Tactical production planning in a hybrid MTS/MTO system using Stackelberg game

In this paper, a pricing and lead time determination problem, is investigated in a hybrid MTS/MTO production environment using Stackelberg game. A manufacturing company with three types of products including MTS, MTS/MTO and MTO and a manufacturing system consists of one common and one differentiation stage is considered. Due to the conflict of interests between these two stages, the decision making sequence is modelled using a Stackelberg game where the differentiation and the common stages are considered to be leader and follower, respectively. The company aims to satisfy the demand of MTS products along with on time delivery of MTS/MTO and MTO orders. The optimal order price and delivery time are suggested to be obtained from a profit maximization model, where the market demand is affected by these two values. Finally, a simulation model is conducted to examine the effectiveness of the proposed model and also to implement a sensitivity analysis on the results.

average response time and average order delay and also having an effective capacity plan are of great concern (Cakravastia and Nakamura 2002). In contrary to MTS systems, an MTO one is completely based on customer desires which means that, in a MTO system, the production process of a specific item is not started, unless a specific order, with clearly defined customization details, received. MTO systems have been largely used in many firms including Dell, McDonald's and etc., with the purpose of enhancing customers' ability to apply their desired characteristics in products' design and production process (Hendry and Kingsman 1989;Holweg and Pil 2001;Dobson and Yano 2002). Despite the strengths of an MTO system, its long delivery time usually decreases customer's utility and leads to a considerable market loss for the company. In order to cope with this drawback, hybrid MTS/MTO systems have been recently investigated to take advantage of both MTS and MTO systems (Yingdong 2001;Soman et al. 2004).
In hybrid MTS/MTO systems the production line is separated into two distinct stages; the common and the differentiation. During the common stage, all products are made to stock through the same process and in a certain period of time. Then the semifinished products, from the common stage, goes through the customization process according to each order's specifications, during the differentiation stage. These two stages are separated from each other in a point called the Customer Order Decoupling Point (CODP). CODP is where the customers' desired specifications enter the production process (Lee and Tang 1997;Olhager 2003;Wikner and Rudberg 2005) (Fig. 1). Due to the complexity of hybrid production systems, Hax and Meal's Hierarchical Production Planning (HPP) structure (Hax and Meal 1975) is applied to MTS/MTO systems (Soman et al. 2004). This HPP consists of three decision making levels including strategic, tactical and operational. In the strategic level, product family formation and determination of CODP for each product family are addressed (Van Donk 2001;Zaerpour et al. 2009;Hemmati and Rabbani 2010;Rafiei and Rabbani 2011;Rabbani et al. 2014), while in the operational level, the most detailed decisions including production scheduling and order sequencing are made (Soman et al. 2006). This paper concentrates on the tactical level decisions which include balancing demand and capacity, lot size determination for MTS products, and also lead time and price determination for both MTO and MTS/MTO orders as two of the most important features affecting an order's acceptance or rejection, by the customer (Wester et al. 1992;So and Song 1998). In the following, different tactical decisions in the literature have been presented.
In 1993, Hendry and Kingsman (1993) addressed an input-output control method in order to accept or reject the new arriving orders. Easton and Moodie (1999) Lin and Chang (2008) suggested a method for order selection and pricing process in an MTO system with limited production capacity. Ebadian et al. (2008) also proposed a five step procedure to tackle the order selection problem in a MTO system. The first two steps of their model are devoted to the process of rejecting undesirable orders and making optimal decisions for the others. Then a linear programming model is presented to determine the price and delivery date of each order. The fourth step is dedicated to customers' decision of acceptance or rejection the announced price and delivery time and at last, another programming model is presented to determine the suppliers and subcontractors for the accepted orders. Kalantari et al. (2011), presented another five step decision support system for order acceptance or rejection in a hybrid production system, where its first step dedicates to customer prioritizing. In the second step, some undesirable orders are rejected according to the available production capacity. In the next step, a mixed-integer programming model is proposed to determine the price and due date of each non rejected order. At the forth step, some instructions are provided related to the organization's negotiation with customers on prices and due dates. Finally, the fifth step addressed the final decision of an order's acceptance or rejection. Rafiei and Rabbani (2012) addressed capacity coordination in a hybrid MTS/MTO system with three types of products including MTS, MTO and MTS/ MTO. They also presented a mathematical programming model to cope with MTS lot sizing problem. Hemmati et al. (2012) proposed a decision making structure for acceptance or rejection of incoming orders in a MTO system. First, they prioritized orders using TOPSIS method and then, they decided about an order's acceptance or rejection based on the rough cut capacity calculation of each order. Manavizadeh et al. (2013), presented a decision support system for order acceptance or rejection in a hybrid production environment, in which, the announced price and due date of each incoming order is calculated via a mathematical programming model. They also consider the bargaining power of the customers. In another research, Ghalehkhondabi et al. (2017), simultaneously addressed the CODP positioning and the pricing problem for each product type, without taking the due dates into account.
To the best of our knowledge, all of the mentioned researches that addressed the price and due date determination problem in a hybrid production system (Ebadian et al. 2008;Kalantari et al. 2011;Manavizadeh et al. 2013) have obtained the optimal price of each order by adding a constant value to the final manufacturing cost, using a mark-up approach, resulted from minimizing the total cost function. In this regard, obtaining price and lead-time through profit maximization and investigating the impact of the determined price and due date on demand is neglected so far. This issue is worth considering because, these two variables, i.e. price and due date, have a considerable effect on a product's demand. Establishing a balance between these two variables not only raises the probability of a suggested set of price and due date to be accepted by the customer, but also guarantees an order's on time production and delivery, according to the company's cost and capacity constraints. Therefore, this paper investigates the problem of determining price and lead time for each incoming order by maximizing the company's profit, taking into account the impact of price and lead-time on demand. It is considered that the probability of an incoming order to be accepted by the customer depends on the announced price and delivery time by the company. The company produces three types of products including MTS, MTS/MTO and MTO using a production system with one common and one differentiation stages. In the common stage, higher fill rate, satisfying the MTS demand and holding the lowest amount of WIP is of concern, while the differentiation stage is interested in satisfying MTS/MTO and MTO orders in their promised due date with minimum amount of delayed orders. When an order enters the company, the common stage declares the order's specifications to the differentiation stage. Next, the differentiation stage announces the customization time and price to the common stage. Then, Final price and due date which is calculated by the common stage will be announced to the customer. The decision made in the differentiation stage, will considerably affect the common stage and the whole company's profit. So, in order to maximize its profit, the differentiation stage would make his decision based on an estimate of the common stage's behavior. Besides, it is obvious that announcing a high price or a long due date by each of these stages would result in a lower percentage of order acceptance by the customer and ultimately would decrease the company's profit. While these effects are mutual and conflicting, a Stacklberg game is suggested to cope with them.
The rest of paper is organized as follows: In Sect. 2, a general framework of the proposed model is presented and the model and its solution are explained in detail. Section 3 dedicates to the simulation process and its obtained numerical results. At last, the conclusion remarks and some future research directions is presented.

The proposed model
In this paper, the order price and delivery time determination problem along with the issue of order acceptance-rejection is tackled. The model considers three types of products including MTS, MTS/MTO and MTO. The considered production system consists of two stages including common and differentiation, where the boundary between these two, is known as a priori. The common and the differentiation part of the production line are managed separately and the relationship between these distinct parts, can be modeled using a Stakelberg game. In a hybrid production system, the MTS part does a routine in a nearly fixed duration with certain amount of resources, but the MTO part's specifications vary according to the order's characteristics. Therefore, the differentiation stage is the point that creates distinction between orders' final prices and due dates. This fact makes the differentiation stage more important and affective than the common stage. So, in this paper, the differentiation stage is considered as leader and the common stage, as follower. This allocation is also due to the fact that, the differentiation stage announces the customization time and price before the common stage do the same about the whole product.
Besides, it is considered that the common stage not only produces generic products but also takes the responsibility of customer side services, such as receiving orders, announcing the determined price and due date of orders and delivering the order to the customers.
Regarding the production constraints, when a specific order enters the system, a rough-cut capacity calculation is done by the common stage, in order to investigate each order's feasibility for the company, using the following inequality.
r 2 1; . . .; R f g , index of production stations; CODP, the customer order decoupling station; g RPT ri , the required processing time for order i in station r; n i , number of products in order i; NQ r , number of waiting MTS/MTO or MTO products to be processed in the rth station; C r , the maximum available capacity of station r; RC MTS , the required capacity for responding MTS demand; RC MTO , the reserved capacity for future incoming MTS/MTO or MTO orders; a i , the priority of order i which varies from 0 (for low priority orders) to 1 (for high priority orders) If the company accepts the incoming order, the customer's specifications will be declared to the differentiation stage who determines the customization price and lead-time. Then the common stage will determine final price and due date based on the price and lead-time of differentiation stage. Consequently, the customer makes the final decision whether to assign this order to the company or not.
The probability of an order to be accepted by the customer depends on the company's suggestion to be satisfying or not. In fact, the company's demand is affected by the announced price and due date, as follows (Easton and Moodie 1999): TM i , the announced delivery time of order i to the customer; PM i , the final price of order i; W i , the estimated work content of order i; k, the cost rate per unit of work content For MTS/MTO orders, the production process in common stage has a constant duration, therefore we have, TM i = TC i ? C, where TC i is the time of customization in differentiation stage and C is a constant. In case of full MTO orders, CODP is located in the beginning of production line and the production process begins form the first station according to the customers' desired characteristics. In this case, C equals to zero and the production time equals the differentiation time. Figure 2, shows a comprehensive scheme of the proposed structure.
Due to conflict of interests between the two parts of the production line, the decision making sequence of this paper is modeled using a Stackelberg game where the differentiation stage is leader and the common stage is follower. In more details, when an order is detected as a feasible one, at first, the differentiation stage determines the customization price and lead-time as the leader and then the common stages follows it, by providing the final price and lead time as the follower. As a summery, the sequence of calculating the decision variables of the paper can be shown as represented in Fig. 3. So, the profit maximization model for both parts of the production line, is as follows: And Z C , the expected total profit of the common stage; Z D , the expected total profit of the differentiation stage; PC i , customization process price (declared by the differentiation stage); g i , the penalty cost for each unit time of tardiness in delivery of order i that the common stage pays to the customer; CM i , production cost of order i for the common stage; CC i , the customization cost of order i for the differentiation stage; c i , the penalty cost that the differentiation stage pays to the common stage for each unit time of tardiness in delivery of order i; tc i , the actual time of customization process of order i by the differentiation stage, f(t ci ) is its density function and F(t ci ) is its distribution function; TC i , the announced delivery time from the differentiation stage to the common stage; P i , the probability of order i being accepted by the customer Fig. 3 The sequence of generating decision variables Tactical production planning in a hybrid MTS/MTO system… 1797 To obtain SPE, the backward induction method is used. First the follower variable, PM i , is calculated by putting the first derivative of Z C equal to zero, as presented in (6). The detailed calculation are presented in Appendix 1. Besides, by taking the second derivative of Z C with respect to PM i the concavity of the objective function is proved in Appendix 2. TM i is also obtained from TM i = TC i ? C, where TC i will be calculated in the next step and C is the constant time of production in common stage.
To calculate the leader variables, PC i and TC i , (4) and (5) are rewritten as (7) by adding the Lagrange multiplier. Therefore, the optimal values are presented in (8) and (9) respectively, where the detailed calculations are presented in Appendix 3.
Having these calculations, a new customer will be provided with an appropriate price and lead-time that leads to the highest possible benefit for both the common and the differentiation parts, simultaneously. In the next section, the effect of our proposed model's adoption is investigated through simulating the understudy company.

Simulation
In this section, the applicability of the suggested structure is investigated using the data of a real industrial case study. The data correspond to Irankhodro which is a car manufacturing company with a production process similar to the proposed model's characteristics, previously mentioned in Sect. 2. The company produces three types of products including MTS, MTS/MTO and MTO. The common stage not only produces generic semi-finished products but also takes the responsibility of customer side services, such as receiving orders, announcing the determined price and due date of orders and delivering the order to the customers. The differentiation stage is responsible for making customized changes to the generic semi-finished product based on the customer desires. Each of the two mentioned stages consist of some manufacturing stations. The company's ideal not only is to on time delivery of its incoming MTS\MTO and MTO orders, but also is to meet the demand of its MTS products.
Due to the restrictions of implementing our suggested approach in this real case, and in order to tackle the uncertainties in demand, order entries and processing times a simulation model is proposed. This model provides an appropriate comparison between the situation before and after model implementation. Formerly, the understudy company used to calculate the final price and due date of its incoming orders in a different way from the suggested model. In the old method, each stage, used to separately calculate its desired price by maximizing its profit and the final price used to come from the sum of these two values. The optimal prices of the common (PG i ) and the differentiation stage (PC i ), in that method used to be obtained from (10) and (11) respectively. Since, the most important criterion for the company is to have the highest percentage of on time delivered orders and on the other hand, minimizing the common stage's WIP that reduces the holding cost is critical for the company, the management team decided to take advantage of a more efficient pricing policy in a competitive environment. Besides, the company used to accept all the incoming orders regardless of its remained capacity, so the rough cut capacity calculation is also suggested to improve company's responsiveness.
The company produces two types of MTS products, two types of MTS\MTO products and two types of MTO products. For each order type the customization process is done by the differentiation stage which takes the generic semi-finished products from the common stage and finishes the production process according to customers' desires. Considering that the time between order arrivals follows an exponential distribution with parameters 1.00, 1.20, 0.8 and 0.7 for MTS/MTO1, MTS/MTO2, MTO1 and MTO2 product families respectively. The obtained results from the Arena10 Rockwell software based on the Stackelberg game and the proposed simulation model are shown in Tables 1, 2, 3 and 4 and Fig. 4, respectively. The simulation period is assumed to be 1 year. CODP for MTS/MTO1 and MTS/MTO2 product families are located in stations 4 and 6, respectively. Therefore, MTS/MTO1 family is produced through 3 stations in the common stage and 4 stations in the differentiation stage. Similarly, MTS/MTO2 family is manufactured through 5 stations in the common stage and 2 stations in the differentiation stage. In each replication of the simulation process, the number of on time delivered orders and average waiting time for each entity will be investigated.
It worth noting that the processing times before CODP are assumed to be constant, while after CODP processing times are distributed normally with already known means and standard deviations.  In Table 5, a comparison between before and after implementing our proposed model is shown, where the before implementation prices are achieved from (10) and (11). The results confirm that application of the proposed model, improves the company's performance, significantly. As an example, because of the rough cut capacity calculation, after implementation results, show less order acceptance percentage by the manufacturer, but the order acceptance percentage by the customer is much higher. Besides, due to the new pricing and delivery time determination system, higher percentage of on time delivered orders is obtained.
In the last step, while the WIP value determination is an important issue in hybrid production systems, the sensitivity analysis of this variable is shown in Figs. 5 and 6, with respect to the total profit and the average order delay during the planning period. According to the analysis, holding 35 units of WIP for product family 1 and 20 units for product family 2, yields to the best total profit and simultaneously minimizes the average order delay (Figs. 5 and 6).
It is worth noting that the proposed simulation model is validated by the real 6 month data of the understudy company. As it is mentioned earlier, the company used to calculate the price and due date of each incoming order from Eqs. (10) and (11). So, in the first step, the real data from the past 6 month of the company, have been tested by putting these two equations into our simulation model modules. By doing this, the similarity between the simulation model outputs and the real data is observed and the model's correctness is verified. After the validation, the mentioned equations have been replaced with the new proposed ones [Eqs. (6) and (9)] in order to obtain the simulation results of the suggested model.

Conclusion
This paper investigates price and lead time determination issue in a hybrid MTS/ MTO production system, where the conflict of interests between common and differentiation stages is considered as a Stackleberg game. The differentiation manager acts as the leader who first suggests the customization lead time and price and the common stage manager acts as the follower, who makes the final bid with the customer. It is also assumed that the company produces three types of products Also, the probability of each incoming order to be accepted by the customer is considered to be affected by the announced price and lead-time. The optimal price and lead-time of each order is calculated by maximizing the profit and a simulation model is proposed to validate the model and to compare the before and after model implementation results. There is much scope in extending the present work. While this paper only addressed the tactical issues of a hybrid MTS/MTO production system, one idea is to develop similar game theoretic approaches to cope with strategic and operational level decisions of these systems, as well. For example, as a strategic decision, determining the best place of CODP along a production line could be tackled using game theory, due to the conflict of interests between different players such as suppliers, different manufacturing stations and customers. Besides, as another idea, considering the customer as a separate player and forming a multi-player game would be of interest, where different strategies of a customer will be the acceptance of the announced set of price and due date or to reject it or even to bargain with the manufacturer in order to reach to a more desired set. In this regard, also considering the company's owner or manager as another independent decision maker with different preferences, could make the multi-player game more similar to the real world cases.

Appendix 1
To calculate the optimal value of follower variables, PM i and TM i , we have: Be À1 , where W (x) is defined to be the Lambert Function satisfying WðxÞe WðxÞ ¼ x.

Appendix 2
To prove the strict pseudo concavity of Z C with respect to PM i for a fixed TM i , the second derivative of Z C is calculated as follows, where we have L ¼ PM i À CM i À

Appendix 3
The optimal PC i is obtained as follows: