 0 000 $456.00 $556.00 0 1 00 $448.00 $548.00 0 2 00 $440.00 $540.00 Figure 5 6. The cost of starting a pharmac ist shift on any given hour of the day PAGE 31 31 Time Period 8 Hour Shift 10 Hour Shift 0 3 00 $432.00 $532.00 0 4 00 $424.00 $524.00 0 5 00 $416.00 $516.00 0 6 00 $408.00 $508.00 0 7 00 $400.00 $500.00 0 8 00 $400.00 $505.00 0 9 00 $400.00 $510.00 10 00 $405.00 $515.00 1 1 00 $410.00 $520.00 12 00 $415.00 $525.00 13 00 $420.00 $530.00 14 00 $425.00 $538.00 15 00 $430.00 $546.00 16 00 $438.00 $554.00 17 00 $446.00 $562.00 18 00 $449.00 $565.00 19 00 $452.00 $568.00 20 00 $455.00 $571.00 21 00 $458.00 $574.00 22 00 $461.00 $5 69.00 23 00 $464.00 $564.00 Figure 5 6.Continued With the cost of starting a pharmacist on a given hour on either of the shifts is calculated the objective function can be completed. The objective function is seen below. (5.1) Where is the number of pharmacist starting an eight hour shift i = 0 or a ten hour shift i = 10 23. The number of pharmacist starting for a particular hour and shift is multiplied times the costs of an eight or ten hour shift variable with an eight hour shift i = 0 or a ten hour shift i = 1, on is from 0 to 23 PAGE 32 32 The constraints are calculated based on the number pharmacist s required per hour to meet the h ospital requirements. These requirements are found in Table 3 4. These constraints are put into standard form as follows ad ding up the eight hour shift pharmacist s from the last seven hours and the ten hour shift pharmacists from the last nine hours plu s the shifts starting during that hour to add up the total number of pharmacist working at that given hour. The total number must equal or be greater than the required number of pharmacists required. The constraint put i nto standard form is as follows (5.2) Where is the number of pharmacist starting an eight hour shift i = 0 or a ten hour hour of the requirement, and k is either the weekday, I = 0, or weekend shifts, i = 1. The is the from Table 3 4 which is the number pharmacist required for that hour of the day where j is the hou r of the day, 0 through 23 There are 24 requirements based on the general form seen in figure 3 3. Each one corresponds to an hour of a 24 hour day. Constraints must be put on the inputs themselves. The variables must be greater than zero and that they must be integers This allows the overall standard form equation to be formed. The requirement that the input variables must be integers equates to a pharmacist must be a whole unit and cannot be negative which reflects PAGE 33 33 reality. The ability to use a pharmacist in a fraction form is not in line with the pharmacist shift requirements. These two requirements put into standard fo rm are seen in the following figure. (5.3) The combined integer programming optimization problem in standard form to be optimized is seen (5.4) The final constraints and variables to add to the stand ard form equation for this optimization problem are those relating to the CVaR constraints. There must be a constraint for loss tolerance shown here (5.5) The loss function is defined in this optimization problem as (5.6) PAGE 34 34 This loss function is converted into a constraint in standard form with the t s variable s introduced to take into account the random distribution of requirements observed for this optimization. The CVaR constraints are (5.7) T he complete CVaR optimized integer problem to be solved with the included CVaR constraints is (5.8) 5.3 The Results of the Case Study To first get a standard integer programming solution as a base for the implem entation of CVaR constraints, the standard form without CVaR constraints based on the average demand per hour was implemented. The solver was used on the PAGE 35 35 objective functi on with the goal of minimizing costs The integer option was selected for all inputs The solver was run for an optimized weekday schedule and weekend schedule each. The following are the results of both optimization calculations. Weekday Schedule: 02 00 1 person starts a 8 hr shift and 1 person starts a 10 hour shift 0400 1 perso n starts a 10 hr shift 0800 3 people start an 10 hr shift 0900 1 person starts an 8 hr shift 1000 1 person starts a 10 hr shift 1100 2 people start an 8 hr shift 15 00 1 person starts an 8 hr shift 1800 2 people start an 8 hr shift and one perso n starts a 10 hr shift Figure 5 5 The schedule results of the minimization solver for the weekday schedule Weekend Schedule: 0300 2 people start a 8 hr shift 0600 1 person starts a 10 hr shift 0700 1 person starts an 8 hr shift 0900 2 people s tart an 8 hr shift 1000 2 people start an 8 hr shift 1100 1 person starts an 8 hr shift 1600 2 people start a 10 hr shift 1800 1 person starts an 8 hr shift Figure 5 6 The schedule results of the minimization solver for the weekend schedule The raw results and spreadsheet answers generated by the solver as well as the set u p of the standard form equations in excel are all found in Appendix B. The number of input factors for the equations is 48 inputs The file used for the optimization is avai lable for further analysis The results of the integer programming were encouraging. The cost of running the ph armacy for a week day with the current schedule is $7,847 .00 and the cost of running the pharmacy during a weekday with the optimized schedule is $6,647.00. The PAGE 36 36 cost of running the pharmacy on a weekend day with the current schedule is $5,412.00 and the cost of running the pharmacy on a weekend day with the optimized schedule is $5,349.00. The cost saving over a year equate to $312,000 for the weekday shift and $6,552 for the weekend shift. The overall yearly saving based on this optimization schedule is $ 318,552 .00 Just as important as costs is that the number of pharmacists is better distributed for the demands of the hospital. With this baseline established, the CVaR constraints were added to the standard form of the personnel scheduling problem. A Monte Carlo method was used to pick from the available data on the pharmacy orders per hour to create ten scenarios used for the implementati on of the CVaR constraints. Working with the pharmacist team the value for w as set at 0.9 The loss function was implemented for each of the ten scenario s with the variables added as constraints The resultin g calculations produced the following results. Weekday Schedule: 0 700 5 pe ople start a 8 hr shift 09 00 2 p eople start a 8 hr shift 1100 2 people start a 8 hr shift 12 00 1 person starts a 8 hr shift 1500 4 people start a 8 hr shift 2300 3 pe ople start a 8 hr shift Figure 5 7 The schedule results of the solver with CVaR Constraints for weekday schedule Weekend Schedule: 0300 1 person starts a 8 hr shift 0600 1 person starts a 10 hr shift 0700 1 person starts a 10 hr shift Figure 5 8. The schedule results of the solver with CVaR Constraints for weekday schedule. PAGE 37 37 0800 1 person starts a 8 hr shift 09 00 2 people start a 8 hr shift 1000 1 person starts a 8 hr shift 1100 1 person starts a 8 hr shift 1500 1 person starts a 8 hr shift 1600 1 person starts a 10 hr shift 1700 1 person starts a 10 hr shift 2300 1 person starts a 8 hr shift Figure 5 8. Continued With the CVaR constraints factored in the costs of running the pharmacy on the weekdays is $ 7144 .00 per day and for the weekends is $ 5464 .00 per day. This show s substantial savings in costs for the weekdays compared to $7,847.00 for the current schedule The weekends costs actually are higher than the current schedule. This is because the current weekend schedule ca uses the pharmacy to get undermanned and behind on orders during some high demand weekends The costs of running the pharmacy on a weekend day is $5,412.00 unoptimized versus $5,464 .00 base d on the CVaR constrained optimized schedule This slight increa se in cost allows much better pharmacy coverage of hospital demands on the weekend s This difference in costs of the new CVaR optimized schedule compared to the current schedule is $182 780 .00 for weekdays and $ 5,408 .00 for the weekends over the span of a year. Overall this equates to $177,372 in saving per year while better covering the needs of the hospital and giving a statistically significant solution This amount is less than the $318,552.00 saving based on integer programming without the CVaR cons traints but ensures that the pharmacy will have coverage during higher demand times while still saving money based on the current schedule. PAGE 38 38 CHAPTER 6 CONCLUSION The res ults of the optimization of the personnel scheduling problem utilizing CVaR constrain ts show an effective application of this new theory T his thesis prove s that CVaR constraints can easily be added to personnel scheduling problem s The possible substantial amount of yearly saving s is a great example of how money can be saved utilizing m athematical programming to find a more optimal way of doing business while at the same time allowing organizations to determine acceptable levels of risk to achieve those cost savings Translating all the constraints and inputs into standard form and esta blishing the integer programming parameters took time and effort but was straight forward and easy to alter once initially setup. The i nteger program with CVaR constraints setup made it very easy to import in more data as demand changed over time This s howed how this model is easy to alter for future needs With a few easy changes the hospital pharmacy can alter how much risk they are willing to take with their pharmacy schedule to better serve their needs In conclusion, integer programming with CVaR constraints was successfully applied to a real world optimization problem in this thesis proving that CVaR is viable and useful method to producing personnel timetable assignments PAGE 39 39 CHAPTER 7 RECOMMENDATIONS The results of the optimization problem are val id and would enhance pharmacist coverage of need ed hospital requirements while saving money. Though the results appear great in this thesis the reality is that the resulting schedule is more complicated than the previous schedule There is a great deal of pharmacist s coming and going as the shifts are very spread out. This would be a major shift in the way business is done by the pharmacy. Upon showing the results of this opt imization problem to the management of the Marietta pharmacy there was great r eluctance to implement such a scheduling change. The results speaks for themselves, the schedule easily shows how many pharmacist end up working at any given hour giving the manager confidence that the sche dule does indeed cover the hospital needs The r eal challenge with this optimization is people. Change can be difficult and it would require significant management push to implement this for the day shift pharmacist s in particular who are typically senior and prefer their current schedule If manageme nt is willing to make the personnel push and work through the employees reluctance then I would recommend implementing this new schedule. As stated previously, this sc hedule would save money, handle the hospital needs more effectively and have the risk o f being short staffed factored into the scheduling decisions PAGE 40 40 APPENDIX A RAW DATA Sunday Monday Tuesday Wednesday Thursday Friday Saturday 65 69 79 55 78 73 69 50 49 66 59 70 56 55 34 35 31 62 73 76 37 44 44 39 48 52 63 53 25 44 58 75 42 46 33 44 55 48 43 63 41 34 57 57 55 37 63 66 46 50 62 53 59 76 58 55 82 142 114 132 110 147 142 111 148 162 182 173 149 118 140 176 153 193 204 198 215 164 228 203 196 202 244 249 217 199 201 185 198 222 237 193 157 208 202 152 186 193 195 204 185 178 178 177 177 139 194 170 217 179 227 199 126 196 167 201 258 202 220 118 191 165 192 174 195 128 85 187 167 190 145 172 112 80 116 138 117 113 147 89 64 106 97 124 113 112 80 77 98 98 93 139 148 65 63 61 92 73 120 131 98 57 87 69 78 110 83 89 Table A 1. One week of raw pharmacy requests broken down by hour PAGE 41 41 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 78 69 56 55 71 74 69 70 49 23 59 90 71 55 73 35 38 62 23 37 37 52 44 41 48 18 48 53 42 44 48 75 64 48 33 63 55 39 43 59 36 34 63 57 38 37 58 46 46 76 62 64 59 91 81 55 110 142 132 132 134 150 142 173 148 152 182 190 180 118 204 176 176 193 212 195 215 202 228 242 196 232 217 249 198 199 227 185 222 199 237 152 157 197 202 210 207 193 178 204 183 178 172 175 177 179 194 178 217 199 232 199 258 196 247 201 192 220 220 174 191 186 192 183 188 128 145 187 169 190 168 181 112 113 116 118 117 159 130 89 113 106 83 124 129 108 80 139 98 85 93 120 112 65 120 61 78 73 101 92 98 110 87 76 78 85 87 89 Table A 2 Raw pharmacy order requests for a second week PAGE 42 42 Time Period Weekdays Weekend 0 67.9 70.25 1 59.2 57.5 2 47.2 45.25 3 44.5 50.5 4 54.4 33.25 5 48.2 43.75 6 51.4 53 7 66.5 59 8 133.5 119 9 166.6 130 10 187.6 193.5 11 218.8 216 12 203.7 22 2.25 13 187.8 182.75 14 183.4 181.75 15 200.7 179 16 208 206 17 185.7 137 18 175.6 113.5 19 127.1 92.75 20 110.2 84.25 21 108.4 86.5 22 88.2 94.75 23 84 86.25 Table A 3. The average pages of requested orders broken down into weekdays and weeke nds. PAGE 43 43 Time Period Sunday Sunday Sunday Sunday Avg VAR 0 65 43 48 55 52.75 90.91667 1 50 44 53 76 55.75 196.25 2 34 40 58 41 43.25 106.25 3 44 44 48 30 41.5 62.33333 4 25 48 38 35 36.5 89.66667 5 44 54 40 25 40.75 144.9167 6 57 56 38 30 45.25 1 79.5833 7 50 66 63 46 56.25 94.91667 8 82 116 113 132 110.75 436.9167 9 111 151 117 136 128.75 333.5833 10 140 185 185 168 169.5 451 11 164 253 204 210 207.75 1326.917 12 217 249 203 224 223.25 370.9167 13 193 121 210 195 179.75 1591.583 14 195 128 183 176 170.5 864.3333 15 139 124 197 198 164.5 1489.667 16 126 87 181 219 153.25 3408.25 17 118 117 160 89 121 856.6667 18 85 83 93 82 85.75 24.91667 19 80 50 60 56 61.5 169 20 64 93 70 64 72.75 190.25 21 77 62 78 57 68.5 112.3333 22 63 55 53 74 61.25 90.91667 23 57 66 60 74 64.25 56.25 Table A 4. Here is the variance of for samples of a Sunday PAGE 44 44 APPENDIX B THE DATA WITH THE OPTIMI ZATION PROBLEM Weekend 8 Hr 0 1 2 3 4 5 6 7 8 9 10 11 0 0 0 2 0 0 0 1 0 2 2 1 10 Hr 0 1 2 3 4 5 6 7 8 9 10 11 0 0 0 0 0 0 1 0 0 0 0 0 12 13 14 15 16 17 18 19 20 21 22 23 0 0 0 0 0 0 1 0 0 0 0 0 12 13 14 15 16 17 18 19 20 21 22 23 0 0 0 0 2 0 0 0 0 0 0 0 Table B 1. The input setup for the weekend solution with the results of the optimizer as the answers. PAGE 45 45 Weekday 8 Hr 0 1 2 3 4 5 6 7 8 9 10 11 0 0 1 0 0 0 0 0 0 1 0 2 10 Hr 0 1 2 3 4 5 6 7 8 9 10 11 0 0 1 0 1 0 0 0 3 0 1 0 12 13 14 15 16 17 18 19 20 21 22 23 0 0 0 1 0 0 2 0 0 0 0 0 12 13 14 15 16 17 18 19 20 21 22 23 0 0 0 0 0 0 1 0 0 0 0 0 Table B 2. The input setup for the weekday solution with the results of the optimizer as the answers. PAGE 46 46 0 Shift 0 0 1 0 0 0 0 0 3 >= 3 0 0 2 0 0 0 0 0 0 0 1 Shift 0 0 1 0 0 0 0 0 3 >= 3 0 0 2 0 0 0 0 0 0 0 2 Shift 0 0 0 0 0 0 0 0 2 >= 2 0 0 0 2 0 0 0 0 0 0 3 Shift 0 0 0 2 0 0 0 0 2 >= 2 0 0 0 0 0 0 0 0 0 0 4 Shift 0 0 0 2 0 0 0 0 2 >= 2 0 0 0 0 0 0 0 0 0 0 5 Shift 0 0 0 2 0 0 0 0 2 >= 2 0 0 0 0 0 0 0 0 0 0 6 Shift 0 0 0 2 0 0 0 0 3 >= 3 0 0 0 0 0 0 1 0 0 0 7 Shift 0 0 0 2 0 0 0 1 4 >= 3 0 0 0 0 0 0 1 0 0 0 8 Shift 0 0 2 0 0 0 1 0 4 >= 4 0 0 0 0 0 0 1 0 0 0 9 Shift 0 2 0 0 0 1 0 2 6 >= 5 0 0 0 0 0 0 1 0 0 0 10 Shift 2 0 0 0 1 0 2 2 8 >= 7 0 0 0 0 0 1 0 0 0 0 11 Shift 0 0 0 1 0 2 2 1 7 >= 7 0 0 0 0 1 0 0 0 0 0 12 Shift 0 0 1 0 2 2 1 0 7 >= 7 0 0 0 1 0 0 0 0 0 0 13 Shift 0 1 0 2 2 1 0 0 7 >= 7 0 0 1 0 0 0 0 0 0 0 14 Shift 1 0 2 2 1 0 0 0 7 >= 7 Table B 3. Weekend constraints PAGE 47 47 0 1 0 0 0 0 0 0 0 0 15 Shift 0 2 2 1 0 0 0 0 6 >= 6 1 0 0 0 0 0 0 0 0 0 16 Shift 2 2 1 0 0 0 0 0 7 >= 7 0 0 0 0 0 0 0 0 0 2 17 Shift 2 1 0 0 0 0 0 0 5 >= 5 0 0 0 0 0 0 0 0 2 0 18 Shift 1 0 0 0 0 0 0 1 4 >= 4 0 0 0 0 0 0 0 2 0 0 19 Shift 0 0 0 0 0 0 1 0 3 >= 3 0 0 0 0 0 0 2 0 0 0 20 Shift 0 0 0 0 0 1 0 0 3 >= 3 0 0 0 0 0 2 0 0 0 0 21 Shift 0 0 0 0 1 0 0 0 3 >= 3 0 0 0 0 2 0 0 0 0 0 22 Shift 0 0 0 1 0 0 0 0 3 >= 3 0 0 0 2 0 0 0 0 0 0 23 Shift 0 0 1 0 0 0 0 0 3 >= 3 0 0 2 0 0 0 0 0 0 0 Table B 3. Continued PAGE 48 48 0 Shift 0 0 2 0 0 0 0 0 3 >= 3 0 0 0 0 1 0 0 0 0 0 1 Shift 0 0 2 0 0 0 0 0 3 >= 3 0 0 0 0 1 0 0 0 0 0 2 Shift 0 0 1 0 0 0 0 0 3 >= 3 0 0 1 0 1 0 0 0 0 0 3 Shift 0 0 1 0 0 0 0 0 3 >= 3 0 0 1 0 1 0 0 0 0 0 4 Shift 0 0 1 0 0 0 0 0 3 >= 3 0 0 1 0 1 0 0 0 0 0 5 Shift 0 0 1 0 0 0 0 0 3 >= 3 0 0 1 0 1 0 0 0 0 0 6 Shift 0 0 1 0 0 0 0 0 3 >= 3 0 0 1 0 1 0 0 0 0 0 7 Shift 0 0 1 0 0 0 0 0 3 >= 3 0 0 1 0 1 0 0 0 0 0 8 Shift 0 1 0 0 0 0 0 0 6 >= 5 0 0 1 0 1 0 0 0 3 0 9 Shift 1 0 0 0 0 0 0 1 7 >= 7 0 0 1 0 1 0 0 0 3 0 10 Shift 0 0 0 0 0 0 1 0 7 >= 7 0 1 0 1 0 0 0 3 0 1 11 Shift 0 0 0 0 0 1 0 2 9 >= 9 1 0 1 0 0 0 3 0 1 0 12 Shift 0 0 0 0 1 0 2 0 8 >= 8 0 1 0 0 0 3 0 1 0 0 13 Shift 0 0 0 1 0 2 0 0 8 >= 7 1 0 0 0 3 0 1 0 0 0 14 Shift 0 0 1 0 2 0 0 0 7 >= 7 Table B 4. Weekday constraints PAGE 49 49 0 0 0 3 0 1 0 0 0 0 15 Shift 0 1 0 2 0 0 0 1 8 >= 8 0 0 3 0 1 0 0 0 0 0 16 Shift 1 0 2 0 0 0 1 0 8 >= 8 0 3 0 1 0 0 0 0 0 0 17 Shift 0 2 0 0 0 1 0 0 7 >= 7 3 0 1 0 0 0 0 0 0 0 18 Shift 2 0 0 0 1 0 0 2 7 >= 7 0 1 0 0 0 0 0 0 0 1 19 Shift 0 0 0 1 0 0 2 0 5 >= 5 1 0 0 0 0 0 0 0 1 0 20 Shift 0 0 1 0 0 2 0 0 4 >= 4 0 0 0 0 0 0 0 1 0 0 21 Shift 0 1 0 0 2 0 0 0 4 >= 4 0 0 0 0 0 0 1 0 0 0 22 Shift 1 0 0 2 0 0 0 0 4 >= 4 0 0 0 0 0 1 0 0 0 0 23 Shift 0 0 2 0 0 0 0 0 3 >= 3 0 0 0 0 1 0 0 0 0 0 Table B 4. Continued PAGE 50 50 S = 10 Overall CVaR Constraint C 19.94218809 0.9 18.71247 <= 20 1 2 3 4 5 6 7 8 9 10 396.387 306.387 191.3874856 0.38749 318.3874856 113.387 199.3874856 244.387 110.3874856 157.3874856 < < <= <= <= < <= <= <= <= t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 ts 22.55508 1.739802 1.409276532 4.393267 1.757499393 1.243737 1.42542377 1.54461 1.237800083 1.348162704 ts Max 22.55508 1.739802 1.409276532 4.393267 1.757499393 1.243737 1.42542377 1.54461 1.237800083 1.34 8162704 Table B 5. Weekday CVaR constraints PAGE 51 51 S = 10 Overall CVaR Constraint C 7.483129122 0.9 19.73049 <= 20 1 2 3 4 5 6 7 8 9 10 275.405 16.4047 265.4046653 193.405 35.40466533 87.4047 54.40466533 148.405 173.4046653 94.40466533 <= <= <= <= <= <= <= <= <= <= t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 ts 0.516578 1.781172 0.519341936 0.522869 0.523697179 0.517978 1.208647597 20.50992 0.53083104 0.58258303 ts Max 0.516578 1.781172 0.519341936 0.522869 0.523697179 0.517978 1.208647597 20.50992 0.53083104 0.5 8258303 Table B 6. Weekend CVaR constraints PAGE 52 52 LIST OF REFERENCES [ 1 ] Pharmacy Department Policy and Procedure Wellstar Health System, DPP #II 01, August 2008 [ 2 ] Pharmacy Department Polic y and Procedure Wellstar Health System, Aug 2008, DPP# I 01 [ 3 ] D. C. Montgomery, Design and Analysis of Experiments 7 th edition, John Wiley & Sons, Inc., New Jersey, USA, 2009 [ 4 ] D. J. Patil, Hospital And Clinical Pharmacy 2 nd edition, Nirali Prakashan Inc, Gultekdi, India, 2008 [ 5 ] D.P. Ahlfeld, P.M. Barlow, and A. E. Mulligan, GWM a ground water management process for the U.S. Geological Survey modular ground water model {MODFLOW 2000}, U.S. Geological Survey Open File Report 2005 1072, 124p. [ 6 ] F. Glover and Comput. & Ops Res. Vol. 13, No.5, pp. 563 573, 1986 [ 7 ] J.K. Karlof, Integer Programming Theory and Practice Taylor & Francis Group, Boca Raton, FL, USA, 2006. [ 8 ] M. S. Baza raa, J.J. Jarvis, and H. D. Sherali, Linear Programming and Network Flows 4 th edition, John Wiley & Sons, Inc, New Jersey, USA, 2010 [ 9 ] M.I. Andreica R Andreica and A. Andreica Minimum Dissatisfaction Personnel Scheduling Proceedings of the 32nd Cong ress of the American Romanian Academy of Arts and Sciences (ARA Congress) pp. 459 463, Boston, USA, 2008 [ 10 ] Office of the Professions, www.op.nysed.gov/prof/pharm/ taken March 2010 [ 11 ] P.D. Causmaecker, P. D emeester, G. V. Berghe, and B. Verbeke Analysis of real world personnel scheduling problems Information Technology Gent, Belgium 2008 [ 12 ] R. A. Wurbs, Computer Models for Water Resources Planning and Management IWR Report 94 NDS 7, Texas A&M, USA, July 1994 [ 13 ] Optimiser, a linear programming spreadsheet InfoWorld vol. 4, no. 20, pp. 34 35, May 1982 PAGE 53 53 [ 14 ] R.T. Rockafellar. and S. Uryasev, At The Journal of Risk Vol. 2, No. 3, 2000, 21 41 [ 15 ] S Agne The I mpact of Pesticide Taxation on Pesticide Use and I ncome in Costa Rica 's Coffee P roduction Publication Series Special Issue No. 2, Pesticide Policy Project, IICA Biblioteca Venezuela, March 2000 [ 16 ] S. Uryasev and P. M. Pardalos, Stochastic Optimization: Alg orithms and Applications Kluwer Academic Publishers Dordrecht, The Netherlands, 2001 [ 17 ] at Risk: Optimization Algorithms and Financial Engineering News vol. 14, February 2000. [ 18 ] V. L. Boginski, C. W. Commander, time Optimization Letters, Vol. 3, pp. 461 473, 2009 [ 19 ] W. L. Winston and M. Venkataramanan, Operations Research Volume One Introduction To Mathematical Pro gramming, 4 th edition, Thompson Learning, California, USA, 2003 PAGE 54 54 BIOGRAPHICAL SKETCH John Robert Stripling IV (Robert) was born in New Orleans LA in 1979 to Catherine Ann Stripling and Dr John Robert Stripling III Robert grew up with a half sister, Mo nica Davison Hertzbach, 14 years his elder and a younger brother, Nathan George Stripling, 18 months his junior. In 1994 Robert and his brother moved up to Atlanta GA for the remainder of their pre college education. Robert and his brother attended one of the top college preparation schools in the Atlanta area, Greater Atlanta Christian School. Robert became one of the top students in high school while also lettering in football, tennis, and wrestling. He became student body president his s enior year and was set to begin college after high school graduation in 1998. Once acceptance letters were received, Robert selected Vanderbilt University in Nashville, TN as the school of choice for his higher education. In August 1998 Robert began his undergraduate degree of undecided. In the second semester of his sophomore year he changed his degree to computer engineering. During these initial years, Robert was active outside of the class room. He became a cheerleader for the u niversity In the second semester of his sophomore year Robert joined the Sigma Alpha Epsilon fraternity the largest frater nity in the country at the time Robert graduated in May 2002 with a Bachelor of Engineering and entered a He found work at web design. After six months of looking for an engineering job, Robert decided to join the Air Force. By July, 2003 he had started ba sic officer training. After basic officer PAGE 55 55 training, he developing network operations capabilities working under the Air Force Information Warfare Center. After three years, his next assignment was to Eglin AFB in Fort Walton Beach, FL. There he managed two offices of government electronic warfare engineers supporting the combat Air Force It is at E glin AFB that Robert began his m degree at the University of Florida pursuing a degree in industrial and systems engineering which began June 2008 