Integrated production-inventory and pricing decisions for a single-manufacturer multi-retailer system of deteriorating items under JIT delivery policy

Int J Adv Manuf Technol DOI 10.1007/s00170-016-9169-0 ORIGINAL ARTICLE Integrated production-inventory and pricing decisions for a single-manufacturer multi-retailer system of deteriorating items under JIT delivery policy Zhixiang Chen 1 & Bhaba R. Sarker 2 Received: 27 April 2016 / Accepted: 11 July 2016 # Springer-Verlag London 2016 Abstract This paper studies an integrated optimization problem of production-inventory and retail pricing decision for a single-manufacturer multi-retailer system of deteriorating items under just-in-time (JIT) delivery environment. The objective of the model is to maximize the total profit of the system which equals to the total revenue minus the total cost. The features of the model lie in three facets: first, revenue is dependent on the demand and selling price, while demand is price sensitive; second, transportation costs between manufacturer and retailers are included in assessing the total cost, and third, customer service level constraint is also considered in the model. Since the model is a complex mixed-integer nonlinear programming (MINLP) model, it is difficult to derive a closed-form solution using classic differentiation methods. In this paper, two meta-heuristic algorithms, particle swarm optimization (PSO) and quantum-behaved PSO (QBPSO) are developed to solve the model. Experiments show that QBPSO is more effective and more efficient than basic PSO. Sensitivity analysis reveals the impact of model parameters on solution, and some important managerial implications on the JIT strategy are summarized. Keywords Pricing . Just-in-time . Production inventory . Deterioration . Transportations . Meta-heuristic algorithm * Bhaba R. Sarker bsarker@lsu.edu Zhixiang Chen mnsczx@mail.sysu.edu.cn 1 Department of Management Science, Sun Yat-Sen University, Guangzhou 510275, China 2 Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803, USA 1 Introduction In reality, fresh and perishable goods, such as milk, fish, and vegetable, usually deteriorate in processes from production to usage or consumption. Deterioration usually leads to degradation of quality and utility, and thereby their usable volume is lost. Since deterioration is a loss for companies, the longer duration the items have, the more loss companies bear. This problem inspires the academic research interests in the study of productioninventory optimization of deteriorating items [27, 29, 32, 39, 43, 48]. This paper focuses on simultaneously optimizing the retail pricing, production lot sizing, and distribution delivery decisions of a single-manufacturer multi-retailer (SMMR) system for deteriorating items under JIT (just-in-time) environment. During the last two decades, some authors have discussed the integrated optimization issues of single manufacturer and multiple distributors or retailers under JIT policy. This kind of model can be viewed as a special instance and extension of SVMB (single-vendor, multi-buyer) model [10, 17, 51] and JELS (joint economic lot sizing) model [35]. In early years of this type of research, most of the SVMB production-inventory models of JIT system dealt with non-deteriorating items, for example, Parija and Sarker [28], Banjerjee et al. [3], and many others researched production-inventory models of single manufacturer multiple retailers system under JIT environment. Law and Wee [22] developed an integrated productiondistribution inventory model considering both ameliorating and deteriorating effects and taking into account of multiple deliveries, partial backlogging, and time discounting. Lin and Lin [24] studied a cooperative inventory model of single manufacturer and single retailer with deteriorating items, in which back-order is allowed. Lin et al. [25] studied the conflict and cooperation in a two-echelon supply chain (one manufacturer and one retailer) for deteriorating items, four cooperative behavior modes (no information sharing, supplier dominant, Int J Adv Manuf Technol retailer dominant, and cooperation) were examined. Wang et al. [40] constructed an integrated inventory optimization model for products with time-sensitive deteriorating rates in a three-echelon supply chain comprising one producer, one distributor and one retailer. Yan et al. [45] developed an integrated production-distribution model for a deteriorating item, in which supplier’s production batch is restricted to an integer multiple of the discrete delivery lot to the buyer. Wu and Sarker [44] developed a joint decision model of single-vendor (manufacturer) multiple buyers model, their model is an extension of the model of Lin and Lin [24]. Later, Sarker and Wu [34] extended their model to consider materials storage cost. Taleizadeh et al. [38] developed a single-vendor multi-buyer model for deteriorating items with pricing and replenishment and production rate, but their decision model is a decentralized decision model using the Stackelberg game approach rather than integration approach. Nonetheless, all above models for deteriorating items do not consider the JIT environment. As far as we know, only a few other authors have researched the SVSB or SVMB models for deteriorating items under JIT environment. Yang and Wee [47] developed a single-vendor multi-buyer production-inventory model for deteriorating item, in which JIT lot splitting concept was applied from raw material supply to production and distribution. Meantime, Rau et al. [30] studied an integrated inventory model of a three-echelon supply chain comprising of supplier, producer, and buyer, in their model; JIT policy was considered in production and warehousing. Chung and Wee [11] developed an integrated deteriorating inventory for single-buyer-single-supplier under JIT delivery policy. Fong and Wee [14] studied a near optimal solution for integrated production inventory supplier-buyer deteriorating model considering JIT delivery batch. Jha and Shanker [20] also studied the single-vendor single-buyer production-inventory model for decaying items under JIT delivery and controllable lead time. Huang and Yao [19] studied a single-vendor multi-buyer production inventory model for deteriorating items, and their model was revised from the model of Yang and Wee [47]; furthermore, both Yang and Wee’s model and Huang and Yao’s model do not consider transportation and shortage cost. Chen and Sarker [8, 9] proposed a multi-vendor integrated procurement and production under shared transportation and just-in-time system and Sarker [33] also made a critical and comparative review on consignment stock policy models for supply chain systems. As above literature review, there are a few authors who research the vendor-buyer inventory system of deteriorating items under JIT environment. However, all these researches do not incorporate customer service level problem in the models (no shortage allowance). Furthermore, all above models do not consider the marketing factors, such as pricing, and only optimize the production-inventory system to minimize the cost. Recently, integrated models of pricing and inventory decision have also attracted academic attention. There are some authors that have discussed the problem of joint pricing, production, and inventory. Here, we review representative literature of joint pricing and production-inventory decisions. Goyal and Gunasekaran [18] developed an integrated production-inventory-marketing model with deteriorating item under different market policies, such as pricing and advertisement. Abad [1] formulated a generalized model of dynamic pricing and lot-sizing by a reseller who sells a perishable good, in which partial backlogging is allowed. Weng [42] discussed the pricing and ordering strategies in manufacturing and distribution alliance to meet price-sensitive random demand with objective of maximizing expected profits of both the manufacturer and distributor. Abad [2] studied a pricing and inventory decision in a single-manufacturer single-supplier system when the manufacturer’s demand is price sensitive and the supplier offers price reduction. Chan et al. [6] studied the joint pricing, production, and inventory policies for manufacturing with stochastic demand and discretionary sales in a multiple-period horizon. They analyzed and compared partial planning or delay strategies. Chung and Wee [11] studied an integrated production-inventory and pricing model for deteriorating items for single-manufacturer single-retailer considering quality inspection and stock-level-depend demand. Li et al. [23] studied pricing and inventory control for a perishable product in an infinite period; they analyzed the optimal solution structure of a twoperiod lifetime problem and developed a base-stock/listprice heuristic policy. Saha and Basu [31] modeled a pricing and inventory decisions for seasonal products. Yu et al. [49, 50] also studied an integrated pricing and deteriorating model for a vendor-managedinventory system, but they did not consider shortage and transportation cost. Maihami and Abadi [26] studied a joint control of inventory and its pricing for noninstantaneously deteriorating items under permissible delay in payments and partial backlogging. Giri and Bardhan [16] investigated an integrated singlemanufacturer single-retailer model of inventory and pricing under decentralized and centralized mechanisms. Although both the SVMB models for deteriorating items and the joint decision models of pricing and inventory optimization have been researched by a number of authors, little attention has been paid to the problem of joint pricing and inventory optimization for SVMB of deteriorating items—especially, there is almost no literature on the integrated model of pricing and SVMB inventory of deteriorating items under JIT delivery mode. In this paper, an integrated optimization model of retail pricing, production lot sizing, delivery for single-manufacturer multi-retailer system of deteriorating items under JIT environment is studied. Since the model is a complex mixed integer nonlinear programming problem Int J Adv Manuf Technol (MINLP), it is difficult to solve it using classic optimization methods. In this paper, we design two meta-heuristic algorithms, i.e., particle swarm optimization (PSO), quantumbehaved PSO (QBPSO) to solve the problem, and we compare the effect of the algorithms. This study differs from previous works in several ways. First, in this study, transportation and shortage cost are involved into the model while most previous works do not consider these two types of cost. Second, customer service level is considered as constraint in the model, whereas no previous works consider this factor. Third, this study simultaneously investigates the combined effect of retail pricing, production lot sizing, and delivery frequency for SVMB supply chain of deteriorating items under JIT environment, while most literature concerns non-JIT environment. Forth, we investigate meta-heuristic algorithms to solve the model and improve the efficiency of computation; the proposed algorithms can solve more complicated models with more constraints. The rest of the paper is organized as follows. Section 2 describes the problem, including JIT production and delivery system, deterioration definition, assumptions, and notations. Section 3 formulates the problem. Section 4 develops the two algorithms, i.e., PSO and QBPSO. Section 5 gives a detailed numerical study to show the effectiveness and comparison of the two algorithms, sensitivity analysis also is conducted to analyze the influence of model parameters and JIT strategies on solution. Section 6 summarizes the conclusions and remarks. 2 Problem description The manufacturer produces single product and delivers it to different retailers using JIT philosophy—small lot size production and delivery. The product deteriorates during production and warehousing processes. This is a SVMB system comprising of single manufacturer and multiple retailers. This model is motivated from a local agricultural production and distribution supply chain (e.g., rice supply chain). 2.1 JIT production and delivery system We suppose the manufacturer and retailers are located in the same city, the distances from the manufacturer to retailers are not more than 50 km (since too long distance is not suitable for JIT delivery); meanwhile, information sharing between the manufacturer and each retailer is available. The manufacturer benefits from this information to implement a JIT production and delivery system with retailers. In order to reduce the loss of deterioration and increase responsiveness to customer demand, the integration of retailers’ marketing (e.g., pricing) and production decisions (e.g., lot sizing) is possible and implementable. The problem background of the model proposed in this paper is shown in Fig. 1. Under this JIT production and delivery environment, this paper models the integrated optimization of retail pricing, production lot sizing, and shipments (transportation) in a SVMB system for deteriorating item and develops effective algorithms to solve the problem. The decisions of the system comprise of two parts: (a) for retailers, there are three decisions: sell price of product, delivery cycle time from the manufacturer, and number of delivery from the manufacturer to retailers per production cycle and (b) for the manufacturer, the decisions are as follows: production cycle time and production lot size. Traditionally, these decisions are made by the manufacturer and retailers independently, while in the environment of e-business and esupply chain, partners in supply chain can synchronize their operations by sharing information and cooperation to reach a win-win position. In this paper, we try to use an integrated decision model to coordinate and synchronize the operations of the manufacturer and retailers. 2.2 Deterioration in production-inventory processes Deterioration is a natural phenomenon for many products, such as volatile liquids, agricultural productions, radioactive substances, films, drugs, blood, fashion goods, electronic and components. These items are subjected to depletion by some natural phenomena rather than demand, for example, spoilage, shrinkage, decay, and obsolescence. [7]. Deterioration can take place in many forms such as chemical changes, physical changes, and biological changes. Major causes of deterioration include growth and activities of micro-organisms (e.g., principally bacteria, yeasts, and molds), activities of natural food enzymes, insects, parasites and rodents, temperature (both heat and cold), moisture and dryness, and ambient conditions (air, in particular, oxygen; light; time, etc). During production and inventory processes, deterioration leads to inventory loss, this loss usually is described as deterioration rate. According to the characteristics of different deterioration processes, there are two types of deterioration rates, one is constant, and another type is dynamic. In literature, most authors assume deterioration rate is constant, such as, Chakrabarti and Chaudhri [4]. Chakraborty et al. [5], Yang and Wee [46, 47], Rau et al. [30], Huang and Yao [19], Chung and Wee [11], Lin and Lin [24], Chen and Chang [7], Jha and Shanker [20], and Yu et al. [49, 50]. However, there are a few researchers such as Covert and Phillip [12], Ghare and Schrader [15], Wee [41], and Wang et al. [40] who assume deterioration rate is dynamic and is a function of time. From those literature, we find that no matter whether the deterioration rate is constant or dynamic, its value is very small, and it ranges from 0.01 to 0.05. In terms of this characteristic, we assume the deterioration rate as a constant value. This assumption is reasonable for some items and does not significantly impact the results, and also it is convenient for computation. Int J Adv Manuf Technol Fig. 1 JIT production and delivery system of a single manufacturer and multiple retailers Demand information online placing system Manufacturer Retailer 1 Retailer j Customers JIT delivery system (Milk-Run transportation system) Retailer n Order delivery information track system 2.3 Assumptions and notations AM The assumptions and the explanations of the notations for formulating model are described in the following. Assumptions: 1. There are one manufacturer and multiple retailers in the system. 2. The demand of different retailers is a function of price. 3. Products deteriorate at a constant rate during production and warehousing. 4. Inventory shortage in retailers is allowed but there is a limit to service level. 5. All retailers’ replenishment cycle times are equal. 6. Production cycle time of the manufacturer is an integer multiple of retailers’ replenishment cycle time. 7. There is no replacement or repair for deteriorated items. 8. A JIT delivery system is implemented and the transportation cost is paid by the manufacturer. Systems parameters: cdj cdM D(pj) HM Hj VT F0 Fx S0 P aj bj πj θR θM Aj Demand of retailer j (j=1,2,..,m), which is the function of the price pj (unit/month) Item holding cost of the manufacturer ($/unit/month) Item holding cost of retailer j (j=1,2,..,m) ($/unit/ month) Transportation capacity of vehicle (unit) Fixed transportations cost of each vehicle ($/unit/run) Per unit transportation cost ($/unit) Allowed average shortage rate of retailers Production rate of the manufacturer (units/month) Market scale factor (coefficient) of retailer j’s demand (aj >0) Price elasticity coefficient of retailer j’s demand (bj >0) Backlogging cost of retailer j ($/unit) Deterioration rate of items in retailers (retailers) Deterioration rate of items in the manufacturer Ordering cost of retailer j ($/order) Setup cost of the manufacturer per run or batch ($/batch) The cost each deteriorated unit at retailer j ($/unit) The cost of each deteriorated unit at the manufacturer ($/unit) Decisions variables pj fj TM T T1 T2 n qj Q Price of retailer j (j=1,2,..,m), ($/unit) Fraction of shortage time to cycle time for retailer j (0≤ fj ≤1) Production cycle for the manufacturer (month), TM = T1 + T2 Interval between two shipments from the manufacturer to retailers (month), T=TM/n Production time in each production cycle for the manufacturer (month) Downtime in each production cycle for the manufacturer (month) Delivery number for all retailers in one production cycle (times) Delivery quantity for retailer j (j=1,2,…, m) (unit/ shipment) Output lot size of the manufacturer in each m production cycle (unit/lot), Q ¼ n ∑ q j j¼1 3 Formulation of the problem The objective of the problem is to maximize the total profit of the system which equals to the total revenue minuses the total cost. The total revenue of the system depends on the price and demand. As assumption, in this model, the demand of retailers is the function of price, and the most common demand function is linear function or exponential demand function [2, 6]. The total cost function is formulated through a series of subtotal functions in progression which are now given below. Int J Adv Manuf Technol Based on these assumptions, the total revenue function for all retailers is 3.1 Demand function and total revenue In this paper, we apply linear demand function, i.e., D(pj) = aj − bjpj, where αj is an initial demand without price fluctuation (i.e., market scale factor of retailer j), and bjis the price elasticity of retailer j. For notational simplicity, we will use D(pi) and Di interchangeably in this paper. Revenue function is constructed based on demand under the following assumptions m

 X TI p j ¼ a j −b j p j p j : 3.2 Total cost of the system In terms of the assumption of the problem that the manufacturer implements JIT shipping policy to deliver items to retailers using small lot size, the inventory profiles of the manufacturer and retailers are shown in Fig. 2. The upper part of Fig. 2 is the inventory profile of the manufacturer, where q is the delivery batch size of the manufacturer in each shipment, which is equal to the total delivery (i) D(pj) > 0 and is continuous for pj > 0; 0 dD p (ii) D j ¼ dpð j Þ < 0, i.e., D(pj) is a non-increasing function j for all pj ∈ (0, ∞); n
o ( i i i ) T h e m a r g i n a l r e v e n u e d p j D p j =dD j ¼
 0 p j þ D p j =D j , is a strictly increasing function of pj and thus 1/D(pj) is a convex function of p j. Fig. 2 Inventory profile for manufacturer and retailer ð1Þ j¼1 n quantity of all retailers in one shipment, i.e., q ¼ ∑ qi . In i¼1 Production and shipment IM(t) Shipment Non-deterioration Non-deterioration With deterioration T T T T T Time T T2 T1 Production cycle (TM) Retailer 1 IR1(t) q1 q1 q1 q1 T T T T q2 q2 q2 T q2 q2 T Retailer m IRm(t) qm qm T q1 q1 T T q1 Time T Retailer 2 IR2(t) T q1 q1 qm q2 q2 T qm T qm q2 T q2 qm qm Time T qm Time T T T T T T Int J Adv Manuf Technol each time interval, T, the manufacturer ships a batch to retailers using one truck with capacity of VT, if the batch size q is larger than VT, it needs more than one truck. The dotted line of Fig. 2 is the inventory without deterioration. During production time of T1, inventory increases since production rate is larger than demand rate, and then it stops production when inventory reaches a peak top. During the downtime,T2, the inventory decreases to zero because of depletion due to demand for which reason, the total production cycle of the manufacturer becomes TM = T1 + T2.
 ¼ T −T f j ¼ T 1− f j ; then, using Eqs. (4) and (5), we can obtain the total inventory and the total shortage of retailer j in one replenishment cycle as follows: I bþ j 3.2.1 Total cost of retailers We assume that all retailers place and receive orders from the manufacturer concurrently, so the interval of placing and receiving orders for all retailers is the same (i.e., T). The inventory replenishment interval is [t i − 1 , t i ], for i = 1,2,…n, where ti is the i-th replenishment period. At the beginning of shipment, an initial replenishment of qj is made to retailer j. The instant demand decreases to zero, the deterioration is denoted as t si , and then from ti − 1 to t si , inventory is positive and is denoted as I bþ j ðt Þ for retailer j. At the time t si , the inventory is zero. During the time inter  val t si ; t i , the inventory level becomes negative, i.e., inventory is backlogged to I b− j ðt Þ units. The inventory of retailer j has a relationship as dI bþ j ðt Þ dt
   þ θR I bþ t i−1 ≤ t ≤ t si ; i ¼ 1; 2; ::n ð2Þ j ðt Þ ¼ − a j −b j p j ; and the shortage of retailer j has a relationship as
 dI b− j ðt Þ ¼ − a j −b j p j ; dt  t si ≤ t ≤ t i ; i ¼ 1; 2; ::::; n ð3Þ  t ¼  a j −b j p j
θR ðts −tÞ   e i −1 ; t i−1 ≤ t ≤ t si ; i ¼ 1; 2; ::n θR and I b− j ðt Þ
¼ a j −b j p j  t si −t  ;  t si ≤ t ≤ t i ; i ¼ 1; 2; ::::; n  ð4Þ ð5Þ Since it is assumed that all retailers have the same delivery   intervals, T, so t i − t i − 1 = T and t si −t i−1 ¼ t i −t i−1 − t i −t si t si a j −b j p j
θR ðts −tÞ  e i −1 dt θR t i−1 t i−1 Z s Z s a j −b j p j ti θR ðts −tÞ a j −b j p j ti ¼ e i dt− dt θR θR t i−1 t i−1 Z s  a j −b j p j θR ts ti −θR t a j −b j p j
¼ e i T 1− f j e dt− θR θR t i−1  a j −b j p j θR ts  −θR ts −θR ti−1  a j −b j p j
i e i −e ¼− e T 1− f − j θR θ2R ¼ ¼  a j −b j p j
θR T ð1− f Þ  a j −b j p j
j e T 1− f j −1 − 2 θR θR I bþ j ðt Þdt ¼ Z ð6Þ and I b− j ¼ Z ¼− ti t si I b− j ðt Þdt ¼ Z a j −b j p j 2 2 T f j: 2 ti
t si a j −b j p j   t si −t dt ð7Þ The order quantity of retailerj equals to the summation of inventory and shortage in a replenishment cycle, b− i.e., q j ¼ I bþ j þ I j . Therefore, using the equations (6) and (7), it is obtained that the order quantity of retailer j is given by qj ¼ where I bþ j ðt Þ is the inventory level of retailer j at the time t while I b− j ðt Þ is the shortage level of retailer j at the instant t. Through integral computation of the above two equations, we can obtain the inventory and shortage of retailer j at the instant t as Z ts
 i bþ −θR t eθR t a j −b j p j dt I j ðt Þ ¼ e t si Z a j −b j p j
θ2R þ  a j −b j p
 j T 1−f j eθR T ð1− f j Þ −1 − θR a j −b j p j 2 2 T f j: 2 ð8Þ and T C Sj denote the inventory cost and Let T C IV j shortage cost of retailer j in one replenishment cycle, respectively. Then, the inventory and shortage cost of retailer j in one replenishment cycle can be calculated as " #  a j −b j p j
θR T ð1− f Þ  a j −b j p j
IV j TC j ¼ H j −1 − T 1− f j e θR θ2R ð9Þ and T C Sj ¼ π j a j −b j p j 2 2 T f j: 2 ð10Þ Int J Adv Manuf Technol Let Qdj denote the quantity of deteriorated items for retailer j in one replenishment cycle, which equals to the instantaneous inventory at the beginning of replenishment cycle i, I bþ j ðt i−1 Þ minus the total demand in the replenishment cycle,

 ts t si ð t Þ−∫ a −b p dt. ∫tii−1 a j −b j p j dt, i . e . , Qdj ¼ I bþ i−1 j j j j t i−1 Also, let T C dj denote the lost-cost of retailer j from deteriorated items, which can be expressed as  Z tsi
 a j −b j p j dt T C dj ¼ cdj Qdj ¼ cdj I bþ j ðt i−1 Þ− ti−1  
 a j −b j p j
θR T ð1− f Þ 
j ¼ cdj −1 − a j −b j p j T 1− f j : e θR m Q= n ∑ q j . In order to calculate the inventory of the manuj¼1 facturer, the profile of accumulated inventory in each production cycle is depicted in Fig. 3. Let I IM ðt 1 Þ, 0 ≤ t1 ≤ T1 and I IIM ðt 2 Þ,0 ≤ t2 ≤ T2 denote the accumulated inventory during production and downtime periods, respectively, which can be expressed as differential equations as follows: . ð13Þ dI IM ðt 1 Þ dt 1 ¼ P−θM I IM ðt 1 Þ; 0≤ t 1 ≤ T 1 and ð11Þ . dI IIM ðt 2 Þ dt 2 ¼ −θM I IIM ðt 2 Þ; 0≤ t 2 ≤ T 2 ð14Þ So the total cost for m retailers, TC R equals to the summation of ordering cost, holding cost, shortage cost, and the lost-cost from deterioration, i.e., I IM ð0Þ ¼ 0, and I IIM ðT 2 Þ ¼ Q ¼ ∑ nq j , one can obtain  m
d S T C R ¼ T1 ∑ A j þ T C IV which can be calj þ TC j þ TC j I IM ðt 1 Þ ¼ j¼1 culated as ( m X m X "
  Þ −1 − a j −b j p j T 1− f e Hj Aj þ j 2 θR θR j¼1 j¼1  m m X a j −b j p j 2 2 X d a j −b j p j
θR T ð1− f Þ  j þ cj e πj −1 T fjþ 2 θR j¼1 j¼1

io : − a j −b j p j T 1− f j 1 T CR ¼ T a j −b j p j
θR T ð1− f j # 3.2.2 Total cost of manufacturer According to the assumptions, the manufacturer produces a batch of Q units in each production cycle time, TM, and ships m one batch of q = ∑ q j units to retailers in each interval, T while j¼1 each retailer receive a lot of qj units for each shipment. There are n shipments in a production cycle, so the production lot size of the manufacturer is a function of retailers’ lot size as I(t) m j¼1 and I IIM ðt 2 Þ ð12Þ Fig. 3 Profile of accumulated inventory in one production cycle Solving the Eqs. (13) and (14) with boundary condition, ¼ P  −θM t1  1−e ; 0 ≤ t1 ≤ T 1 θM n m X ! q j eθM ðT 2 −t2 Þ ; j¼1 ð15Þ 0≤ t 2 ≤ T 2 : ð16Þ In order to make the integral, it is necessary to know T1 and T2 functions with respect to T. Since the derivation of the production time T1 and downtime T2 is a complicated process and it makes the model computationally more difficult, so some researches turn to seek the approximate expressions (see [11, 19, 46, 47]). In this paper, we derived a simple approach to calculate T1 and T2. In Fig. 3, since the production m lot size is Q ¼ n ∑ q j , the input production lot size ! j¼1 0 is Q ¼ m n ∑ qj =ð1−θM Þ considering the deterioration j¼1 rate, and since T1 = Q′/P, so the production time, T1, and Accumulated inventory after deterioration Accumulated inventory without deterioration Accumulated inventory before deterioration Accumulated consumption of retailers T T2 T1 TM T Q= n Time Int J Adv Manuf Technol downtime (a period of pure inventory consumption time), T2, during one cycle are n T1 ¼ m X transportation quantity of each cycle. The total transportation m quantity in one replenishment cycle is ∑ q j , so the transporj¼1 qj j¼1 ð17Þ ð1−θM ÞP tation vehicle number in one replenishment cycle is & ’ m ∑ q j =V T , where, VTis the capacity of each vehicle. So j¼1 the total transportation cost is expressed as and T 2 ¼ nT−T 1 : ð18Þ
T C T n; T ; f Let IM denote the inventory of the manufacturer per production cycle, which can be expressed as Eq. (19). IM ¼ ¼ Z P θM T1 0 I IM ðt 1 Þdt1 þ T1 þ Z e−θM T 1 −1 θM T2 0 I IIM ðt 2 Þdt 2 −T þ m X n qj j¼1 θM  m X j¼1 m m X X qj þ 2 q j þ ::: þ ðn−1Þ qj j¼1  eθM T 2 −1 −T j¼1 ! m X nðn−1Þ qj m X ð19Þ ð20Þ q j: The total cost of the manufacturer includes (a) setup cost, (b) inventory cost, and (c) deterioration cost. Hence, considering TM = nT, the total cost of the manufacturer is given by m X qj
 A HM P e−θM T 1 −1 H M j¼1  θM T 2  M þ þ T1 þ T C M T ; n; f j ; p j ¼ −1 e nT nT θM T θM θM m X qj ðn−1Þ ! m X cdM j¼1 −H M þ qj PT 1 −n 2 nT j¼1 ð21Þ where q j ¼ a −bθ p j j 2 R j
eθR T ð1− f j Þ −1  − a j −b j p j θR where the first item is the total fixed transportation cost which equals to the fixed cost of each vehicle, F0, times the number & ’ m of used vehicle number, ∑ q j =V T , and the total delivery 1 T, 2 j¼1 & m ’ m X . 1 1 X q j V T þ Fx q j ð22Þ ¼ F0 T T j¼1 j¼1 j¼1 j¼1 where the first item denotes the accumulated inventory during production time T1, the second item is the accumulated inventory during downtime T2, the third item is the accumulated inventory consumption by retailers (see the shadowed area in Fig. 3). The total number of deteriorated product per production cycle time is equal to I dM ¼ PT 1 −Q ¼ PT 1 −n j 
 a −b p T 1−f j þ j 2 j j T 2 f 2j . 3.2.3 Total cost of transportation number, in 1 month. The second item is the total variable transportation cost which equals to the per unit transportation cost, Fx, times the total transportation quantity in each replenm ishment cycle, ∑ q j , and total delivery number in 1 month, T1 . j¼1 3.3 Objective function and constraints of the system (1) Objective function The objective function of the system is the total profit which equals to the total revenue minus the total cost, i.e.,

 TP T ; p j ; f j ; n ¼ TI p j −TC where m X
 D p j p j; j¼1


 TC ¼ TCR T ; f j ; p j þ TCM T; n; f j þ TCT n; T ; f j ; ( m " # m 
 1 X X a j −b j p j
θR T ð1− f Þ  a j −b j p j
j TCR T ; f j ; p j ¼ e T 1− f Aj þ Hj −1 − j θR T j¼1 θ2R j¼1 )  m m  
X a j −b j p j 2 2 X d a j −b j p j
θR T ð1− f Þ 
j T fjþ ; πj e cj −1 − a j −b j p j T 1− f j þ 2 θR j¼1 j¼1 TI ¼ m X qj −1 AM H M P e H M j¼1  θM T 2  þ −1 e T1 þ þ nT nT θM T θM θM m X ðn−1Þ qj ! m X cd j¼1 qj ; þ M PT 1 −n −H M 2 nT j¼1 & m ’ m
 1 X . 1 X TCT n; T ; f j ¼ F 0 q j V T þ Fx q j; T T j¼1 j¼1
 TCM T ; n; f j ; p j ¼ n Transportation cost plays an important role in a supply chain system [37]. In this paper, the items from the manufacturer are transported to different retailers using identical trucks with the same capacity; the transportation cost depends on the ð23Þ T1 ¼ m X −θM T 1 qj j¼1 ð1−θM ÞP ; T 2 ¼ nT−T 1 ; and qj ¼ a j −b j p j
θ2R  a j −b j p  a j −b j p
j j 2 2 eθR T ð1− f j Þ −1 − T 1− f j þ T f j: θR 2 Int J Adv Manuf Technol (2) Constraints In order to meet the customers’ satisfaction, we set an average shortage rate, f , of all retailers as the system service level, which is less than a predetermined value, S0, i.e., m 1X f ≤ S0: f ¼ m j¼1 j ð24Þ Besides this constraint, the price of each retailer also needs to meet the constraint of D(pj) > 0, i.e., for linear demand function, the following relationship holds. . p j < a j b j: ð25Þ 4 Algorithms development and solution procedures Since the objective function in the proposed model is a complicated function and it is difficult to prove its convexity or concavity, it is difficult to use traditional differentiation method to solve the model. Therefore, in this paper, we adopt two meta-heuristic algorithms, i.e., particle swarm optimization (PSO) and quantum-behaved PSO (QBPSO) to solve the model. 4.1 PSO algorithm Particle swarm optimization is a population-based stochastic search technique developed by Kennedy and Eberhart [21], inspired by social behavior of bird flocking, fish schooling, and swarm theory. The basic principle of standard PSO can be depicted as follows. Suppose there is a D-dimensional space problem withX = (x1, x2, ⋯ , xj, ⋯ , xD). PSO uses the position of each particle to represent one solution, and the i-th particle of the swarm is represented byXi = (xi1, xi2, .⋯, xij, ⋯ , xiD). The flying of a particle from one position to another position represents a search step and iteration of solution updating. The position vector of the i-th particle at the k-th step search is denoted by Xi(k) = (xi1(k), xi2(k), ⋯ , xij(k), ⋯ , xiD(k)), the velocity vector of the i-th particle at the k-th step search is Vi (k) = (v i 1 (k), v i 2 (k), ⋯ , v i j (k), ⋯ , v i D (k)), where i (i = 1, 2, ⋯ , M) denotes the number of particles, and j (j = 1, 2, ⋯ , D) denotes the dimensions of the problem. During the search, each particle is attracted towards the location of local best position (the fitness of local optimal solution) achieved by itself and the global best position (the fitness of global optimal solution) found by the whole population. After finishing one step of the search process (i.e., an iteration of the algorithm), each particle updates its position and traveling velocity. The core of this Fig. 4 Position moving of particle of PSO algorithm algorithm is updating the formulas of particle position and traveling velocity. The position moving of a particle is shown in Fig. 4. In Fig. 4, Xi(k) is the position of particle i at search step k, and Xi(k + 1) is the new position of particle i at search step (k + 1). pbesti(k) is the so-far-best position of the individual particle i at the search step k, and pbest(k) is the global best position of the whole population. Vi(k) is the velocity of particle i at search step k. The velocity vector updating and position vector updating of particle i can be represented as the following two formulas   vij ðk þ 1Þ ¼ wvij ðk Þ þ c1 ϕ1 pbest ij ðk Þ−xij ðk Þ   þ c2 ϕ2 gbest j ðk Þ−xij ðk Þ ð26Þ xij ðk þ 1Þ ¼ xij ðk Þ þ vij ðk þ 1Þ; ð27Þ and where the first parameter, w is called inertia weight which shows the effect of previous velocity vector on the new velocity vector, and c1 and c2 are positive acceleration constants used to the contribution of the cognitive and social components, respectively [13]; they are also called cognitive learning factor and social learning factor, respectively. ϕ1 and ϕ2 are two random numbers uniformly distributed in (0, 1). In the velocity update formula (26), the inertia weight has important impact on the convergence and search ability of the algorithm. In order to improve the performance of the algorithm, we use a linear inertia weight updating strategy, which is expressed as: . w ¼ wmax −ðk−1Þ  ðwmax −wmin Þ ðk max −1Þ ð28Þ where wmax and wmin are the maximum and minimum inertia weights, respectively; kmax is the maximum iteration number, i.e., k = 1,2,…, kmax. Because the model is a mixed-integer nonlinear programming model, so in each iteration, an additional step of rounding continuous values into integer values for integer variables is added. The algorithm steps can be described as following. Int J Adv Manuf Technol Step 1: Initialize parameters. number of particle, M; maximum iteration number, kmax; learning factors c1 and c2, inertia weight, wmax , wmin; variable number of the model,D; maximum velocity of particle, vmax. Step 2: Generate initial solution (initial position and velocity) for each particle. Set k=1 (initial iteration number), randomly generate the initial position of each particle, Xi (k), and initial velocity, Vi(k), and calculate the initial fitness of each particle: F(Xi(k)). Step 2.1: F i n d t h e b e s t i n d i v i d u a l p o s i t i o n for particlei, pbesti(k = 1) = Xi(1) with F(pbest i (k = 1) = F(x i (k = 1))), Step 2.2: Find the best position for all particles, gbest ðk ¼ 1Þ ¼ argmin1 ≤ i ≤ M ð F ðpbest i ðk ¼ 1ÞÞÞ (for minimization problem) or gbestðk ¼ 1Þ ¼ argmax1 ≤ i ≤ M ð F ðpbest i ðk ¼ 1ÞÞÞ for maximization problem. While (k ≤ kmax) Step 3: Update inertia weight, particle position and velocity using the following equations: w ¼ wmax −ðk−1Þ*ðwmax −wmin Þ=ðk max −1Þ;   vij ðk þ 1Þ ¼ wvij ðk Þ þ c1 ϕ1 pbest i ðk Þ−xij ðk Þ   þ c2 ϕ2 gbestðk Þ−xij ðk Þ ; xij ðk þ 1Þ ¼ xij ðk Þ þ vij ðk þ 1Þ: Step 4: Evaluate the fitness of new solution, and find out the individual best solution of each particle and global best solution of the whole population, i.e., If F(Xi(k)) < F(pbesti(k)) [If the objective is maximization, then, this condition is F(Xi(k)) > F(pbesti(k))] Pbest i ðk Þ ¼ X i ðK Þ gbestðk Þ ¼ argmin1 ≤ i ≤ M ð F ðpbest i ðk ÞÞÞ for minimization problem or gbestðk Þ ¼ argmax1 ≤ i ≤ M ð F ðpbest i ðk ÞÞÞ for maximization problem. Endif k=1+1 End while (stop condition) Step 5: Output the best solution. 4.2 Quantum-behaved particle swarm optimization Although all kinds of revised versions of PSO based on standard PSO have improved the performance, there still exists essential shortcomings in PSO algorithm that have not been overcome, i.e., the search mechanism of PSO strongly dependents on the two vectors—velocity vector and position vector, and the values of these two parameters significantly impact the convergence speed and search ability of the algorithm. In order to significantly improve the algorithm performance, it is necessary to change the search mechanism of particles. Quantum-behaved PSO (QBPSO) is a new particle swarm algorithm which has a search mechanism different from PSO and can largely improve the algorithmic performance. QBPSO is first developed by Sun et al. in 2004 [36], and this algorithm has been successfully applied in a wide range of continuous optimization problems. Unlike PSO, QBPSO has no velocity vector; and the search mechanism is also different from the standard PSO, its global search ability and efficiency have been largely improved compared with PSO. The basic principle of QBPSO can be described as follows. Similar to PSO algorithm, let D denotes the dimension of the problem, and Mdenotes the number of particle. LetX(t) = {X1(t), X2(t), ⋯ , XM(t)}denote the solution of M particles at t-th instant. At instant t, the position (solution) of particle i is   X i ðt Þ ¼ xi;1 ðt Þ; xi;2 ðt Þ; ⋯; xi; j ðtÞ; ⋯; X i;D ðtÞ ; i ¼ 1; 2; ⋯; M ; j ¼ 1; 2; ⋯; D ð29Þ The best individual position of particles is Pi(t) = [pi , 1(t), pi , 2(t), ⋯ , pi , D(t)], and global best position for all particles is G(t) = [G1(t), G2(t), ⋯ , GD(t)]. Let f[⋅]denote the fitness function of the optimization problem, and the fitness at the t-th search step for particle i is f[Xi(t)]. For minimization problem, the best position can be expressed as if f ½X i ðt ފ < f ½Pi ðt−1ފ X i ðt Þ; (30) P i ðt Þ ¼ if f ½X i ðtÞ≥ f ½Pi ðt‐1ފ Pi ðt−1Þ ; The best position globally is decided by the following two equations g ¼ arg min f f ½Pi ðt ފg; 1≤i≤M ð31Þ Gðt Þ ¼ Pg ðt Þ: ð32Þ Let pi , j(t) = ϕj(t)Pi , j(t) + [1 − ϕj(t)]Gj(t), where ϕj(t) is a random number in the range (0, 1), i.e., ϕj(t)∼U(0, 1). The update equation of particle position is   X i; j ðt þ 1Þ ¼ pi; j ðt Þ  α C j ðt Þ−X i; j ðt Þ ⋅ln 1=ui; j ðt Þ ð33Þ where α is an algorithm parameter called contractionexpansion (CE) coefficient, and it can be a fixed value or a dynamic value with iteration. Experiments show that when its value linearly decreases from 1 to 0 during the search, it can Int J Adv Manuf Technol obtain better effect. In this paper, we set it as dynamic value. Suppose the maximum iteration number is Tmax, and the value of α in the t-th iteration can be expressed as α ¼ ð1−0:5Þ⋅ðT max −t Þ=T max þ 0:5: ð34Þ ui , j(t) is a random number range of (0, 1), i.e., ui , j(t)∼U(0, 1), and Cj(t) is average best position of the jth dimension variable for all particles in iteration step t, which is expressed as C j ðt Þ ¼ M 1 X Pi; j ðt Þ: M i¼1 ð35Þ Similar to the PSO, in QBPSO algorithm, integer variables are also obtained through rounding continuous values into integer values. The procedure of QBPSO is described as follows. Begin: Initialize parameters: number of particle, M, maximum iteration Tmax, Contraction-Expansion coefficient,α. Randomly set the current position of all particles, set t=0, Xi(0), and the individual best position Pi(0) = Xi(0),,i.e., P i (0) = [p i , 1 (0), p i , 2 (0), ⋯ , p i , j (0), ⋯ , p i , D (0)] = Xi(0) = [xi , 1(0), xi , 2(0), ⋯ , xi , j(0), ⋯ , Xi , D(0)], and global best position Gð0Þ ¼ Pg ð0Þ ¼ argminfPi ð0Þg1 ≤ i ≤ M . For t = 1 to Tmax (Iteration number) Calculate the Contraction-Expansion coefficient using α ¼ ð1−0:5Þ⋅ðT max −t Þ=T max þ 0:5: Calculate the average best position of M particles using M 1 X C j ðt Þ ¼ Pi; j ðt Þ: M i¼1 For i=1 to M (particle number) For j=1 to D (dimension number) Calculate fitness: fitness[i] = f[xi , j(t)], find individual best position using equation (30) and using (31) and (32) to find best position of all particles. Table 1 5 Model application and managerial implications After the model is developed and the algorithms are designed, examples are used here to show the application of the model and algorithms. The two algorithms are coded in MATLAB program and run at the same computer to test the examples. The programs were ran on Dell Pentium 4 CPU processor: 1.99 GHz, Memory: 512 MB. 5.1 Algorithms test In order to test the performance of the algorithms, we fist use some benchmark functions to verify the validity of the algorithms, four chosen benchmark functions are listed in Table 1: These functions are complex functions and the first three ones have many local optimization points, and function 1 even has noise signal with random variable. We test these four functions by implementing the two algorithms, PSO and QBPSO, and obtain results as reported in Table 2. From the results of Table 2, we find that, for small scale problems, both PSO and QBPSO can obtain high accuracy, but our further test experiments for large scale problems show that the QBPSO reveals its higher performance than PSO; furthermore, QBPSO needs shorter CPU running time for all problems than PSO. Therefore, comprehensively considering the accuracy and CPU running time, QBPSO is better than PSO for continuous domain optimization problems. Benchmark minimization functions No. Name of function 1 Update position of each particle using equation (33) endfor end for end for. Quatic function Function expression Solution space n f 1 ðX Þ ¼ ∑ ix4i þ randomð0; 1Þ, n = 2 Optimal function Optimal solution (−1.28, 1.28) f1 = 0 X = {0, 0} (−5.12, 5.12) f2 = 0 X = {0, 0} (−100, 100) f3 = 0 X = {0, 0} (−4.5, 4.5) f4 = 0 X = {3, 0.5} i¼1 2 Rastrigin 3 Schaffer 4 Beale  f 2 ðX Þ ¼ ∑ni¼1 x2i −10cosð2πxi Þ þ10Š, n = 2  pffiffiffiffiffiffiffiffiffiffiffiffi2 sinð x1 2 þx2 2 −0:5 f 3 ðX Þ ¼ 0:5 þ ð1:0þ0:001ðx 2 þx 2 ÞÞ2 1 f 4 ðX Þ ¼ ð1:5−x1 þ x1 x2 Þ   þ 2:625−x1 þ x1 x32 2 2 2   þ 2:25−x1 þ x1 x22 2 Int J Adv Manuf Technol Table 2 Comparison of algorithms for benchmark functions Algorithm Parameters Best solution (ten runs) CPU time (s) PSO M = 300 f1 = f2 = f3 = f4 = f1 = 11∼12 10∼11 11∼12 11∼12 7∼8 Kmax = 300 QBPSO M = 300 Tmax = 300 f2 = 0, X = (−4.1104E-10, −7.1119E-10) f3 = 4.8452E-11, X = (5.7184E-6, −3.9628E-6) f4 = 0, X = (3.0000, 0.5000) 5.2 Model applications Two examples are framed here to illustrate the solution procedure and results under different algorithms. Data are used as indicated in respective examples. Example 1: Single-manufacturer three-retailer system This is a small scale example with only three retailers, totally, the model has seven variables. The data of model parameters are listed in Table 3. The parameters setting for the two algorithms are as follows: PSO: M = 500 (number of particle), c1 = 1.5, c2 = 1.5, vmax = 0.5(maximum velocity), and kmax = 300(maximum iteration number). wmax = 0.9 and wmin = 0.5. (2) QBPSO: M = 500 (number of particle, Tmax = 300 (maximum iteration number). For both algorithms, after ten runs, obtained best results are as Table 4: From Table 4, we know that, for this example, QBPSO can obtain better result than PSO from the perspectives of effectiveness and efficiency, since the obtained total profit of QBPSO is higher than PSO, and also the CPU time of QBPSO is lower than PSO. Example2: Single-manufacturer six-retailer system In order to verify the performance in different scale problems, we further test another example with six retailers. The Table 3 1.5588E-7, X = (−0.001709, 0.012797) 0, X = (−3.5505E-10, 7.0134E-11) 0, X = (−5.0292E-10, 2.6918E-9) 0, X = (3.0000, 0.5000) 8.8155E-6, X = (−0.04605, −0.02812) 7∼8 7∼8 7∼8 parameters of the model are listed in Table 5. Totally, this problem has a total of 14 direct variables, i.e., T, f j (j = 1,2,…,6), pj(j = 1,2,…,6), and n. All parameters of algorithms are the same with Example 1, and each algorithm also was run for ten times. The best results are listed in Table 6. From the results, we can see that for this larger scale problem, QBPSO also can obtain higher performance than PSO in terms of accuracy level. 5.3 Sensitivity analysis of model parameters In order to analyze the impact of some important parameters on the performance of the system, it is necessary to conduct sensitivity analysis. In this paper, we focus on these parameters: (1) deterioration rates, θM and θR; (2) service level (i.e., shortage rate, S0); (3) holding cost parameters; (4) JIT policy (setup cost reduction and ordering cost reduction); and (5) transportation cost. In the following analysis, we use Example 1 as analysis background and QBPSO as analysis tool. 5.3.1 Effect of deterioration rates (θM,θR) Of all the decision variables, production cycle time TM(=T1+ T2), retailers’ order quantity qj, and delivery number n are most important decision variables which are affected by deterioration Parameters data Notations Values Unit HM 5 $/unit/month Hj (7, 6, 8) $/unit/month Aj AM aj bj P θM (500, 400, 600) 2000 (3000, 2000, 2500) (10, 10, 20) 2500 0.03 $/order $/batch Unit Unit/$ Unit/month None Notations cdM cdj πj F0 Fx VT θR S0 Values Unit 20 $/unit (24, 25, 23) $/unit (26, 27, 25) 250 2.5 200 0.05 0.10 $/unit $/vehicle $/unit Unit/vehicle None None Int J Adv Manuf Technol Table 4 Variable T* f *1 f *2 f *3 p*1 p*2 p*3 Best results for Example 1 Unit Optimal values Variable PSO QBPSO (Month) 0.3220 0.3220 None 0.1377 0.1372 None 0.0331 0.0367 None 0.1293 0.1262 $/unit 150.81 150.82 $/unit 100.89 100.88 $/unit 63.28 63.39 T *1 T *2 T *M n* q*1 q*2 q*3 Q* Efficiency CPU time Objective functional value TP* Dollars 388,791.28 388,791.58 rates, so we focus on the impact of parameters changes on the solution of these variables and the total profit. In order to observe the impact of deterioration rates on total profit, we can take the partial derivation of total profit with respect to the deterioration rates (θM,θb), and then obtain.
 ∂T C R 1 m ¼ ∑ H j a j −b j p j ∂θR T j¼1  θ T 1− f 
 1 m d
e R ð j Þ þT ð1− f j Þ θb T ð1− f j Þ 2 þ c a −b p 1−e þ ∑ 3 2 j j j θR θR T j¼1 j "  θR T ð1− f j Þ eθR T ð1− f j Þ
T 1− f j þ 1−e θ2 Š>0 R θR T and ∂T∂θCRM ¼ ∂TC ∂θR ¼ 0. ∂T C R So, ∂TP ∂θR ¼ 0− ∂θR < 0 indicating that TP decreases with the increasing of θR. Table 5 Unit Optimal value PSO QBPSO (Month) 0.1948 0.1946 (Month) 0.7713 0.7714 (Month) 0.9661 0.9660 Times 3 3 Unit 59.2646 59.3011 Unit 48.3599 48.0028 Unit 49.8306 50.0277 Unit 472.3653 471.99 Second 40.79 36.67 Similarly, we can prove that TP deceases with the increasing θR. As regards the impact of θR and θM on decision variables, since there is no closed-form of solution, so we numerically show the analysis. Table 7 is the results of the sensitivity analysis for deterioration rates (θM,θR). From the table, we can find that with the increasing of deterioration rates of the manufacturer or retailers, the total profit decreases, and this fits our above theoretical analysis. On the other hand, the impact of deterioration rates on decision variables reveals different results for different variables. With the increasing of deterioration rates, the replenishment interval, T* decreases while replenishment number, n* increases. However, when the deterioration rates increase to some extent, i.e., θM ≥ 0.05 and θR ≥ 0.07, T*, n*, and T *M remain unchanged, and q*j and T *2 decrease a little, whereas production cycle T *M and production time T *1 increases. Parameter data for Example 2 (m = 6) Notations Values Unit HM 10 $/unit/month Hj (7, 6, 8, 8, 9, 10) $/unit/month Aj AM aj bj P θM (500, 400, 600, 400, 500, 500) 2000 (3000, 2000, 2500, 3000, 2500, 2500) (10, 10, 20, 15, 10, 20) 1000 0.03 $/order $/batch Unit Unit/$ Unit/month None Notations cdM cdj πj F0 Fx VT θR S0 Values Unit 20 $/unit (24, 25, 23, 25, 23, 25) $/unit (26, 27, 25, 25, 23, 24) 250 2.5 200 0.05 0.20 $/unit $/vehicle $/unit Unit/vehicle None None Int J Adv Manuf Technol Table 6 Best results for Example 2 Variable Unit T* f *1 f *2 f *3 f *4 f *5 f *6 p*1 p*2 p*3 p*4 Optimal values Variable PSO QBPSO (Month) 0.3267 0.3640 None 0.1803 0.1691 None 0.1481 0.1229 None 0.2118 0.1959 None 0.2189 0.2036 None 0 0.2456 None 0.2787 0.2628 $/unit 150.39 150.4000 $/unit 100.28 100.3400 $/unit 62.84 62.9400 $/unit 100.41 100.4400 125.48 125.4600 q*6 Q* 762,666.80 763,080.98 Performance or efficiency CPU time Second $/unit p*5 Objective functional value TP* dollars p*6 T *1 T *2 T *M n* q*1 q*2 q*3 q*4 q*5 5.3.2 Effect of holding cost (HM , Hj) By examining the model (23), we can find that parameters of holding cost (HM , Hj) significantly impact the manufacturer’s cost, retailers’ costs, and the total profit, but their impact on Table 7 Optimal value PSO QBPSO $/unit 62.97 63.0200 (Month) 0.9278 0.7216 (Month) 0.0522 0.0063 (Month) 0.9800 0.7279 times 3 2 Unit 56.48 71.5922 Unit 39.96 52.0543 Unit 44.37 56.5716 Unit 52.66 67.1441 Unit 66.81 52.1438 Unit 39.72 50.4927 Unit 899.8825 699.9973 29.04 29.81 the retailers’ marketing decisions (pricing and revenue) are not significant. We set the changes of holding cost (HM , Hj) at the change ranges as −50 %, 50 %, and 100 %, we analyze how the holding cost parameters, HM , Hj, impact retailers’ decisions on order intervalT, order quantities,qj, the manufacturer’ Sensitivity analysis for deterioration rates θM θR T* 0.03 0.01 0.03 0.05 0.07 0.09 0.12 0.15 0.3321 0.3276 0.3220 0.2500 0.2500 0.2500 0.2500 0.3270 0.3220 0.2500 0.2500 0.2500 0.2500 0.2500 0.01 0.03 0.05 0.07 0.09 0.12 0.15 Unit 0.05 q*1 q*2 q*3 63.6434 61.4066 59.3011 35.7547 35.5499 35.5208 35.4019 61.2563 59.3011 35.2424 35.7835 35.7917 35.3982 35.3574 51.0782 49.5432 48.0028 28.7789 29.1495 29.3741 29.4407 49.5600 48.0028 29.1401 28.8382 28.7723 29.0447 29.0412 53.2547 51.6413 50.0277 30.4196 30.4083 30.3790 30.5770 51.5239 50.0277 30.4664 30.2743 30.2896 30.3181 30.2151 All other parameters are the same as in Table 3. Algorithm: QBPSO n* 3 3 3 4 4 4 4 3 3 4 4 4 4 4 T *M T *1 T *2 0.9996 0.9827 0.9660 0.9998 1.0000 1.0000 1.0000 0.9811 0.9660 1.0000 1.0000 1.0000 1.0000 1.0000 0.2078 0.2011 0.1946 0.1566 0.1569 0.1572 0.1574 0.1968 0.1946 0.1597 0.1633 0.1668 0.1723 0.1781 0.7918 0.7816 0.7714 0.8243 0.8431 0.8428 0.8426 0.7843 0.7714 0.8402 0.8367 0.8332 0.8277 0.8219 TP 389,283.58 389,035.81 388,791.58 388,596.03 388,410.89 388,129.36 387,846.77 389,006.93 388,791.58 388,608.92 388,426.60 388,236.74 387,937.31 387,805.24 Int J Adv Manuf Technol Table 8 Sensitivity analysis for holding cost (HM , Hj) Hj HM T* (7, 6, 8) −50 % Default +50 % +100 % −50 % Default +50 % +100 % 5 q*1 q*2 q*3 0.3333 0.3220 63.8568 59.3180 51.3730 48.0879 53.4769 49.9849 0.2500 0.2494 35.5753 35.1917 29.0432 28.9491 30.2016 30.1274 0.3333 0.3220 65.2113 59.3180 49.2623 48.0879 54.8945 49.9849 0.2500 0.2500 35.0191 34.3379 30.0479 31.0025 29.6069 29.1428 All other parameters are the same as in Table 3. Algorithm: QBPSO decisions on shipment frequency n, production cycle time TM, production time T1, and downtime T2. Tables 8 and 9 report the results of the sensitivity analysis for holding cost. From the result of Tables 8 and 9, we derive the following viewpoints. (a) For retailers, with the increasing of per unit holding cost (either retailers or the manufacturer), there is a decreasing trend of optimal decisions on order interval T* and order quantities, q*j . (b) For the manufacturer, with the increasing of per unit holding cost (either retailers or the manufacturer), there is a decreasing trend of optimal decisions on production time p*1 , production lot size Q*, an increasing trend of optimal decisions on replenishment number n* and downtime T *2 . (c) For the whole system, with the increasing of per unit holding cost (either retailers or the manufacturer), the total profit of the system always decreases, and the total cost always increases. Table 9 Hj (7, 6, 8) −50 % Default +50 % +100 % 5.3.3 Effect of allowed shortage rate S0 Since the complexity of the model, it is difficult to derive the explicit expression of the relationship between allowed shortage rate S0 and solution, so, in order to analyze the impact of allowed shortage rate S0 on the solution, we numerically take sensitivity analysis to examine how the allowed shortage rate S0 impacts the solution. We take the change range of S0 from 0 to 0.5. Table 10 reports the results. From Table 10, we can summarize the following conclusions: (a) With the increasing of allowed shortage rate, S0, the replenishment number n* has a turning point, i.e., when allowed shortage rate is bigger to a certain value (in this case is about 0.10), replenishment number from one value reduces to another value—before and after this turning point, replenishment number keeps unchanged. (b) As the allowed shortage rate increases, the total profit increases whereas the total cost decreases, but when allowed shortage rate is over certain point (in this case it is about 0.40), the total profit and the total cost will not Sensitivity analysis for holding cost (HM , Hj) (continuous) HM −50 % Default +50 % +100 % 5 T *M T *1 T *2 1.0000 0.9661 1.0000 0.9997 1.0000 0.9661 1.0000 1.0000 0.2087 0.1947 0.1564 0.1555 0.2095 0.1947 0.1562 0.1558 0.7913 0.7714 0.8436 0.8422 0.7905 0.7714 0.8438 0.8441 All other parameters are the same as in Table 3. Algorithm: QBPSO Q* n* TP TC 506.1201 472.1725 379.2590 377.0730 508.1045 472.1725 378.6957 377.9329 3 3 4 4 3 3 4 4 389,377.97 388,791.58 387,870.80 386,957.70 390,533.90 388,791.58 387,481.36 386,187.12 13,718.00 14,304.00 15,231.00 16,139.00 12,574.00 14,304.00 15,614.00 16,875.00 Int J Adv Manuf Technol Table 10 S0 Sensitivity analysis for allowed shortage rateS0 T q*1 q*2 q*3 T *M T *1 Q* T *2 n* TP TC 0 0.2500 46.8233 31.1517 38.6930 1.0000 0.1924 0.8076 466.6719 4 387,748.07 15,350.00 0.050 0.2500 39.7602 31.0849 33.8250 1.0000 0.1727 0.8273 418.6803 4 388,340.93 14,757.00 0.100 0.150 0.3220 0.3310 59.3213 58.0324 48.0338 45.0741 50.0491 48.1936 0.9662 0.9929 0.1947 0.1872 0.7715 0.8057 472.2125 453.9005 3 3 388,791.58 389,228.30 14,304.00 13,869.00 0.200 0.3333 54.7811 40.7128 44.8688 1.0000 0.1738 0.8264 421.0828 3 389,549.60 13,549.00 0.25 0.30 0.3333 0.3333 51.2780 48.2418 36.4905 33.1145 41.4667 38.7468 1.0000 1.0000 0.1599 0.1486 0.8401 0.8514 387.7053 360.3092 3 3 389,744.55 389,810.22 13,355.00 13,290.00 0.40 0.3333 48.2359 33.1083 38.7415 1.0000 0.1486 0.8514 360.2570 3 389,810.22 13,290.00 0.50 0.3333 48.2359 33.1083 38.7415 1.0000 0.1486 0.8514 360.2570 3 389,810.22 13,290.00 All other parameters are the same as in Table 3. Algorithm: QBPSO change—this means that there exists a best allowed shortage rate for the highest performance. (c) At a certain value of replenishment, with the increasing of allowed shortage rate, S0, for retailers, optimal order quantity q*j decreases, and for the manufacturer, optimal setup, and H is per unit per time holding cost, it is known that setup cost reduction can lead to small lot size. Using the Example 1 model parameters above, we analyze the impact of setup cost reduction on the solution. We change the setup cost from 2000 to 200 with the change rate from −25 % to −90 %, then the results are shown in Table 11. production time T *1 and production lot size Q*decreases, whereas downtime T *2 increases. From Table 11, we can obtain two points of important conclusions as 5.3.4 The impact of JIT philosophy on the solution (a) When setup cost is less than ordering cost (i.e., AM < Aj), as setup cost reduces, except the replenishment number n* remains unchanged, all other decision variables (replenishment interval, T*, ordering quantity, q*j and pro- The core idea of JIT is waste elimination and cost reduction by continuous improvement. In order to demonstrate the impact of JIT philosophy on solution, we took two important JIT strategies as a basis of analysis, i.e., (1) setup time (cost) reduction and (2) ordering cost reduction by close vendor-buyer relationship. duction cycle time T *M , production time p*1 and downtime,T *2 ) reduce. (b) When setup cost reduces to be smaller than ordering costs of retailers (see the last row of Table 12, at this row, the setup cost is 200, whereas ordering costs of retailers are of 500, 400, 600), retailers’ decisions on replenishment interval T* and ordering quantity, q*j , will increase, whereas 1. Setup time (cost) reduction strategy effect on solution Setup time reduction (or quick changeover) is a very important strategy for JIT implementation. Setup time reduction is the prerequisite of implementing small lot sizing strategy (from economic lot sizing model qffiffiffiffiffiffiffi EOQ ¼ 2DA H , where D is demand, A is setup cost per Table 11 AM −25 −50 −75 −90 a The effect of setup cost reduction on solution T* Defaulta % % % % the manufacturer’s decisions on T *M , production time p*1 , and downtime,T *2 continuously reduce. 0.3222 0.3114 0.29964 0.28809 0.28985 q*1 q*2 q*3 59.3195 55.3424 51.3663 47.5033 48.0966 48.1459 44.8353 41.5880 38.4731 38.9207 50.0984 46.7293 43.3443 40.0288 40.5411 n* 3 3 3 3 2 T *M T *1 T *2 0.9667 0.9334 0.8989 0.8643 0.5797 0.1949 0.1818 0.1686 0.1559 0.1052 0.7718 0.7517 0.7303 0.7084 0.4745 The value of default of AM and other parameters are from Table 3. Algorithm: QBPSO TP TC 388,791.59 389,317.92 389,863.49 390,430.53 390,874.56 14,304.00 13,779.00 13,236.00 12,670.00 12,227.00 Int J Adv Manuf Technol Table 12 Effect of ordering cost reduction on solution Aj T* Defaulta −25 % −50 % −75 % −90 % a q*1 q*2 q*3 0.3220 59.3180 48.0338 50.0491 0.2500 0.2000 35.9244 22.5601 28.8374 18.6381 30.1957 19.5679 0.1667 0.1429 15.9453 11.7780 12.8977 9.3797 13.3901 9.9115 n* TC 0.7715 388,791.58 14,304.00 0.8434 0.8747 390,284.59 391,842.03 12,822.00 11,272.00 0.8955 0.9103 393,800.40 395,257.25 9315.01 7862.40 p*1 T *2 3 0.9662 0.1947 4 5 1.0000 1.0000 0.1566 0.1253 6 7 1.0000 1.0000 0.1045 0.0897 The value of default of Aj and other parameters are from Table 3. Algorithm: QBPSO 5.3.5 The effect of transportation cost on the solution 2. Ordering cost reduction by close vendor-buyer relationship Close vendor-buyer relationship is the most important philosophy of JIT, which emphasize cooperation between vendor and buyer, information sharing. A good JIT cooperation relationship can reduce ordering cost, because JIT vendor-buyer relationship is a long-term contract relationship, and it can reduce unnecessary negotiation and transaction cost, so ordering cost can be significantly reduced. Table 12 shows that ordering cost reduction can increase the total profit and reduce total cost. The transportation cost also has an important impact on the solution. Here, we analyze the impact of unit transportation fee on the solution. We assume the transportation cost increase from original value (default value = $2.5) by an incremental of 25 %, 50 %, and 100 % to the value of 3.125, 3.75, 4.375, and 5.00 dollars. These results are shown in Table 13. From the Table 13, we can see that with the increase of per unit transportation fee Fx, the replenishment cycle time T, and the production cycle TM, decrease, while the profit decreases and total cost increases. We also find that, the increasing proportion of profit is far less than the increasing proportion of total cost. For example, when Fx increases 100 % from 2.5 per unit to 5 per unit, the total profit only decreases 0.245 %, but the total cost increases 6.7 %. From this, we know that, the transportation cost has higher impact on total cost than on total profit. Table 12 shows that ordering cost reduction can increase total profit and reduce total cost; meanwhile, ordering cost reduction can reduce production time and replenishment interval, as well as ordering quantity with more replenishment number. Comparing Tables 11 and 12, we find that at the same change rate, the impacts of setup cost and ordering cost on solution are different. Ordering cost reduction can obtain more increase in profit and more reduction in total cost than the effect of setup cost reduction. From this, we obtain an important managerial implication, that is, as JIT philosophy, the strategy that establishing close relationship between the manufacturer and retailers to reduce ordering cost can benefit the system more than the strategy that setup cost reduction in the manufacturer. Table 13 T* Defaulta 0.3220 0.3160 0.25 0.25 0.25 +25 % +50 % +75 % +100 % 6 Conclusions The JIT production is one important pillar of lean strategy, and the goal of lean production is to reduce waste and ultimately increase enterprise profit. This paper studies an integrated optimization model of pricing, production lot sizing and delivery in a single-manufacturer multi-retailer system for deteriorating Effect of unit transportation fee on solution FX a TP T *M q*1 q*2 q*3 59.3180 59.9171 36.2193 35.4485 35.5030 48.0338 46.2570 28.5038 28.9059 28.8944 50.0491 48.2249 30.1744 30.3875 30.3719 n* 3 3 4 4 4 T *M T *1 T *2 0.9662 0.9479 1.0000 1.0000 1.0000 0.1947 0.1873 0.1565 0.1563 0.1563 0.7715 0.7606 0.8435 0.8437 0.8437 The value of default of FXand other parameters are the same as in Table 3. Algorithm: QBPSO TP TC 388,791.58 388,489.07 388,308.74 388,073.12 387,836.22 14,304.00 14,604.00 14,791.00 15,025.00 15,263.00 Int J Adv Manuf Technol items under JIT production and delivery policy. The objective was to optimize the total profit of the system where one product is produced by the manufacturer and delivered to multiple retailers, and the demand of product is price sensitive. Besides the production and inventory cost, the deterioration cost, and transportation cost, and service level is also considered in the model. Since the model is a complex mixed-integer nonlinear programming (MINLP) model, it is difficult to solve it using classic differentiation optimization methods. Thus, two metaheuristic algorithms, i.e., particle swarm algorithm (PSO) and quantum-behaved PSO (QBPSO), are designed to solve the model. Experiments show that QBPSO is most effective and more efficient than basic PSO. Sensitivity analysis is undertaken to analyze the impact of some important parameters and JIT strategies on the solution. Based on the study, some important managerial implications on the JIT strategy applying in production-inventory for deteriorating items can be summarized as follows. (a) Increase of deterioration rate reduces total profit, but it has different impacts on retailers’ and the manufacturer’s decisions—at certain small value range, with the increasing of deterioration rate, there is a decreasing trend for retailers’ optimal order interval, order quantities, and the manufacturer’s optimal decisions on production cycle time, production time and downtime time. However, when deterioration rates exceed a certain value, both the replenishment interval and production cycle remain unchanged. (b) Holding cost also reveals some different impacts on retailers and the manufacturer. As unit holding cost increases, there is a decreasing trend of optimal decisions on order interval T* and order quantities q*j for the retailers and there is a decreasing trend of optimal decisions on production time p*1 , production lot size Q*, and an increasing trend of downtime T *2 for the manufacturer. But, for the whole system, with the increasing of unit holding cost (for either retailers or manufacturer), the total profit of the system always decreases, and the total cost always increases. (c) Customer-allowed shortage rate constraint also impacts the solution. At a small value range of shortage rate, as allowed shortage rate increases, the total profit decreases and the total cost increases. When the allowed shortage rate is over certain point, the total profit and the total cost will not change. This reveals that there exists a best allowed shortage rate for the highest performance of the production-inventory system. (d) For two important JIT strategies, i.e., setup time (cost) reduction and ordering cost reduction by close vendorbuyer relationship, our numerical analysis shows two points of important implications, i.e., First, ordering cost reduction can always reduce production cycle time, replenishment interval, and ordering quantity with more replenishment numbers. On the contrary, the impact of setup cost reduction on decision variables has relationship with ordering cost, that is, the effect of setup cost reduction on decision variables when AM < Aj is different from the effect when AM > Aj. Second, the strategy that establishing close relationship between the manufacturer and retailers to reduce ordering cost can benefit the system more (profit increase and cost reduction) than the strategy that setup cost reduction in the manufacturer. This reveals the importance of JIT cooperation between the manufacturer and retailers. This study also has limitations and more works can be done to extend it. First, the demand function can be reformulated as other forms, such as products having cross price elasticity or substitution characteristics. Second, deterioration rate function also can be adopted other form to demonstrate the model robustness, such as varying deterioration rate. Third, extending the model into a multi-echelon supply chain is also a challenging topic. Acknowledgments This research is funded by the NSFC (#71372154). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Abad PL (1996) Optimal pricing and lot-sizing under conditions of perishability and partial backordering. Manag Sci 42(8):1093–1104 Abad PL (2003) Optimal price and lot size when the supplier offers a temporary price reduction on an interval. Comput Oper Res 30:63–74 Banjerjee A, Kim S-L, Burton J (2007) Supply chain coordination through effective multi-stage inventory linkages in a JIT environment. Int J Prod Econ 108:271–280 Chakrabarti T, Chaudhuri KS (1997) An EOQ model for deteriorating items with a linear trend in demand and shortages in all cycles. Int J Prod Econ 49:205–213 Chakraborty T, Giri BC, Chaudhuri KS (2009) Production lot sizing with process deteriorating and machine breakdown under inspection schedule. Omega 37:257–271 Chan LMA, Simchi-levi D, Swann J (2006) Pricing, production and inventory policies for manufacturing with stochastic demand and discretionary sales. Manuf Service Oper Manag 8:149–168 Chen T-H, Chang H-M (2010) Optimal ordering and pricing policies for deteriorating items in one-vendor multi-retailer supply chain. Int J Adv Manuf Technol 49(1):341–355 Chen Z, Sarker BR (2010) Multi-vendor integrated procurement and production under shared transportation and just-in-time system. J Oper Res Soc 61(11):1654–1666 Chen Z, Sarker BR (2014) An integrated optimal inventory lot sizing-vehicle routing model for a multi-supplier single-assembler system with JIT delivery. Int J Prod Res 52(17):5086–5114 Chu C-L, Leon VJ (2008) Single-vendor multi-buyer inventory coordination under private information. Eur J Oper Res 191:485–503 Chung C-J, Wee H-M (2007) Optimal replenishment policy for an integrated supplier-buyer deteriorating inventory model considering multiple JIT delivery and other cost functions. Asia Pac J Oper Res 24(1):125–145 Covert RP, Phillip GC (1973) An EOQ model for items with Weibull distribution deterioration. AIIE Trans 5:323–326 Int J Adv Manuf Technol 13. Engelbrecht AP (2009) Fundamentals of computational swarm intelligence. Wiley Publishing, Inc 14. Fong JF, Wee H-M (2008) A near optimal solution for integrated production inventory supplier-buyer deteriorating model considering JIT delivery batch. Int J Comput Integr Manuf 21(3):289–300 15. Ghare PM, Schrader SF (1963) A model for exponentially decaying inventory. J Ind Eng 14(5):238–243 16. Giri BC, Bardhan S (2012) Supply chain coordination for a deteriorating item with stock and price-dependent demand under revenue sharing contract. Int Trans Oper Res 19(5):753–768 17. Glock CH, Kim T (2015) The effect of forward integration on a single-vendor-multi-retailer supply chain under retailer competition. Int J Prod Econ 164:179–192 18. Goyal SK, Gunasekaran A (1995) An integrated productioninventory-marketing model for deteriorating items. Comput Ind Eng 28(4):755–762 19. Huang J-Y, Yao M-J (2006) A new algorithm for optimally determining lot sizing policies for a deteriorating item in an integrated production-inventory system. Comput Math Appl 51:83–104 20. Jha JK, Shanker K (2009) A single-vendor single-buyer productioninventory model with controllable lead time and service level constraint for decaying items. Int J Prod Res 47(24):6875–6898 21. Kennedy J. and Eberhart, R.C., 1995. Particle swarm optimization, In Proceedings of the IEEE international joint Conference on Neural Networks, 1942–1948 22. Law S-T, Wee H-M (2006) An integrated production-inventory model for ameliorating and deteriorating items taking account of time discounting. Math Comput Model 43:673–685 23. Li Y, Lim A, Rodrigues B (2009) Note-pricing and inventory control for a perishable product. Manuf Serv Oper Manag 11:538–542 24. Lin C, Lin Y (2007) A cooperative inventory policy with deteriorating items for a two-echelon model. Eur J Oper Res 178:92–111 25. Lin Y-H, Lin C, Lin B (2010) On conflict and cooperation in a twoechelon inventory model for deteriorating items. Comput Ind Eng 59:703–711 26. Maihami R, Abadi INK (2012) Joint control of inventory and its pricing for non-instantaneously deteriorating items under permissible delay in payments and partial backlogging. Math Comput Model 55:1722–1733 27. Misra RB (1975) Optmum production lot size model for a system with deteriorating inventory. Int J Prod Res 13(5):495–505 28. Parija GR, Sarker BR (1999) Operations planning in a supply chain system with fixed-interval deliveries of finished goods. IIE Trans 31:1075–1082 29. Raafat F, Wolf PM, Eldin HK (1991) An inventory model for deteriorating items. Comput Ind Eng 20:89–94 30. Rau H, Wu M-Y, Wee H-M (2003) Integrated inventory model for deteriorating items under a multi-echelon supply chain environment. Int J Prod Econ 86:155–168 31. Saha S, Basu M (2010) Integrated dynamic pricing for seasonal products with price and time dependent demand. Asia Pac J Oper Res 27(3):293–410 32. Sana SS (2011) Price-sensitive demand for perishable items—an EOQ model. Appl Math Comput 217:6248–6259 33. Sarker BR (2014) “Consignment stock policy models for supply chain systems: a critical review and comparative perspectives. Int J Prod Econ 155(S1):52–67 34. Sarker B, Wu B (2016) Optimal models for a single-producer multibuyer integrated system of deteriorating items with raw materials storage costs. Int J Adv Manuf Technol 82:49–63 35. Siajadi H, Ibrahim RN, Lochert PB (2006) A single-vendor multiple-buyer inventory model with a multiple-shipment policy. Int J Adv Manuf Technol 27:1030–1037 36. Sun J., Xu, W.-B., Feng, B. 2004. A global search strategy of quantum-behaved particle swarm optimization. Proceedings of 2004 I.E. Conference on Cybernetics and Intelligent Systems, Singapore, 111–115 37. Swenseth SR, Godfrey MR (2002) Incorporating transportations costs into inventory replenishment decisions. Int J Prod Econ 77(2):113–130 38. Taleizadeh AA, Noori-daryan M, Cardenas-Barron LE (2015) Joint optimization of price, replenishment frequency, replenishment cycle and production rate in vendor managed inventory system. Int J Prod Econ 159:285–295 39. Teng J-T, Chang C-T (2005) Economic production quantity models for deteriorating items with price-and stock-dependent demand. Comput Oper Res 32:297–308 40. Wang K-J, Lin YS, Yu JCP (2011) Optimizing inventory policy for products with time-sensitive deteriorating rates in a multi-echelon supply chain. Int J Prod Econ 130:66–76 41. Wee H-M (1999) Deteriorating inventory model with quantity discount, pricing and partial backordering. Int J Prod Econ 59:511–518 42. Weng ZK (1997) Pricing and ordering strategies in manufacturing and distribution alliance. IIE Trans 29:681–692 43. Widyadana GA, Wee HM (2012) An economic production quantity model for deteriorating items with multiple production setups and rework. Int J Prod Econ 138:62–67 44. Wu B, Sarker BR (2013) Optimal manufacturing and delivery schedules in a supply chain system of deteriorating items. Int J Prod Res 51(3):798–812 45. Yan C, Benerjee A, Yang L (2011) An integrated productiondistribution model for a deteriorating inventory item. Int J Prod Econ 133:228–232 46. Yang PC, Wee HM (2002) A single-vendor and multiple-buyers production-inventory policy for a deteriorating item. Eur J Oper Res 143:570–581 47. Yang PC, Wee H-M (2003) An integrated multi-lot-size production inventory model for deteriorating item. Comput Oper Res 30:671–682 48. Yu JCP (2013) A collaborative strategy for deteriorating inventory system with imperfect items and supplier credits. Int J Prod Econ 143:403–409 49. Yu, Y., Huang, G.Q., Ren, Z., and Liang, L. 2003. An integrated lotsize model of deteriorating item for one vendor and multiple retailers considering market pricing using genetic algorithm. Proceedings of the 3rd International Conference on Electronic Business, Singapore, December 9–13, 565–573 50. Yu Y, Huang GQ, Hong Z, Zhang X (2011) An integrated pricing and deteriorating model and a hybrid algorithm for a VMI supply chain. IEEE Trans Autom Sci Eng 8(4):673–681 51. Zavanella L, Zanoni S (2009) A one-vendor multi-buyer integrated production-inventory model: the ‘consignment stock case’. Int J Prod Res 118(1):225–232