A New Approach in Determining Lot Size in Supply Chain Using Game Theory

Several seller-buyer supply chain models are suggested which emphasis simultaneously on production andmarket demand. In thesemodels lot size is obtained based on different approaches. In this paper we present a novel approach to determine lot size in a seller-buyer supply chain. There is an interaction between the seller and the buyer, since the seller prefers large production lot sizes and the buyer likes small ones. Therefore for determining lot size, the seller and buyer’s power is illustrated in the models.We consider two strategies for each situation (Seller-Stackelberg, BuyerStackelberg) whether the seller or the buyer as a leader, has more power. The leader can justify or enforce the strategy about the lot size to the follower, or let the follower determine their own lot size. Based on our findings we propose the optimal strategies for each situation. Each strategy’s result will be compared by numerical examples presented. In addition, sensitivity analysis of some key parameters in the models and further research are presented.


Introduction
Seller-buyer supply chain models concern production and market demand decisions simultaneously [7,17,18]. The optimal solution of models are obtained by maximizing their profit under cooperative or non-cooperative seller and buyer's relation. In these models, the term retailer has been used correspond to the buyer. Similarly, the words manufacturer, vendor and supplier have been used interchangeably correspond to the seller. In this paper, words of the seller and the buyer have also been used according to the mentioned classification. The seller-buyer models can be observed from different views: certain or uncertain market demand and various coordination. We briefly summarize these models in order to compare the model with the new approach.
Market demand plays an essential role in the seller and buyer's profit. In other words, if the structure of demand is changed the seller and buyer's optimal policy under constant, variable, stochastic or uncertain demand will also change. Chan and Kingsman [2], van den Heuvel et al. [16], Dai and Qi [15] present the sellerbuyer models with a constant demand. The optimal order (production) cycles and lot size(order quantity) are determined by maximizing the whole supply chain profit in a cooperative structure. However, Yue et al. [20], Sajadieh and Jokar [13] consider some factors such as price or marketing expenditure which influence on market demand in their models. On the other hand, certain demand is avoided in some research. The optimal selling price and lot size are obtained in the models under market demand uncertainty [19].
Moreover, various types of coordination have been discussed in the literature on supply chain [10]. For instance sharing advertising cost is investigated in a supply chain includes a manufacturer and a retailer [14]. The demand is influenced by price and advertisement. Their model is developed by Giri and Sharma [8] with competing the retailers. The centralized and decentralized supply channel are considered by choosing different pricing strategies in the presence of consumers' referenceprice effects [21]. However the multiple manufacturers and a common retailer are considered in a supply chain facing uncertain demand [9].
By reviewing recent publishes, the papers have covered all or some drawbacks of previous works. For example, Yugang et al. [18], introduce models in which the demand is a function of both price and marketing; also, not to assume that lot size is the same as the demand. However, the main question in the supply chain is who determines the lot size or order quantity which has been ignored by previous models. The optimal lot size in these models is determined based on their assumption which specifies who determines the lot size for the whole supply chain, the seller or the buyer, regardless to their power. Therefore, one of the participants determines the lot size while the other one has market power, which doesn't make sense. While the seller prefers large production lot sizes, the buyer likes small ones [11]. In this paper, we apply a novel approach to determine lot size in the seller-buyer model based on the seller or buyer's power. The seller as a manufacturer produces a product and wholesales it to the buyer, who then retails the product to the end consumer. The production rate of the seller is assumed to be linearly related to the market demand rate and demand is sensitive to selling price and marketing expenditure both charged by the buyer. The lot size is determined by either seller or the buyer based on their power. We consider two situations, when the seller dominates the buyer(Seller-Stackelberg), and also when the power has shifted from the seller to the buyer(Buyer-Stackelberg). In the Seller-Stackelberg(Buyer-Stackelberg) model, the seller(buyer) faces two strategies, either determines the lot size for the whole supply chain or lets the buyer(seller) determine their own lot size. The optimal solution of each strategy is obtained and it is shown that in each situation, which strategy for the seller and buyer is the best decision. Numerical examples presented in this paper, including sensitivity analysis of some key parameters, will compare the results between the models considered.
The remainder of this paper is organized as follows. The notation and assumptions are given in Sect. 2. In Sect. 3, the new approach in the seller and buyer's model is presented and compared all situations via optimal solution. In Sect. 4, some computational results, including a number of numerical examples and their sensitivity analysis that compares the results between different models are presented. Finally, the paper concludes in Sect. 5 with some suggestions for future work.

Notation and Problem Formulation
This section introduces the notation and formulation used in this paper. The demand is assumed to be a function of P and M as follows: (1)

• Assumptions
The proposed models are based on the following assumptions: 1. Planning horizon is infinite. 2. Parameters are deterministic and known in advance.
3. The annual demand depends on the selling price and marketing expenditure according to (1) (see [4]). 4. Shortages are not permitted.

Stackelberg Games by Considering Different Lot Sizing Approaches
In the literature, the relationships between the seller and buyer are modeled in the supply chain, by Stackelberg games where the seller and buyer take turn as a leader and follower [6]. In the previous models, the lot size is determined based on the assumption which specifies whether the seller or the buyer determines lot size. Therefore, the follower determines the lot size for the whole supply chain while the leader has market power. This doesn't make sense. In this section, we consider different approaches in seller-buyer models: first we consider the seller as the leader (Seller-Stackelberg).
Regarding to the seller's role, the seller can justify or enforce the strategy about the lot size to the follower (the buyer), or let the follower determine their own lot size. We also consider the same role for the buyer when the buyer is a leader and the seller is a follower (Buyer-Stackelberg). The optimal policy of seller and buyer is obtained in each strategy in sequence as follows:

Seller-Stackelberg
The Seller-Stackelberg model has the seller as leader and the buyer as follower is widely reported in the literature as the conventional form [1,3]. Therefore the seller can enforce the optimal lot size to the buyer, or can let the buyer determine their own lot size. We illustrate each strategy in order as follows:

Optimal Policy of Seller and Buyer Under Seller's Lot Size
Regarding the seller's power, for a given Q s = Q and V of the seller, according to the following model, the buyer obtains the best marketing expenditure M * and selling price P * , Eqs. (4) and (5) [4]: Then, the seller maximizes her profit Π s 1 (V , Q) based on the obtained P * and M * from the buyer's model. Thus, the problem reduces to Substituting the constraints into the objective function, the problem transforms into an unconstrained nonlinear function of two variables V and Q, where the optimal solution can be found using a grid search.

Optimal Policy of Seller and Buyer Under Seller's Lot Size and Buyer's Lot Size
For a given V of the seller, according to the following model, the buyer obtains the best marketing expenditure M * , selling price P * and the buyer's lot size Q * b , Eqs. (8)-(11) [5].
The seller then maximizes their profit Π s 2 (V , Q); Q = Q s based on the pair P * , M * and Q * b . Thus, the problem reduces to By considering Eqs. (2), (8) and (9), (10) and (11) could be changed to: Therefore, the seller's model would be: By obtaining D from the constraint and substituting in the objective function the model transforms into an unconstrained nonlinear function of four variables V , Q, P and M where the optimal solution can be found using a grid search.

Buyer-Stackelberg
In the last two decades, the buyer has increased their power relative to the seller's power [12]. Therefore, the buyer acts as the leader and the seller act as follower that is called Buyer-Stackelberg Model. For example, Wal-Mart effectively uses its power to get reduced prices from its sellers [20]. Since the buyer has more power, the lot size could be determined by the buyer or the buyer could let the seller determine their own lot size. Each situation is investigated in sequence as follows:

Optimal Policy of Seller and Buyer Under Buyer's Lot Size
For a given Q b = Q, P and M of the buyer, according to the following model, the seller obtains the best price V * , Eq. (15) [4].
The buyer then maximizes their profit Π b 3 (P, M, Q) based on V * . Thus, the problem reduces to Substituting the constraint into objective function, the above Buyer-Stackelberg problem reduces to optimizing an unconstrained nonlinear objective function. The optimal solution can again be found using a grid search.

Optimal Policy of Seller and Buyer Under Buyer's Lot Size and Seller's Lot Size
For a given P, M and Q b = Q of the buyer, according to the following model, the seller obtains the best price V * and Q s , Eqs. (18) and (19) [5].
The buyer then maximizes the profit Π b 4 (P, M, Q b ) based on V * . Thus, the problem reduces to Substituting the constraints into objective function, the above problem reduces to optimizing an unconstrained nonlinear objective function of four variables P, M, Q and V . The optimal solution can again be found using a grid search.

Computational Results
In this section, we present numerical examples which are aimed at illustrating some significant features of the models established in previous sections. We will also perform sensitivity analysis of the main parameters of these models. We note that  The corresponding seller's and buyer's profits are Π * s = 1014.7 and Π * b = 3196.7 respectively. The second model, in contrast to the first one, utilizes less marketing expenditure, and has smaller seller's price and selling price charged by the buyer. Therefore, demand of the second model is higher than the first one. As expected from the theory in Sect. 3, the seller's profit here is less than in Example 1, although the demand is high. The reason would be the seller's role in determining a larger lot size for the whole supply chain. When the seller has more power the buyer's profit is less in the first model opposite to the seller's profit. In fact it makes sense that the buyer uses the freedom to choose lower lot size in the second model.

Example 3
The Buyer-Stackelberg model when the buyer determines the lot size for the whole supply chain, produces the following optimal values for the decision variables: D * = 1387.8, P * = 6.5, Q * = 393.1, M * = 0.6 and V * = 2.1. The seller's and buyer's profits are Π * s = 586.5 and Π * b = 4960.7 respectively. By comparing Examples 1, 2 and 3 we find out that the profit of the buyer in the buyer-Stackelberg is more than both models in the Seller-Stackelberg opposite to the seller's profit.

Example 4
The Buyer-Stackelberg model when the seller and buyer each one determines the lot size for themselves, produces the following optimal values for the decision variables: D * = 1387.9, P * = 6.5, Q * s = 420.8, Q * b = 362.5M * = 0.6 and V * = 2.1. The seller's and buyer's profits are Π * s = 586.4 and Π * b = 4962.5 respectively. As expected from the theorem in Sect. 3 the buyer's profit when the buyer enforces the optimal lot size to the seller is less than the buyer lets the seller determines their own lot size in the Buyer-Stackelberg. When the seller has more power, the seller would rather determine the lot size for the whole supply chain although the buyer would like determine the lot size. In addition, when the power shifts from the seller to the buyer, the buyer would like to let the seller determine the lot size, while the seller does not like such a strategy.

Sensitivity Analysis
We investigate the effects of parameters A s , A b , i, and C s on P * , V * , Q * , M * , D * , Π * b and Π * s in scenarios of the Seller-Stackelberg and Buyer-Stackelberg models through a sensitivity analysis. We will fix k = 36080, α = 1.7, β = 0.15, and, u = 1.1 as in the previous examples but allow A b , A S , C s , and i to vary. Results of these sensitivity analysis are summarized in Tables 1, 2 , 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16.  The results in Tables 1, 2, 3, 4, 5, 6, 7 and 8 are also graphically displayed in Fig. 1. In each curve, the numbers 1, 2, 3 and 4 refer to the first, second scenarios in the Seller-Stackelberg and the third, forth in the Buyer-Stackelberg respectively.    Since i and C s and also A s and A b have the same effect on P * , V * , Q * , M * , D * , Π * b and Π * s , we show only the effect of i and A b in sequence on Figs. 1 and 2. As is seen in the figure, by increasing i and C s , selling price, marketing expenditure and seller's price increase which cause the decrease of the demand, seller and the buyer's profits. Note due to increasing production cost and percent inventory holding cost, i and C s , the holding cost will increase. Therefore, the seller and buyer would change their strategies to decrease lot size, which validates our model.
The effect of parameter A s , for each scenario of the Seller-Stackelberg and the Buyer-Stackelberg are graphically displayed in Fig. 2. As A b and A s increase, P * , M * and V * increase. By increasing A b and A s the seller and the buyer increase their lot size to decrease the ordering and the seller's setup cost respectively. As shown in Figs. 1 and 2 the results validate the models.

Conclusion
In this paper, the lot size problem in the seller-buyer supply chain is considered. The lot size in the supply chain is determined based on the seller and the buyer's power. The seller produces a product and wholesales it to the buyer, with the production rate linearly related to the market demand rate. The demand is sensitive to the selling price and the buyer's effort in marketing. We consider the seller-buyer relationship under Stackelberg games: Seller-Stackelberg, where the seller is the leader, and Buyer-Stackelberg, where the buyer is the leader. Regarding to the leader's role, the leader can justify or enforce the strategy about the lot size to the follower(the buyer), or let the follower determines their own lot size regardless to the leader's lot size. Optimal solution for each models is obtained. It is shown that when the seller has power, it would be better if the seller determines the lot size for the whole supply chain and when the buyer has power, it would be better if the buyer let the seller determines their own lot size. Numerical examples are presented which aim at illustrating the used approach. Through a sensitivity analysis, the effect of the main parameters of the model on the seller and the buyer's decisions are also investigated which illustrate our findings.
There is much to be extended the present work. For example, marketing expenditure could incorporate advertising expenditure and the seller may agree to share fraction of the advertising expenditure with the buyer by covering part of it. In this case, we could investigate cooperative games. Also, we have assumed in this paper that the production rate is greater than or equal to the demand rate, in order to avoid having to consider shortage cost. By not making this assumption, the extra cost could be incorporated into future models. Finally, even though the seller or buyer presumably knows their own costs and price charged to the consumers, it is unlikely that their opponent would be privy to such information. This would lead to incomplete knowledge on the part of the two participants and result in bargaining models with incomplete information along the line of models.