(s, S) POLICY FOR UNCERTAIN SINGLE PERIOD INVENTORY PROBLEM

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1 International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 21, No. 6 (213) c World Scientific Publishing Company DOI: /S (s, S) POLICY FOR UNCERTAIN SINGLE PERIOD INVENTORY PROBLEM YUAN GAO State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, 144, China gaoyth@163.com MEILIN WEN Science & Technology Laboratory on Reliability & Environmental Engineering, School of Reliability and Systems Engineering, Beihang University, Beijing, 1191, China wenmeilin@buaa.edu.cn SIBO DING School of Management, Henan University of Technology Zhengzhou, 451, China dingsibo@haut.edu.cn Received 1 May 212 Revised 24 April 213 The traditional single period inventory problem assumes that the market demand is a random variable. However, as an empirical or subjective estimation, market demand is better to be regarded as an uncertain variable. This paper is concerning with single period inventory problem under two main assumptions that (i) the market demand is an uncertain variable and (ii) a setup cost and an initial stock exist. Under the framework of uncertainty theory, the optimal inventory policy for uncertain single period inventory problem with an initial stock and a setup cost is derived, which is of (s,s) type. Also, some expansions are obtained. Keywords: Inventory; (s, S) policy; uncertainty theory. 1. Introduction The single period inventory problem is to find an optimal inventory policy which maximizes the expected profit, or equivalently minimizes the expected cost. The single period inventory problem has two significant assumptions: (i) the market Corresponding author. 945

2 946 Y. Gao, M.-L. Wen & S.-B. Ding demand during the period is non-deterministic, and (ii) items can only be ordered or produced in batch at the beginning of the period. The traditional single period inventory problem also assumes that if any inventory remains at the end of the period, the excess inventory is sold by a discount or disposed off simply; on the other hand, if market demand exceeds the inventory level, some profit is lost. These assumptions are proper in most cases, such as in fashing, sporting and service industries. As a result, after been introduced by Hadley and Whitin, 7 a lot of extensions to single period inventory problem have been developed, such as Bassok et al., 1 Ehrhardt, 4 Fu et al., 5 Lau et al., 1 Sana, 2 etc. At first, the non-deterministic demand of single period inventory problem was regarded as a random variable. So, the above research work employed probability to deal with the non-deterministic demand. In 1996, Petrović et al. 18 used fuzzy set to describe the demand, which opened the door of introducing fuzzy theory into single period inventory problem. Some extension fuzzy models were investigated by Dutta et al., 3 Ishill et al., 8 Ji et al., 9 Li et al., 11 etc. With the development of theories on non-deterministic phenomenon, it is found that some non-deterministic phenomenon cannot be described by randomness or fuzziness, such as the empirical estimate of demand in a future period. In order to describe this type of non-deterministic phenomenon, uncertainty theory was proposed by Liu 12 in 27 and refined by Liu 16 in 21. In 29, Qin and Kar 19 introduced uncertainty theory into single period inventory problem, and they regarded the market demand as an uncertain variable. In their paper, they derived the optimal order quantity which maximizes the expected profit. However, Qin and Kar s model is a simple one with neither a setup cost nor an initial stock. In real life, either the setup cost or initial stock has to be taken into account. Then, the inventory policy must be changed correspondingly. This paper expands Qinand Kar smodel to a morecomplexone, whichconcernswith an initial stock and a setup cost. Still, the market demand in this paper is regarded as an uncertain variable. The main contribution of this paper is to derive the optimal policy of single period inventory problem with an initial stock and a setup cost, which is of (s,s) type. The rest of the paper is organized as follows. In Sec. 2, uncertainty theory is introduced in several paragraphs, and some basic concepts and properties of uncertainty theory are presented. In Sec. 3, single period inventory problem with a setup cost and an initial stock level will be described in detail. In Sec. 4, (s,s) policy is derived as the optimal inventory policy for single period inventory problem described in Sec. 3. Section 5 concludes this paper with a brief summary. 2. Preliminary of Uncertainty Theory In the past, when constructing mathematical models, the empirical or subjective estimation of non-deterministic information, such as about 1 kg, approximately 39 C, big size and young, is described by random variable or fuzzy

3 (s, S) Policy for Uncertain Single Period Inventory Problem 947 variable. However, a lot of surveys showed that it s not suitable. For example, we say the distance between Beijing and Shanghai is about 13 km. Obviously, about 13 km is not a random variable, since it is a constant which we do not know exactly. Can it be described by fuzzy variable? The answer is no. If we employ fuzzy variable to describe the concept of about 13 km, then there exists a membership function, such as a triangular one (12, 13, 14). Based on this membership function, possibility theory will conclude: (i) the distance between Beijing and Shanghai is exactly 13 km with belief degree 1, and (ii) the distance between Beijingand Shanghaiis not13km with beliefdegree1.it isaparadox. In uncertainty theory, this paradox will not happen. Coming from the judgement of manager, the market demand of next business period is also a non-deterministic information like about 13 km. In order to suitably deal with these non-deterministic information, introducing uncertainty theory to describe market demand is necessary. Founded in 27, uncertainty theory is a new branch of mathematics. However, theory and practice have shown that uncertainty theory is an efficient tool to deal with some non-deterministic information, such as expert data and subjective estimate, which appears in many optimization problems. In theoretical aspect, in 28, Liu 13 first introduced uncertain process, a sequence of uncertain variables indexed by time or space. Later, uncertain calculus was proposed by Liu 15 in 29, and Chen and Liu 2 proved the existence and uniqueness theorem for uncertain differential equation. Nearly at the same time, uncertain set theory was proposed by Liu 17 in 21 as a generalization of uncertainty theory to the domain of uncertain sets. In practical aspect, Liu 14 built the framework of uncertain programming, which was soon applied to machine scheduling problem, vehicle routing problem and project scheduling problem. Through the work of Liu 17 and Gao et al., 6 uncertain inference is developed under uncertain set theory. In 21, Liu 16 started the research of uncertain statistics, which gives an empirical uncertainty distribution from expert s experimental data. Meanwhile, Zhu 21 studied uncertain optimal control, and applied it into portfolio selection model. In short, uncertainty theory is researched and used more and more. In this section, we introduce some foundational concepts and property of uncertainty theory, which will be used throughout this paper. Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ L is assigned a number MΛ} [,1]. In order to ensure that the number MΛ} has certain mathematical properties, Liu 12,16 presented the four axioms: normality, self-duality, countable subadditivity, and product measure axiom. If satisfying these four axioms, the set function MΛ} is called an uncertain measure. Definition 1. (Liu 12 ) Let Γ be a nonempty set, L a σ-algebra over Γ, and M an uncertain measure. Then the triplet (Γ, L, M) is called an uncertainty space.

4 948 Y. Gao, M.-L. Wen & S.-B. Ding Definition 2. (Liu 12 ) An uncertain variable is a measurable function ξ from an uncertainty space (Γ,L,M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event. ξ B} = γ Γ ξ(γ) B} The uncertainty distribution of an uncertain variable ξ is defined by Φ(x) = Mξ x} for any real number x. For example, the zigzag uncertain variable ξ Z(a, b, c) has an uncertainty distribution, if x a (x a)/2(b a), if a x b Φ(x) = (x+c 2b)/2(c b), if b x c 1, if x c. Definition 3. (Liu 16 ) An uncertainty distribution Φ is said to be regular if its inverse function Φ 1 (α) exists and is unique for each α (,1). Obviously, zigzag uncertain variable has a regular uncertainty distribution. If Φ is regular, uncertainty distribution Φ is continuous and strictly increasing at each point x with < Φ(x) < 1. We usually assume that all uncertainty distribution in practical application is regular. Otherwise, a small perturbation can be imposed to obtain a regular one. Definition 4. (Liu 12 ) Let ξ be an uncertain variable. Then the expected value of ξ is defined by E[ξ] = + Mξ r}dr Mξ r}dr provided that at least one of the two integrals is finite. Example 1: The expected value of zigzag uncertain variable ξ Z(a,b,c) is E[ξ] = (a+2b+c)/4. 3. Problem Description Generally, there are two equivalent approaches to follow when constructing the objective function for single period inventory problem. One is to maximize the expected value of total profit during the period; another is to minimize the expected value of cost. In this paper, we follow the first one. The notation and assumptions are listed as below.

5 (s, S) Policy for Uncertain Single Period Inventory Problem Notation ξ quantity demand, which is an uncertain variable; y inventory level, which is a decision variable; x initial stock; K setup cost; c purchasing or production cost of per unit; h salvage value of per unit; p selling price of per unit; r(ξ,y) the revenue for demand ξ and inventory y; R(y) the expected revenue for inventory level y, i.e., R(y) = E[r(ξ, y)]; f(ξ,x) the profit for demand ξ and initial stock x; F(x) the expected profit for initial stock x, i.e., F(x) = E[f(ξ,x)] Assumptions (1) K, c, h and p are constant and independent of inventory policy and market demand; (2) Shortages are permitted, and there is no shortage cost other than loss in revenue; (3) The salvage value h is positive, and p > c > h > ; (4) There is no budget constraint Mathematical formulation It is assumed that the initial stock is x, the setup cost is K >, and no item will be ordered during the period. Because the setup cost is positive, any order will lead to a positive cost K. This means that ordering nothing may yield the maximum expected profit in some situation. According to the given notation and assumptions, if order nothing, the revenue during the period is px, if x ξ r(ξ,y) = r(ξ,x) = pξ +h(x ξ), if x ξ. Ordering nothing means no cost happens, then the total profit is just the revenue, that is px, if order nothing and x ξ f(ξ,x) = pξ +h(x ξ), if order nothing and x ξ. If order up to y > x, the revenue during the period is py, if y ξ r(ξ,y) = pξ +h(y ξ), if y ξ.

6 95 Y. Gao, M.-L. Wen & S.-B. Ding Abstracting the cost (c(y x)+k) from revenue, we obtain the total profit py c(y x) K, if order up to y and y ξ f(ξ,x) = r(ξ,y) c(y x) K = pξ +h(y ξ) c(y x) K, if order up to y and y ξ. Obviously, r(ξ, y) and f(ξ, x) are both uncertain variables. Taking the expected value of r(ξ,y) and f(ξ,x), we get R(x), if order nothing F(x) = E[f(ξ,x)] = R(y) c(y x) K, if order up to y, where R(t) = E[r(ξ, t)]. Let Φ(t) be the uncertainty distribution of ξ, which is regular. Qin and Kar 19 proved R(y) = E[r(ξ,y)] = py (p h) y Φ(r)dr. Our objective is to seek optimal inventory policy, which maximizes the expected profit, that is } max R(y) c(y x) K R(x). y>x The following section will derive the optimal policy for this model, which is a special case of the (s,s) type. 4. The Optimal Inventory Policy In order to find the maximum of the objective function, it is better to investigate the extremal property of function (R(t) ct) first. Taking the derivative and setting it equal to zero, we obtain or d(r(t) ct) dt = p (p h)φ(t) c =, Φ(t) = p c p h. (4) It s assumed p > c > h, then p c p h (,1). Since uncertainty distribution Φ is regular, there exists t satisfying expression (4). What is more, Φ(t) is an strictly increasing function on R. Then, there is only one root satisfying expression (4), denoted as t = S. It is easy to verify that (R(t) ct) is decreasing on [S,+ ) and increasing on [,S]. That is, when t = S, (R(t) ct) reaches its maximum. Define s as the smaller value of t satisfying R(t) ct = R(S) cs K. Since K >, we have s S. See Fig. 1. We will derive the optimal policy with the help of Figure 1. It breaks down into three cases.

7 (s, S) Policy for Uncertain Single Period Inventory Problem 951 R(t)-ct K s S Fig. 1. Graph of (R(t) ct). Case 1. Assume x > S. For any given y x, obviously R(x) cx R(y) cy > R(y) cy K. Hence, R(x) > R(y) c(y x) K, where the right-hand side of the inequality is the expected total profit if one ordersup to y, and the left-hand side is the expected total profit if one orders nothing. This indicates that if x > S, order nothing. Case 2. Assume s x S. For any given y x, obviously, R(x) cx R(S) cs K R(y) cy K. Again, we get R(x) R(y) c(y x) K, which still indicates that if s x S, order nothing. Case 3. Assume x < s. From Fig. 1, we obtain } R(y) cy K = R(S) cs K > R(x) cx, max y>x Hence, R(S) c(s x) K > R(x), that is, if x < s, order up to S. This leads to an optimal policy of (s,s) type. Up to now, we can summarize: Theorem 1. The optimal policy for uncertain single period inventory problem with an initial stock x and a setup cost K is if x < s, then order up to S if x s, than order nothing, where the value of S satisfying Φ(S) = p c p h, and s is the smallest value of t satisfying R(t) ct = R(S) cs K.

8 952 Y. Gao, M.-L. Wen & S.-B. Ding Example 2: Assume ξ is a zigzag uncertain variable, i.e., ξ Z(9,1,12), K = 4,p = 1,c = 8 and h = 5. Note that S must satisfy Φ(S) = p c p h = =.4, where Φ(t) is the uncertainty distribution of ξ Z(9,1,12). We can obtain the unique solution: S = 9.8. Because s < S < b = 1, s can be obtained from 1s (1 5) where S s Φ(r)dr 8s = 1S (1 5) s Φ(r)dr = S s S r 9 2 dr = s2 18s, 4 Φ(r)dr 8S 4, r 9 Φ(r)dr = 2 dr = S2 18S = Then, s = The optimal inventory policy is: if the initial stock level x 2.84, order nothing; if the initial stock x < 2.84, order up to 9.8. If K =, the above single period inventory problem degenerates to one with only an initial stock x. Assume the uncertain demand ξ has uncertainty distribution Φ(t). The objective function can be expressed as where max y>x R(y) c(y x) } R(x), R(t) = pt (p h) t Φ(r)dr, t >. For this problem, the optimal ordering policy is if x < S, then order up to S if x S, then order nothing, where the value of S satisfying Φ(S) = p c p h. 5. Conclusion Uncertainty theory provides a new approach to describe some non-deterministic information, such as empirical and subjective estimation. This paper employed uncertainty theory to model single period inventory problem with an initial stock and a setup cost, where the market demand was regarded as an uncertain variable. It showed that the optimal policy for the uncertain single period inventory problem with an initial stock and a setup cost is of the (s,s) type. As a degeneration model, the optimal inventory policy for single period inventory problem with only an initial stock is also given in this paper. This paper just concerned with single period inventory problem with single product. In fact, it can be extended to the problem with multiple products, which is our next research point.

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