Stationary analysis and optimality conditions for (o,S) policies in multi-commodity inventory control problems by Nabil S. Faour

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Stationary analysis and optimality conditions for (o,S) policies in multi-commodity inventory control problems by Nabil S. Faour
Faour, Nabil Sami, 1940-
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Carrying costs ( jstor )
Cost allocation ( jstor )
Cost analysis ( jstor )
Cost efficiency ( jstor )
Cost functions ( jstor )
Delivery costs ( jstor )
Minimization of cost ( jstor )
Operating costs ( jstor )
Unit costs ( jstor )
Variable costs ( jstor )
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Industrial and Systems Engineering thesis Ph. D
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ACKNOWLEIXIEMM'S I would like to ejq^ress my thanks to my advisor. Professor B. D. Sivazlian, for his advice and guidance. I am also grateful to Professors J. F. Bums, R. G. Blake, and Z. R. Pop-Stojanovic for their assistance, encouragement and support. Thanks goes also to Mr. Philip L. Raphals for his fast and excellent typing. Finally, to my wife, goes special thanks for her unending patience and encouragarent. This research was partially st^jported by Project THEMES, ARD-D Contract nAH-CX)4-68 CX)002. 11


TABi:£ OF CXKTENTS Page ACKNOWLEDGEMENTS ii ABSTRACT V Chapter I. INTRC»UCnCN .' 1 1 . 1 Introduction 1 1 . 2 Literature Revisff 6 1.3 Definition of the Problan and an Outline of the C3iapters 20 II. MAIHEMATICAL PRELIMINARIES 26 2 . 1 Introduction 26 2 . 2 The Generalized Convolution 26 2.3 On the Solution of an Integral Equation 36 2 . 4 Other Mathematical Concepts 46 III . MATHEMATICAL FORMULATION 48 3.1 IntroductiCTi 48 3. 2 Mathematical Formulation 49 IV. THE OPTIMIZATIOJ PRDBLEiyi 71 4 . 1 Introduction 71 4 . 2 Characterization of T 72 4.3 Necessary and Sufficient Conditions 76 4.4 Geanetric Reforniolation of the Problem 98 4 . 5 The Functicxi L (x) 100 V. OCMPUTATIOSIAL ASPECTS OF THE PROBI£M AN EXAMPLE 103 5.1 Introduction 103 5.2 Exaitple 104 5 . 3 Ccnputational Aspects 105 5.4 Numerical Exarrple 116 111




Abstrac± of Dissertation Submitted to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirotents for the Degree of Doctor of Philosophy STATIONARY ANALYSIS AND OPTIMALTTY OONDITICNS FOR (o,S) POLICIES IN MULTIHXM^DDITY INVHWORY OCWTROL PRDBLEJ^ By Nabil S. Faour March, 1972 Chairman: Professor B. D. Sivazlian Major Department: Indxastrial and Systems Engineering An m-oonmodity (m >^ 1) inventory control model with periodic review is formulated and studied when a dyadic stationary ordering (a,S) policy in e"' is used. It is assumed that: (1) the ordering decision at the beginning of each of a sequence of periods of time is affected by a single set v^ cost, a linear variable ordering cost and holding and shortage costs; (2) the demands for the items in each period are identically and independently distributed continuous random variables; (3) the delivery of orders is immediate; and (4) catplete badklogging of unfilled demands is allowed. Next the model is analyzed by minimizing the e^qjression for the stationary total expected cost per period. The set of simultaneous equations used to determine the optimal policy parameters are restated in terms of a real valued function in E . Finally, the optimization analysis is restricted to the special case of a twoocmtodity inventory oontrol problem v^ere the danands for the items obey the expcxaential distribution, and the holding and shortage costs are linear. The integral equation vdiich is used in solving for the optimal


policy parameters is cxinveirted into a linear hyperbolic partial differential equation of the second order with boundary conditions. This boundary value prcblan is solved analytically. The solution is then used to determine the optimal policy parameters. In a numerical exanple a corputer program is developed to determine the cptimal policy parameters. VI


CHAPTER I INTRODUCTION 1.1 Introduction Ihe replenishment and control of inventories is a caution practice in the fields of business, eocnanics and management in general. Sinply, an inventory is a stock of physical goods v^ch are held or stored for future use, sale or production. Inventory problems are integrated in nature. Ihey may involve production, scheduling, distribution of catmodities, or oarbination of all. In the past twenty years, much work has been done to formulate and apply the proper mathanatical tools for determining economic decision rules in managing the varioxas inventory systems . Viewed as decision problems , the analysis of such systems essentially proceeds in three steps: (1) the formulation of a model expressing a set of relations between a set of decision variables whose values are to be determined; (2) the establishment of an objective function of the decision variables so as to minimize or maximize an eoononic objective; and (3) a numerical method to determine the values of the decision variables that optimize the objective function. In this introduction we shall restrict ourselves to models in v^ch decisions are made at the beginning of each of a nunber of eqiaal periods of time, and in vArLch the demands are indqpendently and identically distributed random variables from period to period. The decision at the beginning of each period is affected by a fixed set up cost, a


variable ordering cxDst and a holding and shoirtage costs. Backlogging and irtmediate delivery of orders are allcv/ed. IWo principal distinct approaches have Been used in analyzing such inventory models both in theory and practice. In the first aj^roach, the syston is viewed as a multistage decision process and the technique of dynamic progranming is enployed in finding the optimal policy that minimizes the total expected cost over the duration of the process. To ensiire that the optimal policy is of the "sinple type" certain restrictions are inposed on the cost factors that affect the decision rules. More precisely, when the ordering cost is linear and v^en the sum of holding and shortage costs L(x) is convex, the optimal policy is of the "sinple type." However, vrtien the duration of the process is infinite, a second approach is often used: A "sinple type" ordering policy is chosen to be used in each period. Under the chosen sinple policy the sequence of inventory levels at the beginning of each period forms a Markov process, whose stationary behavior can be analyzed. Here, instead of optimizing the total cost over the horizon period, \4iich is meaningless, we minimize the stationary total expected cost per period . The optimal values of the decision variables are called the steady-state or stationairy solutions to the inventory problem. Sane of the advantages of the stationary approach over the dynamic progranttiing ^proach are: 1. The stationary ^proach provides us with information about the dependency of the optimal policies on the many parameters involved in the model and about sensitivity of costs as function of the policies, while this information is not provided by following the dynamic progranming approach.


3 2. As the notiber of ocnttodities increases, the nuirber of the state variables increases drastically, and the use of dynamic progranining techniques in obtaining cptimal policies for the inventory problem will be tims-consutning even vdien using the fastest ccnputers. 3. Frequently, for lew-cost high turnover items, optimal policies are not really required. It is sufficient to obtain sinple analytical approximations to optimal policies, and these are <±)tained from the stationary ^jproach. The main ooncem of this research will be to find the stationary solutions to the m-ocmnodity (m >^ 1) inventory prc±)lem. For the one ocmtodity, the theory of inventory has studied the statiaiary behavior of the system and provided cptimal decision rules for inventory replenishment under a linear ordering cost and a convex holding and shortage costs function. The results of the one-ocnmodity inventory system are valid for multi-ocmnodity inventory system where each comnnodity is treated independently. However, in practice this independency assarption is unocrrtDn and it is more realistic to incorporate the interdependency in the set up order cost of the oortmodities \4iich arises in the nature of the following types of problems. 1. Ordering different ccimodities at the same time from a oamcai vendor may reduce the procurement ordering costs, thus incurring one ordering cost. 2. Several warehouses used as storage and distribution points are si;53plied by a single factory. Whenever an order to replenish a given commodity stored by the warehouses is


made to the factory, a fixed set up oost is incurred whether the warehouses order individually or simultaneously. The configuration of this systan is shown in Fig. (1.1) . The two problans are similar in structure and it is practical to take advantage of the reduced procurement cost in controlling the inventory of the cottnodities . Furthermore, if one can solve the multiocmnodity prcijlem, then the multi-warehouse problem, as given previously, is solved. The procurement cost of ordering £ >^ 0^ units is assumed to be equal to a set ip cost plus a linear purchase cost T C Z = (Cj, Co,..., c ) where c^ >_ 0, i = 1,.. . , m, is the unit purchasing cost of item i. The fixed set up oost is incurred only if an order is made. Thus if z > units are ordered, the fixed cost will be a function of z, say K(z) In general K(z) will take as many different values as the nunber of alternative different ordering decisions. For the two conmodity prcblen K(z) = Ki K2 K if Zi > and Z2 = if zi = and Z2 > if Zj > and Z2 > if Zi = and Z2 = v*iere Ki, K2 and K are all nonnegative, and the inequality max (Kj , K2) 1 K < Ki + K2 is satisfied. Thus , the procurement cost is given by K{z) + c'^z


•^ Pm


1.2 Literature Review Vfe shall first survey the literature on the steacty-state or stationary solutions to the one-camodity inventory system operating \3nder a (s,S) policy; next we shall elaborate on the work done on the stationary solutions to the m-comnodity inventory system operating under a (o,S) policy. For the one-coimodity problem we shall consider a c^amic inventory nvodel. At the beginning of each period, a decision is made to order up to S if the stock falls below s, otherwise, nothing is ordered. The denands, {D.}, in successive periods (i = 1, 2, . . .) are assured to be described by a sequence of non-negative random variables, independently and identically distributed with a joint density function '^'ri ^ * ^ • Delivery of orders is inmediate and total backlogging of unfilled dotiands is assumed. The ordering decision in each period is affected by a set vp cost K, a linear purchase cost cz where z is the quantity ordered and c the unit cost, a holding cost for carrying inventory, and a shortage cost for not meeting demands. L(x) represents the expected inventory holding and shortage oosts for being in stock X at the beginning of the period. Pollowing the pioneering work of Anrcw, Harris, Marchak [1], Karlin [2] studies the steady state for this model operating under a (s,S) policy. With this ordering policy, the sequence of inventory levels Xi , X2 , . . . , X , . . . , v^ere X. is the inventory level at the end of the ith period, forms a Markov process. Karlin shews that the distribution of X^ coverages in the sense of distributions to a stationary distribution with the density f {•) given by


f(x) = f 'P (S X) 1 + 4'(S s) S-s (S s) + I i|;(t) ^(S X t) dt for s < X < S (1.1) 1 + 4'(s s) for -<» < X < s vAiere i|;{x) = f (|>^"Nx) n=l and 'i'(x) = j; -D^^Nx) n=l 4) ( * ) and $ ( • ) are the well-known syntxDlic notations for the n^ fold convolution of (()(•) and $(•), respectively. Itie stationary distribution is clearly independent of any econoTdc consideration. By inposing the assumed cost structure on the process the total expected cost per period denoted by g(s, S-s) is given as K + L(S) + L(S x) i/)(x) dx g(s, s s) = TTim "" "^ r J n (1.2) vdiere Q = S s and \i = E(D) / E(*) is the expectation operator.


Greenberg [5] obtains f (•) , the transient soluticn for the (s,S) inventory system. Using the generating function approach, the stationary distributic^ f ( • ) is given by the sta n da r d Abelian theorem lim " f (x) = (1 z) I z"f^(x) z-^l n=l v^ch results in (1.1) . Iglehart [7] shows that the optinal values of s* and Q* minimizing g(s, S s) are given as the solution to the following equations L'(Q + s*) + L ' (Q + s x) ij; (x) dx = K + L(Q* + s*) + L(s*) = L(Q* + s x) \l){x) dx 1 + "PCQ*) where L ' ( • ) is the derivative of L ( • ) • Another approach in studying the (s,S) inventory model is follcwed by Sivazlian [11] . He defines f (x) to be the total expected cost of operating the system for n periods when no order is placed initially, i.e. , s <_ x < S, h (x) to be the total expected cost of operating the system for n periods v^en an order is placed initially, i.e., -°° < X < s. Then f (x) and h (x) satisfy the following functional equations


'X~S f (x) = L(x) + 1 f .(x t) 4)(t) dt h (x t) 4) (t) dt s < X < S (n > 1) x-s h (X) = K + c(S x) + L(S) + n rS-S f T (S t) ^ (t) dt n-i + 1 h , (S t) (J> (t) dt -» < X < s (n >^ 1) J S-s "~ Starting with these basic functicaial equations and using Howard's ^proadi [6] he deduces the stationary total ei^ected cost per period as given by (1.2) . The optimal values for s and Q = S* s* that minimize the total ejqaected cost per period are given as the soluticais to the set of equations ,* /-.* M(s*,Q ) = K and 9M(s*,x) = x=<3" vihere ,r„/ * VI _ L{L(s*) L(s* + x)} L{M(s ,x)} 1 lU(x)} the operator I. { } is the Laplace transform operator.


10 Several corputational itethods for c±)tairujig values of s and Q* are given and elaborated later by Sivazlian.^ For the case of gartma distributed demand and linear holding and shortage costs, a dimensional analysis is carried to reduce the number of variables. A numerical inversion technique of the Laplace transform e:^ressians using Gaussian quadrature is used to solve for s* and Q* . For large values of Q, Sivazlian [11] and Roberts [3] shew that s* is the solution to the equation ^^ [1 *(v)] dv = hTT-^ I ^' ^ ^ ^ ^^^^1 and Q* /„ y , where y and a^ are the mean and the variance of the randan variable D, h is the unit holding cost, and p the unit shortage cost. Sivazlian [11] also shows that for small Q,s* and Q* are the solutions for the following approximate system of equations L'(S*)=-P L"(s*) =|^+p4,(0) It is inportant to note that an integral equaticn of the renewal type in one dimension is solved by each of the authors in carrying their analysis. In Scarf's work [2] , the density function f (•) as given by (1.1) is a solution of a renewal type integral ^See additional references.


11 equation. Sivazlian [11] in finding the total expec±ed cost per period solves a renewal type integreil equation. For the case v*ien the desriand is discretely distrib\ited, Veinott and Wagner [16] develop the nvethod of stationary or renewal analysis for the ccrputation of an optimal (s,S) policy. The total es^jected cost per period is given by S-s K + L(S) + I MS j) i|)(j) g(s, S s) = — + c^ ^ 1 + H'(S s) which is a discrete version of (1.2) . Ihe algorithm provided for searching the (s,S) policies to find one that minimizes g(s,S s) consists of two steps. Step 1. Determine lower bounds (s,S) and upper bounds (s,S) on s and S . St^ 2. Find the collection of all (s,S) policies that minimizes g(s,S s) over the class of (s,S) policies falling within the bounds defined in Step 1. Any policy in the oollection is optimal. An interval bisection or a Fibonacci search technique is then eitployed if A ju JL to find the optimal values of s and Q = S s . Note that this procedure does not guarantee the optimality of g{s,S s) since, in general, g(s,S s) is not a unimodal function in s and Q.


12 RDr other work done in the area of one-ocninodity inventory, the survey articles of Scarf [9] and Veinott [18] should be noted. Let us now t\xcn to the m-ocrmodity (m >^ 1) prctolem. As itost practical problems in inventory involve more than one product, there has been substantial interest recently in studying the theory of multi-camodity inventory problons. The prcblem we shall consider is the familiar dynamic inventory system with periodic review [17] . Danands, {D.}, for the items over a sequence of successive periods (i = 1,2,...) are assumed to be described by a sequence of nonnegative random variables, independently and identically distributed. No specific restrictions will be placed \;?x3n the statistical properties of the danands for corfttxiLties within a given period; thus, in general, the donand distribution is given by a joint density function j^it) (t > 0) . Ocnplete backlogging and iimiediate delivery of orders are assured, and as previously discussed the ordering decision in each period is affected by a single set up cost, a linear ordering cost and holding and shortage costs. Following Sivazlian [13], we define the following: In if", let n = {x|3« £S} v*iere x = (x^ ,X2,. . . ,xj is the inventory levels of all itars prior to a decision and S = (Si ,82 , . . . ,S^) is a non-negative vector. Let the hypersurface that subdivides the set Q into the subsets a and a^ be defined inplicitly by an equation of m variables, Z( S x° ) = 0, x°efi, v^ere the real-valued function Z( S x) has the following properties:


13 1. Z( S x) is defined and ocaitinuoxos for V xefl 2. Z( S x) < V xeo° 3. Z( S x) > V xea Define r = {xjxe^, Z( S x ) = 0} and u)(x) = {t|t £ x; t, x eo^]; The stationary policy considered is to order all itans for xea . For the case of two ccmnodities , Fig. (1.2) illustrates gecmetrically the decision regions. It is inportant to note that for x^n, the (a,S) policy is not feasible. However, once a stock level reaches a level xefi, the policy is and remains feasible. With this ordering policy, the sequence of inventory levels Xi ,X2 , . . . ,X , . . . , vdiere X. is the inventory level of all items at the beginning of the i^h period, forms a Markov process since the demands are independently distributed frcxn one period of time to the next and since the (a,S) policy depends only on the last state of the process. Sivazlian [10] and [13] studies the stationary behavior of the systan for the special case v^en T is admissible, i.e., rnA(x)= (})(null set) where A(x) = {t|x < t < S , xca^} . If W(^) denotes the distribution function of the stationary distribution of of the stock levels imttediately after a decision, then


14 6M(y) = M if Y = S; \l) {y) *^ if ^ edc v^ere 8<^ = a^ S, M is the probability of placing an order, and \1){Y) satisfies the integral equation ii (y) = M4> ( S y ) + 4) (x) (j) ( x y ) dx


15 o^ = aj(S) Fig. (1.2). (a,S) ordering policy for a to^-cxjimodity inventory system.


16 Sivazlian [10] shows that the solutican of the integral equation is given by (n) 'J^{y) =M I <})^"' ( S y) n=l v*iere (x) with itself and M = 1 + li^(t) dt S-teu (S) The optimal decision variables Si, S2 , . . . ,S and the oonfigioration of the admissible set r are derived by minimizing the total ej^sected cost per period given as m g(a,S) = E(C) = KM + I [c.E(D.) +h.E(Y.) + (h. +p.) E(B.)] (1.3) where C is the total cost per period; and for items i = l,2,...,m c. = variable cost/unit h. = holding cost/ijnit at the end of a period p. = shortage oost/unit at the end of a period B. = unfilled demand at the end of a period Y^ = inventory level at the end of a period


17 E(*) denotes the expectation operator. For the two-oonmodity problem v^en the demands for the items are independently and exponentially distributed with (f* j^ (t . ) = X . e~ """ "^ <_ t. < », X^ > (i = 1,2) , the optimal policy is investigated for the twD given classes of adnissible T Case 1. Here r is the set of points for v^di Xj (Sj xi) + X2(S2 X2) = a vy^iere a is an unknown positive parameter to be determined. Case 2. Here T is the set of points for whioh itax [Xj (Sj Xj) bj , X2 {S2 X2) b2] = v^ere bj and b2 are unknown positive parameters to be determined. For both cases the optimal decision variables are ootputed. Moreover Sivazlian shews by numerical exaitples that the joint (a,S) policy yields a lower operating cost than the case when eaoh item is ordered separately. In another paper Sivazlian [12] considers the multiooninodity control problem operating vinder a stationary (a,S) policy. The joint density function of demand for all items in a given period is given by *(t) = n 4). (t.) < t. < 0°, (i = l,2,...m) i=l 1 1 1


18 viiere 4) (t.) = X.e ^ ^ (i = 1,2, . . . ,in) . T is admissible and here it i 1 1 is the set of points for vghich Xi(Si Xi) + X2(S2 X2) + ... + X^(S^ X J = a a > In this case, let A = lij(t) dt 's-t£a)(S) By generalizing the concept of Dirichlet's multiple integrals, Sivazlian shews that A can be ejq>ressed as a single integral . Next/ similar ejqsressions for E(Y.) and E(B.) are obtained. For the case v^en the variable a takes on large values, explicit analytic solutions for the optimal values of S and a are obtained by minimizing (1.3) . It is also shewn that when m, the nvj±>er of itans, beccres very large and a is finite, the prc±iability of placing an order in any given period tends to 1, and that it is almost optimal to use the "order 14) to" policy. For the twD-ocniTodity problan, Wheeler [20] follows Iglehart's [7] approach for the one-cannodity (s,S) model in showing that , . C (x) g(a ,S ) = n>o° n v^ere C (x) is the total expected cost of operating the system for n periods, given that x is the initial inventory level and an optimal policy is used in each period, and g(a*,S ) is the optimal stationary total e3q)ected cost per period. Pollcwing Greenberg's [5] notations


19 and procedure, Wheeler also atta:npts to find the transient solutions for the probability distribution of the stock levels. Wacker^ formulates and stiidies the multi-comiodity inventory problem with periodic review v^en a stationary m-dimensional (s,S) policy is followed. Following the work of Roberts [3] he gives analytical formulas for determining the asyrptotic reordering region v^en the set up and shortage costs have large values. For the case when the demand D in each period is discretely distributed with probability mass function <\>^P^it) , vrfiere x is the stock level at the beginning of the period, and (J'^^^i/O) < 1/ Johnson [8] uses the policy iteration method to choose the optimal (a,S) reorder policy which minimizes the stationary total expected cost per period. Further he provides a ccnputaticnal algorithm for finding bounds and characterizing the optimal policy. The algorithm is a search procedure and only attenpts to find the ocaif iguraticn of the optimal policy pointwise. Finally, the work of Soland [15] is briefly mentioned. In solving some renewal processes in two-dimensions, Soland solves a oontinxious review inventory problem involving two products for the case v^en the demand is discretely distributed. In minimizing the total expected cost per cycle (defined as the time between ordering) he concludes that the optimal policy is to order both products at the end of a cycle. Here again, as in the one-ocmnodity prcblan, a renewal type integral equation in m-dimensions is solved by each of the authors in carrying their analysis. ^See acMitional references.


20 A more ejqslicit ctefinition of the problem is given next. 1.3 Definition of the Problem and an Outline of the Ch^ters A multi-oamiodil^ inventory system with periodic review operating lander a stationary (a,S) policy is considered. The ordering decision in each period is affected by a single set up cost K, a linear variable ordering cost c = (cj , Cj , . . . ,c ) , and the expected holding and shortage cost function, L(x) , conditional on being in stock level x = (xi , X2,...,xJ at the beginning of a period; L{x) is assuned to be twice dif f erentiable . Danand, {D.}, for the items over a sequence of periods (i = 1,2,...) , is assumed to be independently and identically distributed oontinuous non-negative randan variables with continuous joint density function 4'p,(t) . Delivery of orders is irmediate and total backlogging of unfilled donands is assumed. Under a stationary (a,S) policy, as discussed before, either all items will be ordered to bring the inventory level to S = (Si, Sz, . . . ,S^) if X, the inventory level at the beginning of a period prior to making a decision, is in a (ordering region) or nothing is ordered c if xeo (not ordering region) . The principal concern of this study is to find the optiniality condition for (a,S) policies. This is done by minimizing the ej^ression for the stationary total expected cost per period with respect to the decision variables that characterize the policy being used. The study begins in Chapter II by introducing seme mathematical concepts that are needed in subsequent chapters. First the concept and properties of a generalized convolution operator defined on locally


21 integrable real-val\aed functions of in (m > 1) variables are introduced; next the solution of an integral equaticsi of the renewal type in m dimensions is studied. In Ch^ter III the analysis for deriving an analytical ejqjressicn for the stationary total ejqaected cost per period, proceeds in three steps: (1) the basic functional equations which relate f (x) , the total es^jected cost of operating the system for n periods v^en starting with xea , and h (x) , the total expected oost of operating the system for n periods vrfien starting with xea, are derived; (2) for large n it is shown that f (x) = ng + u(x) and h (x) = ng + v(x) ; \diere g is the n— — n— "^ stationary total ejqaected cost per period and u(x) , v(x) are f\jnctions of the initial stock levels; and (3) an ancilytical ej^ression for g, which is a function of (a,S) , is derived by solving a renewal type integrail equation that results frcm the use of the asyrnptotic ejqaressions for f (x) and h (x) in the fvmctional equations. In Chapter IV, the optimality conditions for (a,S) policies are studied. The minimization of the ejqnression for the stationary total expected cost per period with respect to the decision variables proceeds as follows : First, it is shewn that at optimality r = {x|xeJ^; L(x) C* = 0}, viiere C* is the minimum value of g(a,S) excluding the variable ordering oost. Then the necessary and sufficient conditions for the existence of prcper relative minima of g(a,S) at the pair (S*,C ) are established. In Chapter V, the stucfy considers the two-oomnodity inventory control problem v*iere the demands for each ocmnodity, in a given period.


22 are independent and exponentially distributed, and vAiere the holding and shortage costs for each coimodity are linear. The equations used to solve for the optimal policy parameters are analytically determined. In a numerical exarrple a ootputer program is developed to determine Sj*, S2* and C*. Finally, in Chapter VI recamendations for future research are presented. As carpared to other works, in the m-ocmnodity (m > 1) inventory control problem c^jerating under a stationary (a,S) policy, this research is more general, in the sense that no assurtptions are made about the configuration of the ordering region or the admissibility of r. The caaly restriction on the set r = {x|xefi, Z ( S x) = 0} that subdivides the set fi into the ordering region, a, and the not ordering region, a , is that the real valued function Z( S x) is continuous and defined for all xef2. jybreover the ordering regions must be sinplyconnected. The ordering regions considered by other researchers, in the m-oonmodity problem for oontinuoios demand, are all special cases of the general case considered here. In E^, Fig. (1.3) shows geonetrically the configuration of sane of the regions that are considered in this research. Fig. (1.4) shows some ordering regions that will not be considered. The m-dimensional ordinary convolution operation used by other researchers in studying the system ixnder consideration is a special case of the generalized convolution operation introduced in Ch^ter II. For the case when the set r is admissible, the generalized cxjnvolution of any fxmction reduces to the ordinary convolution and


23 SzJfS Sa^ Si (Fig. (1.3)


24 Fig. (1.4).


25 all the generalized oonvolxition operation properties with the ccmnutative law hold. The analysis \ased in deriving the e3q)ression for the stationary total expected cxDst per period is similar to Sivazlian's [11] approach for the one-ocxtinodity problem when operating \jnder a (s,S) policy. The asynptotic ej^ressions for the costs of operating the system are deduced by ^:pealing to the results of Yosida and Kakutan [22], The set of equations that will be used to determine the c^timal parameters of the adopted policy are similar in form to the set of equations used by Sivazlian [11] in detentdning the optimal values of s* and Q* = S* s* for the one ocrmodity problem.


CHAPTER II MATHEMATICAL PRELIMINARIES 2.1 Introduction In this chapter a number of mathematical concepts that will be needed in subsequent chapters are discvissed. First, the concept and properties of a generalized convolution operator defined cai locally integrable real-valued functions in m(m >^ 1) variables are studied and then the solution of a renewal type integral equation is given. 2.2 The Generalized Convolution Let S be a positive vector in a space of m (m >^ 1) dimensions ; i.e., S > £ <=> Sj^ > 0, i = l,2,...,m. In e'", define the set U = {x I X j^ S} / and let the hypersurf ace that subdivides the set fi into the sx±)sets a and a^ be defined inplicitly by an equation of m variables ; Z ( S-x° ) = , x" efi , where the real-valued function Z ( S-x) have the following properties: 1. Z ( S-x) is defined and continuous for Vxefi; 2. Z (S-x) < Vxea^ 3. Z (S-x) > Vxea Next, define the sets: (a) (ij(S) = {x | xefi; Z ( S-x) < 0}; (b) u (x) = { t I t, xeo) (s_) ; t £ x} ; (c) r = {x I xe^; Z(S2X) = 0}; (d) 0) ' (t ,x) = {v I t

27 (e) r"(x) = {t I t <_x; J^wCS)}; (f) r'{x) = r"(x) a)(x) I^t us make the transformation u = x-t , tew (x) , then the set R(x) is the image of the set w(x) , r^ is the image of T , and Rj ( x-t) is the image of w' (t,x) . In the particular case ra = 2, Fig. (2.1) illustrates geometrically the effects of the transforTtBtion on the above defined sets. For v^R(x) , define the set Rj (u) = (t | 1 t <^ u; teR(x)}. Consider the set of functions D, f (t)eD is a real valued function of ^(2.1) m variables (t, ,t, , . . . ,t ) for which we have * ^ m 1. f (t) = f or t < 2. f (t) is locally integrable on Rj (u) On this set of functions, addition and g-convolution operations are defined as follows: rvaf 1. Addition (f 4g) (t) = f (t) +g(t); Def f 2. g-cJonvolution (f*g) (u) = I f ( u-v) g(Y_) dy verTT^) then the following relations are satisfied 1. f(t), g(t)eD => f(t) + g(t)eD 2. f (t) + g(t) = g(t) + f (t) (Connutative Law) 3. {f(t) + g(t)} + h(t) = f(t) + {g{t) + h(t)} (Associative Law) where the equality sign is used in the ordinary sense. And the set of functions D is an abelian group vdth respect to addition in v^ch the zero element is the functicm f (t) =0 and the inverse of the functiCTi f (t) is f (t) .


28 y


29 Properties of g-Convolution 1. Closure Law If f (t), g(t)eD, then h(u), given by h(u) = f f(u2t) g(t) dt ^teRi (u) belongs to D. Proof ; h(u) is locally integrable on RjCu) since both functions f and g are locally integrable on R , (u) . Moreover h(u) is identically zero for u < . 2. Distributive Law If f , g, h, e D then [f * (g+h)] (u) = (f * g) (u) + (f * h) (u) Proof : By definition [f * (g + h)] (u) = f f(u^ [qiY) + h{^)] 6^ = f f(u2^) g(jr) d^ J^^eRi (u) +f f(ur^) h{Y) ^ ' ^gRi{u) = (f * g) (u) + (f * h) (u) 3. Associative Law If f , g, e D then [f * (g * h)] (u) = [ (f * g) * h] (u) Proof ; By definition [f * (g * h)] (u) = [ f (u-v) I g(v-y) hiY) djr_ dv •'veRj (u) ^Y.^Ri (v) = f [f(u-v) g(v-y) h(Y) d^. dv (2.2)


30 v^ere W = ( {v_,y) | veR (u) , ^eR (v) } vMch is equivalent to the set W. defined as 2 Wj = { iy.'Y) ! Y. <. Y. 1 y.' Z' Ye^i ^H.^ ) Also by definition we can write [{f * g) * h] (u) = I [ f f( u-y-t) g(t) dt] TMy) ^ (2.3) Let t = V Y^, then (2.3) becones [(f * g) * h] (u) = I [ I f(u2v) g(v2V) dv] h(y) d^ VeRj (u) v-yeR^ ( u-y) Since g ( v-y ) = for v-y < £, v-ye R^ ( u-y) implies that yt^^ (Z'H.^ ' where r' (jl»u) ={vl Y.l_vl_UrJ!ERi (u)) • Hence we have [(f * g) * h] (u) = j r I f (u^v) g(v2r^) dv] h(j^) djr ^Rj (u) \cR' (y,u) = 11 f^Hli:) g



32 4. CormTutative Law ; We observe that in general, if f , geD, then (f*g) (u) ^ (g*f ) (u) By definition (f*g)(u) = f f (u t) g(t) dt teRi (u) Let u t = v; then (f*g) (u) = I f (V) g(u:^) dv ?^ {g*f) (u) veRj (u) viiere Ri (u) = {v| u-V £Ri (u) }. Note : the g-oonvolution of feD with itself ccmnutes. Moreover , if we define for n = 1,2, f(l)(u) = f(u) f (n) (E) = (f (n-l)*f) = j f (n-1) ^Hlt) f (t) dt (n ^ 2) teRi (u) then f ran the g-oonvolution associativity law we can write f(n)(ll) = j f(n-l)(y=^) f(t) dt = j f(u^) f(^_i)(t) dt teRj (u) teRi (u) (n >_ 2) If we let u-t = v; then Ri (u) is mapped into R^ (u) arxJ we can write f(n)(ii) = j f(n-l)

33 (Z.X) X2 X2-y2 R{x) >A i• xi-yi Ri ( x-y) H Ri (u) Ri ( x-y) 5 Ri (u) X2 T m Ri (x-y) ^-^toCx^) Xi (Fig. (2.3).


34 Henceforth we conclude that the set of functions D defined by (2.1) forms a ncm-ccmnatative ring with respect to addition and convoluticn operations . Remark 2.1 ; It is important to note that, in the case v^en for all ueR(S) , Rj (u) = {t I < t < u} , the ordinary convolution in m (m > 1) dimensions is defined and all the previous properties with the Ccranutative law hold. Ihus, in a sense, the ordinary convoluticai is a special case of the g-coivolutiai. It nay also be noted that for the case when r^ is admissible the g-CCTivolutiai of any function reduces to the ordinary ccnvolution. Remark 2.2 ; l£t (p{x) be a locally integrable density function of the randan variable D . For every xew (S) , define ^[l)M = I 4)(t) dt x-te nj (x) X f^^^X) = I (j)(t) dt and for n >^ 2 *(n)^2i) = J "^(n-D^^^lt) (t-Ct) dt X-tE h) (x) X ff'^^x) = I f(n-l)(x-t) (t)(t) dt I


35 Fran the above definitions and the definition of the set a)(x) r we have by induction *(n)(x) 1 ^^^hx) (n > 1) Thxjs, ?«I>(n)(x) < It^'^^x) n=l ^ ' ~n=l By definition *(x) = r<|)(t) dt < I ^ L.UCti ti,...,tn,) dti... dt.... dtn, Jo JO •'o ^0 Xi

36 = f ^ $.^^^x.-t.) ()).(t.) dt. =$^^Ux ) Jn^ 1111 1 i i cind ^^^Ux) < i.^")(x^) (n > 1) Henceforth, n=l ^^> n=l ~ ~ n=l ^ ^ Fran renewal theory in one dimension [2] , 00 I i.^^' (x.) converges uniformly to il). (x.) , v^ich is tJie expected n=l ^ "• number of renewals that have occurred in the close interval [0,Xj] . Therefore, the series being dominated by a tmiformly convergent series is itself uniformly convergent for every xew (S) . 2.3 On the Solution of an Integral Equation Consider the set of functions F, f (t)eF is a real valued function of m variables (t ,t , . . . ,t ) for which we have J 2 m 1. f (t) H f or t < 2. f (t) is locally integrable on a)(S)


37 Lonna 2.1 ; Let f , geF and h be a locally integrable function cxi u(S) For xea)(S) , if we denote by a(x) = f fCx^t) [ JtecoCx) Y,^oi (t) g(t-y) hi^) dj^J dt b(x) = [ f(x-y^) g(v) dv] h(^) dy^ '^eco(x) ^V£a)( x-y ) C(x) , = [ [ ^e 10 (x) ' veRi ( x-y) then, a(x) = b(x) = C(x) Proof ; In (2.5) g ( t-y ) = for YL ^ t. Hence we can write (2.5) (2.6) f ( x-y-^) g(v) dv] h(Y) ^ (2.7) a(x) = f(x2t) [ 'tea)(x) '^ew(x) Let t = V + y , then (2.6) beoones g(t-y) h(Y) <^] dt (2.8) b(x) = [ •'^ea)(x) f (x-t) g(t-y) dt] h (y^) d^; t-ye o) ( x-y ) Now since t-ye o) ( x-y) inplies that teu)(x) , we can write b(x) = [ [[ f(x2t) gitr^) dt] h(^) d^ •'^e(i)(x) •'tea)(x) (2.9)


38 The integrals given by (2.8) and (2.9) exist . Itierefore, by Fubini's theorem the iterated integrals are equal. Usance for Xje:u)(S) we have a(x) = b(x) Fran (2.6) since g(v) =0 for v^R, ( x-y) eo) ( x-y) , we have ^M = I f f _ f( x-y-v) g(v) dv ] hi^) d;i_= C{x) "^yeto(x) ^veR, (x-v) And this cxanpletes the proof of the lerma. Note : Let x-y-v = t, then fron (2.7) C(x) = f [ f f(t) g( x-y-t) dt ] h(Y.) d^;; ^ileu(x) ^teRi(x-y) Renark 2.3 : Let (b(t) be a locally integrable density fxinction of the randcra variable D , and let ^. v (t ) , n >^ 1 , be the n-f old g-COTivolution of ({)(t) with itself. From the g-cOTivolution properties we have f _ (t, ( x-y-v) 4, (V) dv= f 4, ( x-y-v) (t)(v) dv veRi(x;^) ^^'^1 htR^{J^ ^^'^f Hence fron Larma 2.1 f

39 Similarly frcm the g-cx>nvolution properties and Lentia 2.1 we have ' tea) (x) *(n-l)(^) iew(t) ' ( t-y) h(i^) &^ dt •'^eu)(x) In E^, Fig. (2.4) illustrates georetrically the sets tuCx) , o) ( x-y) , and Ri ( x-y) . Theorem 2.1 : Let (J)(x) be a locally integrable density function of the randan variable D. Ihen, for every xe(D(S) and ||a(x) | < «> there exists a unique solution to U(x) = A(x) + I ,. (xHt) , (f), . (x-t) is the n-fold g-convoluticxi of £n^(n) n=l (n) (f)(x-t) with itself, and \^{x)\\ = Si;^ |a(x) |. xeo) (S) Proof : To prove solution existence and uniqueness, successive approximation method is used. Consider the sequence of functions defined inductively as follows: Uo(x) = A(x) Ui(x) = A(x) + f Uo(t) (j)(x-t) dt tea) (x)


40 h4


41 and U ^^^ (x) = A(x) + f Uj^(t) ^(x-t) dt (n >_ 0) ^tea)(x) where the subscripts refer to the inductive process. Let A,denotes the collection of all such functions. Aj farms a Beinach space in vtiich the norm of an element is defined by ||U(x)|| = Sup |U(x)| xew (S) To proceed in the proof, first v;e wish to show that the sequence of fimctions {Un(x)} converges to a function U(x) for all xeu)(S). Next, we shall shew that this function U(x) satisfies (2.10), and finally that U(x) is a unique solution to (2.10). From the recurrence relations we can write " ^1 ^^) U^^'^) = f 'l'(^-t) I " (t) U , (t) ] dt (n > 1) n+1 n J^^^(j^) n n-1 — Ta3cing absolute value of both sides we obtain l"n+l(?i) "n^^H ^ f ^(x:!) |Un(t) V^-iit) I dt (n > 1) 'tew(x) Let m = |A(x) I Upon taking norms of both sides, of the above relations, we get for n = 0, 1, 2, . . . |U (x) U (x) I < mf (j)(t) dt = m*, . (x)


42 |Uj (x) "i (x) I < mf $ (x-t) t (t) dt = m* (x) and K^lM -U^(x)| l"ij__^^^i(n)(?ilt) Mt) dt=m^^^^j(x) x-te o) (x) (n > 0) With this inecfuality at our disposal we now show that the sequence of functions (Uj^{x)} is a Cauchy sequence with respect to the norm. The proof is as fellows : Taking norms of both sides, we obtain 00 Since, from Ratiark 2.2, the series J] 1'( j (x) is uniformly convergent n=l " for every xew(S) then the sequence of functions {Uj^{x)} converges to a function U(x) for every xew(S) . Moreover, U(x) is an element of Aj since Aj is a cotplete Banach space. r^t us find an analytical expression for U(x) . Let U(x) be an initial solution for (2.10). Using the results from Latma 2.1, Remark 2.3,


43 and the results of the g-oaivolution properties, we get by induction for N = 1, 2, . . . and Uj(x) = A(x) + U^W = A(x) + = A{x) + = A(x) + teu) (x) ({•(x-t) U(t) dt tew (x) <>>(x-t) [A(t) + I ^t,e(o(t) 4>(t-t,) U(t,) dt,] dt ()>(x-t) A(t) dt teu) (x) «t> (x-t) [ I ^'(tit,) U(t,) dt, ] dt teu (x) tj eu) (x) (t» (x-t) A(t) dt + f p, . (x-t) U(t) dt tew(x) ^tea)(x) ^^' N-1 U..(x) = A(x) + j; f <>in)^^~^^ ^(t) dt ^ n=l^teu)(x) ^"^ I *{N)(2Slt) U(t) dt (N > 2) tea) (x) where ^ , . (x-t) , n > 1 , is the n-f old g-convolutiCTi of ( x-t) with (n) — itself. Therefore, U(x) is given by CO U(X) =A(x) + I f , .(>^) A(t) dt ~ ~ n=l ^teu)(x) ^^' and (2.11) follows.


44 To show that (2.11) satisfies (2.10) we proceed as follows : Using (2.11) we can write , U(x) f (t) (x-t) U(t) dt = A(x) + f \i) (x-t) A(t) dt •'tew(x) ^tew(x) f {x-t) [A(t) + f \l){t-t^) A(tj) dt,] dt ^tea)(x) Jt,ew(t) Applying the g-oonvolution properties, Lemia 2.1, Remark 3.3, and the definition of tj; (t) , we get U(x) f ((»(x-t) U(t) dt = A(x) + y f *, ^ (x-t) A(t) dt ~ ^tea)(x) ~" n=l ^tew(x) ^^' f (|) (x^t ) A (t ) dt + f ((I ( x-t) • 'tea)(x) •'tea)(x) n=l %ea)(t) ^""^ -i —i _ = A(x) + y f <\>, .(x-t) A(t) dt f (j)(x-t) A(t) dt n=l Jtea)(x) ^"' ^teu)(x) n^l itea)(x) <"+!) = A(x) Thus (2.11) satisfies (2.10). Conversely, we can show that if U(x) satisfies (2.10), then U(x) satisfies (2.11). Using (2.10) we can write


45 A(x) + f ^ix-t) A(t) dt = U{x) f (j)(x2t) U(t) dt ''tEw(x) ^te(jj(x) + f ^(x-t) [ U(t) f <|.(t-t,) U(t,) dt,] dt ^teu)(x) ^t eu)(t) ' ~ ~ ~ ^1--^^^^ Using, again, Leaina 2.1, Remark 2.3, and the definition of ^{t} , we get A(x) + I (x-t) A(t) dt = U(x) f » (x-t) U(t) dt tea)(x) ^tEu(x) 00 + n f 't'fn)^'^-^) ^^(^) '^t ] n=l ^tEw(x) ^"^ ~ I J ''>(n+i)(?Elt) U(t) dt n=l 'tea)(x) ^""^^^ = U(x) Thus the solution for (2.10) is given by (2.11). Now if the set T, as defined in Section 2.1, is admissible, then (t> (t) = (t){n) (t) where (|»^") (t) is the n-fold, n > 1, ordinary (n) — — convolution of i>(t) with itself. Thus the solution for (2.10) will be given by 00 . U{x) = A(x) + I (t> ^"^ (x-t) A(t) dt n=l ^tEa)(x) Q. E. D.


46 2.4 Other Mathgratical Concepts As we shall see later the following theorem fron calculus, Oliivstedl will be needed in Chapters IV and V. Theorem 2.2 ; Mean Value Theorem for multiple integrals. If f (t) is a OOTitinuous function on a ccrpact set T for which A(T) = I dt ^ 0, then there exists a point v in the interior ^T of T such that .(T) f f(t) dt = f (V) A( Remark 2.4; Frcm the theory of Linear Operators in Banach Space the follcvdng results will be needed in Chapter III. Define the operation (x) = f a (x-t) (})(t) dt +f a(x) (^(t) dt ^x-te(o(x) ^ x-te r'(x) = f a(x,t) ())(t) dt 'x-ter"(x) = (j) ( x-v) a(x,v) dv xea)(S) 'vet" (X) With ||t"| = 1, t" is the n*^ iterate of T, for n = 1, 2, . . . T is defined for all functions a{v) which are bounded for all ver" (x) , xew (S) . The collection of all such functions forms a Banach (M) Space in which the norm of an elonent is defined to be iSee additional references.


47 Sup Ia{v)| ver" (S) Foo: such integral operation T with bounded density kernel, Yosida and Kakutan [22] showed that tP = Z T. + P" ^^ 1 i=l = B + P" n = 1, 2, . where B is the ergodic part of the operation t" and P^ is the non-ergodic part of '^, k is the number of proper values of T, T^ (i = 1, 2, . . ., k) is a ccmpletely continuous linear operation and p'^ (n ^ 1) is a corpletely oontimxxis linear operaticai (which might vanish) with Ml < -x_^ (l+e)" where r and e are positive constants independent of n.


CHAPTE3^ III MATHEMATICAL FOTMULATION 3 . 1 Introduction An m-comiDdity (m >^ 1) inventory control system operating under a (a,S) policy is considered. At the beginning of each period a decision to order or not to order is made depoxling on the stock level X = (Xj, Xj, . . ., x ) . The ordering decision in each period is affected, as discussed previously in Chapter I, by a single fixed set-up cost K, a linear variable ordering cost c = (Cj, Cj, . . ., cr ) , and the expected holding and shortage cost L(x) conditional on being in stock level x at the beginning of a period. Demand, (Dj^), for the itenis over a sequence of periods (i = 1, 2, ... .) , is assumed to be independently and identically distributed continuous random variables with joint density fimction (i)^{t) , t > 0, t = (t,, t,, . . ., t ). u— — * m Iinnediate delivery of orders and ocrplete backlogging of unfilled demands are assumed for all ccfmodities. As discussed in Chapter I, under the adopted stationary (ci,S) policy, the sequence of stock levels at the beginning of each period forms a discrete-time Markov process. Ttiis chapter is devoted to finding an expression for the stationary total expected cost per period based on the assumed ordering policy. 48


49 3,2 Mathenatical Formulation For an n period prcfclem (n >^ 1) , let X = initial stock level for the items prior to making a decision, f (x) = total expected cost of operating the system for n periods when no order is placed initially, i.e.. h (x) = total expected cost of operating the system for n periods when an order is placed initially, i.e., xea. If no order is placed initially, i.e., xca^, then f (x) satisfies the functional equation f , (X) = L(x) (n = 1) f^(x) = L(x) + I Vl^^^^ ^^^^ ^ x-tew(x) + f h„_i(x-t) {t) dt (n > 2) ^x-ter> (X) ^ ^ ' (3.1) However, if an order is placed initially, i.e., xea, then hu.(x) satisfies the fimctiOTial equation


hi (x) = K + c (S^x) + L (S) h^(x) = K + c (S^Xs + L(S) (n 1) 'L S-tew(S) Vl^— ^ *^^^ ^ + I , h (S^t) 4. (t) dt (n > 2) I ^S-t£r^(S)5o ^ -^ ~ — J Fran (3.1) and (3.2) 50 (3.2) h (x) = K + c (S-x) + f (S) n — n — xea(n > 1) (3.3) To find, ej^licitly, an analytic expression for the stationary total expected cost per period, the followir^ theorem, based on Howard [6] and White^ work, is needed: Theorem 3.1 For large n (a) h^(x) = ng + v(x) xea (b) f^(x) = ng + u(x) xea^ where g is the stationary total expected cost of the process per period and u(x) and v(x) are functions of the initial state of the process . Proof : Fran (3.1) and (3.3) for xea^ fi(x) = L(x) fj^(x) = L(x) + x-tecj (x) + {K+ c [S-(x-t)]} <),(t) dt ''x-terMx) ~ — + Vl^^) J^^^^^i ^^j ^(t) dt (n >. 2)^ (3.4) ^See additional references.


51 Let us denote by (X) = f a(x) = {K + c (S-(x-t))} (f(t) dt 'x-terMx) (3.5) b ,(S,x) = f AS) n-i n-i — x-ter^ (x) (t)(t) dt (n > 2) (3.6) Upon using (3.5) and (3.6) in (3.4) we get fi(x) = L(x) f (x) = L(x) + a(x) + b ,(S,x) + I 4. (x-t) f , (t) dt n — — — n-i . , . n-i — — tew (x) (n > 2) Fran the above relation, on using the g-oonvolution prc^jerties and Remark 2.3, we obtain by induction for n = 2,3,... f2(x) = L(x) + a(x) + bi (S,x) + f3(x) = L(x) + a(x) + b2(S,x) + (^(x-t) L(t) dt tea) (x) teu) (x) * (x-t) X [L(t) + a(t) + bi(S,t) + (t-ti) L(ti) dti] dt •'tjl£U)(t) = L(x)+ a(x) + b2(S,x) + 4> (x-t) [L(t) + a(t) ' tew (x) + bi(S,t)] dt + I (j)(x-t) ' tea) (x) I.. eo)(t) <}.(t-ti) L(ti) dti] dt


52 L(x) + a(x) + b, (S,x) + f 4) ( x-t) [L(t) + a(t) ~ ^tea)(x) + b, (S,t)] dt + f 4) , . (x-t) L{t) dt and n-2 f (x) = L(x) + a(x) +b -(S,x) + y ((,,.. (x-t). "" ~ ^-1-i£l Jt^„(x) <^' [L(t) + a(t) + b ^ . .(S,t)] dt f ^. ,. (x-t) L(t; (t) dt (n > 2) tea) (x) v*iere (j) ( x-t) , n >_ 1 , is the n-f old g-convolution of <(> ( x-t) with itself. On regrouping terms, we get n-1 f_(x) = tL(x) + y tEO) (X) ((». (x-t) L(t) dt 1 n-2 , + [a(x) + y ~ i=l tew (x) (t, (x-t) a(t) dt] n-2 , + [b (S,x) + I 2) (3.7)


53 To proceed in the proof, first, we shall siinplify (3.7) by substituting for a(x) as given by (3.5). Mext, we shall find an analytical expression for f (S) and use it in finding the expression for f (x) , jKa^, for large n. Finally, we shall determine the expression for hyj(x)f xea, for large values of n. Let y = (mi ,y2 f • • '/M ^ r where y is the expected value of '— ni 1 the randan variable D., i = 1, 2, . . . , m. Fron (3.5) and the definition of the sets w (x) and r ' (x) , we have a(x) = f [K + c'^(S-(x2t))] (x-t) dt tew (x) = K + 0*^(3(x-y)) I [K + c'^(S-t)] (x-t) dt '^tew(x) Upon using the g-convolution properties and Lema 2.1 we obtain for i = 1, 2, . . . f (|), (x-t) a(t) dt = f i>, , (x-t) {K + 0*^(5(t-y)) ^tea)(x) t^) 'tew(x) ^^' [ [K + c'r(5-ti)3 6(t-t,) dti) dt 'tie(»)(t)


54 I 4>{xHb) [K + 0*^(3(t-y))] dt tea)(x) f r,)(x-t) [K + c'^(S2t)) dt 'tew(x) and, f (J) ( x-t) a(t) dt= f (|) ( x-t) [K + c'^(S-(t-ij))] dt ^tFMfxl (i) ^trM(x) (i) f r,-xi\(x-t) [K + c'^{S-t)] dt (i > 1) Upon using the above relations, we obtain n-2 x) + y f (}),.. (x-t) a(t) dt = K + c'^(S-x) + c^v i=l ^t£a)(x) ^^' -J {K + c'^(S-t)} (i)(x-t) dt tea)(x) n-2 Ml j (t)(j^)(x-t) [K -tc'^( S-t+y) ] dt tea){x) f 'J'rxix (?^) fK + c'^(S-t)] dt On siitplifying cind regrouping terms we get n-2 a(x) + Y f (J),., (x^t) a(t) dt i=l ^tewCx) ^^^ T ^~^ f T = K + c ^ [1 + y * , . V (x-t) dt ] + c (S-x) i=l ^tEai(x) ^^>


55 " I *(n-l) (?^) f^ + c'^CS^t)] dt (3.8) .' Now, we shall proceed to find an analytical expression for fj^CS), n > 2. Fran (3.4), for x=S, we have f,(S) = L(S) f-CS) = L(S) + f f fS-t) (j)(t) dt ^S-teu)(S) ""^ + f [K + c\] 4,(t) dt ^S-ter»(S) 'S-terMS) f(t) dt + f^ (S) f 4,(t) dt (n > 2) Let us denote by c(S) = f [K + c\] 2) (3.11). 'S-ter'(S) Note T as defined is a bounded linear operation. Substituting (3.10) and (3.11) in (3.9) we get fj(S) = L(S) fn(S) = L(S) + T(f^_]^(t)) (S) + c(S) (n > 2) '(3.12).


56 Fran (3.12), by induction for n = 2, 3, . . . , we have f2(S) = L{S) + c(S) + T(L(t)){S) £3(8) = L(S) + c(S) + T{L(t) + c(t) + T(L(t)))(S) and, n-2 . f„(S) = L(S) + c(S) + I T^{L(t) + c(t)) (S) i=l ~ ~ + T"-l(L(t))(S) (n>2) (3.13) Pron Yosida and Kakutan [22] we have k T^ = I T. + p^ j=l 3 = B +P^ (i = 1, 2, . . .) (3.14) v^ere B is independent of i. The linear operations T^ (j = 1, 2, . . . , k) and P are explicitly defined in Renark 2.4. On substituting (3.14) in (3.13) , we get f^(S) = L(S) + c(S) + [.(B+P») + (B+P2) + . . . + (B+P""^)] L(t)4c:(t))(S) + [B+P""^] (L(t))(S) (n>_2) Upon regrouping terms, we obtain


57 fjj(S) = L(S) + c(S) + (n-l) B(L(t)) (S) n-1 + (n-2) B(c(t))(S) + [ I Pi ] (L(t))(S) i=l n-2 . + t I P^ ] c(t))(S) i=l (n > 2) (3.15) Fran (3.6) and the definition of the sets w(S) and r* (S) , we can write b ,(S,x) = f„ .(S) f (t.(t) dt n-l-n-lJ^_ter»(x) = ^n-l^S) [f (|)(t) dt f (|)(x-t) dt ] tew(x) = fn-l(S) [1 f *(x-t) dt ] (n > 2) ^te(D(x) and for n > 2 n-2 i=l ., (x-t) b , ^ (S,t) dt (i) n-l-i -— tea) (x) = fn_i(S) [1 I (t'Cxit) dt] + tea)(x) n-2 i=l n-l-i f *H\(?Slt) [1 f

58 n-2 = f fS) [1 <}.(x-t) dt] + I f , i(S) . [f „.iv(xit) dt ] (n >2) (3.16) Denote by F^ = f (f)>.> (x-t) dt (i = 1, 2, . . .) (3.17) Jtea)(x) ^^' with F " = 1. Using (3.17) in (3.16) and sinplifying, we get n-2 'tew(x) n-1 ^n-l^^'H) + .1 f , ,*(i)(^) Vl-i(S,t) dt 1=1 tew(x) n-1 • 1 • = y f . (S) [F^"-^ F^] i=l ^-^= y f ,(S) [f"-"^ F^] +f.(S) [f"'^ f"'2] i=l "-1 + f,(S) [F^"^ F^'^1 (3.18) On substiuting (3.15) in (3.18) and expanding v;e can write for n >^ 2 n-2 't£w(x) Vl(S,x) + y I ^ (x^) b^_i_i(S,t) dt 1=1 t£w(X) n-3 = y [L(S) +c(S) + (n-l-i) B(L(t))(S) 1=1 ~ ~ " n-l-i . + (n-2-i) B(c(t))(S) + [ I P^l (L(t))(S) + [ I P-"] (c(t))(S)] [F^" f"-] j=l


59 + L(S) + C(S) + B{L(t)) (S) + P* (L(t)) (S)] [f"~3 f""^] 1-2 ji-1. L(S) [f"-" f"--'] = [L(S) + c(S)] [F" F^ + F^ F^ + . . . + f""^ F^~^ 1-3 n-2 + f""^ f" ] + L(S) [f""2 F^-1] + B(L(t)) (S) [ (n-2) [F" P»] + (n-3) [F» F*] + . . . + 2[F^~^ f""^] +[f""^ F"-2] 1 + B(c(t)) (S) [(n-S) [F" F^]+ (n-4) [F' F^] + . . . + n-2 [f""* f""^] 1 + [ E P3] (L(t))(S) [fO F»] -" j=l n-3 . 2 ^ + [ Z P3] (L(t))(S) [F» F^] + . . . + [ I P-"] j=l j=l (L(t)) (S) [f"""* F^"^] + P' (L(t)) (S) [Fr^-3 f""2] n-3 n-4 . + [ I P^] (c(t))(S) [f" fM + [ I P-"] {c(t))(S) j=l j=l [F» F^] + . . . + PMc(t)){S) [f""^ f""^]


60 On sinplifying, we obtain n-2 ^n-l'^'-'\ilL(x,*<"'-''"-^-^-'-' dt = L(S) + c(S) c(S) F^"2 L(S)F^"^ + (n-2) B(L{t))(S) n-2 . -B(L(t))(S) I F^ + (n-3) B{c(t))(S) i=l n-3 . n-2 . -B(c(t))(S) j; F^ + [ 5; PJ] (L(t))(S) i=l j=l + [ -p""^(L(t))(S)F» p""^(L(t))F2-. . . P>(L(t)){S)F"'^] + [ -P"-^(c{t))(S)F' P^'^(c(t)(S)F' . . . PMc(t))(S)F"'^] + [ I I>h (c(t))(S) = L(S) + c(S) c(S)f""^ L(S)f""-'" + (n-2) B(L(t)) (S) n-2 . -B(L(t))(S) I F^ + (n-3) B(c(t))(S) i=l n-3 . n-2 . -B(c(t))(S) 5; F^ + [ [ P^] (L(t))(S) i=l i=l + [ I V^] (c(t))(S) I P"" ^ '(L(t))(S)F^ i=l i=l n-3 . J _ Jpn-2-i (c{t))(S)F (3.19) i=l


61 Let n-2 n-l D^ -(S,x) = [ ^ P^] (L(t))(S) + t I P"-] (c(t))(S) i=l i=l I p""^"^(L(t))(S)Fi i=l n-3 J . . I p""^'^(c(t))(S)F^ Vn>2) (3.20) i=l 'flien (3.19) becxmes (after substituting for F^, i1 D n-2 I i(x) n-2 / b (S,x) + I 0/.x(x-t) b„ , .(S,t) dt "-1 -i=l ^teu)(x) ^^' ""-^ ^ = L(S) + c(S) c(S) f *, ^^ (x-t) dt ~ " ~ •'tea3(x) ^'^-2) L(S) f , ,.(x-t) dt + (n-2) B(L(t))(S) ^tea3(x) ^""^^ n-2 f + (n-3) B(c(t))(S) -B(L(t))(S) I (|>,.v(x-t) dt ~ ~ ~ ~ i=l ^teco(x) ^^' ~" n-3 , -B(c(t))(S) I 2) (3.21) ' Using (3.8) and (3.21) in (3.7) we get n-1 f„(x) = [L(x) + I f *,..(x-t) L(t) dt] ^~ ~ i=l ^t£a)(x) ^^'


62 + [K + c n-2 , i=l JtEa)(x) ^"-^ + c'^{S-x) f , ,,{x-t) [K + c'''(S-t)l dtl + L(S) + C(S) C(S) te(jj(x) ,„.2,'!^)& tew (x) n-2 + (n-3) B(c(t)){S) -B(L(t))(S) I ~ ~ i=l n-3 tea) (x) L(S) f (J), . . (x-t) dt+ (n-2) B(L(t))(S) * (i\ ( x-t) dt I B(c(t)) (S) y f <|),.x (x^) dt + D^ ,(S,x) i=l ^tea)(x) ^^^ n-2 -(n>_2) (3.22) Next, we shall study the asynptotic behavior of fj^(x) for large n. For 00 n large we have, because of the cx)nvergence of 7 (^ , . ( x-t) to ^ ( x-t) , i=l ^i' (n) (x-t) -» I (|),.,(x-t) -^ il>(x-t) i=l ^^' Moreover, from Rerark 2.4, P^, i >^ 1, is a cx3tpletely continuous linear operation (which might vanish) with P" l< < 1 (l+e) i = 1, 2,


63 where r and e are positive caistants independent of i. Thus, for large n, D _2(S,x) as given by (3.20) converges to a function D(S,x) vAuLch decreases rapidly as n beoomes large. Using these last remarks in (3.22) we get, for large n. f^(x) = L(x) + I iHx-t) L(t) dt + K tew (x) + c\[l + f i|j(x-t) dt] + c'^(5-x) + L(S) ^tetj(x) + c(S) + nB(L(t)+c(t)) (S) 2B(L(t)) (S) 3B(c(t)) (S) B(L(t))(S) f it>(x-t) dt ^tea)(x) -B(c(t))(S) f i|;(x-t) dt + D(S,x) ^t£w(x) Or, f (x) = ng + u(x) xca (3.23) n — — "~ where g = B(L(t)+c(t)) (S) is independent of n, and u(x) is a fianction of X . Now from (3.3) for large n h (x) = K + c'''(S-x) + f„(S) n — " ~ = K + c'^(S-x) + ng + u(S) = ng + v(x) xeo (3.24)


64 where, v(x) = K + c'^(S-x) + u(S) xea (3.25) Q. E. D. Theoran 3.2 ; The stationary total eiqsected cxjst per period, g, is K + L{S) + I L(S-t) i/;(t) dt JC-•t-<^/../C^ 'S-t£a)(S) g = + c\ (3.26) 1 + ij^a) dt 's-teaj(S) where ipit) = I

T v(x) = v(0) -ex xfea 65 (3.29) Using (3.29) in (3.27), we c±)tain ng + u(x) = L(x) + f (n-l)g 4, (t) dt ^x-tcr"(x) + [ u(x-t) ,})(t) dt •' X-tc o) (x) + f [v(0) c'^(x-t)] (|,(t) dt Jx-tcr'(x) = L(x) + (n-l)q f-4)(t) dt + f u(x^) <|,(t) dt 9. ^x-tea)(x) i + [[v(0) c'^(x2t)] 4,(t) dt Jo [ [v(0) c'''(x2t)] 4,(t) dt Jx-teco(x) Siiiplifying, we get g + u(x) = L(x) + v(0) [1 f <^(t) dt] x-tea)(x) 'x-tea)(x) Let x-t = y, then (3.30) beocnes -[c'^(x-t) 4,(t) dt + f c'^(x-t) (j)(t) dt h ''x-tEa)(x) + f u(x-t) (t)(t) dt (3.30)


66 g + u(x) = L(x) + v(0) [1 [ (x-y) d^] ''^G(d(x) I c\ (i)(x-y) dy. + c Y. 4> (x-y) d^ ^0 ^ilew(x) + [ u(^) (t) (x-y) d^ xea^ (3.31) ^yea)(x) Let fT r T c H(x) = L(x) c Y. (^ (x-y) dv + c Y. (t> (x-y) dy xea ^0 Jiiea,(x) (3 32) i(x) = [ (})(X2^) d^. (3.33) ^Y.ew(x) On using (3.32) and (3.33) in (3.31), we obtain aftex transposing g to the R.H.S. u(x) = ^ + H{x) + v(0) [1 $(x)] + [ (l,(x-y) u(v) dy (3.34) Zew(x) Let A(x) = -g + H(x) + v(0) [1 *(x)] (3.35) and (3.34) beocmes u(x) = A(x) + I (&(x-y) u(Y.) ^ xeo*^ (3.36) ''^£a)(x)


67 Each cxitiponent of A(x) as given by (3.32) and (3.35) is bounded over the set (d(S). Thus, A(x) is a bounded function over the oorpact set u,(S). Fran Theorem 2.1 the solution for (3.36) is given by u(x) = A(x) + \l) (x-y) A(^) d^^ xeo^ (3.37) 'yet»»(x) where i()(t) = Y , Jt) . n=l (") ~ Using (3.35) in (3.37) we get u(x) = -g + H(x) + v(0) [1 (x)] + f yji (x-y) [-g + E{y) + v(0)[l i(j^)]] djr This last relation is true for all xea . Set x = S and we get u(S) = -g + H(S) + v(0) [1

(n) Note $ , (S) = $ (S) (i) 68 (b, ,(S-y) dy (n^^l) (3.40) From (3.39) the coefficient of v(0) transposed to the R.H.S. is -1 + [1 i(S)] + f iKSiZ) [1 *(Z)] d^;: •'yeuj(S) ' ilew (S) Fran the definition of J , (S) (n >_ 1) , the g-convolution properties, ^(n) ~ and Remark 2.3, we have for n > 1 f *,„, (§11) *(Z) = f (n) (Sil) [ I ^ *(Zlt) dt] ^ = f [ f d.(n)( S-t-y)

69 Or, g = K + c^ + H(S) + I it)(S-y) H(y) djr ^vea)(S) 1 + f ii;(S-y) d^ (3.43) Let M = [1 + f t|)(S-y) d^]"'. Substituting in (3.43) for H(t) as ^yew(S) defined by (3.32), aives g = M {K + c^S + L(S) + I c\ (KS^x) dy; ^ileo)(S) F c'y (t) (S-y) dy. + f »HS-y) (L(j^) ^0 ^Ycui(S) f c'^t (t) (y^) dt f c\ d) (^) dt] dy ^trM(v) ^n — 'tew(v) dy} Upcsi sinplifying and arranging terms, we get g = M {K + 0*^5 + L(S) + f \l) (S-y) L(v) dv 0*^5 ^S-ye(D(S) + c u + I c\ (i> (S-y) dy + f H;(S-y) Kr^..(c,^ ^ii£w(S) 'yea)(S) T [ [ c t (t)(y-t) dt ] d^ + c\ iJ;(S-y) d^ •'tew(v) •'j^ea)(S) f c\r\){S^ dv } ^VEa)(S) v*iere ]i^ is the expectation vec±or. Siirplifying further we get


70 g = M {K + L{S) + f i>{S:X^ UZ) ^}+ c\ ' S-yeu) (S) + f i|;(S-y) [ f <{)(y-t) c\dt] dy 'Yew(S) ^teo)(^) I c y i|j (S-y) d^ } = M {K + L(S) + ji (S-y) l(y.) djr } + c y •'S-yeb)(S) + M { I c'''^ ^ ( S-y) ^ Yea)(S) n=l ^^eai(S) I f c:T^*(n)<^^ ^ ^ n=l Veu(S) ^"' 'yew! Note the last result follows frcm the definition of ^ (t) , the convolution properties, and Larrra 2.1. Upon simplifying the last relation and substituting for M, (3.26) follows. Q. E. D. As given by (3.26) the stationary expected cost per period, g, is a function of (a,S) , and will be written as g(cr,S) .


CHAPIER IV THE OPTIMIZATICN PROBLEM 4.1 Introduction In this ch^ter we shall provide necessary and sufficient conditions for the existence of an optimal (a,S) policy that miminizes the expression for g(a,S) as given by (3.26) . The unknowns to be determined are the decision variables S. (i = 1, 2, . . . , m) and the set r. In Section 4.2 we shall first determine the configuration of the set r and then reduce the minimizaticai of g(a,S) to finding optimal values for S and an additional variable C, vtiere C will be identified. The purpose of Section 4.3 will be to determine the necessary and sufficient conditions for the existence of the pair (C* ,S) that minimizes g (a ,S) . The necessary conditions will be given for the general m-ooitniodity inventory prc±)leni, wiiile the sufficient conditions will be given only for the two-commodity problem with separable L(x) . Ihe set of equations satisfied by (C*,S*) will be restated in terms of a real valued function M(C*,x) , where M(C*,x) will be defined. Section 4.4 will give a geotvetrical formulation of the cptimizaticn problem in terms of the function M(C*,x) and its inherent prc^jerties . Finally in Section 4.5 the linear form for LCjc) , the conditional expected holding and shortage cost function, will be considered. 71


72 4t2 Characterization of r In this section we shall first characterize the set r, r = {xjxen; Z (S-x) = 0}, by deterndning explicitly the relation Z (S-x) = up to a constant. This then will be used to reduce the miniirdzation problan into a problem in differential and integral calculus. Theorem 4.1 ; Under an optimal (a*,S*) policy, g(a*,S*), the minimum stationary total expected cost per period, satisfies the following relation: g{o*,s*) = LCx") + c\ x'er* (4.1) where jj^ is the expectation vector of the random variable D with joint density function A (t) . Proof: The proof is based on the definitions of h^(x) , xea^ and f^(x) , Xeo*^, as given in Chapter III. At optiirality we must have for x'eF* h (x") = f (x") (n > 1) n — n — — We note that this last relation if explicitly established will determine the configuration of r*. For large n, f (x) as given by (3.1) can be written f (x) = L(x) + f [(n-l)g + u(x-t)] ^{t) dt " ~ ^x-teaj(x) + f[(n-l)g + v{x-t)] {t) dt


73 -J [(n-l)g -tv(x-t)i(t>(t) dt xea^ (4.2) 'xHtew(x) V*iere g is the stationary total expected cost per period and u(.) and v(.) are well defined in Chapter III. For x"er*, and the definition of w(x), the set a)(x°) is an enpty set. Hence, from (4.2) we can write fj^Cx") = L(x'') + F [(n-l)g + vCx^-t)] <^(t) dt, x'el* (4.3) using (3.25) and the asynptotic ejqpressiOTi for f (x') in (4.3) we obtain T ng + K + c (S*-x'') + u(S*) = L(x'') + r [(n-l)q + K + c ( S*-x +t) + u(S*)] <(.(t) dt ^0 ~ ~ ~ = L(x'') + (n-l)g + K + c'^( S*-x° ) + c'^^vi + u(S*) siitplifying we obtain (4.1). 0. E. D. Ihe result of Theorem 4.1, as we shall see later, will be of great inportance in finding the necessary and .sufficient conditions for the ecistence of an optimal policy that minimizes the expression for g(a,S) as given by (3.26). An explicit expression for L(x'), x^eP*, can be iirmediately derived by ccnparing (3.26) and (4.1) to yield


74 K + L(S*) + f L( S*-t) if)(t) dt ^S*-tca)(S*) L(x») = j^ (4.4) , 1 + \ij(t) dt ^S*-tea)(S*) which is sijnilar to Iglehart's result [7] for the single ccmodity case operating under the ( 3;S) policy. For notational purposes v;e shall sonetimes represent L{x'') by the symbol C*, C* refers to the value of the total expected cost per period excluding the variable ordering cost. Corollary 4.1 : r* = {x|xGn; L(x) C* = 0} (4.5) Proof ; Fran Section 2.2 r* = {x|xen; Z ( S*-x) = 0} Fran Theorem 4.1 L(x) c* = xer* Hence, Z ( S*-x) = L(x) C* = xer* and (4.5) follows. Q. E. D. Let us make the transformation x = S*-t ; then (Chapter II) the image of the set a)(S*) is R(S*) and the image of r* is To* = {t|t > 0; L( S*-t) C* = } (4.6) The sets R(S*) and a)(S*) are functions of C* and often will be referred to as R(S*»C*) and w(S*,C*).


75 Wfe may rewrite the expression of L(x ) , x"er*, as K + L(S*) + f L( S*-t) \l)(t) dt ^R(S*,C*) L(x^) = -. (4.7) I + I ^^(t) dt R(S*,C*) •nie relation given by (4.1) , then, may be written T. K + L(S*) + f L( S*-t) ij^(t) dt ^R(S*,C*) g{a*,s*) = -. + c"y 1 + \l,{t) dt ^R{S*,C*) Fran this last relation it nay be noted that to minimize g(o,S) relative to the set a and S is equivalent to K + L(S) + I L{Fr-t) i!»(t) dt Rf^ C) Min { g, (S,C) = 7 } (4.8) 1 + ijj(t) dt ^R(S,C) which is a problem in integral and differential calculus. Let M = [1 + f ipit) dt ]~\ < M < 1, ^R(S,C) then the minimization problan as given by (4.8) is equivalent to: min {g,(M,C,S) = M [K + L(S) + f L(S-t) (^(t) dt] } * ^r> /o r>\ 'R(S,C) S.T. {g3(M,C,S) = M [1 + f \\){t) dt] 1 = } ^R(S,C) (4.9)


76 4.3 ^tec:essary and Sufficient Conditions The Lagrange func±ion, G{M,C,S,X), for (4.9) is given by G(M,C,S,X) = g2(M,C,S) + Xgj(M,C,S) = M{K + L(S) + f L(S-t) i|;(t) dt} ^R(S,C) + X{1 M Mf ,J;(t) dt } (4.10) 'R(S,C) By definition a point (M*,C*,S*) is a proper relative minimum for g2(M,C,S) if g2(M*+^, C*+AC, S*+AS) g2(M*,C*,S*) > or equivalently AP = G(M*+AM, C*+AC, S*+AS ,X*-fAA) G(M*,C*,S*,X*) > (4.11) To determine the necessary and sufficient conditions for the existence of a relative minimum for (4.9) , we shall first obtain an analytical expression for AG as defined in (4.11). On using (4.10) in (4.11) the expression for AG can be written as AG = (M*+AM) {K^+ L( S*+AS) + f L( S*+AS-t) t(;(t) dt } ^R( S*+AS ,C*+AC) + (X*+AX) (1 (M*+AM) (M*+AM)f r\,{t) dt } ''r(S^+AS,C*+AC) M* {K + L(S*) + f L( S*-t) \l){t) dt } ^R(S*,C*)


77 -X* {1 M* M* i|»(t) dt > ^R(S*,C*) Upon recirranging terms we obtain AG = M* {L( ^4A5) L(S*) + f [L( S*-»As-t) X*] i^ (t) dt ^R(SMAS,C*4AC) f [L( S*-t) \*] ^:t) dt + AM (k + L( S*-fAS) A* ^R(S*,C*) + f [L( S*4AS-t ) X*] i|)(t) dt } ^R( S*-fAS >C*-fAC) + AXd (M*+AM) (M*-fAM)| ilfCt) dt } (4.12) ^R( S*-fV\S ,C*-fAC) If we denote by AR(S*,C*) the incremental set between R( S*-fAs ,C*->AC) and R{S*,C*) , then we can write f [LfS^+AS^t) X*] \l)(t) dt 'r(s*+as,c*+ac) ^R(S*, ^R(S*,C* [L( S*-t ) X*] ii;(t) dt C*) [L( S*-t-AS-t) L( S*-t) ] ^{t) dt ) + f [L( S*+AS-t) X*] rl)(t) dt ^AR(S*,C*) (4.13) and


78 J [L( S*+AS-t ) A*] i){t) dt = f [L( S*H-AS-t) A*] ii;{t) dt ^R(S*,C*) ~ — + [ [L( S*+AS-t) A*] ti;(t) dt (4 14) Using (4.13) and (4.14) in (4.12) we obtain /G = M* {L(S^N^) L(S*) + f [L( S*-t-AS-t) L(S*-t)] ij;(t) dt ^R(S*,C*) — + f [L(S^+^-t) A*] Mt) dt } ^tf^(S*,C*) ~ + ^ {K + L(S^+AS) A*[l + [ ,|,(t) dt] ^R(S*,C*) ~~ + I L(S*+AS-t) i|;(t) dt ^R(S*,C*) ~ ~ + f [L( S*+AS-t ) A*] nt) dt} •'ar(s*,c*) — + AA {1 {M*+m) (M*+/!M) f \p{t) dt } (4.15) ^R( S*+AS ,C*+ac) Let t''eAR(s*,C*) and let A{m) denotes the area of the increniental set Z1R(S*,C*). Then from TheorOT 2.2 J [L( S*+AS-t) A*] i|j(t) dt m(s*,c*) ~ — = [L( 5*+^-t °) -A*] Mt") Aim using this in (4.15) v;e obtain


79 AG = M* {L(S*+AS) L(S*) + f [L( S*^-AS-t) L( S*-t) ] ii;(t) dt ^R(S*,C*) + [L( S*^-AS-t ) X*) \l)it^) A(AR)} + AM {K + L( S*-t-AS ) X*[l + f t|;(t) dt] + f L( S*-t-AS-t ) \l)(t) dt ^R(S*,C*) •'R(S*,C*) + [L(S*+AS-t'') X*] 'j/(t'') A(AR)} + AX {1 (M*+AM) (M*+AM)| 'Mt) dt } (4.16) •'r(s*+as,c*+ac) Let t^eV^*. Expandinq !.(•) in a Taylor series, with a rerrainder, rM'), about (S*,C*), the expression for AG as given in (4.16) becones m 3L(S) I , m m 3*L(S) AG = M* { J: I ASi + 5" I I AS . AS . i=i 8s, s^s* ^ ^ 1=1 j=i as^aSj ' s=s* ^ => , m 3L(S-t) [ I :=— I AS 'r(S«,C*) 1=1 3S. &=S* . m m 3^L(S-t) 1=1 j=l 3Sj^9Sj &=S*


80 m 3L(S-to) I + {L( S*-t ) \* + I I AS. i=l 3S. S=S* 1 m 8L( S*-t ) + I At^} i|;(to) A(AR)}+ AM (K + L(S*) i=l 3tjL t=to m 3L(S) I + I I AS. \* [1 + ^(t) dtl i=l 9S^ &=S* ^R(S*,C*) m 9L(S-t) I + [L( S*-t ) + I I AS.] ,J;(t) dt ^R(S*,C*) i=l 9Si S=S* m 3L(S-to) I + [L( S*-tJ X* + ^ I AS. i=i as^ &=S* m 9L( S*-t) + I At.] i|;(t^) A(AR)} + AX {1 (M*+AM) i=l at. t=to (M*+^) [ ^it) dt } ^R(SM^,C*+AC) + RMS*,C*,M*,X*; S*+AS >C*-fAC,M*+AM.X*+AX) (4.17) The above expression for AG will be used to determine the necessary and sxifficient conditions for gj (£,C) to have a proper relative minimum at (S*,C*) .


81 Necessary Conditions We shall first find the set of simultaneous conditions that (S*,C*) will satisfy and then restate these conditions in terms of the function M(C*,x) which will be defined later. The following Lama is needed. Lgma 4.1 ; At cptimality K + L(S*) + f L( S*-t) ^(t) dt ^R(S*,C*) X* = = C* > n (4.18) 1 + f ti;{t) dt ^R(S*,C*) Proof ; The partial derivative of G(M,C,S,X), as given by (4.10), with respect to M, evaluated at (M*,C*,S*,X*) , yields = K + L(S*) + f L( S*-t) ij;(t) dt and 3M (M*,C*,S*,X*) 'R(S*,C*) X* [1 + f t|>(t) dt] = Jr(s*,c*) K + L(S*) + f L( S*-t) ^^(t) dt •'r(s*,c*) * 1 + f i,{t) dt Jr(s*,c*) Fran this last result and (4.7), (4.18) follc*^. Q. E. D.


82 Theorgn 4.2 ; nie optimal values of C > and S > , C* and S* , vtiich ndnimize g^ {S,C) , satisfy the follodng set of equations: 9L(S) ^ptQ* 9L(S-t) as. S=S* 'R(S*,C*) 3S^ ^S* C* L(S*) + 'R(S*,C*) ii;(t) dt = (i = l,2,...,m) (4.19) [C* L( S*-t) ] i|)(t) dt = K (4.20) Proof: The partials of G(M,C,S,X) with respect to S^ (i = 1, 2, . . . ,m) are detained frcm (4.17) 9G 3S^ (M*,C*,S*,X*) = M*[ 3L(S) as . s=s* 1 aL(s-t) 'R(s*,c*) as^ ^=s* ^{t) dt aA(AR) + [L( S*-t o) X*] i|;(to) ] = ^\ (i = 1, 2, . . .,m) where the point toePo*. Smce by (4.6) and (418) X* =C* = L(S*-t,). t,er/, (4.19) follows. To prove the second part of the theorem we have frcm (4.18)


C* [1 + f ,|;(t) dt] = K + L{S*) + f L( S*-t) r\){t) dt ^R(S*,C*) ^R(S*,C*) And this last relation can be written as C* L(S*) + f [C* L( S*-t) ] iKt) dt = K R(S*,C*) 0. E. D. Theorem 4.3! Let the function M(C*,x) satisfy the integral equation M(C*,x) = C* L(x) + f M(C*,X;;t) (^ (t) dt (4.21) •'r(x,C*) = C* L(x) + f (t) (x-t) M(C*,t) dt (4.22) ~ ''a)(x,C*) Suppose that C* aixl S* exist, which minimize g, (C,S) subject to C > and S > 0. Then, if S* > and C* > 0, (C*,S*) is a solution to the set of equations M(C*,S*) = K 3M(C*,x) I 3x . JC=S* (4.23) = (i = 1, 2, . . ., m) (4.24) Proof: Define the functions 3L(x) . 3L(K-t) M, (C*,x) = + rl){t) dt (i = 1, 2, ..., m) 3x^ 'r{x,C*) dxi (4.25) 1


84 M(C*,x) = C* L(x) + f [C* L(x-t)] ii)(t) dt (4,26) ''r(x,C*) = C* L(x) + f [C* L(t)] i) (x-t) dt (4.27) •'u(C*,x) Let X = S* in (4.25) and (4.26) then ccnparing the results with (4.19) and (4.20) we obtain M(C*,S*) = K (4.28) M(C*,S*) = (i = 1, 2, . . ., m) (4.29) From Theoran 2.1 the solution of (4.22) is given by (4.27). Thus M(C*,x) = M(C*,x) (4.30) Hence from (4.28), (4.23) results which proves the first part of the theoran. To prove the second part of the theoran we can write from (4.26) m{C*,x) = M(C*,x+Ax) M(C*,x) = C* L( x+Ax) + f [C* L( x+Ax-t) l i>{t^ dt ^R(x+Ax,C*) [C* L(x) + f [C* L(x2:t)] iKt) dt] ^R(x,C*) If we denote by AR(x) the incratental set between R( x-fAx ,C*) and R(x,C*) , then the expression for AM(C*,x) can be written as


85 Z*1(C*,x) = [L( x+ X) L(x)l + f tC* LC x+Ax-t) ] ,j;(t) dt ^R(x,C*) + f [C* L( x-»-/y-t )] \i,{t) dt 'AR(x) ^R(x, (C* L(x-t)] Mt) dt C*) Let t^eARCx) arri let A(AR(x)) denotes the area of AR(x). Using the Mean Value Theoran for multiple integrals, TTieorejn 2.2, we can write /^(C*,x) = [L(x+^) L(x)] f [L( x-fAX-t) L(X2t)li|;(t) dt JR{X,C*) + [C* L( x+Ax-t P)] ,|^(t») A(AR(x)) (4. 31) Expanding L{x+Ax), L( x+Ay-t °) in a Taylor series, with a ranainder, R' ( x-»-AX ,x;C*) , about x, the expression for ^(C*,x) becomes m 3L(x) f m aL(x-t) mc*,x) = I AX. I ^Ax. V;(t) dt i=l 3x^ ^ Jr{x,C*) i=l 3x^ m 9L(x-t°) + [C* LCx-t") I AX^l ij;(t») A(AR(x)) i=l 3x^ + R* (x+Ax,x;C*)


86 As Ax , t " tjeFo (x) . Hence lijn (Lfx-t") C*> = 0. Dividing Ax-^n AM(C*,x) by Ax^ and lettijig Ax^ -^ we get 3M(C*,x) 3L(x) 3L(x-t) "(j;(t) dt ] (4.32) 9Xj^ 8x^ jR(x,C*) 3x^ (i = 1, 2, . . ., m) Copnparing (4.25) and (4.32) we cistain 3M(C*,x) = M. (C*,x) (i = 1, 2, . . ., m) 3x. Hence frcm (4.29), (4.24) follows. 0. E. D. Sufficient Conditions So far the necessary ocnditions for the determination of the optimal values of C and S v/ere discussed. To study the sufficient conditions that have to be satisfied, vre shall assume that the function L(x) is separable, that is, m L(x) = y L. (x.) where L. (x.) for all i is twice differentiafale. Now, since gs (M*+AM,C*+AC, S*+AS) = 0, and since the necessary conditions, derived above, require that each derivative of G(M,C,S,X) vanish (4.17) reduces to


AG = G(M*+AM,C*+AC,SM^,x*+AX) G(M*,C*,S*,X*) 87 T m m aL(S-t) u u , AS. AS. j=l i=l 3S.8S. S=S* ^ J 1 3 -J m m 9L(S-t) | * J ill AS. AS. ^{X) dt ^R(S*,C*) j=l i=l 9S.8S. S=?* ^ 3 1 : m aL(S-tJ I + {L(S*-to) A* + j; =^ I AS. i=l 3S^ ^S* ^ m 3L(S*-t) 1=1 9t^ ^=tc At^ ]'<''t^) A(AR)] + R»(S*,C*,M*,X*;S;»+AS,C*+AC,M*+AM,X*+AX) (4.33) **iere to = (tjo, tj^, . . ., t_^) eP/, rM«) is the remainder terms mo of the Taylor series expansion, A(AR) is the area of AR{S*,C*) , and L\(t^), L'\(t^), (i = 1, 2, . . ., m), are the first and second derivatives of L. (t.). 1 1 Since L( 5*-t p) = A* = C*, we have fron (4.33) on taking the second partial derivatives, for i, j=l, 2, ...,m a^G =M*[L "(S*) + f L."(S*.-t.) ^{t) dt *.r*.^*.\*\ ^ 1 ^R(S*,C*) ^ ^ ^ ~ "~ 3S^2 (M*,C*,S*,A*) + L.'(S.*-t.,) K,(to) aA(AR) 3S^ (C*,S*) (4.34)


88 3S.3S. (M*,C*,S*,X*) = M* ^ ^i'(V-^io) ^(^)^(c*,s*) {ifij) (4.35) 5^G ' 3S.aC (M*,C*,S*,X*) = M* L '(S *-t ) i^(t ) ^^^^^ \ ^^i ^iO^ '*^^io' 9C {C*,S*) (4.36) 9C^ (M*,C*,S*,X*) = M* 'l'(to) 8A(AR) 'Zl 9C (C*,S*) (4.37) Note the value of tg is not the same in (4.34) , (4.35) , (4.36) , and (4.37) . A sufficient condition for (4.9) to have a proper minimum a (M*,C*,S*) is that the matrix A given belov, is positive definite.


89 9C^


90 v*iere the point t*, t* = (t/,tj*)cr5*, has different values for (i=l,2) Proof : For the tvro c cjii ii o dity problem we must have (a) > 0, and this iirplies frcm (4.37) that 3C2 9a (Ar) I 9C (C*,S*) 3'g 9'g (b) 3e 3Si = > 9V, (4.39) 3c3Si > and fron (a) this inplies 3^0 3Si (M*,C*,S*,X*) > (4.40) Hence fron (4.34), since M* > 0, (4.38) f or i = 1 follows. 3^0 a^G a^G (c) 3^G (3S2' 3e 9Si' 3c3Si )'} 3^G { 3^G 3'g 3'g 3'g 3Si3S2 3^ 3S,3S2 3c3Si 3c3Sj 3'g 3'g 3^G d^G 32g + { . 3C3S2 3c3Si 3Si3s2 3c3S2 3Sj^ Upon regrouping terms we obtain } >


91 d^G a^G 3^0 d^G { ( ) 2 } a^G 32G a^G a^G d^G + 2 . ( ) 9Sj3S2 9C3Si acaSj 3C2 9Si9S2 a^G gJG J •( ) > as, 2 acaSj Upon using (a) and (b) we get a^G a^G a^G a^G ^ a^c a^G a^G { — • ( ) } + 2 . . > aSj* 3c2 3s,2 acas, as,as2 acas, acasj (4.41) New a sufficient condition for aA(AR) > 3C (C*,S*) is that, for any toero*» L.'(S.*-t. ) < (i=l,2). The proof proceeds as follows: r * = {t IL (S *-t ) + L (S *-t ) = C* } '— ' 1 1 10 2 2 20 ' In particular if A = (t ,t )er *, Fig^ (4.1), then 10 2 C* = L (S *-t ) + L (S *-t ) 1110 2 2 20 If we increment C by AC, then for B(t ,t +At ) and D(t +At ,t ) 10 20 20 10 10 20


92 and Thus C*+AC = L, (Si*-tio) + L2(S2* (t2o+At2o)) =L, (S,* (t,o+At,o)) +L2(S2*-t2o) AC = -L2'(S2*-t2o)At2o = -L, •(S,*-tio)At,o IP tc = -L2'(S2*-t2o) ; ^ -L,'(S,*-t,o) A*^2 Atjo ^ tc > if L2'(S2*-t2o) < n; > if L,'(Si*-t,o) < ^^20 At,o And henoe 3A(AR) I >0 9C (C*,S*) if Li'(Si-tio) < n (i=l,2) for all toePo*, which proves the first part of the theorem. Using this condition and the fact that 9A(AR) I I > (i=l,2) 9SjL (C*,S*) vre get from (4.35) and (4.36)

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93 g^G a^G < 0; < 0; asjasj (M*,c*,s*,x*) acas, (m*,c*,s*,x*) 9'G acasj (M*,c*,s*,x*) < (4.42) tioTt Fig. (4.1)

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94 Hence, from (4.42) 3*G a^G a^G 2 . . < and (4.41) inplies that a^G a-2G a^G a^G 2 { — • ( ) } > aSj^ ac2 aSj2 acas, Frcm (b) we have a^G a^G a^G ^ . -{ ) > 8c2 as, 2 acas, •Rierefore fron (4.43) it follows that a^G aSj^ (M*,C*,S*,A*) and (4.38) for i=2 is proved. Theorem 4.5: > n (4.43) 0. E. D. If (S,C*) is a solution to the set of equations (4.19) and (4.20), then (S*,C*) is a proper relative minima of gj (S,C) if (i) L.'(S.*-t.o) < S Vt = (tio, t,,)ET,* (i=l,2)

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95 (ii) M{C*,x) is a strictly concave fxinction of x, where M(C*,x) is given by (4.21) Proof ; Expanding L( x+AX) t L( x-»-/yc-t ») in a Taylor series with a ranainder R^ (x+^,x;C*) , fibout x, the expression for AM, in E*, as given by (4.31) beocmes 2 2 AM(C*,x) ={ };l. '(xJAx. + ^ L."(x.)Ax.*} i=l 1 1 1 i=l { 2 [ I L.'(x.-t.)Ax. R(x,C*) i=l 2 + I L."(x.-t.)Ax.2] i^(t) dt} + {C* L(x-t'') i=l 2 y L. •(x.-t.'')Ax.} il;(t'') A(AR(x)) 1=1 + p2(x,xj7^;C*) (4.44) v*iere AR(x) is the incremental set between R(x+^,C*) and R(x,C*), A(AR(x)) is the area of AR(x), and t"eAR(x) . Let to = (tio,t2o)Gro(x). Then from (4.44) we can write 32m{C*,x) , Z_= {L."(x.) + L."(x.-t.) li^(t) dt 3x.* ^ ^ iR(x,C*) 1 ^ ^ 1 —

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96 3A(AR(x)) + L.'(x.-t.J ^(t,) > (i=l,2) 9x. (x,C*) (4.45) 3^M(C*,x) 9x.9x^ 1 J = -L.'(x.-t.^)^(t,) 9A(AR(x)) 9Xj {x,C*) iT^j (i,j=l,2) (4.46) Now, M(C*,x) is a strictly concave function of x if and only if the Hessian is negative definite. Hence we irust have a^M(c*,x) (a) < 0; (b) ^^ 3'm(C*,x) 3^M(C*,x) W 8x^9x^ a^M(C*,x) 9^M(C*,x) 9x, 9x. 1 2 9x. or, frcm (4.45) for i=l, (a) L,"(x.) + f L,"(x,-ti) iMt) dt * ^R(x,C*) > + Li'(x,-tio) 'I' (to) 9A(AR(x)) 9xi (x,C*) > (4,47) and 3*M(C*,x) 9^M(C*,x) 92m(C*,x) 2 (b) • { } > 9x, 9X2 9x|9x2 from vMch

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97 d^mc*,x) 9^M(C*,x) -=. -=^ > 9xi^ 3x2^ Hence, frcn (a) and (4.45) for i = 2, we have L2"(Xi) + f L2"(X2-t2) ^Pit) dt •'r(x,C*) + L2'(X2-t2d l'(tp) ^^^^ (^^c*) ^ ° ^^-^^^ Set X = S* in (4.47) or (4.48) , then for (i = 1,2) L."(S.*) + 1 1 J L."e*-t.) li'(t) dt R(S*,C*) ^ 1 ^ ^ T . /c * 4^ , ^4^ 9A(AR) ' + L.'(S.*-t.o) ^(tp) -^3— (5^^^*j > 0, t;, = (tio, t2o)ero* merefore, condition (4.38) will hold if M(C*,x) is a strictly concave fvnction of x. Q. E. D. Ifrif ortunately , as stated, the sufficient conditions are not usable in practice.

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98 4.4 Gectnetric Reformulation of the Prc±>lem At this point it is of interest to use the inherent characteristics of the function M(C*,x) in giving gecnetric reformulaticai of the problan for determining the local minima at the point (S*,C*) in E^. If S* and C* satisfy the necessary and sufficient conditions for a proper relative minima, then as we have seen in Section 4.3 determining S* and C* is equivalent to constructing the function M(C*,x) v^ose equation is given by (4.22) with the following properties : (a) for x^er* M(C*,x°) = 9M(C*,x) ' ' = -L.'(x ") (i = 1,2) 9x. x=x° 1 1 1 (b) M{C*,x) achieves a maximum value of K at x = S*, i.e., M(C*,S*) = K Graphically, this reformulation is illustrated in Fig. (4.2). M(C*,x) is plotted for various values of the parameter C*. The function

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99 M(C*,x) f a = a(K) Li(Xi) + L2{X2) = C* / / ^/ /^ ' I I I r K 1'-/ S* Fig. (4.2). The Function M(C*,x)

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100 a = a(K) is the locus of all points S* at which the function M(C*,x) attains a maximum value of K for various values of C*. 4.5 The Function L(x) Fran new on we shall consider the case when L(x) has the form m L(x) = I L. (X.) i=l ^ ^ where each L^(x^) is continuous and twice dif f erentiable . The function L(x) is the cost charged over a given period of time excluding the ordering cost and in general it is the holding and shortage costs. Consider the case when the holding and shortage costs for each item are linear. Let for item (i = 1, 2, . . . , m) h^ = Unit holding cost at the end of the period p^ = Unit shortage cost at the end of the period Then L(x) is the total ej^jected holding and shortage costs Measured at the end of the period and is given by m 1*0 /Xj^ foo L(x) = ^h ... ... (X -t ) (j)(ti,...,t. ,...,t Jdti...dt....dt i=l^-'o-'o '0^^ ^ ^ ^ " m |«) ,oo -oo (4.49)

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101 If 4). (•) and $. (•) denote respectively the marginal density fi3nc±ion and marginal distribution function of the darand for itan i (i = 1, 2, . . . , m) in a given period, then from (4.49) we <±)tain LW = I i=l hfl^ (^i-^i) h^h^ ^^i ^ Pif (v^^i) ^i^^i) *^i (4.50) Setting L. (x.) = h. XX.' X ^ (x.-t.) *. (t,) dt. + p, J (t.-x.) 4>i(t.) dt. •'x. = (h. + p.) fi *. (t.) dt. + p. (Mi-x. ) (i = l,2,...,m) (4.51) we get from (4.50) L(x) = I L. (X.) i=l ^ ^ L. (x.) denotes the marginal expected holding and shortage costs for item i measxxced at the end of a period. Frcm (4.51) it is clear that the function L. (x.) is twice different iahle at all points for v^di 1). (x.) is ccntinuous. In fact for (i = 1, 2, ,m)

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102 Li'(xi) = (hi+Pi)'I>i(xi) Pi L^"(x^) (h.+p.)(t.^(x.) and L. (x.) is strictly convex for all points x. for which <{> (x.) y 0. Hence , L (x) , being the sum of strictly convex functions , is itself a strictly convex function.

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CHAPTER V OOMPOTATIOqAL ASPECTS OF THE PRCBIiM AN EXAMPLE 5.1 Introduction In this chapter the analysis shall be restricted to the ccnputational aspects of the optimization problem for the special case of a two-oomTodity problem v^ere the demand for the items obeys the exponential distribution, and the holding and shortage costs are linear. In Section 5.2 we will present the two-ccmnodity problem under consideration and obtain explicit expression for the function L. (x.) , (i = 1,2) , as given by (4.51) , and its first and second derivatives. Ihe purpose of Section 5.3 will be to determine analyticcil e^ressicffis for the set of equations used to determine the optimal policy parameters. The integral equation (4.22) will be converted into a partial differentieil equation of the second order. Riemann method will be used to solve this boundary value problan at the point S. Finally in Section 5.4 a numerical exanple will be considered. Nvmerical methods will be used to solve for Sj*, S2*/ and C*. 103

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104 5 . 2 Exaitple Consider a two-cxmrodity inventory control problem v*iere the demand, (Dj), for the items over a sequence of periods (j = 1,2,...), is assured to be independently and identically distributed continvious randcm variables with joint density function ^(t) = Mt) = x^x^e~^i^^~^2t2 The marginal probability distributiai function is given by ^j^i\) = I ^ At) dt = 1 e'^i^^i (i = 1,2) Fran (4.51) we can write L. (X.) = (h. + Pi) I ^ 4>. (t.) dt. + p. (u. X.) = (hi^p.) I" (l-e-^iti)dt.^p.(l,-x.) = (h,.p,)(x^.^ e-iXi_^),p.^^_j 1 1 On taking the first and second derivatives of (5.1) , we get V (x.) = -(h. + p.) e'Vi + h. (i = 1,2) (5.2) r2) (5.1)

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105 I^"(x ) = X^Chi+Pi) e > (i=l,2) (5.3) Using (5.1) in (4.5) we get hj+p, -XjXj hj+Pz -X2X2 r* = {x|xen; h,x, + h^Xj + e + e Xi Xa h, hz = C*+ — + — } (5.4) X, X2 5.3 Ccciputational Aspects From Theorem 4.2, Si*, S2*, and C* are the solutions for the set of equations given by (4. 19) and (4.20). Using (5.1) and (5.2) in (4.19) and (4.20) we get h. (h.4p.) e ^ ^ + f [hi (hi+Pi) •'r(s*,c*) -Xi(Si*-ti) e ] 4,(t) dt = (i=l,2) (5.5) C* 2 (h.+p.) -XiS^* hi y [ -±-±e " ^ + h.S.* ^ 1 + [C* i4i Xi Xi Jr(s*,c*) 2 (h.+p.) -Xi(Si*-ti) I [ -^_i_e +hi(Si*-ti) i=l X^ hi _] ,|,(t) dt = K (5.6) X^

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106 where '<'(t) = I it) n=l ^^' Fran (5.5) we can write -^iti Or f (hi+Pi)e ^ ^ 4;(t) dt -Xj^S^* •'r{s*,c*) [h. (hi+Pi) e ] = J (i=l,2) 1 + ip(t) dt ^R(S*,C*) e ii;(t) dt 1 ^R(S*,C*) hi Si* = — In [ —7 ] (i=l,2) Xi ^(t) dt R{S*,C*) hi+Pi The cdtplexity of findinq an analytical expression for ti^(t) dictates abandonix>g the direct solution of (5.5) and (5.6) for Si*,S2*, and C*. Now frcm Theorem 4.3 the values of S* and C* are the real positive solutions to the set of siimiltaneous equations given by (4.23) and (4.24), i.e. 8M{C*,x) M(C*,S*) = K; I = (i=l,2) ,. ^. 3X^ 3F=S* v*iere M(C*,x) is given by (4.21).

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107 To solve the set of equations given by (5.7) , we shall first find an expression for M(C*,x) and then use this excression to determine Si*, Sj*, and C*. One way to solve for M(C*,x) is to convert the integral equation (4.22) into a partial differential equation of the second order. Theorgn 5.1 ; In E^ if ())(t) = XjXj e (A^ > 0, t^ >_0; i=l,2) and L(t) = Lj(tj) + ^(tj), where L^(t^), i=l,2, is given by (5.1), then M(C*,x) satisfying the integral equation (4.22), i.e. M(C*,X ) = C* L(x) + 1 »(x-t) M(C*,t) dt (5.B) ''a)(x,C*) is the solution for the boundary value problem 9*M(C*,x) 3M(C*,x) 3M(C*,x) + Xj + Xj = XjXj [C* hjXj hjXj] 9Xj5Xj 3Xj 9Xj and for x^eF* M(C*,x'') = 3M(C*,x) -X.x," = -L.'(x.°) = (h.+p.)e ^^ h. (i=l,2) 9X; X=x'' "1 _ _ (5.9)

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108 Proof ; Before deriving the partial differential equation, we will note that for i,j=l,2 a^ft) = -\^{t) (5.10) 3ti 3'<(>(t) atj^atj = XiXj(t.(t) (5.11) The partial differential equation will be derived directly from the integral equation using again the method of varicible increment on AM as in Chapter IV. AM = M(C*,x+Ax) M(r*,x) = C* L(x+Ax) + J (x-t) M(t) dt] ''(i)(x,C*) adding and subtracting ({> ( x+Ax-t) M(t) dt to the right hand ^a)(x,C*) side we get eifter sirrplif ication AM = {L( x-f-Ax) L(x)} + I I*(x±^Slt) (fi (x-t) ] M(t) dt ^a)(x,C*)

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109 + ] » ( x+Ax-t) M(t) dt w(x+Ax,C*) f (j) ( x-t-Ax-t ) M(t) dt (5 12) ^a)(x,C*) ~ ~ Let Aa)(x,C*) denote the increiTental set between a)(x+Ax,C*) and u)(x,C*), A(Aw(x)) denote the area of Aw(x), and t^tdjuiM. Using the Mean Value Thearem for irultiple integrals, fTheorem 2.2), we can write J ({> ( x-HAx-t ) M(t) dt 'aj(x+Ax,C*) (j)(x+AxHb) M(t) dt (x)(x,C*) = 4'( x-^Ax-t °) MCt") A(Aw(x)) (5.13) Using (5.13) in (5.12) we obtain AM = {L( x+Ax ) L(x)} + f [(t)( x-t-Ax-t ) (t'(x-t)] M(t) dt ^a){x,C*) + (((( x^Ax-t ") MCt") A(Aa)(x)) Expanding L( x-fAx ) and 4) ( x-t-Ax-t ° ) in a Taylor Series about the point x, the expression for AM can be written as 2 2 AM = { ^ Li'(Xi)Axi + I Li"(x^)Ax.M i=l i=l 2 34)(x-t) ] [ I Ajc ^w{x,C*) i=l 9x^ ^

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110 , 2 2 8^(l)(x-t) + j I I Ax.AX^ ] M(t) dt + {^(x-t") j=l i=l ax^9x^ ' 2 a(f,(x-t°) + y AX.} MCt") A(Aa)(x))i+ RMx+Ax,x;C*) i=l 3x. " ~ 1 (5.14) where R ( x+ax ,X;C*) denotes the remainder term of the expansiai. On using (5.2), (5.3), (5.10), and (5.11) in (5.14) we get -XiX, -XjXj AM={(hj+Pi)e -hi}Ax,-{(h2+P2)e h2> Ax2 Xj(h,+p,)e Ax,* X2(h2+P2)e Ax^^ + I [ -Xj(x-t°)Ax. X,<|)(x-t°)Ax,} . M(t') A(Aw(x)) + R*(x+^,x;C*) (5.15) Figure (5.1) shows gecitietrically the incremental areas due to positive increments in x, and x^. Now fron (5.15) we have for i=l,2 9M(C*,x) -X.x. -A.xr = (h^+Pi)e ^ ^ hi ^i J 't'(x-t; t) M{t) dt ax^ a)(x,C*)

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Ill 9A(Aa)(x)) + <(>(x-t ) M(tj,) toeAwfAx^) or e3q>licitly 3M(C*,x) -XjX, , = (h,+Pj)e hj X, <}'(x-t) M(t) dt 3x. ^u)(x,C*) 8A(Aw(x)) + ((.(Cx^-tjg) M{x,,tj,) (5.16) 3x, where, from Fig. (5.1), t^^e [Xj,X2]. 3M(C*,x) -XjXj , = (hj+P2)e ^2 ^2 » (x-t) M(t) dt 3x, u)(x,C*) 9A(Aa)(x)) + (t.(x,-t,5,0) M(t,o,Xj) (5.17) Ax, 2 •vtjere, from Fig. (5.1), t,je[xj,x,]. 3*M(C*,x) = XjXj (f> (x-t) M(t) dt 3Xj3Xj w(x,C*) 3A(Aa)(x)) Xj(}>(0,Xj-t2 5) M(Xi,t2j) 3x J 3A(^)(x)) X\(|)(Xi-t,5,0) M(tjo,X2)

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112 •t»^J •P X Cm

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113 + ,^(0,0) M(x,,x^) ^5.^3j v^ere f ran Fig. (5.1), t^^t[x„x^] and t^e[x^,x^]. Using (5.16) and (5.17) and the explicit expression for M(x) = l(0,X2-t2 5) M(x,,tjo) 3x, 3A(Aa)(x)) Xi
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U4 A2 "2 3M(C*,x) 9M(C*,x) = ^2 —^^ ^1 —^ + A2X2[C* hiXj h2X2] Hence the partial differential equation given in (5.9) follows. Since for x°er* the set w(x°) is an enpty set, we obtain from (5.8) M(C*,xO) = C* L(xO) = and 3M(C ,x) dx. 1 = L.' (x.O) (i = 1,2) On substitxiting for L.' (x.° ) as given by (5.2) the boundary conditions follav. Q. E. D. Theorem 5.2 ; The solution of the boundary value problem (5.9) at the point S is given by M(C*,S) = f Q-^l(Si-x-^)-X^{S2-X2) Iq[2AiX2(Si Xj) (S2 X2) ] [(hi + pi) e"^i^i hi] dxi

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115 + ^^2 J [C* h,x, hjXj] . a)(C*,S) -X,(Si-Xj) XjfSa-Xj) e l^(2/X,X2(Si-x,) (S2-X2)] dxidxj (5.19) v^ere 1^[ ] is the rxriified Bessel function of the first kind of zero arder. Proof ; In solving the boundary prc^lem, which is a linear hyperbolic particil differential equation of the second order, we will use a method due to Ri€iTann» Miller^. The value of the function at the point S is given by { f "(S,x) .,x, ^r* -3M(C*,x) M(C*,S) «= ( I U(S,x) dxi XjXj ^r* -3x, + I X,Xj[C* h,x, hjXjl U{S,x) dx ) (5. 20) ^w(C*,S) where U(S,x) is the Green function for the problem satisfying the following boundary problem 9x,8x2 9xj 3x2 9U = X U v/hen x^ = S^ ax, 'See additional references. (5.21)

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9U_ 9X2 = X2U v*ien Xj = S] 116 (5.21) U = A1A2 when Xj = Si, X2 = S2 A solution for (5.21) is given by (see [25]) U(x,S) = A1A2 er^i(Si-Xi)-A2(S2-X2) ip[2 AiA2(Si -Xi)(S2 -X2) ] (5.22) using in (5.20) expression (5.22) and the value of M (C*,x) 1 xer' i = 1,2 as given in (5.9) , we obtain (5.19) Q. E. D. 5.4 Numerical Exanple In this nxjmerical exanple, we shall make use of the parameter values considered by Sivazlian [13] . Let us caisider the case when Xi = X2 = 1 Pi = P2 = 20 hi = h2 = 1

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117 K = 5.0 Using these values in (5.1), (5.2), (5.4), and (5.19) we get ^i L (x ) = +21e + Xi 1 (i=l,2) i i ~^i Li'(Xi) = -21e + 1 (i=l,2) -X, ^j r* = {x|Xj + 21e + Xj + 21e = C* + 2 } and at the point S = (Sj,S2) -(Sj-Xj) (Sj-Xj) M(C*,S) = e I,[2/(S,-k,)(Sj-3
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118 Lj (Xj) + ^^(Xj) = C* for the case v^ien C* = 8 w(C*,S) Xj -X2 r* = {xjxi + Xj + 21e + 21e = 10} 2 3 S^ x^ Fig. (5.2)

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119 Numerical Analysis : In carrying out the numerical analysis for this particular exairple, the following points were taken into consideration: (i) at optimality Sj* = S2* (ii) for a given value of C , T is synmetric about the line Xj = X2 (iii) the resiiLts obtained by Sivazlian [13], table 1, for this particular exaiiple under different (a,S) policies (iv) Sivazlian [13j shows that ^ p e"^"l"'"2) io[2,^I^] dui du2 = a{l [e~^^ Io(2a) + e~^^,(2a)]} (a > 0) (v) Sivazlian [13] further shows that f f u.e"^^i'*"^2)ij2/;iru7] dui du2 = alLi^al ^ _ j^-2a ^^^^a) + e'^^ Ii(2a)]} (i = 1,2) (vi) for this particular exanple, M(C*,x), as given in (5.8) is a non-decreasing function of C*.

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120 TaJdng the above points into consideration, we initiated the search procedure by choosing C* = 8.2 and Sj = Sj = 3.8. In ccmputing the double integral in (5.23) for a particular value of C*, Sj and S2 , the region ti
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121 x,/

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122 which is defined over the closed subr^ions A2, A3 and Ai^ is formulated by cutting each subregion into very anall squares each of area AA. , (i = 2,3,4) , where any two of these squares have at nost one side in acrman. Henoe for (i = 2,3,4) ) (C* xi X2) e-<^i-^i)-^S2-X2)i^^2 As,-xi)(S,-x,) 3 dx Ai = lim f (c* xi X2 ) e"^^i-^i.^-(S2-X2.) AA^->0 j=l j j ^3 Io[2.^(Sl Xi ) (S2 X2 )] AA. 3 j ^ v^iere (xi .,X2.) is any point in A. and n. is the maxiimm nxjtiber of squares of area AA. in A. . By syninetry, the integral over Pi^ is equal to the integral over A2. (3) The line integral of the function ^-(S,-Xi)-(S2-X2)j^^^^ (g^ -x,)(S2-X2) ] . {21e-^-l} viiich is defined over the smooth arc T* is formulated by dividing the arc T* into N arcs by inserting (N 1) points Q^(xii,x2i) . Thus, J -(Si-x,)-(S2-X2),^^,^ ^^^ xi) {S2-X2) ]{21e-^l-l]} dxi

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123 N^ i=l Axii-M) Ax2i-K) Numerical Results : To obtain nunerical resiiLts, a caTpatational algorithm in Fortran IV (Appendix 5. A) was written to solve for the cptimal policy variables. The algorithm proceeds in the following steps: Step Oxose initial values for C and the increment in C*,AC ; Choose initial values for S, S_= Sq, and an increment in S, AS . The range of vali>es of S , Sq f. S <^ S° , for a given value of C*, are chosen to assure that M(C*,S) achieves a maxinon in that range; Choose M as an x:?3per liirat on K M(C*,S) . Step 1 Carpute M{C*,S) . Step 2 Check the difference between M(C*,S) and K, then a. If the difference is greater than AK, set AC* = ^ , C* = C* AC*, S = S and go to Step 1.

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124 b. If the difference in absolute value is greater than AK, go to Step 3. c. If this is the first tirns the difference in absolute value is less than AK; set K* = M(C*,S) , S* = S, Cj = C* and go to Step 3. Otherwise, go to Step 3. Step 3 Set S = S + AS. If the new value of S is in the range of values of S, go back to Step 1. Otherwise, if K M(C*,S) > AK over all the range of values of S, set AP* AC* = --1, C* = C* + AC* , S = S snd go back to Step 1 ; else the algorithm ccaiverges on the cptiitial values of C* and S*. For the particular exanple we are considering, v^en K = 5.0, we initiated the iterative search procedure as follcws: The values of C* were chosen over the range of values [8.22, 8.27] in increment of 0.01; Sj = S2 values ranged over the closed interval [3.8, 4.08] in increment of 0.02. The number of "steps" used in the numerical integration N^ were selected to be 60 and 100. For the case N^ = 100 we have C* = 8.2625 and S* = S* = 4.02 M(C*, S*) = 5.00694 and 8M(C*,S) = for (i = 1,2) S = S* 9S. 1 which corpare favorably with the results obtained by Sivazlian [13] when follcvdng different optimal policies.

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125 Grafiiically, the cjoiputed values (J^pendix 5. A) of the function M(C*,S) are plotted in Figs. (5.4) and (5.5) for the cases when Nj = 60 and 100 respectively. As an e3q)eriinental observation, it may be noted that as the number of "steps" used in carrying out the nunerical integration increases the curves beoane siroother. The accuracy of the ocnpubed results is subject to two sources of error, truncation and roundoff. In coiputing the modified Bessel function Iq (x) / only the first ten terms of the convergent series were sunned. Hcwever it was found experimentally that the rouncJoff error is the factor of significance and it is not easy to examine it analytically, since it is a function of the oorputer used and the size of the "steps" used in carrying out the nunerical integration.

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126 Nj= 60 5.0M(C*,S) 4.94.BC*=8.?5 C*=8.24 C*=8.?.3 C*=^.22 3.76 3.7^3 Fig. (5.4)

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127 Ni= 100 / 5.0-

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CHAPTER VI CXX^CLUSICWS AND RECC*ME1«1DATI0NS 6.1 Conclusions A stationary multi-connodity inventory problan with periodic review has been formulated and studied vs^en a dyadic (o,S) policy is used. At the beginning of each of a sequence of periods of time the stock level of each item is reviewed and a decision to order or not to order is made. Ihe cost elanents that affected the ordering deciin each period are: the ordering oosts, the holding cost and the shortage cost. The ordering oosts are assured to be ccttposed of a purchasing cost which is proportional to the quantity of each item ordered and a single set Lp cost which is independent of the quantity ordered. The holding and the shortage costs are assured to be charged on the basis of the stock levels at the end of the period. Demand, for the items, in each period of time is described by a continuous randan vector, with a joint density function, independently distributed from period to period. Immediate delivery of orders and corplete backlogging of all unfilled demands are assumed. The inventory syston just described was treated by many researchers under different optimal (ct,S) policies. This research, as catpared with what has been done in this area, is more general in the sense no assunptions were made about the configuration of the ordering region and no specific joint density functions were considered. 128

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129 The m-dimensional convolution cperation v^ch was introduced in Chapter II and used in the study is a generalized concept of the ordinary convolution operation. For the case v^^en T , the hypersurface that subdivides the ordering region and the not-ordering region, is admissible, the g-oonvolution of any function red\x:es to the ordinary convolution and all the g-oonvolution operation properties with the carmutative law hold. The g-convolution prcperties, in particular, were used in the solution of an m-dimensional integral equation of the renewal type, which is similar in form to the equation solved by other researchers in this area but with no restrictions on the configuration of r. Ttie analysis used in deriving the analytical expression for the stationary level expected oost per period is similar to Sivazlian's [11] approach for the one oormodity problan when operating under a stationary policy of the (s,S) type. The asynptotic results for the problem were deduced by appealing to the results of Yosida and Kakutan [22] in the theory of linear operations. At optimality it was shown that the set T* that subdivides the policy space, Q, is given by r* = {x|x e Q; L(x) C* = 0} v*iere C* is the minimum value of the stationciry total ejqsected cost per period excluding the variable cost, and L(x) is the conditional

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130 expected holding and shortage oost function. By using this result, the minimization problesn was reduced to finding the decision variables Sj* (i = 1,2, ,m) and C* that minimize the stationary total expected oost per period. S* and C* were given as the real positive solutions to the set of simultaneous equations M(C*,S) = K (6.1) 3M(C ,x) 9x. 1 ^ = (i = 1,2,. ..,m) (6.2) x=S vMch are siitdlar in fonn to the set of equations used by Sivazlian [11] in determining the optimal values s* and Q = S* s* for the one cciitnodity problem. M(C*,x) satisfies the integral equation M(C*,x) = C* L(x) + M(C*,X2t) (J)(t) dt (6.3) R(x,C*) in which (p is the joint density function of the demand. For the case of a two-oorrmodity problem viiere the demand for the items obeyed the exponential distribution and the holding and shortage costs were linear, it was feasible to convert the integral equation (6.3) into a hyperbolic partial differential equation of the second order with boundary conditions. Analytical solution of the boundary value problan was then determined and used in (6.1) and (6.2) to determine Sj*, S2* and C .

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131 6.2 Extension ar>d Recxjimendations Several interesting directions for future research and develc^xnent are suggested as a result of this research. The ccnputational aspects of the prc±)lan need further investigation. The necessary and sufficient conditions, established in this study, for the existence of a relative minima are valid for any continuous density function and any conditional expected holding and shortage cost function v\iiich is twice differentiable. In the case of a two-oomiodity problem, v^ere the daiiand for the itans obeys the ej{ponential distribution and the holding and shortage costs are linear, it has been feasible to convert the integral equation into a partial differential equation of the second order with boundary conditions. However, what about the case when the dariands for the two items are given by, say, a gaitma distribution? Is it then feasible to convert the integral equation into a partial differential equation? And, if so, vAiat ccnputational procedure must be followed to solve for the policy parameters? Another qiiesticn also arises, how to solve for the optimal policy parameters for the two-oontnodity prcfclon, when the demand for the items are exponentially distributed and the holding and shortage costs are not linear but, say, quadratic. As we have seen the determination of the cptimum policy parameters is a ccrplex task even in the case of a sinple demand density function.

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132 As an alternative to approximating these parameters by nutverical means, the approach used by Roberts [3] could be follaved to give analytical formulas for determining the asynptotic reordering region for the case when the holding and shortage costs are linear. In order to derive the asymptotic expressions, for the set of equations that is used to determine the policy parameters, an approximation for the renewal function i){t) dt and similar extensions of the type discussed Ir(s,c) — in Smith [14] are required. The work of Yahov and Bickel [21] and FarreH [4] in investigating the asymptotic behavior of the renaval function in two and higher dimensions must be noted. However, the challenging prc±)lem is to find an analogous theorem to Smith's theorem involving the renewal density function in two and higher dimensions. As noted in Chapter I, under the adopted (a,S) policy, the sequence of stock levels at the beginning of each period forms a discrete Markov process. Greenberg's [5] approach could be followed to determine the transient distributions of the stock levels prior to ordering. For the stationary distributions Karlin [2] approach oould be followed. The results of this research, especially the properties of the g-convolution operation, are hoped to be of great use in determining the transient and stationary distributions of the stock levels prior to making ordering decision. T3iis study treated "the Patient Cus toner Case" v^iere all unfilled

PAGE 140

133 demands were hacklogged. For the lost case, the analysis can proceed along the same procedures vised in Chapters III and IV.

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GLOSSAFOr CF NOTATIGN m the nunber of cxrmodities cxnsidered E m-dimensicxial (m >_ 1) Euclidian-space D a random vector representing the demand D. a randan variable representing the deinand of oonmodity i * the pn±)abilitY distribution of D *. the probability distribution of D. (ji a density fxmction of D ()>. a density function of D. y the mean vector of D $ the n-fold ordinary convolution of $ $ . the n-fold ordinary convoluticxi of $ . .(n) '(n) the n-fold ordinary convolution of ^ with itself the n-fold generalized convoluticn of 4) with itself \\) i{;(t) = J 4), V (t) , the renewal density function j^i W a the ordering region a the not -ordering region r the set of points that sepcirates a and a S_ the point up to which we order S. the i^ conponent of the vector S_ K the set up cost C. the unit purchase cost of product i h. the holding cost per unit of product i on hand at the end of a period 134

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135 p. the shortage cx>st per unit of ooriTodity i on hand at the end ^ of a period L(x) the conditional e^^jected shortage and holding cost measured at the end of a period g(a,S) the stationary total ej^jected cost per period I (x) the modified Bessel function of order n n * starred syrobols denote conditions at optimality C* the value of the total expected C30st per period excluding the variable ordering cost

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Algorithm Flowchart 137 Choose initial values for C* and Ac* Choose a range of values for S, SefSofS"] , so that for a given value of C*, M(C*,S) achieves a inaxiinum in that range. Choose an initial value for AS ChooseAK as an upper limit on K-M(C*,S) Conpute M(C*,S) Compute AM=M(C*,S)-K Set AC*=AC*/2 C*=C*-AC* S=So YES Set S=S+AS YFS — KAM>AK am|>ak" NO S>S NO YES YES Set K*=M{C*,S) S*=S Ci=C* ^J0 STOP The algorithn _v converges to "^ S* and C*=Ci

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138 > t-O SD (M LU " I/) t— O t/) •• 3 LU LU .-< Q. >— •-• I/' 5! 13 fi: • o a ui cj Lj 2: i>^ — a n OO (_) »— .-. U "-• o Q UJ O X •LU a: Q. O u. q: "— • a: LU •-' yu m cc a:. Q. UJ o oo > Oj ij f\j -• ^ O LU t/1 > nf < LL C O n UL ;' o I/) I/) o LU UJ rD z) > o o -• o cc lU C 5: UJ ^ o UJ ^ X a. z t»-i IL) O •-' > O uLL q: a O yUJ LU o o < ~ z :^ o < < oo ro ot (a: •-' — r r^ >O t(^ q: a: UI LU (z z o < < — •fM — 2' 3: Cll oJ uo o O (/) LU LU ro >— rr > > ^ — LO < — < q: o 1/1 lu •• o o ^ C< O^ vO O LO 2 -jlululuooluqc^ _j X ^ a: O X 00 --(— ii-^(Niia (NJ f-< O II • *—» o ^ II — •4cc X hfM 00 * ca •& * 00 og — . fvj r«» I— cr (\i •o c ~. ^ r>o: r^ 00 1-0-0 II o ^~ r-. — o a: tQ I— •— a. VQC w o o h»hOD -» oooooooooouo

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139 o o CM X o >

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140 o o X o X

PAGE 148

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143 NUMBER OF SUBDIVISIONS Nl = 100 OPTIMU*^ VALUES GF C AND S CO = 0.826399993896480 01 ITERATION NO. = 4 M(S,C) S1=S2 C 0.500069A9910199D 01 .40 19996643G664D 01 0. 82624999401 1 60D 01

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144 ITERATION NO. = 1 0.^99298689329910 01 . 39 199991226I96E 01 0.826399993896^80 01 ITERATION NO. = I 0.50025237931040D 01 . 393999B6267090E 01 0. 826399993896A80 01 ITERATION NO. = 2 0.^98617315300050 01 . 39 19999 1226 196E 01 . 826l9999A0
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145 ITERATION NO. = 3 0.30018918310^950 01 D . 39999971389771E 01 0. 82629999397323D 01 ITERATION NO. = 3 0.5001^8781866140 01 . 40 199966A30664E 01 0.8262 99993973230 01 ITERATION NO. = 4 0.498800582182030 01 . 39 199991226196E 01 0. 82624999401 1600 01 ITERATION NO. = 4 0^.499763824692570 31 . 39 399906267090E 01 0. 82624999401 1600 01 ITERATION NO. = 4 0.499769408400690 31 3 . 39 59993 I307983E 01 0.826249994011600 01 ITERATION NO. = 4 0.500003905046990 01 . 39 799976348877E 01 0.826249094011600 01 ITERATION NO. = 4 0.499904571616230 31 . 3999997138 9771 E 01 0.826249994011600 01 ITERATION NO. = 4 0.5aCV069499101990 31 .40 199966430664E 01 0.826249994011600 01 ITERATION NC. = 4 0.499933712128080 01 .40 39996 147 1 558E 01 0.826249994011600 01 ITERATION NO. = 4 0.499334954327350 31 .40 59995651 245 IE 01 0. 82624999401 1600 01 ITERATION NO. = 4 0.499084532973770 01 .4079995 1553345E 01 0. 82624999401I60D 01 ITERATION NO. = 4 0.49828455150016D 01 .40999946594238E 01 0.826249994011600 01

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BIBLIOGRAPHY 1. Arrow, K. J., T. Harris and Y. Maichak, "Optimal Inventory Policy," EconoT^trica , IPSl, 19, 250-272. 2. Arrow, K. J., S. Karlin and H. F. Scarf, Studies in the r^thematical Theory of Inventory ctnd ^oduction , Stanford University Press, Stanford, California, 195R. 3. ArrcM, K. J., S. Karlin and H. Scarf (eds.). Studies in Applied Probability and •lanaagnent Science , Stanford University Press, Stanford, California, 1062. 4. Farrell, R. H. , "Limit Theorems for Stoooed Random Walk III," Ann. Math. Statist ., 1966, 1510-1527. 5. Greenberg, H. , "Stock Level Distribution for (s,S) Inventory Problems," Naval Research Logistics Quarterly , 1964, 11. 6. Howard, R. A. , Dynamic PrograiminQ and Markov Process , The MIT Press, Cambridge, Massachusetts, 196n. 7. Iglehart, d., "Dynamic Programning and Stationary Analysis of Inventory Problems , " Chapter 1 in Multistage Inventory ^todels and Techniques , H. Scarf, D. Gilford, and M. Shelly (eds.), Stanford University Press, Stanford, California, 1963. 8. Johnson, K. J., "Optimality and Computation of (a,S) Policies in the Multi-Item Infinite Horizon Inventory Problem," Management Science , 1967, JJ, No. 7, 475-491. 9. Scarf, H. , "A Surve^' of ;\nalytic TeKrhniques in Inventory Theory," Chaoter 7 in Multistage Inventory 'Videls and Techniques , H. Scarf, D. Gilford, and M. Shelly (eds.), Stanford University Press, Stanford, California, 1963. 10. Sivazlian, P. D., "Inventory Control of a Multi-Product Systan \^tb Interacting Procurenent Cost," vrorking paper No. 58, Operations Research GrouD, Case Institute of Technology, Cleveland, Ohio, 1966. 11. Sivazlian, B. D., "Optimality and Ccmputation of the Stationary (s,S) Inventory Control Problem," Technical Report No. 7, Project Themis, Dept. of Ind. & Sys. Enor., the Univ. of Fla. , May, 1968. 146

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147 12. Sivazlian, B. D. , "Asyrrptotic J^roximations to Ordering Policies in a Stationary (a,S) Inventory Problem," Siam J. P^ppl. Math. , 1970, 19, No. 1, 155-166. 13. Sivazlian, B. D. , "Staticaiary Analysis of a Multicarmodity Inventory System with Inter ac±ing Set-Up Cost," Siam J. i^pl. Math. , 1971, 20, No. 2, 264-278. 14. Smith, W. , "Renewal Theory and Its Ranifications , " Royal Statistics Society , Ser. B. , 20, 243-302. 15. Soland, R. M. , "On Sane Renewal Processes in One and Two Dimensions," Ph.D. Thesis, MIT, 1964. 16. Veinott, A. F. and H. M. Wagner, "Ccnputing Optimal (s,S) Inventory Policies," Management Science , 1965, 11, No. 5, 525-552. 17. Veinott, A. F. , "Optijnal Policy for a Multi-Product, Dynamic Non-stationary Inventory Problesn," Management Science , 1965, 12, No. 3, 206-222. 18. Veinott, A. F. , "The Status of Mathematical Inventory Theory," Management Science, 1966, 12, No. 11, 745-777. 19. Watson, G. N., A Treatise of the Bessel Functions , Cambridge, England, University Press, 1952. 20. Wieeler, A., "Multi-Product Inventory Models with Set-Up," Ph.D. Thesis, Stanford University, 1968. 21. Yahav, J., and Bickel, P., "Renewal Theory in the Plan," Ann. Math. Statist. , 1965, 36, 946-955. 22. Yosida, K. , and Kakutan, S. , "Operator Theoretical Treatment of Markoff Processes and Mean Ergodic Theorem," Ann. Math. Statist., 1941, 42, 188-226. Additional Pteferences Miller, K. S. , Partial Differential Equations in Engineering Problgts , Prentaoe-Hall, Englewood Cliffs, N. J., 1953. Olmsted, J. M. , Advanced Calculus , ;^letcin-Centuxy-Crof ts , New York, 1959.

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148 Sivazlian, B. D. , "Dimensional and Ccrputational Analysis in Stationary (s,S) Inventory Problanns with Gairma Distributed Demand," Management Scienoe , 1971, 1]_, No. 6, 307-311. Sivazlian, B. D. , and P. E. Valisalo, "The Utilization of a Hybrid Cdtputer Systan in Solving for Optimal Policy Variables in Inventory Problans," Proceedings Ccnferenoe on H^rid Oatputation , 1970, 62-66. Wacker, W. D. , "Approxinations to Optimal Policies in a Stationary Multioonmodity Inventory Model with Set-Up Cost," Ph.D. Thesis, Washington University, 1971. Vhite, D. J., Dynamic Progranming , Oliver & Boyd, Edinbur^ & London, 1967.

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BIOGRAPHICAL SKETCH Nabil S. Faour was bom in Palestine on December 25, 1940. In Septennber, 1947, he became a refugee. He attended schools in Beirut, Lebanon and graduated from high school in June, 1958. Mr. Faour 's undergraduate education was taken at the Ainerican University of Beirut, where in June, 1962, he received a Bachelor of Mathannatics degree. Between Septerijer, 1962 and July, 1965, Mr. Faour worked with the Arabian Anerican Oil Conpany, Dhahran, Saudi Arabia, in a senior position as an Electronic Systenvs Analyst. The author entered the Industrial and Systems Engineering Department of the University of Florida in S^tertoer, 1967. In Augxist, 1968, he received a ^faster of Operation Research degree. In July, 1970, he joined the American Can Conpany, Greenwidi, Connecticut, and he is presently an Associate Industrial Engineer. Mr. Faour is married cind hcis a son. 149

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I certify that I have read this study and that in my c^inion it conforne to acceptable standards of scholarly presentation and is fully adequate, in scope and qxiality, as a dissertation for the degree of Doctor of Philosophy. / / /,/ / Boghgs D. Sivazlian, Cttainnan Assoc. Prof, of Ind. and Sys. Eng. I certify that I have read this study and that in my opinion It conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree Of Doctor of Philosophy. F. Bums 3SOC. Prof, of Ind. and Sys. Eng. I certify that I have read this study and that in my opinion It conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Robert G. Blake Assoc. Prof, of Math. I certify that I have read this study and that in my opinion It conforms to acceptable standards of scholarly presentation and is fully adequate, ui scope and quality, as a dissertation for the degree of Doctor of Philosophy. , — Pop stoja^ovic/ . Prof, of Math. Zoran Assoc This dissertation was submitted to the Dean of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy March, 1972 ian. College of Engineering Dean, Graduate School

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