مدل موجودی تولید یکپارچه با ملاحظات تکمیل مجدد مواد خام در یک زنجیره تامین سه لایه ای

Most of the existing research has focused on a two stage single-vendor single-buyer supply chain. However, in reality, supply chain networks are more complex and involve more than just a vendor and a buyer. This paper deals with the joint economic lot sizing problem (JELP) in the context of a three stage supply chain consisting of a single supplier, single manufacturer and multi-retailers. The objective is to specify the timings and quantities of inbound and outbound logistics for all parties involved such that the chain-wide total ordering, setup, raw material and finished product inventory holding costs are minimized. In developing the model, the cycle time at each stage is set to be an integer multiple of that for the adjacent downstream stage. To bear a better resemblance to practice, shipments from a particular lot are allowed to take place during production and not after producing the whole lot. We employ derivative-free methods to derive a near closed form solution for the developed model. A numerical example is presented for illustrative purposes and a comparison to models established in the literature is also provided.

For most companies, providing the customers with a better service at a reduced cost is one of the ultimate strategic goals. The production of highly diversified products with short life cycles such as computer parts, fashion clothes and some food items among many others as well as the remarkably high levels of competition pushes the different companies towards the integration of different production and inventory related decisions. Consequently, companies are realizing the necessity of having elevated levels of mutual understanding and better collaboration with their customers and suppliers alike. To remain competitive, firms can no longer operate as individual and autonomous entities but rather as an integral part of the supply chain. The area of supply chain management (SCM) has gained a lot of interest from researchers as well as practitioners in the industry. In particular, the integrated single-vendor single-buyer problem, also called the joint economic lot sizing problem (JELP), has received a lot of attention in recent years as it represents the building block for the wider supply chain. Essentially, the retailer (buyer) observes a deterministic demand and orders lots from the manufacturer (vendor). The vendor satisfies this downstream demand through manufacturing the requested product in lots, where each produced lot is shipped to the buyer in batches. The problem is to find the number of shipments and size of each batch such that the joint manufacturer and retailer cost is minimized. For a vertically integrated supply chain owned partially or jointly by the same company, such coordinated production–shipment policy provides valuable insights and optimal decisions that lead to global optimization. On the other hand, when individual entities are owned separately, such policy may not benefit all parties equally as some may encounter an increase in their costs and hence become less eager to depart from their locally optimized policies. In such situations, sharing those benefits resulting from the coordinated approach becomes a major issue. By using effective incentive systems such as accounting methods, transfer pricing schemes, quantity discount, etc., the objective of each partner can be aligned to that of the supply chain as a whole (Ganeshan, 1999, Li and O’Brien, 1999 and Agrawal et al., 2004). Most of the work related to JELP has been conducted in the context of a two layer supply chain consisting of a single vendor and a single buyer. Goyal (1977) suggested a lot-for-lot policy with the assumption of infinite production rate for the manufacturer. Later, Banerjee (1986) maintained the lot-for-lot policy for the more realistic case of a finite production rate. The lot-for-lot assumption was relaxed by Goyal (1988) where he assumed that the vendor ships the lot in a number of equal size shipments. Goyal (1995) developed a policy where the shipment sizes increase by a factor increasing geometrically. Hill (1997) generalized the latter model through considering the geometric growth factor as a decision variable. The optimal solution to the problem in its general form (i.e., without any assumptions regarding the shipment policy) was obtained by Hill (1999). Goyal and Nebebe (2000) considered a policy where the first shipment is small and the following shipments are larger and of equal size. For a comprehensive review of the JELP, the reader is referred to Ben-Daya et al. (2008). More recently, this problem has been extended to the case of a three layer supply chain. Khouja (2003) was the first to consider a three stage supply chain with one or more firms at each stage. He discussed three inventory coordination mechanisms among the members of the supply Chain: (1) The equal cycle time mechanism where all parties involved share a common cycle time. This corresponds to the lot-for-lot policy mentioned earlier. (2) The integer multipliers of the cycle time mechanism where the cycle time at each stage is set to be an integer multiplier of the cycle time of the adjacent downstream stage. (3) The integer power of two multipliers mechanism in which the cycle time at each stage is an integer power of two multiples of a basic cycle time. Khouja found out that the savings in going from the first mechanism to the second one is more significant as compared to the savings obtained once going from the second to the third coordination mechanism. Recently, Ben-Daya and Al-Nassar (2008) applied the idea of Lu (1995), which calls for making shipments from a production lot as it is being produced, to the three stage multi-customers supply chain using the integer multiplier of the cycle time mechanism. The authors showed that their policy results in a lower total cost as compared to the corresponding policy suggested by Khouja. Lee (2005) added a new dimension to the single vendor single buyer problem by setting the number of raw material shipments received by the vendor per cycle to be a decision variable. Thus, the raw material ordering cost was considered explicitly in the model. According to Lee (2005), models incorporating the raw material procurement and manufacturing setup are called integrated procurement-production (IPP) systems. The purpose of this paper is to incorporate Lee’s idea suggested in a two layer supply chain in the model developed by Ben-Daya and Al-Nassar (2008) for three layer supply chains. The supply chain we are dealing with consists of a single supplier, single manufacturer and multi-retailers. The raw material ordering cost for both the supplier and the manufacturer are included in the model, and the number of raw material shipments received by both parties in every cycle is a decision variable. A cost minimization model is derived along with an efficient solution algorithm that is based on the algebraic approach. The remainder of this paper is organized as follows. Section 2 states the problem definition, notations and assumptions. The derivation of the mathematical model is detailed in Section 3 while the analysis of the model and the solution procedure are both provided in Section 4. The numerical example along with sensitivity analysis for key problem parameters is given in Section 5. Finally, Section 6 concludes the paper and highlights future research directions.

This paper presented a three layer supply chain model in which the supply chain includes a supplier, a manufacturer and multi-retailers. The inventory coordination mechanism applied is the integer multiplier of the cycle time policy which sets the cycle time at each stage to be an integer multiple of that for the adjacent downstream stage. Since the developed model incorporates both the raw material procurement and the manufacturing setup decisions, it is one of the so-called integrated procurement-production (IPP) models. The paper has also presented a derivative-free solution procedure to derive a near optimal solution to the model at hand. Such method has proven successful in inventory management literature and it provides a remedy for dealing with function involving integer variables, in which case classical differential calculus techniques might not be directly applicable. As it turns out, the proposed model outperforms the model that does not attempt to optimize the shipment policy of raw materials (i.e., where raw materials are delivered to the supplier and the manufacturer in a single shipment) and results in savings that might amount to 4% in case of low ordering costs. This work might be extended in several directions. One may consider similar problem settings under stochastic rather than deterministic demand. Also, the shipments made to a downstream stage might assume different sizes that follow a different pattern instead of equal shipments. An alternative extension might be to incorporate the quantity discount issue into the model in case the constraint of having vertically integrated supply chain is no more valid. Introducing the effect of quality in these models is yet another promising avenue for future research.