American Journal of Oper ations Research, 2011, 1, 4650 doi:10.4236/ajor.2011.12007 Published Online June 2011 (http://www.SciRP.org/journal/ajor) Copyright © 2011 SciRes. AJOR Optimal Policy and Simple Algorithm for a Deteriorated MultiItem EOQ Problem Bin Zhang, Xiayang Wang Lingnan College, Sun YatSen University, Guangzhou, China Email: bzhang3@mail.ustc.edu.cn, wangxy@mail.sysu.edu.cn Received March 29, 201 1; revised April 19, 2011; accepted May 12, 2011 Abstract This paper considers a deteriorated multiitem economic order quantity (EOQ) problem, which has been stu died in literature, but the algorithms used in the literature are limited. In this paper, we explore the optimal policy of this inventory problem by analyzing the structural properties of the model, and introduce a simple algorithm for solving the optimal solution to this problem. Numerical results are reported to show the effi ciency of the proposed method. Keywords: Inventory, EOQ, Deterioration, MultiItem 1. Introduction Multiitem inventory problem with resource constraints is an important topic of inventory management [1]. These constrained inventory models are still hot topics in aca demic and practice fields, for example, see [26]. The re source constraints typically arise from shipment capacity, warehouse capacity or budgetary limitation. Since some items such as fruits, vegetables, food items, drugs and fashion goods will deteriorate in the shipment or storage process, many works have been done for investigating inventory problems for deterioration items [711]. Since items’ deterioration often takes place during the storage period, some researchers have considered economic order quantity (EOQ) models for deteriorating ite ms, f or ex am ples see [12,13]. Recently, Mandal et al. [14] present a constrained multiitem EOQ model with deteriorated items. In [14], the model is firstly formulated as the transcendental form and the polynomial form, i.e., without and with trunca tion on the deterioration terms. These two versions of the model are both solved by applying nonlinear program ming (NLP) method (Lagrangian multiplier method). As [14] points out, the polynomial form is an approximation of the transcendental form. Secondly, the transcendental form is converted to the minimization of a signomial expression with a posynomial constraint, which is so lved by applying a modified geometric programming (MGP) method. However, we argue that the studied problem can be solved using a simple algorithm without any model approximation or conversion. In this paper, we prove that the deteriorated multiitem EOQ model is a special convex separable nonlinear knapsack problem studied in [15], which is characterized by positive marginal cost (PMC) and increasing marginal losscost ratio (IMLCR). PMC requires p ositive marginal cost of decision variable, and IMLCR means that the marginal losscost ratio is increasing in decision v ariable. Following [15], we explore the optimal policy for the problem, and develop a simple algorithm for solving it. The main purpose of this paper is twofold: 1) to explore the optimal policy of this inventory problem by analyz ing the structural properties of the model; 2) to introduce a simple algorithm for solving the optimal solution to this problem. The reminder of this paper is organized as follows. We formulate the problem in the next section. In Section 3, we explore the structural properties of the problem, and provide the optimal policy and algorithm. Numerical results are reported in Section 4, and Section 5 briefly concludes this paper. 2. Problem Formulation Consider a multiitem EOQ problem with a storage sp ace constraint, in which all items (1, ,in) deteriorate after certain periods. Before presenting the model, we list all notation as follows. Notice that the same notation used in [14] is presented.
B. ZHANG ET AL. Copyright © 2011 SciRes. AJOR 47 n = Total number of items Qi = Order quantity c0i = Purchasing cost c1i = Holding cost per unit quantity per unit time c3i = Setup cost i = Constant rate of deterioration (01 i ) wi = Required storage quantity per unit time Di = Demand rate per unit time Ti = Time period of each cycle 1 (,, ) n TC TT= Total average cost function W = Available storage space Following [14], we set 2 01 3iiiiiii i acD cDc , 2 010 iiiiiii bcD cD , and 1iiii ccD . Then the deteriorated multiitem EOQ model can be expressed as follows (denoted as problem P). We refer the reader to [14] for the details of this model. 1 11 min (,,) () ii n T nn i iii i ii ii TC TT ae Tbc TT , (1) subject to 11 () ((1)) ii nn T i ii i ii i D Twe W , (2) 0 i T, 1, ,in. (3) The order quantity is given by (1) ii T i ii D Qe , 1, ,in. The first and second order derivatives of () ii T,1, ,in, and the first order derivative of () ii T,1, ,in, are calculated as follows: 2 ()(1 ) ii T iii iii ii df TabeT dT T , (4) 222 23 () 2(22) ii T iii iiiii ii dfTa beTT dT T , (5) () ii T ii ii i dg T we dT . (6) 3. Structural Properties and Algorithm In this section, we first establish structural properties of problem P, and then we present an efficient procedure for solving the optim al solution to problem P. 3.1. Structural Properties Before presenting the structural properties of problem P, we give two basic equations, which will be used in our proofs. Since Taylor expansion of exponential function is 1 1! ii kk Tii k T ek , then we have 22 11 2 ii T iiiii i TTTe , (7) for 0 i T, 1, ,in . By comparing the definitions of ai and bi, we have ii ab , 1, ,in . (8) Considering the objective function of problem P, we have the following proposition. Proposition 1. The cost 1 (,, ) n TC TT is strictly convex. Proof. Since the function 1 (,, ) n TC TT is separable, we only need to prove that () ii T is strictly convex in i T, 1, ,in . According to Equations (7) and (8), we have. 22 2222 44 2(22 ) 221 1 222 ii T ii iiii iiii iii ii iiii abeT T TTbT bb TT . Substituting the above equation into Equation (5), we have. 244 4 23 0 2 2 ii ii iii i ii dfT bT bT dT T for 0 i T. Thus, () ii T and 1 (,, ) n TC TT are strictly convex. QED. The strictly convexity of 1 (,, ) n TC TT ensures that the optimal solution to problem P is unique. This prop erty has also been indicated in [14] by using a more complicated proof procedure. Denote ProductLog( )z as the principal solution for x in zxe, which has the same function name in Ma thematica to stand for the Lambert W function. Let ˆi T, 1, ,in , be a value such that () 0 iii dfTdT . Denote problem CP as problem P without the constraint in Equa tion (2), then the following proposition characterizes the optimal solution to problem CP. Proposition 2. The optimal solution to problem CP is 1 ProductLog ˆ i ii ii a be T , 1, ,in. (9) Proof. Since ()0 ii i dfTdT at the point ˆi T, we have ˆˆ (1) 0 ii T ii ii abe T . This equation can be rewritten as ˆ1ˆ (1) ii T iii i aeT be , and hence we have 1 ProductLog ˆ i ii ii a be T . From Equation (8), we know 1i iii a ebe . According
B. ZHANG ET AL. Copyright © 2011 SciRes. AJOR 48 to [16], we know that ProductLog( )z is an increasing function for 1 i ze , then we have ProductLog( ) i ii a be 1 ProductLog( ) 1 i e . Substituting this equation into ˆi T, we have hence ˆ0 i T, which satisfies the positive constraint in Equation (3). Thus, ˆi T, 1,,in, is an optimal solution to problem CP. Since the optimal solu tion is unique, we know that the optimal solution to problem CP is ˆi T, 1,,in. QED. Following [15], we define the marginal losscost ratio of item i (1,,in) as. 2 () () () ii ii T ii iiii ii ii iii i df T dTa ebTb rT dg TDwT dT . (10) Then we have the following proposition. Proposition 3. () ii rT is strictly increasing in ˆ (0, ] ii TT, 1, ,in. Proof. From Equation (7), we know 1ii T ii Te for 0. i TThe convexity of () ii fT guarantees ii T ii abe (1)0 ii T for ˆ (0, ] ii TT. Using ii ab in Equa tion (8) and the above two equations, we have (2) (2) [(1)][(1)] [(1)][(1)]0 ii ii ii ii ii T iiiiii TT iiiiiiii TT iiii iii beT aT abeTaT be abeTbT e , for ˆ (0, ] ii TT. Hence we have 3 ()[(2 )(2 )]0 ii ii TT iiiii iii ii dr TebeTaT dT DT w . Thus, () ii rT is strictly increasing in ˆ (0, ] ii TT, 1, ,in. QED Since this proposition illustrates that () ii rT is strictly increasing in ˆ (0, ] ii TT, 1, ,in , we let () ii Tr , (,0] i r, be the inverse function of () ii rT . Although it is difficult to write () ii Tr in a closed form, the strict monotony of () ii Tr ensures that () ii Tr can be easily determined by applying a bisection search procedure over ˆ (0, ] ii TT, for any given (,0] i r . 3.2. Optimal Policy and Algorithm We now demonstrate that the deteriorated multiitem EOQ model is a special convex separable nonlinear knapsack problem studied in [15]. Firstly, Proposition 1 illustrates that P is a convex problem; Secondly, from Equation (6), we know () 0 ii i dg T dT , for 0 i T,1, ,in , which means there are positive marginal costs in problem P; Finally, Proposition 3 ensures that the marginal losscost ratio () ii rT is increasing in ˆ (0, ] ii TT, 1,,in . Therefore, the theoretical results and solution procedure proposed in [15] are both applicable for problem P. Denote by * i T, 1, ,in , the optimal solution to problem P. By directly applying the theoretical results in [15], we can summarize the optimal policy for the dete riorated multiitem EOQ problem in the following pro position. Proposition 4. The optimal policy of problem P is (a) *ˆ ii TT , 1, ,in , if 1 ˆ () n ii i TW ; (b) * 1() n ii i TW and ** () () ii kk rT rT, , 1,,ik n , specify the optimal solution * i T, 1, ,in , if 1 ˆ () n ii i TW . This proposition is obtained by directly applying the theoretical results in [15] to problem P, since problem P has PMC and IMLCR. Based on Propositions 24, the idea of the algorithm proposed in [15] can be used for solving problem P. The basic idea of the algorithm is as follows: If the constr ain t in Equ a tion (2) is inactive, i.e. , 1 ˆ () n ii i TW , then the optimal solution to problem P equals to the optimal solution to the unconstrained prob lem, i.e., *ˆ ii TT , 1, ,in ; Otherwise, Proposition 4(b) means that obtaining the exact value of ** * () () ii kk rrT rT, , 1,,ik n , is the key to solving the optimal solution to problem P. The optimal value * r can be searched by applying a binary search method over the interval (,0]rM , where is a sufficient large value such that () 1(( 1)) ii nTM i i ii D we W . Main steps of the above solution procedure for solving the optimal solution to problem P are summarized in the following algorithm. The Algorithm 1 * ˆ 1: Solve , 1,,, from Equation (9); ˆ 2: If (), t h e n ˆ let , 1,,, go to 8; 3: Let ,0; 4: Le t ()2; 5: Calculate i n ii i ii LU LU ii StepT in Stepg TW TTin Step SteprMr Steprr r StepT T 1 1 (), 1,,; 6: If (), then let ; If (), then let ; n ii L i n ii U i ri n Stepg TWrr gT Wr r
B. ZHANG ET AL. Copyright © 2011 SciRes. AJOR 49 * ** Go to 4; 7: Let , 1,,; 8: Calculate (,,), and output. ii in Step StepTT in StepTC TT In comparison with the two methods in [14], there are two main advantages of our algorithm: 1) It is a polyno mial algorithm of O(n) order, which ensures that the al gorithm is applicable for largescale problems; 2) It does not need any approximation or conversion of the original model, thus it always solves the optimal solution to prob lem P. 4. Numerical Results In this section, numerical experiments are provided to show the efficiency of the proposed algorithm for solv ing problem P. The instances of problem P are all ran domly generated. We use the notation x ~ (,)U to denote that x is uniformly generated over [,] . The parameters of test instances are generated as follows: i w~ (1,10)U, 0i c~ (1,10)U, 1i c~ (0.5,1.0)U, 3i c~ (40,100)U, i ~ (0.01,0.10)U, i D~ (200,500)U, 1, ,in, and 100Wn. In this numerical study, we set n=100 and 1000, re spectively. For each problem size n, 100 test instances are randomly generated. The statistical results on number of iterations of the binary search and computation time (in milliseconds) are reported in Table 1, where 95% C.I. stands for 95% confidence interval. From Table 1, we can conclude that the proposed al gorithm can solve largescale deteriorated multiitem EOQ models very quickly in few iteration times. Since the ranges of parameters are large, the standard devia tions of number of iterations and computation time are quite low, reflecting the fact that the algorithm is quite effective and robust. We also use our algorithm to solve the illustrative exam ple studied in [14], which outputs the optimal solution: * 10.2899T, * 20.2176T, * 1102.6397Q, * 298.6801Q, *2587.1382TC with *= 0.5925r. Unfortunately, this result cannot be directly compared with that solved by [14], because there is something wrong with the values of * i T and * i Q, 1,2i, shown in Tables 2 and 3 of [14], since they violate the basic equation Table 1. Performance of our algorithm for solving the ran domly generated instances. Number of iterations Computation time Problem size n 100 1000 100 1000 Mean 27.9 31.8 217.0 2619.3 Std. Dev. 1.7 1.7 8.9 38.8 95% Lower27.6 31.4 215.2 2611.6 C.I. Upper28.3 32.1 218.8 2627.0 (1) ii T i i i D Qe , 1, 2i . For example, in the Table 2 of [14], when * 10.2414712T and * 20.2419020T, (1) ii T i ii D Qe gives 185.3365Q , 2109.7828Q . In addition, their mistake can also be verified by our proved optimal pol icy ** ** 11 22 () ()rT rT. For example, the values of ** () ii rT , 1, 2i for the MGP solution presented in Table 3 of [14] are ** 11 ( )0.6017rT , and ** 21 ( )0.5857rT , which does not satisfy ** ** 11 22 () ()rT rT. From the above analysis, we illustrate that the solution provided in [14] does not satisfy the optimal policy proved in this paper, which is easily used to verify the optimality of a solution to problem P. Thus, the numeri cal results in [14] are incorrect. Since some comparison of NLP and MGP given by [14] were established based on the numerical results, especially the results in Tables 2 and 3 of [14], we argue that the comparison of NLP and MCP in [14] are questionable. 5. Conclusions In this paper, we explore the structural properties of de teriorated multiitem EOQ model and propose a simple algorithm for solving the optimal solution by proving that the studied problem is a special convex separable nonlinear knapsack problem. In addition, it is obvious that the basic idea and obtained results in this paper can be simply modified for solving the classical constrained multiitem EOQ problem. 6. Acknowledgements The authors would like to thank the two reviewers for their insightful comments, which helped to improve the manuscript. This work is supported by national Natural Science Foundation of China (No. 70801065), the Fun damental Research Funds for the Central Universities of China and Natural Science Foundation of Guangdong Province, China (No. 10451027501005059). 7. References [1] G. Hadley and T. M. Whitin, “Analysis of Inventory Systems,” PrenticeHall, Englewood Cliffs, 1963. [2] B. Zhang, X. Xu and Z. Hua, “A Binary Solution Method for the MultiProduct Newsboy Problem with Budget Constraint,” International Journal of Production Eco nomics, Vol. 117, No. 1, 2009, pp. 136141. doi:10.1016/j.ijpe.2008.10.003 [3] S. Sharma, “Revisiting the Shelf Life Constrained Multi Product Manufacturing Problem,” European Journal of Operational Research, Vol. 193, No. 1, 2009, pp. 129139. doi:10.1016/j.ejor.2007.10.045 [4] B. Zhang and S. F. Du, “MultiProduct Newsboy Prob
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