M Y. Jaber*1, M Bonney2, A L. Guiffrida3
1
Department of Mechanical and Industrial Engineering Ryerson University, Toronto, ON, M5B 2K3, CANADA
2
Nottingham University Business School University of Nottingham, Nottingham, NG8 1BB, UK 3
Department of Management and Information Systems Kent State University, Kent, Ohio 44242 USA
(* corresponding author. E-mail: mjaber@ryerson; Fax: 416-979-5265)
ABSTRACT
This paper investigates a three-level supply chain (supplier-manufacturer- retailer) where the manufacture undergoes a continuous improvement process represented by a learning curve. To attain this continuous improvement the manufacturer may need to coordinate its orders with its customer for finished products and with its supplier of raw material. A mathematical model is developed to determine the joint lot sizes for this supply chain and numerical examples illustrating the solution procedure are provided. The results suggest that the manufacturer should produce in smaller lots more frequently.
INTRODUCTION
The modern market for products is dynamic, global and competitive. This environment imposes pressures on companies to deliver quality products at competitive prices at the time required. As product life cycles shorten, companies need to be responsive to market changes and to reduce the time from concept to market. The pressures compel companies to be responsive, efficient and flexible. Effective supply chain management involves the integration of functions such as production, purchasing, materials management, warehousing and inventory control, distribution, shipping, and transport logistics. This integration is needed within the operations of specific supply chain members and, more importantly, across all members of the supply chain. To maintain sustainable competitiveness, operations within the supply chain will benefit from continuous improvement programs that include fostering organizational learning. Historically, learning curve theory has been applied to a diverse set of management decision areas such as inventory control, production planning and quality improvement. These decision areas exist both within the individual organizations of the supply chain and, as a result of the interdependencies among chain members, across the supply chain as a whole. By modelling these learning effects, management may use established learning models to utilize capacity, manage inventories and coordinate production and distribution better throughout the chain.
The lot sizing problem with learning and forgetting effects in production has received considerable attention from researchers and a detailed review of this literature is found in Jaber and Bonney (1999). Jaber and Bonney (2003) also investigated the effects that learning and forgetting in set-ups and in product quality have on the economic lot-sizing problem. In recent years, the lot sizing problem with learning and forgetting has been investigated using various assumptions (e.g., Balkhi 2003; Chiu et al. 2003; Chiu and Chen 2005; Alamri
and Balkhi 2007; Jaber and Guiffrida 2007) and to a lesser extent, in conjunction with the Joint Economic Lot-Sizing Problem (JELSP), by Nanda and Nam (1992, 1993).
The JELSP forms the basis of a two-level supply chain with order coordination between the chain members. Nanda and Nam (1992) developed a joint manufacturer-retailer inventory (two-level supply chain) model for the case of a single buyer. Production costs were assumed to reduce according to a power form learning curve with forgetting effects caused by breaks in production. A quantity discount schedule was proposed based on the change of total variable costs of the buyer and manufacturer. To meet the demand of the buyer, the manufacturer considers either a lot-for-lot (LFL) production policy (e.g., Banerjee 1986), or a production quantity that is a multiple of the buyer's order quantity (Lee and Rosenblatt 1985). Nanda and Nam (1992) assumed a LFL policy, and did not specify the form of the forgetting curve. They extended their work in a subsequent paper (Nanda and Nam 1993) to include multiple retailers.
This paper integrates the work of Nanda and Nam (1992) and Jaber and Bonney (2003) to investigate a joint replenishment inventory model for a three-stage (supplier-manufacturer-retailer) supply chain with the manufacturer having learning effects in setups, production, and product quality. Munson and Rosenblatt (2001) were the first to model a three-level supply chain. They assumed that all parameters were deterministic and that:(i) the retailer orders a single product according to its economic order quantity (EOQ), (ii) the manufacturer optimises its lot-sizing policy according to a lumpy order pattern, which is an integer multiple of the retailer’s order quantity, and (iii) the supplier orders according to the resulting lumpy ordering pattern of the manufacturer, which is an integer multiple of the manufacturer’s order quantity. Munson and Rosenblatt (2001) further assumed that the manufacturer is the most influential player in the supply chain who offers quantity discounts to the retailer to entice him/her to order in larger quantities than the retailer’s economic order quantity. Quantity discounts were computed in the model (e.g., $/unit) as the difference in holding and ordering costs between the retailer’s old ordering (no coordination) and new ordering (with coordination) policies divided by the annual demand. Jaber et al. (2006) extended the work of Munson and Rosenblatt (2001) by assuming a price discount approach, a price dependent demand and profit sharing scenarios. This paper adopts a centralized decision-making process for coordinating the supply chain model by integrating order quantities. When players in a supply chain coordinate their order quantities, it is possible that some players in the supply chain could benefit more than others; indeed some could lose. To overcome this, the paper assumes that the benefiting (dominant) player, the manufacturer, compensates the losing players by offering quantity discounts.
METHODOLOGY
This paper investigates the coordination of order quantities amongst the players in a three-level supply chain with a centralized decision process. A mathematical model for a three-level supply chain (supplier-manufacturer-retailer) is developed. Integrating order quantity models among the three levels is a way to achieve coordination (e.g., Goyal and Gupta 1989). This paper assumes a uniform demand on the retailer that results in lumpy demand on the manufacturer and the supplier. It also assumes (1) a single product that requires g units of raw material, (2) no shortages, (3) zero lead-time, (4) imperfect
quality items, (5) unlimited storage capacity and (6) an infinite planning horizon. The manufacturer’s total cost per cycle is the sum of its set-up cost, raw material procurement cost, labour cost, holding cost and quality cost. It is assumed that the manufacturer undergoes a continuous improvement program characterized by learning [measured by the learning rate (LR)] in setup (LR-stp), in production (LR-prod) and in process quality (LR-qlty). The total supply chain cost (objective function) is the sum of the cost functions of the retailer, the manufacturer and the supplier. This is written as a mathematical programming problem of the form
Minimize
ψ
i,sc(λ
s,λ
m,Qr,)
=ψ
r( )
Qr +ψ
i,m(λ
m,Qr)
+ψ
s(λ
s,λ
m,Qr)
Subject to:Order quantity: Qr ≥ (y1D/(1−b))1/b
Multipliers:
λ
s,λ
m,which are integersThe terms
ψ
i,sc,ψ
r,ψ
i,m, andψ
s respectively represent the total supply chain, the retailer (customer), the manufacturer, and the supplier costs per unit of time for the manufacturer’s production cycle i, whereλ
sandλ
mare integer multipliers to adjust the order quantity of the retailer, Qr, to that of the manufacturer (m)and the supplier (s), and D being the demand rate. The continuous improvement learning curve is assumed to be of a power form as advocated by many researchers, i.e., y(x) = y1 x-b (Wright 1936) with y(x) being the time to manufacture the xth unit, y1 the time to produce the first unit, b (0 b < 1; corresponding to 100% > LR-prod > 50%) the production learning curve exponent and x is cumulative production. Due to the limit on the number of pages, we refrain from explicitly discussing the mathematics of the terms
sc i,
ψ
,ψ
r,ψ
i,m,andψ
s.However, readers may refer to Jaber and Bonney (2003) and Jaber et al. (2006) for background on the mathematical modelling of the problem presented above.The developed mathematical programming model was solved for various values of the input data for the cases of coordination and no coordination of orders among the players in the three-level supply chain using the EXCEL SOLVER optimization tool enhanced by VISUAL BASIC code.