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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.12 No.2 pp.85-94
DOI : https://doi.org/10.7232/iems.2013.12.2.085

Supply Chain Contract with Put and Call Option: The Case of Non-Linear Option Premium Price

Huynh Trung Luong*, Chirakiat Saithong
Department of Industrial and Manufacturing Engineering, School of Engineering and Technology, Asian Institute of Technology, Pathumthani, Thailand
Department of Industrial Engineering, Faculty of Engineering at Si Racha, Kasetsart University Si Racha Campus, Chon Buri, Thailand
(Received: January 31, 2013 / Revised: May 30, 2013 / Accepted: May 30, 2013)

Abstract

This research investigates the supply chain contract between a distributor and a supplier in which the selling period isrelatively short in comparison with long production lead time. At the first stage, supplier who is a Stackelberg leaderoffers the distributor a contract with a set of parameters, and subjected to those parameters, the distributor placesthenumber ofinitial orders as well as options. In order to purchase the option, the distributor pays non-linear option premiumprice with respect to the number of purchased options. At the second stage, based on realized demand, the distributorhas the right to exercise option as either put or call which is limited up to the number of purchased options.The wholesale price contract is used as a benchmarking contract. This research has confirmed that the supply chaincontract with a non-linear option premium price can help to coordinate the supply chain.

12-2-2_85-94_ Chirakiat Saithong and Huynh Trung Luong.pdf2.37MB

1. INTRODUCTION

In the global economy, many organizations rely on an effective supply chain in order to compete in the global market. In a supply chain, there is a management need to cooperate organizations in the supply chain to work together so as to maximize efficiency. These organizations are suppliers, manufacturers, distributors, logistic providers and retailers. A highly efficient supply chain is defined as the supply chain that is able to satisfy demand with right place, right quantity, and right time. Moreover, highly efficient supply chain derives benefits from customer satisfaction whereas poorly efficient supply chain does in the opposite way. Customer satisfaction leads to organization’s profit in a supply chain. One of the criteria used to measure efficiency of the supply chain is “coordination.” The coordination occurs if and only if the organizations in the supply chain make decisions in such a way that the supply chain’s profit is maximized. In other words, a coordinated supply chain is a network that works as a single organization and helps to optimize the supply chain’s profit. However, a noncoordinated supply chain can occur if different individual incentives of organizations exist in the supply chain (Cachon, 2003). Contracts are useful ways to assist coordination of the supply chain. A contract contains a set of parameters in a specific context. Roughly, a contract captures three types of flows, i.e., material flow, information flow and financial flow. There are many types of contract which have been proved to help the coordination of the supply chain and to allocate be-nefit arbitrarily such as buy back contracts, revenue-shar-ing contracts, quantity-flexibility contracts, sale-rebate contracts, and quantity-discount contracts.

 In the business practice in the case of no contract agreement in a supply chain, the relationship between the supplier and the distributor typically operates following a wholesale price contract mechanism. However, this contract usually leads to low supply chain’s efficiency because both parties desire to maximize their individual profit. As a result, the global maximum profit of the supply chain is not reached because of the double marginalization effect (Spengler, 1950). For some kinds of product with long lead time, short selling season and high fluctuation of demand, in order to satisfy customer need,the supply chain must have high agility to serve demand. Thus, the distributor must decide on the order quantity based on forecasted information. As a result, either shortage of inventory or excessive inventory may exist in the supply chain (Huang et al., 2011; Wang and Tsao, 2006; Zhao et al., 2010). Supply contract with option is a way to cope with this problem. In option contract, the planning horizon of the contract can be divided into two stages. In the first stage, the distributor agrees to place an initial order as well as the amount of options. In the second stage, the supplier allows the distributor to modify his orderafter the realization of demand.

 Option in the supply chain contract can be classified into three types by direction of exercise option. Upward adjustment, or call option, allows the distributor to increase the order quantity only after realization of demand. On the other hand, downward adjustment, or put option, allows the distributor to decrease the order quantity after realization of demand. The last type of option contract, bidirectional adjustment, i.e., put and call options, allows the distributor to either decrease or increase the order quantity after realization of demand. This research studies bidirectional adjustment which provides more flexibility to the supply chain in order to cope with the fluctuations of demand.

2. LITERATURE REVIEW

In their study of supply chain coordination, Giannoccaro and Pontrandolfo (2009) examined the revenue sharing contract and demonstrated that the percentage of profit sharing plays an important role in coordination of supply chain. They used simulation approach to examine how the negotiation process of revenue sharing contract works. The result showed that the best scenario which helps to achieve supply chain coordination is low contractual power for all parties. Xiao et al. (2010) studied supply chain coordination in which customers are allowed to return products. Buy back and customer return product policies were integrated in the framework of this study. The result showed that at low variation of demand, full-refund policy contributes higher profit than no refund whereas no refund policy contributes higher profit than full-refund policy at high level of variation of customer’s demand. Huang et al. (2011) studied the designing coordination mechanism in reverse supply chain to solve profit allocation between supplier and retailer in the case of false failure returns. They designed a quantity discount contract to solve the problem by exerting costly effort to retailer if there is too many false failure returns. Chen and Bell (2011) also studied coordination of reverse supply chain. They proposed two buy back prices for left over and returned items. Furthermore, they proved that the reverse supply chain with two buy back prices policy can assist coordination of the supply chain. Chen (2011) also showed that not only wholesale price discount contract but return-discount contract can help improve supply chain’s efficiency. The benefit of this contract was that it will not only incentivize retailer’s order quantity but also increase supply chain efficiency towards operation as centralized supply chain.

 Related to supply chain contract with option, Gomez_Padilla and Mishina (2009) studied on option contract with bidirectional option. One supplier-one retailer and many suppliers-one retailer models were analyzed in this paper using both wholesale price contract and buy back contract as benchmarking contracts. The result showed that applying supply chain contract with bidirectional option increases not only individual party’s profit but also supply chain profit. Wang and Tsao (2006) also studied supply chain contract with bidirectional option by focusing only on buyer’s perspective. The result from this paper showed that applying supply chain contract with bidirectional option increases buyer’s profit. However, this paper did not consider the coordination issue in supply chain. Zhao et al. (2010) demonstrated how supply chain contract with call option assists coordination of supply chain with Pareto-improvement in comparison with wholesale price contract. They used cooperative game approach to examine coordination issue in the newsvendor model using option contract. The coordinating performance of the contract depends on both individual risk preference of each party and negotiation power. Zhang et al. (2010) proposed a supply chain contract with bidirectional option and buy back policy for perishable product. Wang et al. (2011) demonstrated the risks associated from introducing supply chain contract with call option. Two important parameters, risk indicator and probability of such risk, were proposed in this paper. Lian and Deshmukh (2009) studied the contract under the condition that the buyer could order goods at discounted price if the order was placed in advance. The more advance period is committed, the more discount is provided.

 In summary, there were a number of studies which examined coordination of supply chain such as Giannoccaro and Pontrandolfo (2009), Xiao et al. (2010), Huang et al. (2011), Chen and Bell (2011), Chen (2011), Huaet al. (2008), and Zhao et al. (2010). However, none of those studies examine coordination of supply chain with bidirectional option in the case of the supplier who is the leader of the system. Wang and Tsao (2006) studied supply chain contract with bidirectional option from the standpoint of buyer only and they use linear relationship between number of purchased options and option premium price. This means that if distributor purchases either large number or small number of options, the unit option premium price is exactly the same. This might lead to the situation that the distributor will reserve an unnecessary large number of options. Based on this observation, the research proposes a supply chain contract with bidirectional option in which unit option premium price dramatically increases as the number of purchased options increases, and furthermore, this research not only considers supplier’s interest but also distributor’s interest. In addition, this research also examines whether the supply chain contract with bidirectional option in the case of non-linear option premium price can assistthe coordination of supply chain.

3. MATHEMATICAL MODEL DEVELOPMENT

In a supply chain, if there is no agreement, each party operates using traditional contract, i.e., wholesale price contract. However, wholesale price contract leads to inefficiency of the supply chain due to the double marginalization effect (Spengler, 1950). Thus, this research uses the wholesale price contract just as a benchmarking contract in order to compare efficiency with the proposed supply chain contract with bidirectional option (SCBO). The following notations are used throughout this research:

= unit wholesale price (in wholesale price contract)
w = unit price of initial order (in SCBO)
Q = number of orders in wholesale price contract
Q0 = number of initial orders (in SCBO)
ξ = actual demand
vb = effective salvage value
p = retail price
μ = expected demand
gd = unit shortage cost for distributor
gs = unit shortage cost for supplier
g = total unit shortage cost of the supply chain
c = unit production cost of the supplier
fD( ⋅ ) = probability density function of demand
FD( ⋅ ) = cumulative distribution function of demand
D = random variable representing actual demand
πSC( ⋅ ) = supply chain’s profit function in wholesale price contract
π( ⋅ ) = distributor’s profit function in wholesale price contract
πS( ⋅ ) = supplier’s profit function in wholesale price contract
πtiD( ⋅ ) = distributor’s profit function at time ti (in SCBO)
πtiS( ⋅ ) = supplier’s profit function at time ti (in SCBO)
x = realized demand
qe = number of exercised options
q0 = number of purchased options
w0 (q0) = total option premium cost if the purchased option is q0
wec = unit price if exercising option as call
wep = unit price if exercising option as put

3.1 Wholesale Price Contract

 Let S(Q) be the expected sales, I (Q) the expected excessive inventory and L(Q) the expected shortage inventory. Distributor’s profit function can be derived as:



After some manipulation, distributor’s profit function can be expressed as: 

 

Similarly, supplier’s profit function can be derived as:

 

Or equivalently:



 Profit function of the whole supply chain can then be derived as the sum of (1) and (2):

 

 In a centralized system, both parties are considered as a single entity. The optimal number of ordered products in the centralized supply chain, QSC*, can be derived from the first order derivative of πSC(Q) with respect to Q as follows:

 

 However, in a decentralized system, the optimal number of ordered products is decided by distributor and it is derived from the first order derivative of π(Q) with respect to Q as follows:

 

 For the best interest of supplier, she always set > c in order to make profit. As a result, from (4) and (5), it is no doubt that  and supply chain coordination cannot be reached.

3.2 Supply Chain Contract with Bidirectional Option and Non-linear Opton Premium Price Mechanism

 The mechanism of option contract can be seen in Figure 1. At the beginning of the planning horizon, t0, supplier offers the distributor a contract with a set of parameters, w, w0, wec , wep. Then, the distributor determines Q0 and q0 accordingly. Attime t1, the beginning of the selling season, the distributor can realize the demand based on retailer’s order, and he determines the optimal number of exercised options, . Then, the supplier delivers the amount Q0 + qe to the distributor. It should be noted that the number of exercised options, for both put and call options, cannot exceed q0. Finally, at the end of the selling season, i.e., at time t2 , the distributor’s revenue can be fully determined.

Figure 1. The mechanism of supply chain contract with option.

Wang and Tsao (2006) and Zhang et al. (2010) usea linear relationship between q0 and w0 which may cause difficulty for the distributor to adopt the contract. In the case of supplier is the Stackelberg leader of the system, she offers the distributor a contract with a set of parameters and, subject to those parameters, distributor decides whether he will adopt the contract. After that, if the contract looks interesting for the distributor, he will determine the number of purchased options as well as the number of initial orders. It should be noted that the contract will be adopted by distributor if the incremental profit is attractive for him. As a result, coordination of supply cannot be achieved without agreement of both parties. To deal with the above problem, this research proposes a non-linear relationship between the number of purchased options and the total option premium cost as follows: 

 

 Where: α > 0, β ≥ 1 .

 For a two-stage supply chain, the expected profit will be formulated and solved by using a backward dynamic programming due to the fact that the problem has two stages in which the decision at the previous stage affects the subsequent stage. Thus, all profit functions of the second stage must be firstly formulated following the first stage. This research assumes that the demand is stochastic and follows uniform distribution over [γ − n, γ + n].

 In the second stage, the distributor has to decide on the number of exercised options, qe , in order to maximize his profit subject to known Q0, q0 and x. The optimal number of exercised options, , can be determined as follows:

 
The corresponding distributor’s optimal profit can be derived as follows:


Where: 

 

 On the other hand, at this stage, supplier’s profit function is:


 Where:

 

 

 In the first stage, distributor has to decide on Q0 and q0. Distributor’s profit function can then be formulated as follows:

 

Where:

 

and hence,

 

 In order to determine the optimal values of Q0 and q0 in this stage, the first derivatives of πt0 (Q0, q0) with respect to Q0, and q0, are required.

 

and

 

 Solving (11) and (12) simultaneously,  andcan be derived. Next, after optimal values of and  are derived, the corresponding supplier’s profit function can be determined as follows:

 

Where:

 

4. NUMERICAL EXPERIMENTS

 In this section, numerical experiments are conducted in order to evaluate the profit of both the distributor and the supplier in the proposed option contract with nonlinear option premium price. In principle, each party’s profit has to be increased in comparison with the wholesale price contract; otherwise, the option contract will not be used by both parties. Furthermore, because the supplier is the Stackelberg leader of the system, the incremental profit should be interesting for the distributor in order to urge the distributor to adopt the contract. It should be noted that the following assumptions have to be satisfied:

1. wep ≤ w ≤ wec to ensure that the supplier will not be much suffered from the possibility that the distributor might exercise put option and will be refunded. In addition, distributor has to pay higher for the units exercised as call.
2. vb < wep to ensure that distributor has incentive to exercise as put rather than keeping unsold products.
3. p + gd > wec to ensure that it is worth for distributor to exercise options as call.
4. vb < c to ensure that the supplier will not gain profitability from excessive production.

 It should be noted that demand is assumed to follow uniform distribution over [γ − n, γ + n]. Contract’s parameters used in this section are p = 200, w = 100, gd = 40, vd = 30, wep = 60, wec = 180, c = 35, γ =1000, n = 200. With the given contract’s parameters for the base case, the following results are obtained:

· Supply chain’s profit in wholesale price contract: 159,999.67
· Distributor’s profit in wholesale price contract: 90,666.67
· Supplier’s profit in wholesale price contract: 69,333.00
· Global maximum of supply chain’s profit in centralized system: 164,023.81

 It should be noted that the profit of the supply chain in the wholesale price contract is lower than the global maximum of supply chain profit by 2.52%. This is because of the double marginalization effect. The above results will be used as benchmark values for comparison purpose to examine if the option contract with nonlinear option premium price mechanism can help coordination of supply chain in the next sections.

4.1 Bidirectional Option with Linear Option Premium Mechanism

 In the linear option premium mechanism, a constant cost of additional purchasing option is paid to the supplier by the distributor. However, it is interesting to examine which mechanism looks interesting for the distributor between the linear and the non-linear option premium mechanism.

 From Figure 2, it can be seen that profit of the distributor in the case of non-linear option premium mechanism contributes higher profit than the case of linear option premium mechanism for most of the number of options purchased. The implication is straightforward; by adopting non-linear option premium mechanism, distributor who is the follower of the system will get higher profit in comparison with linear option premium memechanism. As a result, distributor would prefer non-linear mechanism rather than linear mechanism and it is interesting to examine whether the non-linear mechanism can help the supply chain to be coordinated. In the next section, this research will examine whether using nonlinear option premium mechanism assists coordinatio of supply chain.

Figure 2. Comparison of distributor’s profit between the two mechanisms.

4.2 Analysis with Respect to α and β Values in Case of Non-Linear Option Premium Price

 In this section, values of α and β are varied in order to examine the allocation of profit to individual party in the proposed option contract as well as the total profit of the whole supply chain. The value of α is varied from 0.05 to 15.0 and the value of β is varied from 1.1 to 2.0. The results are presented in Figures 2 and 3.

Figure 3. Distributor’s profit at different values of α and β.

From Figure 3, it can be seen that as value of both α and β increase, distributor’s profit will be dramatically reduced. Furthermore, when both α and β are quite high, distributor’s profit comes closer to distributor’s profit in case of wholesale price contract. This is as expected since the increase of both α and β will discourage the distributor to adopt option contract because of too high cost of purchasing option. In other words, the increase of both α and β tends to increase the value of initial order, Q0, whereas it tends to decrease the number of purchased options q0 as can be seen in Tables 1 and 2. 

Table 1. The optimal value of initial order

Table 2. The optimal number of purchased options

 From Figure 4, it can be seen that at a specified value of β , when value of α increases, the supplier’s profit will increase and then decrease. The increase of supplier’s profit is caused by higher cost of purchasing option which is profitable for her. However, if cost of purchasing option is too high, it will affect the distributor’s decision, i.e., the number of purchased options is decreased, as can be seen in Table 2. It should be noted that there are two interesting points. Firstly, at the extremely high value of α , supplier’s profit is close to supplier’s profit in case of wholesale price contract for all values of β . For example, at α = 30.0 and β = 2.0, the distributor almost does not employ the option contract and the initial order quantity is the same as optimal order quantity in wholesale price contract, as can be seen in Tables 1 and 2. This is as expected because too expensive cost of purchasing option will discourage the distributor to use option contract. Secondly, the increase value of β may lead to the reduction in supplier’s profit when α is high. This is because β contributes high effect on cost of purchasing option.

Figure 4. Supplier’s profit at different values of α and β.

Figure 5 illustrates optimal supply chain’s profit at different values of both α and β . For a specific β , there exists a value of α at which the supply chain’s profit is maximized. For example, supply chain’s profit is maximized at α = 5.0 when β = 1.1. It is interesting to notice that when supply chain’s profit reaches the maximum, it is also the maximum profit for the centralized system, i.e., 164,023.81. Further more, at the point of maximum supply chain’s profit, parties, supplier and distributor, gain profit. As a result, adopting supply chain contract with put and call option in the case of nonlinear option premium price can help to coordinate supply chain. 

Figure 5. Supply chain’s profit at different values of α and β.

4.3 Analysis with Respect to wec and wep Value

 In this section, both wep and wec are varied in order to see the possibility of the proposed option contract to allocate the supply chain’s profit when supply chain coordination is reached. It should be noted that the values of α and β are fixed at 10 and 1.5, respectively, because this is a combination of contract parameters which assists coordination of supply chain.

 The results are presented in Figures 6–8.From Figure 6, it can be seen that when wec increases, distributor’s profit decreases and when wep increases, distributor’s profit increase. This makes sense because distributor will be refunded with wep if he exercises as put whereas he has to pay higher exercise cost if wec increases. In addition, when wec is high and wep is low, the distributor’s profit becomes close to his profit incase of wholesale price contract. This trend is reasonable because high wec and low wep will discourage the distributor to employ option contract.

Figure 6. Distributor’s profit at different values of wec and wep.

 From Figure 7, it can be seen that for all values of wep , supplier’s profit gradually increases before reaching the maximum profit and start decreasing when the value of wec increases. It is no doubt that the increase of wec is beneficial to supplier and, therefore, the supplier’s profit will firstly increase when the value of wec increases. However, when the value of wec is so high, the distributor will decide to reduce the number of purchased options, and hence, the supplier’s profit will be reduced. Related to the effect of value of wep, it is interesting to note that at a fixed value of wec, when the value of wep increases, the supplier’s profit also increases. This trend looks counter-intuitive. However, it is understandable due to the fact that when the value of wep increases, the distributor has motivation to increase the number of purchased options, and hence, this decision will help to improve not only supplier’s profit but also distributor’s profit.

Figure 7. Supplier’s profit at different values of wec and wep.

From Figure 8, it can be seen that higher value of wep contributes higher supply chain’s profit. The increase of supply chain’s profit, when the value of wep increases, is understandable because distributor has motivation to purchase more options, and hence, the possibility of shortage will be reduced. Related to the effect of value of wec , when value of wec increases, supply chain’s profit will firstly increase then reaches a maximum profit before decrease. It should be noted that different values of wep give different the values of maximum supply chain’s profit. For example, at the value 40 of wep, the maximum supply chain’s profit is 160,284.42 at the value 100 of wec whereas at the value 100 of wep , the maximum supply chain’s profit is 161,154.41 at the value 120 of wec. The reason for the increase of supply chain’s profit, when value of wec increases, mainly comes from the increase of supplier’s profit. However, when wec is so high, the distributor will try to limit his purchased options, and hence, it is not good for the whole supply chain. 

Figure 8. Supply chain’s profit at different value of wec and wep.

4.4 Analysis of Profit Sharing

Supply chain is considered to be coordinated if and only if supply chain’s profit is maximized, i.e., the supply chain’s profit achieves the profit of centralized system, whereas individual profit, at least, has to equal the profit in case of wholesale price contract. However, the interesting point is that even if the supply chain is considered to be coordinated, profit sharing from applying the contract plays an important role in making decision of each party in the supply chain. In this section, this issue will be examined. The results are presented in Table 3 and Figure 9, respectively. It is noted that the values of α and β are selected such that the global maximum profit of the whole supply chain is achieved. 

Table 3. Profit sharing to each party in coordinated supply chain (at wec = 180, wep = 60, w = 100)

 In Figure 9, x-axis represents cost of purchasing option which is paid by distributor to supplier and y-axis represents percentage of increase in individual profit in comparison with profit in case of wholesale price contract. The values of wec and wep are set at 180 and 60, respectively. It should be noted that supply chain’s profit in case of centralized system is 164,023.81 which is an increase by 2.52% in comparison with profit in case of wholesale price contract.

Figure 9. Profit sharing to each party in coordinated supply chain.

 From Table 3 and Figure 9, it can be seen that the increase of option cost contributes to higher supplier’s profit whereas distributor’s profit is decreased. This is because supplier gains more from the increase number of purchased options by distributor whereas distributor’s profit behaves in the opposite way.

5. CONCLUSIONS

 This research examines the supply chain contract with bidirectional option in which option premium price is non-linearly correlated with the number of purchased options to see whether it assists the coordinate supply chain. Furthermore, this research also examines the allocation of incremental profit to each party. Through the numerical experiments and analysis section, which examines how the contract’s parameters affect the individual profit as well as the supply chain’s profit, it can be concluded that the supply chain contract with bidirectional option in the case of non-linear option premium price can assist the supply chain to operate as the centralized system, i.e., the supply chain can be coordinated, and incremental profit can be arbitrarily allocated. In addition, this research found that the bidirectional option contract with non-linear option premium mechanism provides higher distributor’s profit in comparison with linear option premium mechanism. This implies higher possibility of the bidirectional option contract to be adopted because, in this case, the distributor is the follower and he is the decision maker on whether the contract will be adopted.

Regarding to the value of wep, there is no doubt that the increase in value of wep will increase not only individual profit but also supply chain’s profit. As a result, the supplier should offer the distributor the high value of wep, this is a win-win scenario. For values of wec, it should be noted that there exists the optimal value of wec for each value of wep, this optimal value should be determined and used in practice. It should also be noted that the values of both α and β should be selected so that the supply chain’s profit is maximized.

 This research assumes the demand is realized at the second stage and has not yet studied the associated risk when applying the supply chain contract with bidirectional option in the case of non-linear option premium price. It should be noted that the expected supply chain’s profit as well as the expected individual profit is evaluated at the beginning of the planning horizon. However, at the second stage, after the demand is realized, the expected profit may be distorted from the first stage. As a result, further research can be conducted by evaluating the associated risk which may happen to both parties under bidirectional option contract in the case of non-linear option premium price.

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