## 1. INTRODUCTION

The double marginalization effect causes firms to derive sub-optimal profit and supply chain profit also does not reach its maximum (Spengler, 1950;de Kok and Graves, 2003). Supply contracts can be considered as a mechanism to eliminate the double marginalization effect and help coordinate a supply chain by motivating all firms in the supply chain to make a decision, such that all firms increase their profit compared to the corresponding profits of a standard wholesale price contract. Applying an appropriate supply contract, all members receive an incremental profit and the profit of the supply chain is maximized, which is the profit of a centralized system. In such a case, it can be said that the supply contract helps coordinate the supply chain (or achieve channel coordination). The appropriate supply contracts are determined by the characteristics of the trade product.

Considering a type of product characterized by demand volatility, a long replenishment lead time, and a short selling season, the buyer prefers more flexibility to modify orders after receiving an updated demand. However, ordinarily, modifying the order is unfavourable to the supplier. Therefore, a supply contract with a bidirectional option is the appropriate mechanism that allows the buyer to modify the order (either increasing or decreasing the initial order) after receiving an updated demand and then compensating the supplier with an option premium price. Such a supply contract is the focus of the current research. It has been proved that this supply contract is appropriate for the above-mentioned product characteristics and not only the supplier but also the buyer gains an incremental profit from adopting this supply contract (Zhao *et al*., 2013). In practice, a supply contract with options exists in real supply chain management applications; for example, Hewlett-Packard used an option contract to procure 35% of its total procurement value (Chen *et al*., 2014). Furthermore, this supply contract could be applied in many industries including with toys, electronics, and the aerospace industry (Wang *et al*., 2019), and agricultural products (Yang *et al*., 2017).

Despite the merits of option contracts, there is little evidence in the literature of addressing the issue of coordination of a supply chain using a bidirectional option supply contract. Zhao *et al*. (2013) showed that this supply contract helps coordinate the supply chain. Similarly, Saithong and Luong (2013) showed that using a bidirectional option supply contract with a non-linear option premium price mechanism, the supply chain can be coordinated. However, it is noted that such research assumed the demand at the beginning of the selling season as a realization. Wang and Tsao (2006) considered unrealized demand; however, they were interested in only the buyer’s perspective, so that supply chain coordination was not the issue of interest. Wan and Chen (2015a) and Wan and Chen (2015b) investigated supply chain coordination using this supply contract by taking the effect of inflation into consideration and a realized demand was also assumed. To the best of our knowledge, there has been no previous investigation into supply chain coordination using the bidirectional option supply contract in the case where demand is unrealized at the beginning of the selling season.

The main research objective was to investigate whether the supply contract with a bidirectional option helps coordinate the supply chain in the case of unrealized demand. In order to achieve this aim, first, the expected profit of both the supplier and distributor at the beginning of the selling season will be analytically formulated. Then, using a backward procedure, the expected profit of the supplier, distributor, and supply chain at the beginning of the planning horizon will be derived and numerical experiments will be carried out to examine the research objective.

The remaining sections are organized as follows. In the next section, existing literature related to the option supply contract will be reviewed. After that, the problem definition will be addressed, followed by mathematical model formulation. Then, numerical experiments will be conducted, with the conclusions presented in the last section.

## 2. LITERATURE REVIEW

A supply contract with options can be classified as having either single directional or bidirectional options (Zhao *et al*., 2016). For single directional option contracts, Barnes-Schuster *et al*. (2002) illustrated how using a supply contract with an option provides flexibility for the buyer to respond to customer demand. Furthermore, some conditions were proposed to help achieve channel coordination. Wang and Liu (2007) studied the coordination of a supply chain when the retailer was a Stackelberg leader and they provided conditions under which channel coordination could be achieved. Zhao *et al*. (2010) examined the channel coordination by considering supply chain members’ risk preferences and negotiating power. Rabbani *et al*. (2015) used the supply contract with an option as the strategy to manage the supply disruption problem. Wang *et al*. (2016) studied the channel coordination comprised of a supplier and two competing retailers. Pan *et al*. (2017) proved that using this supply contract, the supply chain could be coordinated when the members had fairness concerns. Chen *et al*. (2018) investigated the use of an option contract in the face of asymmetric cost information and the results revealed a large effect of asymmetric information on the option contract design.

For a bidirectional option supply contract, Wang and Tsao (2006) studied such a supply contract based on the buyer’s point of view. The main contribution of Wang and Tsao’s (2006) work was the derivation of closed-form expressions for the optimal initial order and option quantities under the conditions of unrealized and uniformly distributed demand. It should be noted that even though the unrealized demand at the beginning of the selling season was assumed, their study did not concern with the issue of supply chain coordination. Zhao *et al*. (2013) also derived a closed-form solution for the optimal retailer’s order strategy and they proved that channel coordination could be achieved by using this supply contract when the demand follows a general distribution. Furthermore, Saithong and Luong (2013) studied the coordination of the supply chain when the cost of purchasing option is non-linear in the number of options. Wan and Chen (2015a) and Wan and Chen (2015b) investigated the coordination of a two-stage supply chain comprised of a supplier and a buyer under the effect of inflation. The conditions under which the channel coordination can be achieved were discussed. Chen *et al*. (2017) illustrated the coordination of a supply chain using this supply contract under a service requirement constraint. Yang *et al*. (2017) illustrated the application of a bidirectional option contract in an agricultural supply chain. Wang *et al*. (2019) investigated the effect of customer returns and the option supply contract on a firm’s optimal decisions. The results showed that this supply contract helps reduce the effect of customers’ returns and enhanced the firms’ profit.

None of the existing literature has investigated bidirectional option supply contract in the coordination of a supply chain where demand is unrealized at the beginning of the selling season. It should be noted that the degree of demand realization could influence the distributor’s costs (both shortage and overage costs). Those costs are realized in the face of realized demand; however, they are unrealized in the presence of unrealized demand. Therefore, it is important to examine whether this supply contract still helps coordinate the supply chain in the face of unrealized demand.

## 3. PROBLEM DESCRIPTION AND MATHEMATICAL MODEL FORMULATION

Before addressing the problem definition, Table 1 below provides the notations used in this research.

The study considers a supply chain consisting of a distributor and a supplier, where the distributor sells one type of merchandise at the unit price of *p* in the face of volatile demand, which is assumed to follow a uniform distribution with a probability density function that can be represented as *f _{X}* (

*x*) = 1 / (2

*n*);

*z*−

*n*≤

*x*≤

*z*+

*n*. Due to the highly uncertain demand problem, the distributor prefers to delay placing the order as late as possible in order to receive an updated demand. On the other hand, the lead time of the supplier is very long and so the supplier must request the distributor to place an order as early as possible. This contradiction of the point-in-time placing of the order places both the distributor and supplier in a difficult situation. Besides, the short selling season means the distributor is unable to place an additional order during the selling season or, in other words, there is only one opportunity for the distributor to place an order. For such a problem, the supplier offers the distributor a bidirectional option contract in order to resolve the difficult situation. In this contract, the distributor agrees to purchase not only the initial order

*Q*

_{0}at the unit cost

*w*, but also a number of options

*q*

_{0}at the unit cost

*w*

_{0}at the beginning of the planning horizon, i.e.,

*t*

_{1}. After receiving the amount for both the initial and option orders, the supplier will produce the number of products

*Q*

_{0}+

*q*

_{0}at the unit cost

*c*. Purchasing the number of options, the distributor is allowed to modify the order upon receiving an updated demand

*X*=

*x*at the beginning of the selling season, i.e.,

*t*

_{2}, with either a downward adjustment where the distributor exercises a put option, or an upward adjustment where the distributor exercises a call option. At the point in time

*t*

_{2}, if the put option is exercised, the distributor receives the number of products

*Q*

_{0}+

*q*

_{e}for which

*q*< 0 and the supplier has to refund the option cost to the distributor at

_{e}*w*per unit option exercised as a put. On the other hand, if the call option is exercised, the distributor receives the number of products

_{ep}*Q*

_{0}+

*q*for which

_{e}*q*> 0 and the distributor has to pay the supplier

_{e}*w*per unit option exercised as a call. It should be noted that the distributor cannot exercise option

_{ec}*q*more than the number of options purchased at the beginning of the planning horizon (

_{e}*t*

_{1}), i.e., -

*q*

_{0}−

*q*≤ +

_{e}*q*

_{0}. The shortage inventory will cost the distributor

*g*per unit of shortage. Both supplier and retailer will receive

*v*per unit of unsold inventory. The actual demand of a selling season after receiving the updated demand

_{b}*X*=

*x*also follows a uniform distribution. However, the demand variation at that point in time is lower than the demand variation the distributor receives at the beginning of the planning horizon. The demand random variable follows a conditional probability density function ${f}_{D|x}(\xi )=1/(2m);\hspace{0.17em}x-m\le \xi \le x+m$.

In order to formulate the profit functions of the distributor, supplier, and supply chain, the timeline will be divided into two points at which the distributor makes decisions and the mathematical expressions of the expected profit functions will be derived according to a backward procedure. Let *π _{i,j}* denote the profit of

*i*at the point in time

*j*;

*i*={

*S*,

*D*,

*SC*} and

*j*= {

*t*

_{1},

*t*

_{2}} when

*S*,

*D*,

*SC*denote the supplier, distributor, and supply chain, respectively. At the beginning of planning horizon

*t*

_{1}, the distributor sees volatile demand, which is assumed to follow a random variable

*X*;

*X*∼

*Uniform*[

*z*−

*n*,

*z*+

*n*] , and places both

*Q*

_{0}and

*q*

_{0}. Then, at the beginning of the selling season

*t*

_{2}, the distributor gets an updated demand

*X*=

*x*and decides whether to exercise the option as a put or call, including the number of exercised options

*q*. The derivation of

_{e}*q*

_{e}^{*}was based on Wang and Tsao (2006) and a draft derivation is shown in Appendix A. The expression of

*q*

_{e}^{*}can be summarized as follows:

Then, substituting ${q}_{e}{}^{*}$ into ${\pi}_{D,{t}_{2}}$ and ${\pi}_{S,{t}_{2}}$, as shown in Appendix A, we can derive ${\pi}_{D,{t}_{2}}$ and ${\pi}_{S,{t}_{2}}$ for different values of updated demand *X* = *x* at *t*_{2}. After that, the expected distributor and supplier profit at *t*_{1}, i.e., ${\pi}_{D,{t}_{1}}$ and ${\pi}_{S,{t}_{1}}$, respectively, can be derived as follows:

A simpler form of ${\pi}_{D,{t}_{1}}$ is shown in Appendix B. Regarding ${\pi}_{S,{t}_{1}}$,

A simpler form of ${\pi}_{S,{t}_{1}}$ is also shown in Appendix B. According to (2) and (3), the supply chain profit at *t*_{1}, $({\pi}_{SC,{t}_{1}})$ can be derived as ${\pi}_{SC,{t}_{1}}={\pi}_{S,{t}_{1}}+{\pi}_{D,{t}_{1}}$. At *t*_{1}, the distributor is supposed to determine the optimal initial order quantity ${Q}_{0}*$ and the optimal number of options ${q}_{0}*$. Therefore, ${Q}_{0}*$ and ${q}_{0}*$ can be determined by solving these two equations simultaneously: $\frac{\partial {\pi}_{D,{t}_{1}}}{\partial {Q}_{0}}=0\hspace{0.17em}\text{and}\hspace{0.17em}\frac{\partial {\pi}_{D,{t}_{1}}}{\partial {q}_{0}}=0$. Undoubtedly, these expressions are too complicated to analyze; so, in order to examine whether this supply contract helps achieve channel coordination in the case of unrealized demand, as the main objective of this research, numerical experiments will be carried out.

## 4. NUMERICAL EXPERIMENTS

Before conducting the numerical experiments, the following conditions are assumed to avoid trivial cases (Wang and Tsao, 2006;Saithong and Luong, 2013): *w*_{0} ≤ *w*, *w _{ep}* ≤

*w*≤

*w*,

_{ec}*p*+

*g*≥

*w*+

_{ec}*w*

_{0},

*v*+

_{b}*w*

_{0}≤

*w*,

_{ep}*v*≤

_{b}*c*. Then, the following values of parameters are referred to as the base case (Saithong and Luong, 2013):

*p*= 200,

*w*= 100,

*c*= 35,

*g*= 40,

*v*= 30,

_{b}*n*= 200,

*z*= 1000,

*w*

_{0}= 10,

*m*= 100,

*w*= 180,

_{ec}*w*= 60. Referring to Saithong and Luong (2013), the distributor, supplier, and supply chain profit in the cases of both wholesale price and a centralized system are shown in Table 2. The formula employed to compute the optimal order quantity in both cases can be found in the existing literature.

_{ep}As noted earlier, the channel coordination is achieved if the supply chain profit equals the profit of a centralized system while the expected profit of all members increases compared to the corresponding members’ profit in a wholesale price contract. Therefore, the information in Table 2 implies the conditions under which the supply chain coordination is achieved. More precisely, in order to conclude that the supply chain is coordinated, the supply chain profit must achieve $164,023 while the distributor and supplier must have a profit not less than $90,667 and $69,333, respectively.

First, we will examine the case when demand is realized at *t*_{2}. It should be noted that the degree of demand realization directly relates to the parameter *m*. The degree to which *m* approaches zero represents the degree to which the demand is realized at *t*_{2}. In this research, we assume 0.00001 of the value of *m* as a realized demand case at *t*_{2}. Table 3, Table 4, and Table 5 illustrate the profit of the supply chain, distributor, and supplier, respectively, in the case of realized demand.

The optimal number of initial orders and the optimal number of options in this case can be found in Appendix C. According to Table 3, Table 4, and Table 5, there exists a set of values of *w _{ec}* and

*w*that helps achieve the centralized supply chain profit while all members gain an incremental profit from adopting this supply contract, e.g.,

_{ep}*w*= 200 and

_{ec}*w*= 58. Therefore, we can conclude that the supply contract helps achieve channel coordination. This conclusion is drawn based on the assumption of realized demand at the beginning of the selling season and this conclusion is consistent with existing literature dealing with the coordination of the supply chain using a bidirectional option supply contract.

_{ep}Next, we will examine the effect of unrealized demand at the beginning of the selling season on supply chain coordination. As there exists variation in the actual demand at *t*_{2}, it is assumed that *m* =100 while other values of parameters are kept intact. It should be noted that the expected value of the updated demand at *t*_{2} is exactly the value of the realized demand, as it was realized in the previous case. Table 6, Table 7, and Table 8 illustrate the profit of the supply chain, distributor, and supplier, respectively, when *m* =100.

Table 6 illustrates that, most of the cases, the increase in value of *w _{ep}* causes the marginal improvement of supply chain profit before the decrease. The reason of the increase in the supply chain profit is the increase in distributor profit, as seen in Table 7. On the other hand, the decrease in supply chain profit is a result of the decrease in supplier profit, as seen in Table 8. It can be noticed that, in a special case (very high value of

*w*and very low value of

_{ec}*w*, e.g.,

_{ep}*w*= 200 and

_{ec}*w*= 40), the distributor is discouraged from purchasing option and, therefore, the profit of both members approaches to the profit of a wholesale price contract. Values for

_{ep}*Q*

_{0}

^{*}and

*q*

_{0}

^{*}for this case are shown in Appendix C.

According to the results, adopting the bidirectional option contract for the case of unrealized demand fails to achieve the maximum supply chain profit. Therefore, regardless of the profit levels of supply chain members, it can be concluded that the supply chain cannot be coordinated if the demand is unrealized at the beginning of the selling season. Additionally, various values of parameters can be examined in Appendix D, and the results are consistent for this case. In summary, the supply contract with a bidirectional option helps coordinate the supply chain when the demand is realized at the beginning of the selling season only. In the case of unrealized demand, this supply contract cannot help the supply chain achieve maximum profit and, therefore, the supply chain is not coordinated.

## 5. CONCLUSIONS

In this research, we investigate the coordination of a simple two-stage supply chain for a single type of product characterized as having volatile demand, a long replenishment lead time, and a short selling season. According to the product characteristics, a supply contract with a bidirectional option is appropriate to help achieve channel coordination. This supply contract provides flexibility for the distributor to respond to customers’ demand while compensating the supplier with an option premium price. Adopting this supply contract helps coordinate supply chain; however, while this claim has been made in the existing literature, these other works assumed the customers’ demand is realized at the beginning of the selling season. The current research investigated supply chain coordination where demand is unrealized at the beginning of the selling season. Assuming the customers’ demand as a uniform distribution, then applying the bidirectional option supply contract cannot help coordinate the supply chain if the customers’ demand is unrealized at the beginning of the selling season. In another case, using the same values for the parameters as for the unrealized demand case, this supply contract helped coordinate the supply chain if the customers’ demand is realized at the beginning of the selling season. These findings should caution the members in a supply chain against adopting such a contract if there is still demand variability at the beginning of the selling season. Furthermore, assuming an unrealized demand as realized demand (by using the expected value of the demand at the beginning of the selling season) leads to incorrect conclusions regarding supply chain coordination.

For further research, the assumption of uniformly distributed demand could be relaxed and instead a general distribution could be considered. This would help confirm the effect of unrealized demand on supply chain coordination.

## APPENDICES

### Appendix A. Derivation of ${q}_{e}*$

Wang and Tsao (2006) derived ${q}_{e}*$ and a brief derivation is provided for readability purpose. The decision regarding the optimal exercised option depends on the updated demand *x* received at the beginning of selling season *t*_{2}. For a given value of *x*, the expected distributor and supplier profit functions at *t*_{2}, i.e., ${\pi}_{D,{t}_{2}}$ and ${\pi}_{S,{t}_{2}}$ respectively, when the distributor exercises option as put, are:

On the other hand, the expected distributor and supplier profit functions for a given value of *x*, when the distributor exercises the option as call, are:

Then, the optimal exercised option, ${q}_{e}{}^{*}$, can be determined from $\frac{\partial {\pi}_{D,{t}_{2}}}{\partial {q}_{e}}=0$. In the case where the distributor exercises the option as a put, taking the derivative of (A-1) with respect to *q _{e}*, we have

Applying the Leibniz integral rule (Woods, 1926), we have

After some mathematical operations,

In the above case for ${q}_{e}*$, it should be noted that ${q}_{e}*=x-{Q}_{0}+A<0\hspace{0.17em}\text{and}\hspace{0.17em}{q}_{e}*=x-{Q}_{0}+A>-{q}_{0}$. Thus, ${q}_{e}*$ is applicable only when ${Q}_{0}-{q}_{0}-A<x<{Q}_{0}-A$. If $x\le {Q}_{0}-{q}_{0}-A$, the distributor has to exercise all options as put, i.e., ${q}_{e}*=-{q}_{0}$. Also, ${q}_{e}*=-{q}_{0}$ if $x+m<{Q}_{0}-{q}_{0}$.

Regarding ${q}_{e}*$ in the case of exercising option as a call, taking the derivative of (A-3) with respect to *q _{e}*, ${q}_{e}*$ can be determined in the same way.

The above ${q}_{e}*$ is applicable only when ${q}_{e}{}^{*}=x-{Q}_{0}+B>0$ and ${q}_{e}{}^{*}=x-{Q}_{0}+B<{q}_{0}$, i.e., ${Q}_{0}-B<x<{Q}_{0}+{q}_{0}-B$. If $x\ge {Q}_{0}+{q}_{0}-B$, the distributor has to exercise all options as a call, i.e., ${q}_{e}*=+{q}_{0}$. Also, ${q}_{e}*=+{q}_{0}$ if $x-m>{Q}_{0}+{q}_{0}$.

Lastly, if ${Q}_{0}-A\le x\le {Q}_{0}-B$, ${q}_{e}*=0$. Therefore, ${q}_{e}{}^{*}$ can be summarized as shown in (1).

The above ${q}_{e}*$ is applicable only when ${q}_{e}{}^{*}=x-{Q}_{0}+B>0$ and ${q}_{e}{}^{*}=x-{Q}_{0}+B<{q}_{0}$, i.e., ${Q}_{0}-B<x<{Q}_{0}+{q}_{0}-B$. If $x\ge {Q}_{0}+{q}_{0}-B$, the distributor has to exercise all options as a call, i.e., ${q}_{e}*=+{q}_{0}$. Also, ${q}_{e}*=+{q}_{0}$ if $x-m>{Q}_{0}+{q}_{0}$.

Lastly, if ${Q}_{0}-A\le x\le {Q}_{0}-B$, ${q}_{e}*=0$. Therefore, ${q}_{e}{}^{*}$ can be summarized as shown in (1).

### Appendix B. A simpler form of ${\pi}_{D,{t}_{1}}$ and ${\pi}_{S,{t}_{1}}$

A simpler form of ${\pi}_{D,{t}_{1}}$ is as follows.

Regarding ${\pi}_{S,{t}_{1}}$,