ORMAT A Contract Model for a Decentralized Assembly System
Xuxia Zou, Shaligram Pokharel, Rajesh Piplani
Centre for Supply Chain Management
School of Mechanical and Production Engineering
Nanyang Technological University
50 Nanyang Avenue
Singapore 639 798
SINGAPORE
Abstract
Synchronization of an assembly system facing random demand requires coordinating the supply of various components for the final assembly. However, component suppliers, due to their cost structures, may choose to deliver components such that their own profit is maximized. This creates a mismatch of components and increases inventory costs since the number of products that can be finally assembled depend on the supplier who delivers the fewest components. Also, when a long-term relationship is expected between the assembler and its suppliers, they would look for multi-period contracts, in which the impact of changes in various parameters on the optimal ordering/delivery decisions can be reviewed based on supply and demand situation in the earlier periods.
In this paper, we develop a two-period contract model focusing on a two-echelon decentralized assembly system. We show that the proposed contract model, characterized by wholesale plus buyback prices, is able to realize channel coordination. A numerical study is also presented. It is expected that the proposed model would be suitable for companies seeking proper price and profits adjustment with their long-term partners.
Keywords: supply contract, channel coordination, decentralized assembly system.
i
1. Introduction
Supply contracts have been the focus of study for quite sometime due to the necessity to coordinate suppliers, usually spread globally, in order to increase the supply chain efficiency. Such coordination, referred to as “channel coordination”, helps to align individual incentives towards global optimization by managing the delivery. However, due to demand and lead time uncertainty in such a situation, both suppliers and the assembler have to work in collaboration instead of a one-off work venture. Such collaboration may be brought about by involving the supply chain players in more than one contract period. Based on this, we have proposed a two-period supply contract model for channel coordination.
2. Relevant Literature
Supply chains operate in a centralized or a decentralized control mode. In a centralized mode, one decision-maker determines the order quantity for the benefits of the entire system. In a decentralized mode, all the players can make independent or negotiated decisions.
2.1 Centralized Control Mode
The systems operated with centralized control are expected to achieve the best performances (Lee & Billington, 1993; Lariviere, 1999; Tsay, et al., 1999). The most relevant literature assuming centralized control is on multi-echelon inventory management. The work of Clark & Scarf (1960) is regarded as the first research on multi-echelon inventory management in which stochastic demand was considered and a concept of echelon stock was proposed (Rosling, 1989; Tsay, et al, 1999). Further extension to this model for continuous review with constant demand over an infinite horizon was made by Schwarz & Schrage (1975); for periodic review by Federgruen & Zipkin (1984); and for random customer demand and proportional production and holding costs by Roslin (1989). Schmidt & Nahmias (1985) studied a two-echelon, multi-period MRP-type assembly system (i.e., demand for components arises only through orders placed for the end product; thus, the number of different components always matches) with a stochastic market demand. The literature reviewed above shows how optimal or near-optimal inventory policy can be determined in a multi-echelon supply chains over finite (or infinite) horizon operated with centralized control.
For relevant review papers on multi-echelon inventory management, the readers can refer to Lee & Nahmias (1993); Muckstadt & Roundy (1993); Van Houtum, et al. (1995); Diks, et al. (1995).
2.2 Decentralized control mode
Incentive conflicts and information asymmetry are identified as two commonly observed problems associated with decentralized control (Tsay, et al., 1999; Lariviere, 1999). The two kinds of contract used in the proposed decentralized assembly system are the price-only contract and buyback contract (or referred to as returns policy). While studying price-only contract, Lariviere & Porteus (2001) conclude that this contract alone cannot coordinate the decentralized distribution channel due to “double marginalization”.
Pasternack (1985) shows that a wholesale price plus a return policy can achieve channel coordination in a multi-retailer environment. Gerchak & Wang (2004) find that both revenue-sharing plus surplus-subsidy operated in a vendor managed inventory (VMI) setting and wholesale price plus buyback contract can realize channel coordination.
Apart from buyback contracts, other coordinating mechanisms include revenue-sharing contract (Dana et al., 2001; Cachon & Lariviere, 2004; Wang et al., 2004; Zou et al., 2004), quantity flexibility contract (Eppen & Iyer, 1997; Iyer & Bergen, 1997; Tsay, 1999; Tsay & Lovejoy, 1999), quantity discount (Jeuland & Shugan, 1983; Monahan, 1984; Lee & Rosenblatt, 1986) and joint pricing-production decision-making (Weng, 1997, 1999; Zhao & Wang, 2002).
3. Model Formulation
For simplicity, we consider a two-echelon decentralized assembly system with the following assumptions.
- Market demand in each period is independent and identically distributed (IID) with common cumulative distribution function and probabilistic distribution function.
- All the cost parameters are stationary over the entire horizon except for the wholesale prices which are renegotiated every period.
– Costs and revenue are proportional to the number of units involved.
– Leftover inventories at the end of a period are automatically transferred to the next period incurring corresponding holding costs; excessive market demand is completely backlogged in each period.
– There are only two component suppliers and component usage rate equals to 1.
– All the players are risk neutral and are assumed to be profit maximizers.
– All the delivery/ordering decisions are made before the realization of market demand.
Notations for the proposed model are listed in Table 1.
Table 1 Notations for the two-period contract model
/ Subscript, indicate the period where there are periods left/ Subscript, indicate the coordinated decentralized setting with buyback contract
/ Subscript, , the assembler; , the component suppliers
/ Market demand in period
/ Probability distribution function of market demand
/ Cumulative distribution function of market demand
/ Mean market demand
/ Player ’s production/ordering quantity in period of the coordinated decentralized setting with buyback contract
/ Player ’s initial inventory level at the beginning of period
/ Unit selling price
/ Unit inventory holding cost for player
/ Unit production cost for player
/ Unit shortage cost
/ Supplier ’s unit wholesale price in period of the coordinated decentralized setting with buyback contract
/ The critical order-up-to level in the centralized control setting in period ,
/ The critical order-up-to level in the coordinated decentralized setting of period
/ Player ’s expected system profit in period of the coordinated decentralized setting with buyback contract
The two-period models are formulated with backward stochastic dynamic programming.
3.1 Coordinated Decentralized System
A centralized system is assumed to have a coordinated channel as the system performance (measured in terms of expected system profits and final assembly of the products) is the maximum in such a case. A model for a centralized coordinated system is provided by Zou et al. (2005). Zou et al. (2004) show that in a decentralization mode, system performance reduces due to incentive conflicts in the proposed assembly system.
3.1.1 Last-period problem
To simplify the representation, define , which represents the expected backorder penalty and holding cost at an inventory level of . Hence, with buyback contract, the assembler’s expected profit in the last period is,
1)
It can be derived from that which represents the assembler’s optimal order-up-to level without constraint.
Moreover, supplier ’s () expected profit for the last period can be formulated as,
2)
Assume . Further, it can be derived that there is an order-up-to level (denoted as ) at which supplier 1’s expected profits is maximized. Specifically, . By analogy, there is an order-up-to level (denoted as ) at which supplier 2’s expected profits is maximized. Specifically, . If , and ( are three optimal order-up-to levels derived in the centralized control setting) could be satisfied, it is implied that . Hence, supplier 1’s optimal production policy can be described as given in (3) next,
3)
Further, supplier 2’s optimal production policy is,
4)
Based on the suppliers’ optimal production policy, the assembler’s optimal ordering policy is,
5)
Hence, the optimal production/order policy for the entire system is summarized as follows.
Lemma 1
Suppose , the optimal production/ordering policy for the last period problem of the decentralized assembly system with buyback contract is,
1. If , then .
2. If , then .
3. If , then .
4. If , then .
5. If , then .
6. If , . ■
This is identical to the centralized setting proposed in Zou et al. (2005) under the condition that . Therefore, we have the following proposition.
Proposition 1
Channel coordination can be achieved in the decentralized control system with the proposed two-parameter buyback contract, .
Further, the coordinating parameters satisfy the following equations,
6)
3.1.2 Two-Period Problem
Without loss of consistency, it is assumed that . However, the optimal policy for the last period would only be derived based on the condition of (since it has been proved in the centralized setting that, with , it is optimal for suppliers to produce such that supplier 1’s on-hand inventory level at the beginning of the last period is lower).
Proposition 2
Suppose are three optimal order-up-to levels satisfying. Moreover, it is assumed that. If , and , the proposed buyback contract can be applied to coordinate the decentralized assembly system in a two-period setting.
4. Numerical Study
Two component suppliers (denoted as and ) and an assembler (denoted as) are considered here. Market demand is exponentially distributed with =1000. Initial conditions are listed in Table 2.
Table 2 Initial conditions
Entity / Initial Conditions/ 0.6 / --- / --- / 0.3 / 1000
/ 0.5 / --- / --- / 0.2
/ 1.0 / 8 / 1.2 / 0.4
4.1 Numerical Study for the Last Period
Without loss of generality, it is assumed that . Ten groups of data are tested (see Table 3) to show how changes in one of the coordinating parameters (i.e., ) would affect the allocation of system profits. The total expected system profit attained (i.e., ) is the same as that attained in the centralized setting.
Table 3 The impact of on system performance in a coordinated setting (group 3)
values / / 2367 / / 1674 / / 1345No. / Coordinating Parameters / Profits
1 / 0.5 / 1.0063 / 9.3187 / 7.3908 / 1295.7 / 1955.6 / 1237.3 / 4488.6
2 / 1.0 / 1.4125 / 9.0375 / 7.1829 / 1421.3 / 1925.0 / 1142.2
3 / 1.5 / 1.8188 / 8.7562 / 6.9749 / 1547.0 / 1894.4 / 1047.2
4 / 2.2 / 2.2250 / 8.4750 / 6.7669 / 1672.6 / 1863.8 / 952.2
5 / 2.5 / 2.6313 / 8.1937 / 6.5590 / 1798.3 / 1833.1 / 857.1
6 / 3.0 / 3.0375 / 7.9125 / 6.3510 / 1923.9 / 1802.5 / 762.1
7 / 3.5 / 3.4438 / 7.6312 / 6.1430 / 2049.6 / 1771.9 / 667.1
8 / 4.0 / 3.8500 / 7.3500 / 5.9350 / 2175.3 / 1741.3 / 572.0
9 / 4.5 / 4.2563 / 7.0687 / 5.7271 / 2300.9 / 1710.6 / 477.0
10 / 5.0 / 4.6625 / 6.7875 / 5.5191 / 2426.6 / 1680.0 / 381.9
4.2. Numerical Study for the Two-Period Problem
Similarly, the purpose of the numerical study for the two-period problem is on how variations in coordinating parameters would affect profit allocation among all the players (see Table 4).
Table 4 The impact of on system performance in a coordinated setting (group 8)
values / / 3684 / / 2664 / / 2270No. / Coordinating Parameters / Profits
1 / 1.0 / 2.5 / 1.5865 / 6.0922 / 13.2295 / 596.8 / 3040.4 / 6392.4 / 10030
2 / 1.2 / 1.3586 / 6.3160 / 13.3269 / 776.1 / 2936.0 / 6317.5
3 / 1.4 / 1.1307 / 6.5398 / 13.4243 / 955.4 / 2831.6 / 6242.7
4 / 1.6 / 0.0928 / 6.7636 / 13.5217 / 1134.6 / 2727.2 / 6167.8
5 / 1.8 / 0.6749 / 6.9873 / 13.6191 / 1313.9 / 2622.9 / 6092.9
6 / 2.0 / 0.4470 / 7.2111 / 13.7165 / 1493.2 / 2518.5 / 6018.0
Further, it is assumed that and is assigned to be 2.5. The total expected system profit attained (i.e., ) is the same as that attained in the two-period centralized control setting as given by Zou et al. (2005). Hence was concluded that the proposed model is able to coordinate a decentralized assembly system.
5. Conclusion
The model developed here shows that a two-period contract model for the assembly system can coordinate a decentralized assembly system to obtain the performance of a centralized setting by adjusting the buyback and wholesale prices. The proposed model is suitable for an assembly system with products having relatively long lifecycle. The model is developed only for a two-period setting to minimize the analytical requirements. However, we believe that the analytic concepts presented here can be extended further to a multi-period setting with multiple players.
Reference:
Anupindi, R. and Bassok, Y. (1999). Supply contracts with quantity commitments and stochastic demand. Tayur, S., Ganeshan, R. and Magazine, M. (Editor), Quantitative Models for Supply Chain Management. Boston: Kluwer Academic Publishers.