Minimax Analysis for. Finite Horizon Inventory Models. Guillermo Gallego. Department of IE & OR, Columbia University. Jennifer K.

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1 Minimax Analysis for Finite Horizon Inventory Models Guillermo Gallego Department of IE & OR, Columbia University Jennifer K. Ryan School of Industrial Engineering, Purdue University David Simchi-Levi Department of IE & MS, Northwestern University Submitted: November 1996 First Revision: October 1998 Second Revision: April 1999 Abstract We consider stochastic nite-horizon inventory models with discrete distributions that are incompletely specied by selected moments, percentiles, or a combination of moments and percentiles. The objective is to determine an inventory policy that minimizes the maximum expected cost over the class of demand distributions satisfying the specications described above. We show that many inventory models of this form can be solved by a sequence of linear programs. 1 Introduction In this paper we consider discrete demand, nite-horizon inventory models where the demand distribution is unknown except for a nite number of parameters, such as its mean and variance. The problem is to determine an inventory policy that minimizes the maximum expected cost over all demand distributions with the given parameters. We make no further assumptions on the form of the demand distribution, allowing us to deal with situations in which the exact demand distribution is unknown or changes over time. For example, we may have sucient data to accurately estimate the mean and variance of demand, but may be unable to condently estimate higher moments or the shape of the distribution. As Gallego and Moon (1993) point out, this is frequently the case in the fashion and sporting goods industries. Research supported in part by ONR Contracts N J-1649 and N , and NSF Contracts DDM DMI , and SBR

2 2 To our knowledge, the rst work on distribution free models is due to Scarf (1958), who considered the single period newsvendor problem. Scarf determines the order quantity that maximizes the minimum expected prot over all continuous demand distributions with a given mean and variance. Kasugai and Kasegai (1960) present a dynamic programming approach to the distribution free, multi-period newsvendor problem when demand is assumed to belong to a known closed interval; no additional assumptions are made on the demand distribution. Kasugai and Kasegai (1961) also consider the minimax regret ordering principle for the distribution free newsvendor problem, and compare these results to the minimax policy. More recently, Gallego and Moon (1993) provide a concise derivation of Scarf's single period results and consider various extensions of the problem. Gallego (1992 and 1996) considers a minimax approach to the innite horizon continuous review (Q; R) inventory model with incidence oriented backorder costs (Gallego, 1992) and time weighted backorder costs (Gallego, 1996). The thrust of this research has been to obtain distribution-free cost bounds that reveal the eects of randomness in the worst case, and distribution-free heuristics that are robust to the specication of demand. Moon and Gallego (1994) apply similar techniques to the innite horizon continuous and periodic review models with backorders and lost sales, while Moon and Choi (1997) do so for Assemble-to- Order, Assemble-in-Advance, and composite policies. For a more detailed discussion of previous research on distribution free procedures and other strategies that have been used to deal with inventory models in which the demand distribution is not specied, see Ryan (1997). The approach taken in this paper diers from previous work on distribution free procedures in several ways. First, most previous work has focused either on the single period problem or the innite horizon problem, i.e., models for which one can write a closed form expression for the expected cost. However, a distribution free procedure is particularly justied when limited information is available regarding the distribution of demand; it is more likely that, over a short nite, horizon, we would be unable to gather sucient information to accurately estimate the distribution of demand. Therefore, our focus is on the nite horizon problem. Our approach also diers in that we assume that demand is a discrete random variable, taking values in a known countable set. As we shall shortly see, this assumption allows us to formulate and solve many of the models as linear programming problems. Finally, and most importantly, our approach is dierent than the approaches suggested thus

3 3 far in that we present a general model that allows for a variety of constraints on the demand distribution. Most previous work has specically assumed that the demand mean and variance are known; these works cannot be easily modied to include additional information or constraints. The models presented here, on the other hand, can incorporate any nite number of linear constraints on the demand distribution. For example, the model presented here can easily incorporate any number of constraints on the fractiles of the demand distribution. This exibility is the major contribution of our approach. In Section 2 we present a general formulation for the problem of determining the inventory policy that minimizes the maximum expected cost over all distributions satisfying a variety of linear constraints. Section 3 demonstrates the applicability of our model in several special cases representing the most common single period inventory models, including the single period newsvendor model with and without set-up cost, inventory models with convex production costs and joint demand distributions, as well as methods for incorporating data into the minimax analysis. In Section 4 we consider in detail the multi-period inventory problem with set-up cost. We show that an (s k ; S k ) policy is optimal in each period, and provide an algorithm for nding the optimal (s k ; S k ) policy by solving a series of linear programs. To test the eectiveness of the minimax strategy, we conducted a computational study comparing the performance of the minimax policy to that of a policy based on a normal approximation of the demand distribution. This study demonstrates that if the critical fractile,, satises 0:5 0:8, then the minimax policy can perform quite well relative to a policy based on the normal distribution. The results are discussed in Section 5. Finally, in Section 6 we discuss a number of additional extensions that demonstrate the exibility of our approach. 2 General Formulation In this section we present a general formulation for the single period minimax problem. The objective is to determine an inventory policy to minimize the maximum expected cost over all demand distributions satisfying a set of linear constraints. We rst dene our notation. Let D denote demand which is random with support in a countable set of distinct real numbers, D = fd 1 ; D 2 ; : : :; D n g, where n is a positive integer. Thus, demand D must be a discrete random variable taking a nite number of values. We note that the demand D is a random variable, while the values D 1 ; : : :; D n are constants which represent possible values the random variable D can take.

4 4 The column vector d = (D j ) n represents the demand vector. The probability mass function p j = P (D = D j ); j = 1; : : :; n; is not completely specied, and belongs to the feasible set P = fp = (p j ) n : Ap = b; p 0g; where A is an l n matrix and b is an l-dimensional column vector, where l n. The set P may consist, for example, of all probability mass functions with support in D that have common moments specied by b. This is accomplished by setting, for every i = 1; 2; : : :; l, and j = 1; 2; : : :; n, a ij a i (D j ) = D i?1 j, and b i E[D i?1 ], for all i = 1; : : :; l, and E[D 0 ] 1: Of course, other interpretations are possible. For example, we could let a i (D j ) = I(D j > i ); for some i, i = 2; : : :; l; where I is the indicator function. In this case b i would be the desired percentiles of distributions in P: Alternatively, P could incorporate both moments and percentiles. The rst constraint should always be taken to be P n p j = 1; which reects the fact that p is a probability mass function. This constraint is already embedded into the moment formulation, and can be incorporated into the percentile formulation by setting 1 =?1 and b 1 = 1. Let c(y; D j ) be the cost function when we select inventory level y and the demand is D j. Then c(y) = (c(y; D j )) n is the cost vector under y and c(y)0 denotes the transpose of c(y). Given a probability vector p, p 2 P; and an inventory level y, the expected cost is c(y) 0 p: The problem is to nd y so as to minimize c(y) 0 p against the worst possible distribution in P: That is, our objective is to solve Problem P : min y0 max p2p c(y) 0 p: Observe that, given y, the dual of the inner maximization in Problem P is min 0 b subject to 0 A? c(y) 0 0: Therefore, Problem P can now be written as the following optimization problem, referred to as Problem D: min y0; 0 b

5 5 s.t. 0 A? c(y) 0 0: In general, Problem D is not a linear program. However, for many inventory models c(y) is the maximum of piecewise linear parts, e.g., c(y) = maxff l y + g l g; l2l where L is the set of piecewise linear parts, and f l and g l are n-dimensional column vectors. In this case, Problem D can be written as the following, possibly semi-innite, linear program: min y0; 0 b s.t. 0 A? yf 0 l g0 l l 2 L; with an associated dual, Problem GLP : max p 1 ;:::;p L s.t. X l2l X l2l g 0 lp l Ap l = b? X l2l f 0 l p l 0 p l 0 l 2 L; where the p l are substochastic vectors, i.e., each p l, l 2 L, is a vector of probabilities associated with a subset of demands, and hence the sum of the probabilities in the vector is less than or equal to one. Notice that this formulation, where the entire cost function is dealt with at once, avoids having y as part of the objective function and allows us to solve a single linear program to nd both the worst case distribution and the optimal inventory P policy. The optimal inventory level is the dual variable associated with the constraint? l2l f 0 l p l 0.

6 6 3 Examples We now illustrate how to apply the minmax linear program to the newsvendor problem with and without setup costs, to production with piecewise linear increasing convex costs (diseconomies of scale), and to yield management models. In the next section we consider in greater detail the multi-period inventory model with set-up cost, while in Section 6 we discuss a number of other extensions. 3.1 Single Period Newsvendor Problem Consider the classical newsvendor problem where the cost of ordering y units is cy, the demand is D, the selling price is q > c, and the salvage value for unsold units is v < c. Revenues consists of sales plus salvage value. If y units are ordered and D j units are demanded, the revenue is r(y; D j ) = 8>< >: vy + (q? v)d j if y > D j qy if y D j : Let r(y) = (r(y; D j )) n. With this notation we can write c(y) = c(y; D j) n as c(y) = cye? r(y) = maxff 1 y + g 1 ; f 2 y + g 2 g; where f 1 = (c? v)e; g 1 = (v? q)d, f 2 = (c? q)e, e is an n-dimensional column vector of ones, d = (D j ) n is a column vector of demands, and g 2 = 0 is an n-dimensional column vector of zeros. Clearly, each component of c(y) is convex in y. Problem GLP is then given by max p 1 ;p 2 (v? q)d 0 p 2 s.t. Ap 1 + Ap 2 = b? (c? v)e 0 p 1? (c? q)e 0 p 2 0 (1) p 1 0; p 2 0: Notice that p 1 + p 2 is the worst case distribution and that p 1 is the vector of probabilities

7 7 associated with the event D y. Then, using e 0 (p 1 + p 2 ) = 1, the second constraint reduces to the classical condition P r(d y) = e 0 p 1 (q? c)=(q? v): Notice also that the dual variable of the second constraint is precisely the optimal order quantity. It is common, especially when considering multi-period problems, to charge a holding cost, say h +, per unit of ending inventory in excess of demand, and a penalty cost, say h?, per unit of shortage. The reader can verify that the above formulation of the newsvendor problem remains valid, with the following modications: f 1 = (c+h + )e; g 1 =?(h + +q)d; f 2 = (c?(q +h? ))e; g 2 = h? d. With these modications, the optimality condition becomes P r(d y) = e 0 p Single Period Problem with Set-Up Cost q + h?? c q + h? + h + : (2) In this example we consider the single period inventory problem when there is a xed order, or set-up, cost. Suppose that the cost of increasing the inventory from its initial level y 0 to y; y > y 0, includes a xed cost C plus a linear part proportional to the order quantity y? y 0. Then the total cost vector, starting with inventory y 0 and ordering up to the inventory level y; y y 0, is given by c(y; y 0 ) = C(y? y 0 )e? cy 0 e + c(y); where (u) = 1 if u > 0; (0) = 0, and c(y) is as dened in Example 1. Unfortunately, this cost function cannot be directly reduced to the piecewise linear form max l2l ff l y + g l g: For a given y, dene G(y) = max p2p c(y)0 p = min 0 b; : 0 Ac(y) and notice that c(y) = max l2l ff l y + g l g ensures that G(y), as the solution of a linear program, is convex in y. We can now write the problem min yy0 max p2p c(y; y 0 ) 0 p as?cy 0 + min yy 0 [C(y? y 0 ) + G(y)]:

8 8 The convexity of G(y) together with G(y)! 1 as jyj! 1, implies that an (s; S) policy is optimal (Veinott, 1965), where S is the least minimizer of G(y) and s is the largest value of y smaller than S such that G(y) = C + G(S): Clearly, G(S) is just the solution to Problem GLP in Section 3.1, and S is the dual variable associated with constraint (1). We can nd the reorder point, s, by minimizing y subject to G(y) C + G(S) and y S. Let z d (y) = G(y)? cy, and notice that z d (y) is the maximum expected cost when the initial inventory is y and we do not place an order. Notice that the condition G(y) C + G(S) is equivalent to z d (y) C + c(s? y) + z d (S). Interestingly, we can nd s by solving a single additional linear program referred to as Problem D s : max ys; 0 b s.t. 0 A? y ~ f 0 1 g0 1 0 A? y ~ f 0 2 g b C + c(s? y) + z d (S): where ~ f 1 = f 1? ce =?ve; ~ f2 = f 2? ce =?qe, and g 1 ; g 2 are as before. Proposition 3.1 The optimal y found by solving Problem D s is the reorder point, s. The proof of Lemma 3.1 can be found in Appendix A. Intuitively the Lemma can be explained as follows. Dene z o (y) C + c(s? y) + z d (S) to be the worst case expected cost when we start with initial inventory y and order up to S. Then, since z d (y) is the worst case expected cost when we have initial inventory y and do not order, we can interpret Problem D s as follows: If y < s, we have z d (y) > z o (y), so it is optimal to order. If s < y < S, we have z d (y) < z o (y), so it is optimal not to order. Therefore, s is just the point where z o (y) = z d (y). In other words, s is the point where the cost if we order just equals the cost if we do not order. Also note that nding the optimal order-up-to point, S, can be viewed as nding the point at which the marginal revenue of an additional unit of initial inventory just equals the marginal cost

9 9 of a unit of inventory. Indeed, S = y is the point at which the line z o (y) is tangent to the curve C + z d (y). The slope of the line z o (y) is?c, the marginal revenue of one unit of inventory. The slope of z d (y) is the marginal cost of an additional unit of initial inventory. Therefore, S = y is just the point where marginal revenue equals marginal cost. This interpretation will be useful in the analysis of the multi-period problem performed in the next section. 3.3 Piecewise Linear Convex Production Costs In practice, production cost is often linear within the capacity of the least expensive resource used e.g., regular labor. Beyond that, the production cost is linear within the capacity of the next available resource, e.g., overtime. When overtime capacity is exhausted, the production cost is linear within the capacity of the next available resource, e.g., subcontracting, etc.. This results in a convex production cost of the form: pc(y) max c l (y); l2l where c l (y) = a l + c l y, where the a l 's are non-increasing constants with a 0 = 0 and the c l 's are non-decreasing positive constants. First, we note that in the absence of initial inventory and setup costs the cost function is c(y) = pc(y)e? r(y) = max c l (y)e? r(y) l2l = maxff 1l y + g 1l ; f 2l y + g 2l g; l2l where r(y) is as dened in Section 3.1, f 1l = (c l? v)e, g 1l = a l e + (v? q)d, f 2l = (c l? q)e; and g 2l = a l e: The problem is now in cannonical form, and the optimal production quantity can be obtained by solving a single linear program. Next, suppose that we start with an initial inventory level y 0 and that the cost of increasing the inventory from y 0 to y > y 0 includes a xed cost, C, plus pc(y? y 0 ). Then the total cost is c(y; y 0 ) = C(y? y 0 )e + pc(y? y 0 )e? r(y):

10 10 Let g(y; y 0 ) = pc(y? y 0 )e? r(y): Since convex combinations of convex functions are convex, it follows that G(y; y 0 ) = max p2p g(y; y 0) 0 p = pc(y? y 0 ) + max p2p?r(y)0 p = pc(y? y 0 ) + z d (y) is also a convex function of y, where z d (y) is as dened in Section 3.2. To determine the optimal inventory policy for this problem, let S(y 0 ) be the least minimizer of G(y; y 0 ). Then, since?g(y; y 0 ) is unimodal and G(y; y 0 )! 1 as jyj! 1, we have lim G(y; y 0) > C + G(S(y 0 ); y 0 ): (3) jyj!1 Therefore, it is optimal to order up to S(y 0 ) if and only if y 0 s(y 0 ); where s(y 0 ) is the largest value y S(y 0 ) such that G(y; y 0 ) = C + G(S(y 0 ); y 0 ): Notice that both s(y 0 ) and S(y 0 ) depend on the initial inventory y 0 ; and that Proposition 3.2 S(y 0 ) is a nondecreasing function of y 0. S(y 0 ) does not increase as fast as y 0, or more precisely y 0? S(y 0 ) is nondecreasing in y 0 : Proof. The proof is based on the well known monotone optimal selection theorem (Topkis, 1978 and Veinott, 1965). A special case of this theorem asserts that S(y 0 ), the least minimizer of G(y; y 0 ), is non-decreasing in y 0 if the mixed partial derivative of G(y; y 0 ) with respect to y 0 and y is nonpositive. Clearly since pc() is convex. Next, y 0 =?pc 00 (y? y 0 ) 0 G # (y; y 0 ) = G(y 0? y; y 0 ) = pc(?y) + z d (y 0? y);

11 11 and notice # (y; y 0 =?z 00 d (y 0? y) 0 since z d () is convex. This implies that y 0? S(y 0 ) is nondecreasing in y 0, as asserted. Next, we will show that the optimal policy for this problem is a generalized (s; S) policy. If we include the restriction y y 0 in minimizing G(y; y 0 ), then the least minimizer of G(y; y 0 ) subject to y y 0 is S(y 0 ) max(y 0 ; S(y 0 )): We now prove Proposition 3.3 S(y 0 ) = y 0 for all suciently large y 0. Proof. We must show that G(y 0 ; y 0 ) = z d (y 0 ) pc(y? y 0 ) + z d (y) = G(y; y 0 ) (4) holds for all y y 0 and y 0 suciently large. This is true since z d (y) is convex and lim y 0!1 z0 d(y 0 ) =?v >?c 1 >?pc 0 (): Consequently, (4) holds for all y 0 s where s is the smallest y 0 such that z 0 d (y 0)?c 1 : On the other hand, assume that there is an y 0 such that C + G(S(y 0 ); y 0 ) < G(y 0 ; y 0 ): (5) Condition (5) states that there is at least one initial inventory y 0 for which it is better to order than not to order. We would expect condition (5) to hold when y 0 = 0: Let s be the largest value of y 0 for which (5) holds. Clearly s exists and is bounded above by s. Therefore, the optimal policy is a generalized (s; S) policy where S(y 0 ) = y 0 for y 0 s and S() non-decreasing on the interval (?1; s). S() has supremum S S(s) > s. Notice that on the region y 0 < s, the order-up-to level S(y 0 ) is nondecreasing in y 0 with limit S > s, reecting the fact that the marginal and total cost of producing up to S increases as the initial inventory y 0 decreases. See Karlin (1958) for an analysis of the single stage problem

12 12 with convex costs, and Porteus (1971) for a discussion of generalized (s; S) policies in multi-period problems with concave costs. 3.4 Joint Distributions Frequently, the expected cost depends on the joint distribution of two random variables, where all that is available is limited information on the marginal distributions. To illustrate this, we consider two examples. The rst example is the classical newsvendor problem where demand at the salvage value v < q is a random variable, V. In this case the expected cost can be written as C(y) = cy? qe min(y; D)? ve min[(y? D) + ; V ]: Notice that in the traditional newsvendor problem P (V = 1) = 1: As a second example, consider the case where y is xed and q < v: We can think of y as the seat capacity of a plane, q as the supersaver fare, and v the full coach fare. When there is more than enough capacity, y, to meet the expected demand at the full coach fare, it can be protable to market the excess capacity at the supersaver fare. Since tickets at the supersaver fare are sold rst, the problem is to decide how much capacity to allocate to that fare, reserving the balance of the capacity for sale at the full coach fare. If y 1 seats are made available at the supersaver fare, then min(y 1 ; D) are sold at that fare and y? min(y 1 ; D) are available for sale at the full coach fare. The problem is to nd y 1 2 [0; y] to minimize C(y 1 ) =?qe min(y 1 ; D)? ve min[y? min(y 1 ; D); V ]; where D is the demand at the supersaver fare and V is the demand at the full coach fare. Both of the above problems are relatively easy to solve when D and V are independent random variables with known distributions. Here we will assume that D and V are random variables with support on the countable sets D = fd 1 ; : : :; D n g and V = fv 1 ; : : :; V n g of distinct real numbers. Note that we do not assume that D and V are independent. Let P be the set of allowable marginal distributions for D and V, P = f(p; t) : p = (p j ) n ; t = (t k ) n k=1; A 1 p = b 1 ; p 0; A 2 t = b 2 ; t 0g:

13 13 where A 1 and A 2 are l n matrices and b 1 and b 2 are l-dimensional column vectors. Let c(y; D j ; V k ) denote the cost function when the decision variable is y and the demands are D j and V k : Let the joint distribution of D and V be x jk = P (D = D j ; V = V k ): The expected cost is then C(y) = k=1 c(y; D j ; V k )x jk : The problem is to minimize C(y) subject to the worst possible distribution in P: This results in the primal problem: min y max p;t s.t. p j = k=1 t k = k=1 c(y; D j ; V k )x jk x jk ; x jk ; A 1 p = b 1 A 2 t = b 2 p 0; t 0; j = 1; : : :; n k = 1; : : :; n and the dual problem: min y; 1 ; b b 2 s.t. 0 1 A 1? A 2? 0 0 ( j + k )? c(y; D j ; V k ) 0 8j; k: This problem can be linearized provided that c(y; D j ; V k ) can be written as the maximum of linear functions of y: This is clearly the case for both of the problems discussed above. Indeed, for

14 14 the newsvendor problem with uncertain demand at the salvage value, we have c(y; D j ; V k ) = 8>< >: (c? q)y if y D j (c? v)y + (v? q)d j if D j < y D j + V k cy? qd j? vv k if D j + V k < y: Similarly, for the seat allocation problem with uncertain demand at both fares, we have c(y; y 1 ; D j ; V k ) = 8>< >: (v? q)y 1? vy if y 1 D j ; y? y 1 V k?qy 1? vv k if y 1 D j ; y? y 1 V k (v? q)d j? vy if y 1 D j ; y D j + V k?qd j? vv k if y 1 D j ; y D j + V k : 3.5 Incorporating Data The development of distribution free inventory policies is based on the assumption that limited information or data is available regarding the true distribution of demand. The use of these policies does not make sense when there is sucient data available to make a reasonable estimate of the shape of the demand distribution. It may be the case, however, that while initially there is limited or no demand information available, as time progresses more demand data is observed and a better estimate of the demand distribution can be developed. Therefore, we consider the problem of determining an optimal inventory policy for a multi-period model in which we incorporate observed demand data into the worst case analysis in order to improve our estimate of the distribution of demand. Specically, we consider a T period model in which we have an opportunity to observe demand for the rst m periods before committing to a nal order quantity. To incorporate these demand observations, we add a set of constraints to the basic model that require the worst case distribution to t the observed data within some signicance level. These constraints are in the form of condence intervals for p j : ^p j? j p j ^p j + j j = 1; : : :; n; where ^p j is the maximum likelihood estimate of p j based on the observed data, and (^p j? j ; ^p j + j ) is a conndence interval with the prescribed signicance level, obtained using any of a number of

15 15 statistical techniques, e.g., generalized likelihood ratios. For more details on the solution of the distribution free, multi-period model with data, see Ryan (1997). 4 Multi-Period with Set Up Costs Consider now an m-period problem with set-up cost. Given an initial inventory and a discount rate 2 [0; 1] the problem is to nd an inventory policy that minimizes the maximum -discounted m-period expected cost against the worst possible sequence of probability mass functions of demand over the m periods, where in each period the probability mass function is in the set P = fp : Ap = b; p 0g, where A is an l n matrix and b is an l-dimensional column vector. When analyzing a multi-period problem, one must rst decide whether to consider a stationary or nonstationary model, i.e., whether or not to require the worst case demand distribution to be the same in each period. We rst note that the stationary model can be formulated as a single integer program, with n m continuous variables representing the joint demand distribution and m binary variables indicating whether or not an order is placed in each period, where n is the number of demand points and m is the number of periods. The stationary model will obviously be more appropriate in situations in which we believe the demand distribution is the same in each period. If, however, the distribution of demand may change from period to period, or if the constraints dening the feasible region change from period to period, then the nonstationary model may be more appropriate. Finally, note that the nonstationary model obviously provides a conservative policy. That is, given an inventory policy, the worst case expected cost for the stationary problem will be less than or equal to the worst case expected cost of the nonstationary problem. In this section, we consider the nonstationary, multi-period distribution free problem. While we assume, to simplify notation, that the feasible region, P, remains the same in each period, this assumption is not necessary. See Section 6.1 for further discussion of models with a nonstationary constraint set. In this section, we show that the optimal policy is of the (s; S) type and develop a dynamic programming algorithm to determine the optimal (s k ; S k ) values for k = 1; 2; : : :; m. To do this, we follow the convention of letting the index k represent the number of remaining periods, e.g., k = 1 refers to the nal period, and k = m refers to the rst period. Let y k denote the on-hand

16 16 inventory, before ordering, at the start of period k, k = 1; : : :; m. To formulate the dynamic program, let z k(y d k) be the -discounted k-period expected cost when starting with y k units in period k if no order is placed in that period and the system is managed optimally thereon. Also, let z k (y k ) be the optimal -discounted k-period expected cost when starting with y k units in period k. In the remainder of this paper, \expected" refers to the expectation with respect to the worst case distribution. Let h + be the holding cost per unit of ending inventory in excess of demand, and let h? be the cost per unit of shortage. Then we can write the single period cost when there are y k units of inventory on-hand and demand is D j as r(y k ; D j ) =?qy k + q max(0; y k? D j ) + h + max(0; y k? D j ) + h? max(0; D j? y k ): It follows that z k d (y k ) = max p = max p f r(y k ) 0 p + z k?1 (y k e? d) 0 p j p 2 Pg f?qy k + q p j max(0; y k? D j ) + h + p j max(0; y k? D j ) X +h? n p j max(0; D j? y k ) + p j z k?1 (y k? D j ) j p 2 Pg; where r(y k ) = (r(y k ; D j )) n and zk?1 (y k e? d) = (z k?1 (y k? D j )) n are n dimensional column vectors. Call this Problem P M k. Note that the formulation of problem P M k allows for a dierent worst case distribution in every period, i.e., the worst case distribution for period k, found by solving Problem P M k, will generally not be the same as the worst case distribution for period k? 1, found by solving problem P M k?1. Given the function zd k(y k), the optimization problem that needs to be solved is: z k (y k ) = min fc(y? y k ) + c(y? y k ) + z k d (y)g; (6) yy k Dene the function G k (y) = cy + z k d (y); and observe that if an order placed in period k raises the inventory level to y > y k, then the -discounted k-period expected cost is C? cy k + G k (y). To show that the optimal policy is of the (s k ; S k ) form, it is enough to verify that G k (y k ) is

17 17 C-convex and G k (y k )! 1 as jy k j! 1, for each period k, k = 1; 2; : : :; m. Since G 1 (y) is convex in y, and grows without limit as y! 1, the analysis and results in Veinott (1966), see also, Bramel and Simchi-Levi (1997), can be applied to show: Lemma 4.1 The function G k (y k ) is a C-convex function of y k, for all k, k = 1; : : :; m. In addition, G k (y k )! 1 as jy k j! 1. The Lemma thus implies that (Scarf, 1960): Corollary 4.2 The optimal inventory policy solving equation (6) is an (s k ; S k ) policy for all k = 1; 2; : : :; m. 4.1 Solving the Dynamic Progamming Equations We now show how to solve the dynamic programming model, that is, how to compute the optimal (s k ; S k ) policy for every k, in time which is polynomial in n, the number of demand points, but exponential in the number of periods, m, and the number of constraints in the matrix A, l. For this purpose, we show, Lemma 4.3 The functions z k(y) and d zk (y) are piecewise linear in y with O(n (l+1)k?1 ) breakpoints, i.e., points where the functions change their slopes, for k = 1; : : :; m. Proof. The proof is constructive in nature, and is designed to provide an algorithm for generating the functions z k d (y) and zk (y). The function z 1 d (y) is piecewise linear in y since it is the optimal solution to a linear programming problem. Similarly, by the denition of z 1 (y), see equation (6), it is also piecewise linear in y. The number of constraints? dening the feasible region, Ap = b, is l. Therefore, the number n of basic solutions is l, the number?? of breakpoints in z 1 (y) is at most n d l, and the number of breakpoints in z 1 n (y) is at most l + 1. We proceed by induction on k. Assume the Lemma holds for i = 1; 2; : : :; k? 1; for some k 1, and assume that we have determined all the breakpoints of the function z k?1 d (y). This implies that we can determine s k?1, and that for every j, j = 1; 2; : : :; n, the quantity z k?1 (y?d j ) and the slope of the function z k?1 () at the points y? D j are known. Let b 1 ; b 2 ; : : :; b B be all the breakpoints of

18 18 the function z k?1 () and let I 2 ; : : :; I B be the corresponding subintervals, where I u = [b u?1 ; b u ), for u = 2; : : :; B. To characterize the function z k d (y), choose any value y. For every j, j = 1; 2; : : :; n, let u j be the index of the subinterval the quantity y? D j belongs to, i.e., y? D j 2 I uj this interval, the function z k?1 () has a slope denoted by k?1 j. = [b uj?1; b uj ). On For every j, j = 1; 2; : : :; n, let j = b uj? (y? D j ). That is, j is the amount by which the value of y can be increased without changing the slope of the function z k?1 at the point y? D j. Let 0 = min j j : The quantity z k d (y k) for y k 2 [y; y + 0 ) can be found by solving the dual of Problem P M k, referred to as Problem DM k : min 0 b s.t. 0 A?(q + h? )(y + )e + h? d + z k?1 (ye? d) + k?1 ; 0 A h + (y + )e? (q + h + )d + z k?1 (ye? d) + k?1 ; where k?1 = ( k?1 j ) n is an n dimensional column vector, and = y k? y. Linear programming theory tells us that the optimal solution to Problem DM k is a piecewise linear function of 2 [0; 0 ) and hence z k d (y k) is a piecewise linear function of y k, y k 2 [y; y + 0 ). Thus, z k d (y) is a piecewise linear function of y and equation (6) implies that zk (y) is also piecewise linear in y, with at most one additional breakpoint. most? Since Problem DM k is a linear program, the function z k() will have at n d l breakpoints in the interval [y; y+ 0 ). If we construct intervals [y; y+) such that they cover the entire x coordinate, the number of such subintervals is no more than Bn, where B is the number of breakpoints Bn? of z k?1 (). Hence, the total number of breakpoints of the function z k() is no more than n d l, which is equal to O(n (l+1)k?1 ). Observe that the above proof suggests a procedure for nding the function z k d (y). Indeed, choosing any initial y and repeatedly solving Problem DM k makes it possible to generate the entire function z k d (y), and thus the function zk (y). It remains to show how one uses the functions z k d (y) in constructing the optimal (s k; S k ), for k = 1; 2; : : :; n. We start by computing the optimal policy for period k = 1, and recursively calculating the functions z k d (y). These functions are used to determine the optimal (s k; S k ) for each

19 19 period k = 2; : : :; m. The functions z 1(y) and d z1 (y), as well as the optimal (s 1 ; S 1 ), can be found as described in Section 3.2. Given the functions z j (y) and d zj (y) and the optimal (s j ; S j ), for j = 1; 2; : : :; k? 1, and the function zd k(y), we now show how to determine the optimal (s k; S k ) values. As in the single period model, let zo k (y k ) be the expected cost given that we start in period k with on-hand inventory of y k units, order up to S k in period k, and act optimally in the remaining k? 1 periods. That is, let z k o (y k ) = C + c(s k? y k ) + z k d (S k ): The optimal S k will be the value of y that minimizes G k (y). Alternatively, the optimal S k will be the point at which the function zo k (y k ), a line with slope?c, is tangent to the curve C + z k d (y k). Since z k(y d k) is a piecewise linear function of y k, the optimal value S k is at a point at which z k(y d k) changes slope and a line of slope?c supports the function zd k(y k) from below. Given S k, we nd s k by solving a single additional linear program. Recall from the single period problem that s k is the smallest point at which the line zo k(y k) intersects the curve z k(y d k). That is, s k is the smallest value of y k that solves z k d (y k) = C + c(s k? y k ) + z k d (S k): To determine s k eciently, we use the functions z k o (y k) and z k d (y k) to identify the subinterval, i.e., the breakpoints of zd k(y k), containing that point of intersection, and the slope of zd k (y) at the point of intersection. Let this subinterval be [s 0 ; s 00 ). We determine the exact value of s k by solving a linear program that maximizes z k d (y k) subject to the constraints z k d (y k) z k o (y k ) and s 0 y k s 00. That is, by Lemma 3.1, we solve max ; s.t. 0 A?(q + h? )(s 0 + )e + h? d + z k?1 (s 0 e? d) + k?1 ; 0 b 0 A h + (s 0 + )e? (q + h + )d + z k?1 (s 0 e? d) + k?1 ; 0 b C + c(s k? s 0? ) + z k d(s k ); s 0 + min(s 00 ; S);

20 20 0: Let be the optimal in the above linear program, then s k = s 0 + : Figure 1 shows the functions z 2(y) and d z2 o(y) and the values of s 2 and S 2 for a simple example with four demand points. 4.2 Structural Results We are now interested in determining structural properties for the optimal values (s k ; S k ), k = 1; 2; : : :; m. Unfortunately, very little is known even in the case where the demand distribution is completely specied. Iglehart (1963), however, has shown that, for the stationary, multi-period inventory problem with set-up cost and known demand distribution, S 1 S k ; k = 2; 3; : : :; m. A similar result can be proved for the multi-period, distribution free inventory problem with set-up cost and specied mean demand. Theorem 4.4 Assume that the set of constraints dening the set of feasible distributions, Ap = b, contains the constraint P n pj D j =. Then the optimal policy solving equation (6) satises S k minfs i g; k = 2; : : :; m: ik The proof of Theorm 4.4 is contained in Appendix A. Finally, it is appropriate to point out that the multi-period inventory problem without a set-up cost can be viewed as a special case of the multi-period problem with set-up cost, and therefore can be analyzed in the manner described earlier. In this case, it is easy to show that the expected cost function for the remainder of the planning horizon is convex in y for each period k; therefore, a critical number policy will be optimal in each period. That is, the optimal inventory policy in period k is to order up to S k if y k < S k, otherwise do not order. The method of determining the optimal S k values is identical to the method described above for the multi-period problem with set-up cost. We can also state the following corollary to Theorem 4.4. Corollary 4.5 The optimal policy for the distribution free, multi-period inventory model without set-up cost satises S 1 S k k = 2; 3; : : :; m:

21 21 5 Computational Study The most common criticism of the minimax approach is that it generates inventory policies that perform well only in the worst case, and that, on average, these policies may perform quite poorly. An alternative approach that has frequently been suggested, and is commonly used in practice, is to assume that demand is normally distributed and use an inventory policy that is optimal under that assumption. To examine the robustness of the minimax approach we performed a computational study of the single period newsvendor model without set-up cost. In this study we compare the minimax approach to the alternative based on the normal distribution. In this section, we rst describe the methodology used and the models considered. We then present results from the study and discuss their implications. In particular, we demonstrate that if the critical fractile,, satises 0:5 0:8, then the minimax policy can perform quite well relative to a policy based on the normal distribution. 5.1 Methodology In this study, we assume that the true demand distribution is unknown, but that the retailer has some demand data that can be used to estimate the mean,, and variance, 2, of demand. The retailer, however, does not have sucient data to determine the exact shape of the distribution. Given the estimates of the mean, ^, and variance of demand, ^ 2, obtained from the observed data, the retailer determines an inventory policy in one of two ways. He may assume that demand is normally distributed and use a policy that is optimal under that assumption, or he can use the minimax approach presented in Section 3.1. The performance of these inventory policies will depend on the true distribution of demand. For the computational study, we considered the following demand distributions: 1. Gamma with = 2 = Gamma with = 30; 2 = Exponential with = 20; 2 = 400. In the rst case, the distribution is symmetric and bell-shaped. Therefore, we would expect

22 22 the policy based on the normal distribution to be a good approximation of the true optimal policy. In this case, our goal is to determine how poorly the minimax policy performs. In the second and third cases, the distribution is skewed. Therefore, we would not expect the policy based on the normal distribution to perform well. In these cases, our goal is to determine if the minimax policy performs better than the policy based on the normal distribution. In each case, the analysis consisted of the following steps: 1. We randomly generated ten points from the true demand distribution to simulate situations in which only limited information is available regarding the true distribution of customer demand. Note that this is sucient data to obtain an estimate of the mean and variance of demand, but is not sucient to approximate the shape of the distribution. 2. The demand points were used to estimate the mean (^) and variance (^ 2 ) of demand, using the sample mean and variance. 3. We determined the optimal inventory level under the assumption of normality using the estimated mean and variance: y N = ^ + z ^, where z satises P (Z z ) =, Z is a standard normal random variable, and is the critical fractile in the newsvendor problem. In other words, we choose y N such that P (D y N ) =. 4. We determined the minimax inventory level, y M, by solving the dual of Problem GLP in Section 3.1, with constraints on the mean and variance of demand: P n i=1 D i p i = ^ and P n i=1 D 2 i p i = ^ 2 + ^ 2. To solve this problem, it was necessary to specify the set of possible demand points, D 1 ; : : :; D n. We chose D i = i=100, for i = 0; : : :; We computed the optimal inventory policy, y, based on the true demand distribution and the true mean and variance. 6. We computed the expected single period cost for each policy based on the true demand distribution. 7. We repeated steps (3) through (6) for 60 values of the critical fractile,, ranging from very low (.08) to very high (.95).

23 23 8. We repeated steps (1) through (7) ten times, each with a dierent randomly generated set of demand points. 5.2 Results We are interested in comparing the expected costs for each of the three inventory policies, y N ; y M, and y. Let G(y) denote the expected single period cost for the true demand distribution when the inventory level is y. We made two types of comparisons: 1. We compared the expected cost for the optimal policy with the expected costs for the minimax and normal policies. In other words, we compared G(y ) with G(y M ) and G(y N ): 2. We compared the expected cost for the minimax policy with the expected cost for the normal policy. In other words, we compared G(y M ) with G(y N ): These results are presented in Tables 1 and 2. Table 1 provides information about (1) the true distribution of demand, (2) the % of cases, out of the 600 instances (60 values of the critical fractile times ten simulation rounds), in which the minimax policy outperformed (i.e., had lower expected costs) the normal policy, (3) the average % between G(y M ) and G(y N ) for all cases for which the minimax outperformed the normal policy. The next two columns provide the same information for the instances in which the normal policy outperformed the minimax policy. When evaluating the eectiveness of minimax inventory policies, one useful measure is the expected value of perfect information (EVPI). EVPI is the dierence between the expected costs for the optimal policy, which is determined using complete knowledge of the demand distribution, and the expected costs for an inventory policy which is determined using only limited knowledge regarding the demand distribution, in this case ^ and ^ 2. In other words, we calculate the expected value of perfect information for the minimax policy as EV P I M = G(y )? G(y M ): Similarly, we calculate the expected value of perfect information for the normal policy as EV P I N = G(y )? G(y N ):

24 24 Table 2 presents the expected value of perfect information, calculated as a percent dierence in costs, for both the minimax policy and the normal policy. Specically, the table compares the costs of the minimax policy with the costs of the optimal policy (2nd column) and compares the costs of the normal policy with the costs of the optimal policy (3rd column). Table 1: G(y M ) vs G(y N ) % of cases in which Average % % of cases in which Average % Distribution G(y M ) < G(y N ) Dierence G(y M ) > G(y N ) Dierence Gamma( = 50; 2 = 50) 39 < 1 61 < 1 Gamma( = 30; 2 = 300) Exponential( = 20; 2 = 400) Table 2: G(y M ) and G(y N ) vs: G(y ) Ave % Dierence Ave % Dierence Distribution G(y M ) vs: G(y ) G(y N ) vs: G(y ) Gamma( = 50; 2 = 50) 1 1 Gamma( = 30; 2 = 300) 12 7 Exponential( = 20; 2 = 400) 8 5 As pointed out by Scarf (1958), it is expected that when the critical fractile is near its extreme values, i.e., near 0 or 1, the minimax policy will not perform well. Indeed, in the course of the analysis we noted that the minimax policy performs particularly well when the critical fractile, P (D y) =, satises 0:5 0:8. Therefore, we also present comparisons of the expected costs for the minimax and normal policies for the 220 (of 600) cases we considered in which the critical fractile falls in this range. These results are presented in Tables 3 and 4.

25 25 Table 3: G(y M ) vs: G(y N ), 0:5 0:8 % of cases in which Average % % of cases in which Average % Distribution G(y M ) < G(y N ) Dierence G(y M ) > G(y N ) Dierence Gamma( = 50; 2 = 50) 43 <1 57 <1 Gamma( = 30; 2 = 300) Exponential( = 20; 2 = 400) Table 4: G(y M ) and G(y N ) vs: G(y ), 0:5 0:8 Ave % Dierence Ave % Dierence Distribution G(y M ) vs: G(y ) G(y N ) vs: G(y ) Gamma( = 50; 2 = 50) 1 1 Gamma( = 30; 2 = 300) Exponential( = 20; 2 = 400) Implications The computational results presented here suggest: If the distribution of demand is symmetric and bell-shaped, then both the minimax policy and the normal policy perform quite well. The dierence between the expected costs of the two policies is minimal, i.e., less than 1%. In other words, if the distribution is approximately normally distributed, then the retailer does not lose much by using a minimax policy rather than the policy based on the normal distribution. Both the minimax policy and the normal policy perform fairly well when the distribution of demand is skewed. The average percent dierence between the expected cost of the optimal policy and the expected cost of the minimax policy is about 10%, while the average percent dierence between the expected cost of the optimal policy and the expected cost of the normal policy is about 6%. In addition, the normal policy outperforms the minimax policy about 60% of the time. If the distribution of demand is skewed, and we consider only the cases in which the critical fractile,, is between 0.5 and 0.8, then the minimax policy performs better than the normal

26 26 policy between two-thirds to three-quarters of the time. Note that if the distribution is not skewed, then the dierence between the expected costs of the two policies is minimal, i.e., less than 1%. This nal result indicates that if the retailer has a cost structure such that the critical fractile is between 0.5 and 0.8, then he may want to consider using a minimax policy rather that the normal policy. Note that this criteria, i.e., whether or not the critical ratio falls within a given range, is quite simple for the retailer to check. In addition, we note that Scarf (1958) demonstrated that the continuous version of the minimax policy performs quite well relative to the optimal policy when demands are normally distributed and 0:1 0:9. Finally, it is appropriate to point out that if the mean and variance of demand are known exactly, the inventory policy based on the normal distribution performs quite well. This is true for a variety of demand distributions, particularly those that are not dramatically skewed. 6 Summary and Extensions As discussed above, the approach presented in this paper is quite exibile. It allows the incorporation of various levels of information about the demand distribution. Indeed, moments, fractiles, or condence intervals can be included without changing the complexity of the algorithm. For example, when the mean demand is estimated based on observed data, we may not want to force Pi p id i to be equal exactly to the estimated mean demand. Instead, as we have seen in the computational study, it is possible to require that this quantity be contained in a certain interval around the estimated mean. This approach, which allows us to capture the uncertainty in the estimated mean, turned out to be the most eective approach in our computational study. In addition, it is important to point out a number of extensions and alternative formulations that can easily be included in our model. These are discussed below. 6.1 Alternative Assumptions on the Demand Distribution The results of Section 4 also hold for the dynamic, distribution free, multi-period inventory problem. The proof of the optimality of an (s k ; S k ) policy and the procedure developed to nd the optimal policy do not depend on the assumption that the constraints dening the feasible distributions are

27 27 the same in every period. For instance, our results hold for the distribution free problem when the mean, k and variance, 2 k are allowed to vary. 6.2 Including Capacity Constraints We can also include capacity constraints of the form y?y k C into the multi-period model without set-up cost. If the ordering capacity in any period is C, then it is easy to show that the optimal policy for period k will be order min(c; S k? y k ) if y k < S k ; otherwise do not order. For more details, see Ryan (1997). We remark that this property may fail in the presence of setup costs. 6.3 Including a Service Level Constraint We have included a shortage cost h? in our model; alternatively, for the P single period model, we may want to include a service level constraint of the form P r(d > y) = i:d i >y p i ; for some service level ; 0 1. This constraint can easily be added to Problem GLP in Example 1 above. For more details on the multi-period model with service level, see Ryan (1997). 6.4 Other Models The inventory models analyzed in this paper represent a small subset of the inventory models that can be analyzed using our approach. Other models, including innite horizon models, continuous review and periodic review models, may be amenable to a similar approach. This is a topic for possible future research. 7 References Bramel, J. and D. Simchi-Levi (1997), The Logic of Logistics: Theory, Algorithms and Applications for Logistics Management, Springer-Verlag, New York. Gallego, G. (1992), A Minmax Distribution Free Procedure for the (Q; R) Inventory Model. Operations Research Letters 11, pp. 55{60. Gallego, G. (1996), New Bounds and Heuristics for (Q; R) Policies. To appear in Management Science. Gallego, G. and Moon, I. (1993), The Distribution Free Newsboy Problem: Review and Extensions. J. Opl. Res. Soc. 44, pp. 825{834. Iglehart, D. (1963) Optimality of (s; S) Policies in the Innite Horizon Dynamic Inventory Problem. Management Science 9, pp. 259{267.