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1、Inventory/distribution control system in a one-warehouse/multi-retailer supply chainChumpol Monthatipkula,1, Pisal Yenradeeb,*AbstractThis paper proposes a new inventory control system called the inventory/distribution plan(IDP)control system for a one-warehouse/multi-retailer supply chain.In the sy
2、stem,a proposed mixed-integer linear programming model is solved to determine an optimal IDP control system,a proposed mixed-integer linear programming model is solved to determine an optimal IDP that controls the inventories of the supply chain.The efficiency of the IDP control system is compared t
3、o that of the echelon-stock R,s,S control policy,where R is a periodic review interval,s is a reorder point,and S is an order-up-to level,at various fillrates.The experimental results show that when the system faces non-stationary demands,the IDP control system significantly outperforms the echelon-
4、stock R,s,S control system because it can give lower total costs for all ranges of fill rates.Keywords:Inventory/distribution plan; One-warehouse/multiple retailers; Supply chain；Mixed-integer linear programming1.Introduction In Supply Chain Management(SCM),the inventory control problem is very comp
5、licated and challenging because the planner needs to consider several factors,for example,supply chain structures,coordination levels,and information sharing processes.The inventory control policy used by each entity is also an important factor because it affects the inventory replenishment process
6、of the upstream entity.The upstream demand may be distorted and far from the actual demand faced by the downstream entity.This phenomenon is known as the Bullwhip Effect,which is presented in Forrester(1961).Many classical inventory control systems(s,Q,s,S,R,S,R,Q,and R,s,S,etc.where s,Q,S,and R den
7、ote reorder points,reorder quantities,order-up-to levels,and periodic review periods,respectively)are still used in the supply chain environment.However,determination of their control parameters is very difficult.For SCM,all entities in the supply chain should be planned and controlled simultaneousl
8、y to obtain goodcontrol parameters and low inventory costs.The aim of this paper is to develop a new inventory control system called the inventory/distribution plan(IDP)control system that determines optimal product flow through a one-warehouse/multi-retailer supply chain under both stationary and n
9、on-stationary uncertain demand situations.The IDP control system controls each supply chain member using the optimal IDP operating under a one-period rolling horizon planning strategy.The optimal IDP is obtained by solving a proposed mixed-integer linear programming model.The performance of the prop
10、osed IDP control system is compared with that of the echelon-stock R,s,S control policy since both systems are similar in manyaspects.The main contribution of this paper is the development of IDP control system thathas good performances under both stationary and nonstationary uncertain demand situat
11、ions.This paper also proposes a practical way to determine appropriate safety stock factors at the warehouse and retailers that yield relatively low total costs while maintaining relativelyhigh fill rates.1.1.Literature reviewThis paper involves two main streams of research:(a)proposing new inventor
12、y control systems and(b)determining suitable control parameters of classicalinventory control systems.Therefore,the literature review is presented in two main subsections.(a)Literature review proposing inventory control systems:So far,many researches including this paper have focused on proposing ne
13、w inventory control systems or improving classical inventory control systems,and also comparing their proposed/improved systems to a classical inventory control system.Yoo et al.(1997) proposed an improved Distribution Resource Planning(DRP)method using concepts of installation-stock s,Q and R,S sys
14、tems.The order quantities and order points are dynamically determined to meet the demand in a just-in-time concept and minimize the out-of-stock probability.In the improved system,regional distribution centers can make a decision to reduce the related costsorder only the amount available or postpone
15、 the ordering. Based on experiments,the proposed system out performs he classical DRP.Another relevant research work belongs to De Kok andFransoo(2003).The authors proposed the so-called Synchronized Based Stock(SBS)policy and compared its performance to the LP-based system under the backlogging mod
16、el.The SBS, which uses the base-stock-policy concept,raises the inventory position to meet the adjustable target level in every inventory review.It is concluded that the SBS outperforms the LP-based system considerably because it uses more sophisticated ordering method and rationing rule.Ganeshan et
17、 al.(2001)studied two inventorycontrol systems,namely,DRP and Reorder Point systems in a four-echelon network.TheDRP system refers to a calculation of upstream reorder quantities and reorder intervals by aggregating all downstream demands and offsetting them by related lead times.The Reorder Point s
18、ystem refers to a situation that manufacturers forecast needs at the distribution centers.The authors concluded that the DRP system gives higher service levels and lower cycle times.A s,S system where s and S vary with states(SMART s,S system)was proposed by Giannoccaro and Pontrandolfo(2002).A thre
19、e-stage serial supply chain is formulated as a semi-Markov decision process model and then solved by SMART(semi-Markov average rewardtechnique)algorithm.The authors compared the SMART s,S system to the echelon-stock R,Ssystem and then summarized that the SMART s,S system gives lower total costs and
20、is more robust if demand undergoes only slight changes.Wang et al.(2004)proposed just-in-timedistribution requirements planning(JIT-DRP)which aims to pull material through a multi-warehouse/multi-retailer supply chain effectively.The JIT-DRP gives optimal solutions under deterministic conditions.Lit
21、erature review determining suitable control parameters of classical systems:The classical inventory control system in supply chains can be broadly divided intotwo types,namely, installation-stock and echelon-stock inventory control systems.The difference between them is mainly based on information u
22、sed to make an inventory replenishment decision.The echelonstock inventory control system allows the planner to utilize the network information, while the installation-stock control system allows the planner to use only local information.Details of the two categories can be seen in Axsater and Rosli
23、ng(1994). Many research works try to determine control parameters of classical inventory control systems in supply chains.They aim to minimize the total costs including inventory costs(ordering,holding,and shortage costs)and transportation costs(transportation and in-transit holding costs).Schneider
24、 and Rinks(1991) and Schneider et al.(1995)provided a good approximation for parameter setting for the echelon-stock R,s,S control policy.The model is developed based on the backlogging model of a one-warehouse multiple-stores supply chain.Ganeshan(1999)studied a non-linear programming model which a
25、ccounts for inventory and transportation costs.Solving the model by the Newton or the conjugate gradient method,the planner obtains near-optimal control parameters of the installation-stock s,Q system.Abdul-Jabar et al.(2003)studied inventorycontrol systems of a one-warehouse/N-retailer network unde
26、r centralized and decentralized policies.The authors obtained inventory control parameters(replenishment times and reorder quantities)by solving various types of mathematical models concerning holding and ordering costs.Another determination of control parameters on a one-warehouse/N-retailer networ
27、k belongs to Axsater(2003).The author proposed a technique to approximate optimal reorder points.The model considers holding costs at all locations and backorder costs at retailers.Yokoyama(2002)studied a multi-DC/multi-Retailer model controlled by the installation-stock R,S system.The target invent
28、ory and the transportation amountare determined so as to minimize the sum of transportation,holding,and shortage costs.Tagaras(1999)considered the installation-stock R,S system in a one-warehouse/N-retailer network.Order-up-to quantities are calculated by solving a mathematical model concerning hold
29、ing,shortage,and transshipment costs,if the transshipment between retailers is allowed.Optimal stock levels in general divergent networks under the echelon-stock R,S system was studied by Heijden(2000).The objective is to achieve target fill rates and to minimize total holdingcosts of the entire net
30、works.Recently,control parameters of the traditional R,S system for serial supply chains were improved by Xie et al.(2006).The authors propose a two-level supply chain coordination algorithm to adjust the values of R and S,which are regularly determined by the local optimization.Some numerical exper
31、iments have been conducted and it is found that the supply chain performance is increased due to the new values of R and S.Al-Rifai and Rossetti (2007)proposed an efficient heuristic optimization algorithm to determine the control parameters of the s,Q system.Their model is a one-warehouse/multi-ret
32、ailer supply chain.The goal is to minimize the inventory investment that is affected by average annual ordering frequency and expected number of backorders.The continuous review s,Q policy were also studied by Seifbarghy and Akbari Jokar(2006).The authors study a onecentral-warehouse/multi-identical
33、-retailer supply chain,which faces independent Poisson demands.When the batch size is given,the related reorder point at each facility is determined bya proposed approximate cost function. The remainder of this paper is organized as follows.Section 2 describes a supply chainmodel.The IDP control sys
34、tem and safety stock policies are presented in Sections 3 and 4,respectively.Section 5 contains the echelon-stock R,s,S control system and an approach to determine its control parameters.The experimental design is shown in Section 6.Section 7 discusses the experimental results.Finally,the results ar
35、e concluded in Section 8. 2.A supply chain model A supply chain under consideration comprises of one-warehouse and multiple identical retailers as depicted in Fig.1.It faces uncertain demand of a single product.When the demand is not satisfied,it is considered as a lost sale.The retailers replenish
36、their inventories from the central warehouse,which in turn replenishes its inventory from an incapacitated vendor outside the concerned supply chain.It is assumed that all storage and transportation capacities are unlimited.Transportation occurs after orders have been placed from the destination.Lat
37、eral transshipments between retailers are not allowed.The related costs are ordering,holding,intransit holding,transportation,and lost-sale costs. If insufficient products are sent to fulfill customer demands,the system would have lowfill rate and high lost-sale cost.In contrast,if excessive product
38、s are sent instead,the systemwould have high holding cost.Moreover,place and time of delivery also affect the system fill rate and total cost.Sending products to a wrong place in an inappropriate period may result in both low fill rate and high cost.Thus,it is interesting to optimize the product flo
39、wthrough the concerned supply chain,such that the system can give the desired fill rate withthe lowest total cost.Therefore,the relevant decision in this model is the determination of shipping quantity sent from the warehouse to each retailer in each period.In this paper,the concerned supply chain i
40、s controlled by the IDP which is obtained by solving the proposed mixed-integer linear programming model presented in the next section. 3. The IDP control system The IDP model is formulated as a mixed-integer linear programming model.It aims to minimize the sum of ordering,holding,holding in transit
41、,transportation,and lost-sale costs.Theconstraints are safety stock policies and material balance equations.Notations are defined asfollows:Indicest period index(1,y,T)i node index(0,y,N),Note that the warehouse has index i?0,the retailer has index i40.Decision variablesTti shipping quantity sent fr
42、om a source node to node i at the beginning of period tBti binary variable representing whether an order is placed by node i in period tki safety stock factor at node iLSti lost-sale quantity at node i at the end of period t(note that lost sales may occur atthe retailers,but not at the warehouse)Par
43、ametersM a large positive numberAi fixed ordering cost at node ihi holding cost per unit period at node ihiti per period unit holding cost in transit from a source node to node igi unit transportation cost from a source node to node isltiunit lost-sale cost at node i in period tdti mean customer dem
44、and of the product at node i in period t(note that customer demandoccurs only at the retailers,but not at the warehouse)Sti standard deviation of demand over the lead time plus the review period at node i in period tLi constant transportation lead time for transportation to node iIEt=0i initial inve
45、ntory at node i Objective function The objective function is to minimize the total expected cost comprising of ordering cost,holding cost at the warehouse/retailers,holding cost in transit,transportation cost,and lost-sale cost. The first term in Eq.(1)is the ordering cost.The ordering cost is obtai
46、ned from the multiplication of ordering cost Ai and binary variable Bti.The second andthird terms express the holding costs at both stages.At the warehouse(i?0),the holding cost is calculated based on the ending inventory because the inventory reduces at the beginning of a period and remains constan
47、t during the period.At the retailers(i40),it is calculated from the average of beginning and ending inventories because the inventory reduces due to the arrival of customers which may occur any time during the period.The fourth term is the holding cost in transit.The fifth term is the transportation
48、 cost and the last term represents the lost-sale cost.Constraints1. Safety stock policy constraints:The ending inventory at each node must not be lower than the required safety stock level as presented in constraint(2)It is noted that the safety stock factor ki is a decision variable.It is determine
49、d by a simulation-search approach as presented in Section 4.The safety stock factor ki is very important because it has a large effect on system performance.An inappropriate number of ki can result in poor system performance(low system fill rate with high system cost).2. Inventory balance constraints:The beginning inventory level is equal to the ending inventory level of last period plus the total incoming quantity.The ending inventory level is equal to t
