Coordination in a triple sourcing supply chain using a cooperative mechanism under disruption

Computers & Industrial Engineering 101 (2016) 194–215

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Coordination in a triple sourcing supply chain using a cooperative mechanism under disruption Narges Mohammadzadeh a, Seyed Hessameddin Zegordi b,⇑ a b

Industrial Engineering, Tarbiat Modares University, Tehran, Iran Department of Industrial Engineering, Faculty of Engineering, Tarbiat Modares University, P.O. Box 14115 143, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 18 September 2015 Received in revised form 31 July 2016 Accepted 2 September 2016 Available online 6 September 2016 Keywords: Supply chain Disruption Uncertainty Reliability Coordination Cooperation Game theory

a b s t r a c t In this study, we take into consideration a two-level supply chain consisting of a manufacturer and three suppliers under supply disruption from an unreliable supplier. The paper aims to determine optimal ordering policy for manufacturer as well as optimal pricing and production capacity for two reliable suppliers in situations where there is supply disruption and demand uncertainty for manufacturer. For this purpose, two competitive bi-level models have been developed in which the manufacturer is leader in games and reliable suppliers are followers. In these two models, reliable suppliers determine their optimal prices and production capacities in both competitive and cooperative game, respectively. Finally, in order to improve coordination among manufacturer and reliable suppliers under supply disruption, a cooperative approach among manufacturer and reliable suppliers have been developed. The obtained results indicate that cooperative approach under supply disruption, causes improvement in manufacturer’s and reliable suppliers’ profits. Other features of models and the optimal policies’ structures have been explained through appropriate numerical problems and sensitivity analyses. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction In the last two decades, manufacturers have concentrated more on industry trends such as global sourcing, outsourcing, reducing their inventory level and reliance on their suppliers so that they can expedite their production process, decrease costs, improve efficiency and achieve competitive advantages (Li, Wang, & Cheng, 2010; Narasimhan & Talluri, 2009; Tang, 2006). Following these trends will increase length and complexity of supply chains (Harland, Brenchley, & Walker, 2003; Kleindorfer & Saad, 2005) and consequently make increase in vulnerability of organization from unplanned and unpredictable events that are resulted due to peripheral uncertainties. These events are called disruptions in supply chain literature. A supply chain disruption is unexpected and unpredictable event that interrupts the normal flow of raw materials and final products within the supply chain and finally leads to supply chain risks (Li et al., 2010) due to uncertainties created in supply, demand, cost etc. The organizations affected by such risk, will experience special and unusual situations as compared with their normal business activities (Wagner & Bode, 2006).

⇑ Corresponding author. E-mail addresses: [email protected] (N. Mohammadzadeh), [email protected] (S.H. Zegordi). http://dx.doi.org/10.1016/j.cie.2016.09.004 0360-8352/Ó 2016 Elsevier Ltd. All rights reserved.

In order to reduce their costs of purchasing the raw materials, organizations would rather to choose only one source and try to maintain their mutual relationships with selected supplier (Davarzani, Zegordi, & Norrman, 2011); although by depending merely on one supplier, the manufacturer deals with risks of losing the total or part of the order. For instance, Philips’ semi-conductor plant is interrupted by a fire accident for two weeks in 2000. Both Ericson and Nokia Companies had ordered their required components from this supplier. As a result, Erikson suffered about 400 million US $ loss due to its single sourcing policy, but Nokia, on the other hand, could decrease its loss to a minimum degree for their dual sourcing policy (Silbermayr & Minner, 2014). Providing all the orders from one source might incur lower costs, however in case of any disruption for the supplier and the loss of all or part of the orders will certainly bring irreparable damages and losses for the manufacturer such as interrupting production line and losing the market demand (Davarzani et al., 2011). The more dependency of the manufacturer on the supplier, the more break would be caused within the supply chain and more problems would arise in its performance. Supply disruption is a kind of disruptions which are caused by some interruptive events such as financial restrictions, bankruptcy, human source problems, natural events, political issues, and war. Negative impacts which are resulted from supply disruption can affect performance of an organization; so it is reasonable for the

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organization to consider this fact that its main supplier may not be always available. Actually, single sourcing strategy increases the supply risk which is caused by a disruption. Different factors can lead to the unreliability of suppliers for a buyer. These factors are in fact sources of disruptions in supply chain that are presented as follows:
Problems due to human resources like human mistakes or strikes in supplier organization, which can lead to delay in supply or imperfect supply of orders to the buyer.
Problems due to environmental variations, which can make some events like flood, earthquake, and storm. Under such conditions, supplier would be unavailable completely and the buyer loses all orders.
Problems due to political instability and legal restrictions, which can avoid cooperation of buyer with the supplier or limit their cooperation. Under such conditions, buyer may lose its supplier completely or a part of orders may be supplied for the buyer.
Problems related to shipping goods from source (supplier) to destination (buyer) such as lack of required equipment to transport parts and issues due to roads of shipping, laws and permissions for shipping, which can result in uncertainty of buyer about receiving orders from the supplier completely (Kleindorfer & Saad, 2005; Vilko & Hallikas, 2012). Above mentioned factors can result in absolute or partial unreliability of the supplier for a buyer. This fact leads the manager to select more than one source in order to decrease risk of loosing its orders (Kleindorfer & Saad, 2005; Silbermayr & Minner, 2014). One of the sources leads to supply chain disruption is political restriction (Friesz, Lee, & Lin, 2011; Kleindorfer & Saad, 2005). A disruption named sanction is caused by political restrictions which forbid companies in various countries to collaborate and cooperate with each other. In some cases, the sanction disruption can increase the possibility of unpredictable price increases and loosing main supplier. For instance, financial sanctions made interruption for Iranian automotive industry in receiving supplied parts from some unreliable resources (Davarzani et al., 2011; Zegordi & Davarzani, 2012). In much of the researches performed in this field, one reliable and one unreliable supplier were considered for the buyer; however it seems that in such conditions working with several reliable suppliers would be much more preferable: When the manufacturer work only with one reliable supplier, in case of the possibility of any disruption event, it is likely to suffer from costs due to the rise of the reliable supplier’s price. Because of reducing negative impacts come from disruption, the manufacturer’s dependency on reliable supplier is unavoidable. The reliable supplier is aware from the supply disruption and manufacturer’s dependency on its. So, when the disruption probability goes up, the reliable supplier increases its price. In this case, selection of two reliable suppliers could decrease the dependency of manufacturer on one supplier and also make the price competition between two reliable suppliers for achieving more shares of manufacturer’s orders (Jin & Ryan, 2012; Meena, Sarmah, & Sarkar, 2011). These results lead to reducing power of reliable suppliers in bargaining over price and finally make reduction in their prices. The second status is conceivable when a reliable supplier has insufficient capacity to supply all the manufacturer’s orders. In such cases, manufacturer prefers to select two or more reliable suppliers for its orders. The above considered situations are highly probable in some real cases. For instance, it is very likely that a manufacturer prefers to order its required parts from a global source because of its lower cost and higher quality. But sometimes a global source may not be

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fully reliable for manufacturer because of some restrictions and probable events which have been mentioned before. In this case the manufacturer may confront with supply risk due to disruptive events. Also, the manufacturer intends to keep working with unreliable global source because of some advantages like as quality. As a result, the manufacturer decides to choose one or more local sources a long with the global source because of their reliabilities. In one hand, selection of two or more reliable local source has much more advantages for manufacturer which have been mentioned before. In other hand, when the manufacturer decides to work with two or more reliable suppliers, occurrence of two different situations is probable: first, it is probable that reliable suppliers compete with each other to gain more orders from manufacturer according to their competitive factors. In this situation, competition among reliable suppliers avoids irregular increasing in selling prices (make reduction in selling prices) and hence is profitable for manufacturer. Second, it is highly probable that reliable suppliers have some information about unreliability of the main supplier of manufacturer. With this account it is also possible that reliable suppliers decide to make coalition and cooperate with each other against the manufacturer to achieve additional benefits from probable supply disruption; because cooperation among reliable suppliers makes increase in their selling prices and hence makes loss and reduces profit for manufacturer. With this account, it is beneficial for manufacturer to apply an effective and useful technique to prevent reliable suppliers from making coalition against itself. As a result, in this article we propose a cooperative contract between manufacturer and reliable suppliers in supply disruption situation. In this study, a two-level supply chain consisting of a manufacturer and three suppliers have been considered. There is a possibility of disruption for the first supplier with given probability, so the manufacturer has considered two other reliable suppliers for its orders besides the first supplier which become unreliable now. Each of reliable suppliers has a primary capacity, so they will compete or cooperate with each other to gaining much more of the manufacturer’s orders and to determine their sales prices and capacities. To extract optimal policies at first, the problems of manufacturer and reliable suppliers have been investigated separately and optimal policy of each member has been elicited by considering fixed policies of the others. Then, optimal ordering policy of manufacturer and optimal policies of reliable suppliers are determined by developing two bi-level models based on Stackelberg game theory. Along with multi sourcing mitigation strategy, a cooperative approach has also been used to improving the coordination among the manufacturer and reliable suppliers. The remaining of this article is organized as follows: In Section 2, multi sourcing strategy and coordination under disruption are studied and existing approaches for managing disruption in terms of these two contexts are reviewed. Then contributions of this study in comparison with the others are mentioned. In Section 3, problem of this study is described and then problems of manufacturer and reliable suppliers are formulated based on defined assumptions and notations. The ‘‘CmCm”, ‘‘CmCo” and ‘‘CCo” models are investigated in Sections 4–6 respectively. Then computational study, sensitivity analysis and also limitations of the study and some directions for suture researches are illustrated in Section 7. Finally conclusions are mentioned in Section 8 with a brief summary of this study.

2. Literature review Considering the focus of this paper, the literature review is conducted in terms of two aspects: multi sourcing strategy and coordination as two mitigation strategies for supply disruption management. He, Huang, and Yuan (2015) classified

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characterization of supply uncertainty in literature by three approaches: ‘‘the first includes researches that consider supply uncertainty by a random yield model; it means that the fraction of quantity ordered from supplier is delivered. The second class includes studies that consider supply uncertainty in which all the amount of orders delivered to buyer or nothing. This approach indicates supply disruption in models. The last includes papers that model supply uncertainty by using a stochastic lead time or capacity” (He et al., 2015). In this study we consider supply disruption as the second approach which has been noted by He et al. (2015) using a Bernoulli probability distribution. The multi sourcing strategy in researches related to supply chain disruption has been defined and modeled with two approaches. In the first approach prior to occurrence of any supply disruption and just for reducing possible negative impacts, the buyer chooses multi sourcing strategy. In such occasions, some sources are reliable and some unreliable. This approach is defined as a mitigation strategy in disruption management literature. In the second, the buyer uses one or more backup sources after occurrence of any supply disruption. The last approach is defined as a contingent strategy in disruption management in supply chains literature (Snyder et al., 2012; Tomlin, 2006). Recently multi sourcing strategy has been used in industry and academic studies as a mitigation strategy in supply chains. This paper illustrates the fact that two different approaches can be found in researches using multi sourcing mitigation strategy when a manufacturer deals with supply disruption. The first approach, in which the possibility of occurrence of a disruption is considered for all available suppliers, is reviewed in the works of Burke, Carrillo, and Vakharia (2007) and Federgruen and Yang (2008) who considered a special rate of reliability for each supplier. Each supplier provides purchased parts to the extent of its reliability rate, i.e. supplied parts of each source is less than the amount of orders placed by buyer. Li et al. (2010), Meena and Sarmah (2013), Xanthopoulos, Vlachos, and Iakovou (2012) and Yan and Liu (2009) have considered the possibility of supply disruption for all sources. Silbermayr and Minner (2014), also recognized the same event for all sources within specific periods of time, it means that each source might be unavailable in a given time period. Xiaoqiang and Huijiang (2009) have considered one buyer and two symmetric suppliers, which exposure to disruption. They have found that optimal quantity of orders in multi-sourcing strategy is more than single-sourcing strategy and this can cause inventory reserve and reduction of risk. Moreover, for demand function with uniform distribution, the expected income from multi-sourcing strategy is more than single-sourcing strategy. Finally, they have noted that multi-sourcing strategy is optimal policy for disruption management in organizations. Zhu (2015) has considered a supply chain with a local supplier and an overseas supplier which both suppliers are exposed to disruption. In this study, effects of source of disruption and availability of information on costs have been analyzed. They have found that disruption in local supplier can cause more loss in cost efficiency than that of overseas source. Also, the information about local supplier is more important than that of overseas supplier. Ray and Jenamani (2016) have presented a Mean-Variance (MV) framework to allocate orders of a risk-averse buyer among several unreliable suppliers. They have found that the more the degree of risk aversion is, the less order quantity would be. However, amount of order from supplier with less variability is more than others. Sawik (2014) has studied selecting supplier and customer orders scheduling by considering disruption risks for two single and multi-sourcing strategies. Suppliers are at different geographical zones and are involved with different types of disruptions. The aim of the study has been minimizing worst case cost or maximizing the worst case customer service level. Obtained results from

modeling indicated that multi-sourcing strategy can better mitigate risk of high costs or low service levels than single sourcing strategy. Silbermayr and Minner (2016) have studied effect of dual sourcing strategy on risk reduction in cases where both suppliers are prone to disruption. In this study, it has been mentioned that saving resulted from dual sourcing strategy is considerable if demand is distributed optimally between two sources. They have found that even if suppliers are symmetric, symmetric allocation of demand between them is not optimal. Some other papers, however, studied the second approach in using multi sourcing mitigation strategies for supply disruptions. In these papers, the possibility disruption occurrence is regarded, not for all, but for some sources (Chen, Zhao, & Zhou, 2012; Dada, Petruzzi, & Schwarz, 2007; Hou, Zeng, & Zhao, 2010; Hu, Lim, Lu, & Sun, 2013; Qi, 2013; Tomlin, 2006). In these researches, in order to reduce the risk and destructive results for buyer, one or several reliable suppliers are considered along with some unreliable supplier(s). Tomlin (2006) assumed one inexpensive unreliable supplier and one costly reliable supplier. Both suppliers have restrictions in terms of capacity; however, capacity of the reliable supplier is flexible. In his research, Tomlin proved that nature of disruption and probability of unavailability of unreliable source are two key factors in determining the optimal policies for disruption management. Dada et al. (2007) assumed a supply chain with several suppliers. There are two statuses for each supplier, either fully reliable or absolutely unreliable. They concluded that even in cases of higher reliability, the supplier with higher prices wins no order at all. The amount of orders specified for each selected supplier is based on their rate of reliability. According to the analyses conducted in this paper, the total orders of buyer while some suppliers are unreliable, is more than the case where all suppliers are reliable. Chen et al. (2012), however, assumed the unreliability of a supplier for a specified period of time. In their study, multi sourcing strategy is considered as a contingent strategy; it means that the buyer places order to the reliable supplier only when the unreliable supplier has become disrupted. Davarzani and Norrman (2014) have considered one unreliable and two reliable suppliers, which produced parts of unreliable supplier, have higher quality than the reliable ones. The purpose of this study has been decision making on optimal ordering policy under different situations and disruption probabilities. They have found that the optimal policy is very sensitive to setup costs. Dual sourcing strategy make lower setup cost but higher risk and triple sourcing strategy make higher setup cost but lower risk. Moreover, in very high or very low probabilities, single-sourcing strategy with different suppliers is optimal. All studies conducted so far aimed to determine optimal ordering policy of buyer/manufacturer from several unreliable and reliable sources under various restrictions. This study is distinguished from others in considering simultaneously optimal pricing policy and production capacity of reliable suppliers and optimal ordering policy of manufacturer under supply disruption. There are a few researchers who studied suppliers pricing policy under such conditions include Babich, Burnetas, and Ritchken (2007) and Li et al. (2010). Their works differ from ours in that; all suppliers were regarded as unreliable and there was no reliable source for buyer. Thus, these researchers approach is not disruption management by considering reliable source, they just seek to determine optimal pricing policy for suppliers encountered with disruption and optimal ordering policy of buyer from its unreliable sources. In this paper, in order to create a competition among reliable suppliers and optimally allocating their shares of orders, an allocation function is defined. In this function two factors include of price and capacity determine the optimal share from manufacturer orders

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for each reliable supplier. Allocation functions, in the literature, have been used to divide the amount of demands/orders among several retailers/suppliers. Parlar and Weng (2006), Gallego, Huh, Kang, and Phillips (2006), Benjaafar, Elahi, and Donohue (2007), and Jin and Ryan (2012) have used allocations functions with different structures and competitive factors to divide orders. In literature, cooperation has been defined as ‘‘the process of coordinating goals and actions” and as ‘‘a special type of coordination” (Drechsel, 2010). Coordination and cooperation among members of a chain have been suggested as one of the mitigation strategies in risk and disruption management in previous researches. This fact has been significantly considered particularly in recent researches done on supply chain disruption (Chen & Xiao, 2009; Hou et al., 2010; Xiao & Qi, 2008; Xiao, Qi, & Yu, 2007; Zhang, Fu, Li, & Xu, 2012). Each of these papers has suggested one or several coordination mechanisms to improve the performance of supply chain. Some of these studies used multi sourcing and coordination strategies simultaneously for disruption management. Hou et al. (2010) and Hu et al. (2013) in their works assumed a supply chain with one buyer/manufacturer, an unreliable supplier and a reliable one. Hou et al. (2010) modeled and explained a buyback contract between manufacturer and reliable supplier. Also Hu et al. (2013), integrated two risk sharing and buyback contracts in order to create coordination between buyer and reliable supplier. In these papers, the optimal parameters of contracts were determined in such way that the optimal policies in decentralized decision making situation are equal with centralized supply chain. In this study, a cooperative approach between manufacturer and reliable suppliers has been modeled under supply disruption. Thus, optimal quantity of orders, prices and capacities has been determined with the aim of maximizing their total profit functions. Finally we applied Nash Bargaining Game to determine the optimal profit share of each member in cooperative approach. Li et al. (2010) also used this method to determine the optimal profit shares of cooperative suppliers in a non-cooperative supply chain. If a manufacturer makes coalition with reliable suppliers to cooperate with them, it can prevent reliable suppliers’ cooperation and increasing their selling prices. So, in this article, these three different situations are defined and modeled: first, it is assumed that two reliable suppliers compete with each other, then making coalition among them to cooperate with each other is studied and finally the manufacturer join the coalition with reliable suppliers to cooperate with them. It is hypothesis of this study that the last situation is an effective method to manage disruption in reviewed situations and reduce negative impacts of supply disruption in manufacturer’s performance. This paper took contributions regarding the use of multi sourcing strategy to manage the supply disruption in supply chain, which is going to be investigated in terms of three different aspects. First, none of these researches conducted studied the optimal pricing policy for reliable suppliers by consideration of the supply disruption from unreliable supplier. The competition over increasing production capacity between the reliable suppliers was not also put forward in them. Second, no research mentioned above assumed allocation function defined in this paper. Third, in none of the papers which considered supply disruption, multi sourcing and coordination strategies at the same time, the cooperative approach have not been used to improve the coordination under disruption. 3. Problem description and formulation The supply chain in this study is a two level chain consisting of a manufacturer and three suppliers. It is assumed that the manufacturer places an order to the first supplier for one of its required

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components used in final product and is been informed from potential occurrence of disruption in the work of its first supplier. In other words, because of occurrence a kind of disruption such as sanction with a given probability it presumes not receiving supplied components from the first source. Since failure to receive the required component leads to the fail of production line, the manufacturer selects two other reliable suppliers, other than the first unreliable one, for cases when a disruption takes place and so that it can meet part of its market demand. It is notable that quality of components produced by unreliable supplier is higher than the reliable ones. So, the manufacturer wishes to keep purchasing from the first supplier. The configuration of the supply chain and structure of the problem which have been defined in this paper is presented in Fig. 1. This paper aims to allocate manufacturer’s orders optimally among each of suppliers and decide on optimal pricing for reliable suppliers by considering supply disruption for the manufacturer. For this purpose, we developed three mathematical models under various conditions: the first two models (‘‘CmCm” and ‘‘CmCo” models) are based on game theory’s rules, bi-level and competitive models with Stackelberg equilibrium solution in which, the manufacturer is leader and reliable suppliers are followers. The difference in the ‘‘CmCm” and ‘‘CmCo” models lies in the type of game determined among reliable suppliers in the second level of the game. In this sense, in the first model, a competitive game with Nash equilibrium solution and in the second, a cooperative one were considered to determine the optimal prices and capacities among reliable suppliers. In order to determine the optimal share of orders placed by manufacturer to each of reliable suppliers as well as to create competition among them aiming at reducing prices and increasing capacities, a linear allocation function was developed based on two factors consisting of price and capacity. According to the defined function, the reliable supplier with lower price and higher capacity will receive more orders from the manufacturer. Along with a multi sourcing strategy aiming at reduction of negative impacts caused by disruption, a cooperative approach in order to improve coordination between the manufacturer and reliable suppliers under disruption is used. Thus, the third model (‘‘CCo” model) is a cooperative model in which the optimal values of all variables are determined with the aim of maximizing the total profit of manufacturer and reliable suppliers. To determine the optimal profit share of manufacturer and each of reliable suppliers we used Nash bargaining game. In the next part, hypotheses, input parameters, random variables and decision variables used in models will be explained. 3.1. Assumptions and notations The developed models in this paper are based on the following assumptions: 1. Developed models are single-period with single product. 2. The manufacturer’s problem is formulated based on Newsvendor problem. 3. The market demand for manufacturer has been considered exogenous and stochastic with a normal distribution function. Demand for manufacturer is considered probabilistic. 4. All decisions have been taken before initiation of the sales season and before the manufacturers’ information about actual demand or disruption probability have been completed. 5. To produce the final product, manufacturer orders one of its required components to suppliers whose consumption is only one in the final product.

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Unreliable Supplier

Reliable Supplier 1

Reliable Supplier 2

CmCm

Vertical Competition

Optimal Decision Making about Quantity of Orders from Suppliers

Cooperation

Manufacturer

Cooperation

CmCm and CmCo

Horizontal Competition

CmCo

CCo

Supply Disruption

Optimal Decision Making about Selling Prices and Production Capacities

Demand Uncertainty

Market

Fig. 1. Configuration of the considered supply chain in described problem.

6. The quality of produced components by reliable suppliers is lower than those of unreliable sources. So, the cost of such low quality is incurred on the manufacturer side. 7. Final product’s sale price and component’s purchase price from the first unreliable supplier are constant and the last is definite for manufacturer due to previous contracts. On this account, the first supplier does not play a role in the first game. 8. Unreliable supplier has unlimited capacity. 9. Reliable suppliers have a primary capacity and they compete/cooperate with each other over determining equilibrium prices and capacities to gain more orders from the manufacturer. On this account, in the objective functions of second and third suppliers, pricing costs and capacity increasing cost are taken into consideration. 10. Reliable suppliers have the same quality of production and degree of reliability. 11. The amount of orders made by manufacturer to reliable suppliers is distributed according to defined allocation function. In the developed models, the manufacturer’s orders are allocated based on suppliers’ prices and capacities. Actually, allocation function in these models has been defined as Eq. (1):

ðK i =wbi Þ  hi ¼
K i =wbi þ K j =wbj

ð1Þ

Allocation function is defined in such a way that the supplier with more capacity and lower price receives more orders from the manufacturer. 12. According to defined allocation function, it can be understood that the supplier who increases its prices, will receive lower amount of orders. Although the increase in prices leads to increase in supplier’s profit; it consequently causes the decrease in receiving orders and finally in profit. Thus, increasing in proposed price has an indirect effect on supplier’s profit. Parlar and Weng (2006) in their research, assumed direct effect of increase in prices, besides the indirect one, on reducing the supplier’s profit by inserting an

additional cost function in their model. They mentioned that if a supplier decides to propose a higher price it has to pay more advertisement costs. In developed models, in this paper, we used cost function defined by Parlar and Weng (2006). The defined function is quadratic function to the price variable. Terminologically, this is called Pricing Cost (Parlar & Weng, 2006). Parameters and variables used in these models are explained bellow: Symbol

Parameter

P

unit sales price of the manufacturer unit production cost of the manufacturer unit production cost of the unreliable supplier unit production cost of the reliable supplier i unit shortage (lost sales) cost unit holding cost unit quality cost supply disruption probability at the main supplier unit wholesale price from the main supplier the constant coefficient of pricing cost of reliable supplier i primary production capacity of reliable

Cp Cm C bi

i ¼ 1; 2

Cl h Cq

a

wm

ai

k0i

i ¼ 1; 2

i ¼ 1; 2

N. Mohammadzadeh, S.H. Zegordi / Computers & Industrial Engineering 101 (2016) 194–215

S:T: Q m ;

(continued) Symbol

supplier i the constant coefficient of capacity increasing cost of reliable supplier i mean of the demand distribution function standard deviation of the demand distribution function

l r

Symbol Qm

Decision variable order quantity from the main supplier sum of orders’ quantity from reliable suppliers unit wholesale price from the reliable supplier i production capacity of reliable supplier i

Qb

i ¼ 1; 2

wbi

i ¼ 1; 2

Ki

Symbol x f ðxÞ FðxÞ  Y¼

0 1

If disruption occurs If disruption does not occur

ð3Þ

The expected profit function of the manufacturer is seen in Eq. (2) concerning defined parameters and variables. In Eq. (2), Bernoulli variable Y is used to calculate the profit of manufacturer in two modes of occurrence or nonoccurrence of disruption. If the value of Y is zero (indicating occurrence of disruption) variable Q m (which is the amount of order from unreliable supplier) is omitted from function and coefficient a (probability of disruption occurrence) is multiplied by profit function. If Bernoulli variable Y equals 1 (which means non-occurrence of disruption), both variables Q m ; Q b (sum of orders from reliable suppliers and quantity of orders from unreliable supplier) remains in the function. Then coefficient ð1  aÞ that is the probability of disruption nonoccurrence is multiplied by profit function. Finally, expected profit function of manufacturer is calculated by summing these two modes. Since there is uncertainty in market demand, manufacturer incurs inventory cost for purchasing all components from suppliers which are exceeded to the real demand. The manufacturer also bears the cost of losing market and sales for shortage of components purchased in comparison with the real demand. The only constraint of the manufacturer’s problem is the non-negativity of its variables.

Parameter

ci i ¼ 1; 2

Qb P 0

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Proposition 1. The optimal quantities of manufacturer’s order from suppliers are:

 8 1 > ð A Þ; 0 ; if wm < D F > > >  
opt opt  <  1 1 1 Qm ; Qb ¼ F ðAÞ  F ðC Þ ; F ðC Þ ; if D 6 wm 6 E > >   > > : 0; F 1 ðBÞ ; if wm > E

Random variable normal variable of demand, density distribution function, cumulative distribution function Bernoulli variable of disruption occurrence

ð4Þ where


 p  cp  wm þ cl  A¼
p þ h þ cl  cp

ð5Þ




 ðK 1 þK 2 Þ p  cp  cq þ cl  wKb12 wwb2þK
b1 1 wb2 B¼ p þ h þ cl  cp

3.2. The manufacturer’s model formulation The purpose of the manufacturer is to determine the optimal amount of orders from unreliable supplier and each of reliable suppliers in order to reach possible maximum of expected profit. The expected profit function of manufacturer is obtained from the sales income minus production cost, lost sales cost (shortage cost), inventory cost for additional components ordered as compared with real demand, cost of purchasing components from suppliers and quality cost of orders produced by reliable suppliers. In order to determine the optimal share of manufacturer’s orders from reliable suppliers as well as to create competition between suppliers with the purpose of reducing their proposed prices, we used allocation function defined earlier. So, the cost of purchasing from reliable suppliers in expected profit function of the manufacturer has been considered by using the allocation function. The problem of manufacturer is presented as follows:

pM ðQ m ;Q b ; wb1 ; wb2 ; K 1 ;K 2 Þ

Max

1 X

¼

!

½ð1  Y Þa þ ð1  aÞY 

Y¼0

Z

YQ m þQ b

 Z þ 

  P  C p x  hððYQ m þ Q b Þ  xÞ f ðxÞdx

0 1

YQ m þQ b











  P  C p ðYQ m þ Q b Þ  C l ðx  ðYQ m þ Q b ÞÞ f ðxÞdx  YQ m wm

 wb1 wb2 ðK 1 þ K 2 Þ Q b  Q bCq K 2 wb1 þ K 1 wb2

!

ð2Þ

 
 
  ðK 1 þK 2 Þ p  cp þ cl þ ð1a aÞ  wm  a1  wKb12 wwb2þK þ cq 1 wb2 b1
 C¼ p þ h þ c l  cp 

1 ð1  aÞ

D¼

ð6Þ

ð7Þ





 wb1 wb2 ðK 1 þ K 2 Þ a  þ cq  p  cp þ cl K 2 wb1 þ K 1 wb2 ð1  aÞ ð8Þ

 E¼

wb1 wb2 ðK 1 þ K 2 Þ þ cq K 2 wb1 þ K 1 wb2

 ð9Þ

Proof. Considering concavity of profit function (please see Appendix A) and linearity of constraints in manufacturer’s problem, we can elicit optimal quantity of orders from first-order derivatives of profit function with respect to the variables presented in (10) and (11). Then optimal quantity of orders is obtained by adding Lagrangian Multipliers based on Kuhn-Tucker conditions into (10) and (11). (For more explanations about Kuhn-Tucker conditions please see (Colson, Marcotte, & Savard, 2007)).


 @ pM Q m ; Q b ; wbi ; K i ; wbj ; K j @Q m 

 ¼ ð1  aÞ  cp  p  h  cl F ðQ m þ Q b Þ þ p  cp  wm þ cl ð10Þ

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@ pM Q m ; Q b ; wbi ; K i ; wbj ; K j @Q b
 ¼ cp  p  h  cl ða  F ðQ b Þ þ ð1  aÞ  F ðQ m þ Q b ÞÞ
 wb1 wb2 ðK 1 þ K 2 Þ þ p  cp  cq þ cl  K 2 wb1 þ K 1 wb2

ð11Þ

So, the manufacturer’s problem can be solved as follow:


 @ pM Q m ; Q b ; wbi ; K i ; wbj ; K j þ k1 ¼ 0 @Q m

ð12Þ


 @ pM Q m ; Q b ; wbi ; K i ; wbj ; K j þ k2 ¼ 0 @Q b

ð13Þ

k1 Q m ¼ 0

ð14Þ

k2 Q b ¼ 0

ð15Þ

Q m ; Q b ; k1 ; k2 P 0

ð16Þ

By solving the system of Eqs. (12)–(15) and considering constraint (16), we should analyze four possible cases: (i) If k1 ¼ 0; k2 > 0, then Q m P 0 and Q b ¼ 0 according to (14) and (15) respectively. So, from (12), we get 
 Q m ¼ F 1 ðAÞ. According to (13), k2 ¼  cp  p  h  cl
 w w ðK 1 þK 2 Þ ð1  aÞ  F ðQ m Þ þ p  cp  cq þ cl  Kb12 w b2þK 1 w  should be b1

compared with the cost of purchase from reliable suppliers. Based on this proposition, when disruption probability equals to 1, all required components are ordered from reliable sources for any unreliable supplier’s price.
Based on (8) D can get a negative or positive value. If D gets a negative value, any value of selling price of unreliable source is greater than D. In this case even if selling price of unreliable source is too small compared with reliable sources, the optimal ordering policy of manufacturer is not to order just from unreliable source. In case of positive value of D, it is probable that the manufacturer orders just from unreliable source based on  ðK 1 þK 2 Þ its selling price. Finally, when cq > a p  cp þ cl  wKb12 wwb2þK , 1w b1

b2

D gets a positive value and ordering only from unreliable source can be an optimal policy for manufacturer if wm < D.
It is important to say that the quality cost is a critical parameter for manufacturer to decide about quantity of orders from unreliable supplier and reliable ones along with their prices in a given probability of disruption.
If selling price of unreliable supplier is smaller than D, the optimal policy of manufacturer is single sourcing strategy with an unreliable source. If selling price of unreliable supplier takes place between D and E, optimal policy of manufacturer is triple sourcing with an unreliable and two reliable sources. Finally, if selling price of unreliable supplier is more than E, optimal policy of manufacturer is dual sourcing with two reliable sources.

b2

positive. Consequently, if F ðQ m Þ > ð1B aÞ which is equal to

3.3. The reliable supplier’s model formulation

1

wm < D; Q m ¼ F ðAÞ and Q b ¼ 0 is an optimal solution. (ii) When k1 > 0; k2 ¼ 0, in one hand Q m ¼ 0 and Q b P 0 should be satisfied based on (14) and (15) respectively. In other hand, it is obtained that Q b ¼ F 1 ðBÞ from (13). According  

 to (12), k1 ¼  ð1  aÞ  cp  p  h  cl F ðQ b Þ þ p  cp  wm þ cl Þ should be positive. So, if F ðQ b Þ > A which is equal to wm > E; Q m ¼ 0 and Q b ¼ F 1 ðBÞ is an optimal solution. (iii) When k1 > 0; k2 > 0, both Q m and Q b must be equal to zero according to (14) and (15) respectively. So, k1 ¼ h

 and k2 ¼  p  cp  cq þ  ð1  aÞ  p  cp  wm þ cl i ðK 1 þK 2 Þ that should be positive based on to (12) cl Þ  wKb12 wwb2þK 1w b1

b2

and (13) respectively. It is clear that both k1 and k2 can not be positive. So, Q m ¼ 0 and Q b ¼ 0 can not be an optimal solution. (iv) If k1 ¼ 0; k2 ¼ 0, then Q m P 0 and Q b P 0 according to (14) and (15) respectively. So, if D 6 wm 6 E, then based on (12) and (13) we get Q m ¼ F 1 ðAÞ  F 1 ðC Þ; Q b ¼ F 1 ðC Þ that is an optimal solution. For more resolutions about this proposition we present a numerical example in Appendix D. From Proposition 1, the following view points are extracted assuming fixed reliable suppliers’ prices and capacities:
Proposition1 shows that optimal quantity of orders depends on the selling price of unreliable supplier. According to this Proposition, if the selling price of unreliable supplier takes place between two calculated bounds D and E, manufacturer has to order from unreliable supplier and reliable ones simultaneously. It is notable that when this price is smaller than D, all of the manufacturer’s order must be supplied by unreliable source. Also, if the price is larger than E, all of the orders will be supplied by reliable suppliers. In this case the quantity of orders is independent of the disruption probability.
According to this proposition it is mentioned that the ordering policy of manufacturer depends on the unreliable supplier’ price

Each of reliable suppliers aims to determine the optimal pricing policy and the capacity of its production in order to reach its possible maximum profit. Relations between two reliable suppliers can be competitive or cooperative. So, problem of reliable suppliers in cases of competition and cooperation will be investigated separately as follows: 3.3.1. Competitive reliable suppliers In competitive game, each of reliable suppliers has to determine the optimal price and capacity so as to maximize its profit. It is assumed that both reliable suppliers make decisions simultaneously about their policies. So, the optimal prices and capacities in competitive game among reliable suppliers are determined based on Nash equilibrium solution. According to above explanations, problem of competitive reliable suppliers has been presented as follows:

max ¼




pBi Q m ; Q b ; wbi ; K i ; wbj ; K j



K i wbj Q ðwbi  C bi Þ  0:5ai Q b w2bi K j wbi þ K i wbj b  0:5ci ðK i  k0i Þ

2

8i – j;

i; j 2 f1; 2g

ð17Þ

S:T wbi P C bi

i 2 f1; 2g 8i – j

i 2 f1; 2g 8i – j K i wbj Ki P Q i 2 f1; 2g 8i – j K j wbi þ K i wbj b

K i P k0i

ð18Þ ð19Þ ð20Þ

The profit function of each reliable supplier presented in Eq. (17), is obtained through the components’ sale income minus production cost, pricing cost as well as capacity increasing cost which is inserted as a quadratic function with respect to the capacity variable in the model. Such cost is calculated by considering each unit increased with comparison to the primary capacity. Pricing cost is also inserted as a quadratic function with respect to the selling price variable in the objective function of each of reliable suppliers.

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Each of reliable supplier’s problems has 3 constraints that are shown in (18)–(20). The first indicates that the proposed price of each reliable supplier is at least the same as their production cost. In the second and third constraints, two low limits are determined for the proposed capacity of each reliable supplier. The first low limit is primary capacity and the other is the manufacturer’s order form that source. Because each of sources has to totally supply the manufacturer’s order from itself. In order to obtain the optimal policies, first concavity of profit function for each of reliable suppliers should be proved with respect to the variables. Concavity of this profit function with respect to both price and capacity variables depends on the value of two parameters a and k. Therefore, the ranges of these parameters should be determined for each reliable supplier in such a way that its profit function is concave. Obtaining this needs to performing some complicated computations. In order to simplify proving concavity of profit function for each reliable supplier, it is assumed that both reliable suppliers are symmetric. So, in Appendix B, concavity of profit function for each of reliable suppliers has been proved assuming the condition that both reliable suppliers are symmetric. Also, obtaining optimal price and capacity of each reliable supplier while they are asymmetric requires to solving a set of sixth-order polynomial equations. Thus, due to simplify solving such equations, the optimal policies are obtained assuming that condition (two reliable supplier are symmetric) by using first-order condition of profit function for each reliable supplier regarding its variables. Consequently, in the following, parameter C b represents production cost of each component for reliable suppliers, parameter a represents constant coefficient of pricing cost, K 0 shows primary capacity, c is constant coefficient for capacity increasing cost, w is selling price and K represents the increased capacity of each supplier. The meaning of symmetric suppliers is that coefficients of marginal cost and fixed costs are same for them. If coefficients of costs are different, two suppliers are not symmetric. Symmetric suppliers are probable in reality; For example, competitive industries, in which all members have same conditions and technology in production (Hsieh & Kuo, 2011; Jin & Ryan, 2012). Proposition 2. The optimal price and capacity of symmetric competitive reliable suppliers is:


 @ pBi Q m ; Q b ; wbi ; K i ; wbj ; K j ¼ @K i

Q b K j w2bi wbj  Q b K j wbi wbj C bi
2 K i wbj þ K j wbi

!

 ci ðK i  K 0i Þ ð23Þ   w þ Cb  aw þ k1 ¼ 0 4w Qb

ð24Þ

  w  Cb  cðK  k0 Þ þ k2 ¼ 0 4K

ð25Þ

ðwb  C b Þk1 ¼ 0

ð26Þ

   1 k2 ¼ 0 K  max K 0 ; Q b 2

ð27Þ

wb P C b 1 K P maxðK 0 ; Q b Þ 2

ð28Þ

k1 ; k2 P 0

ð29Þ

By considering (18), (28) and (29) and solving the system of Eqs. (24)–(27), we should analyze four possible cases: (i) If k1 > 0; k2 ¼ 0, then w ¼ C b and K ¼ k0 according to (26)   b and (25) respectively. Based on (24), k1 ¼ aw  wþC 4w   should be positive. So, if a > 2C1 ; k1 is positive. Furtherb

more, when Q b 6 2k0 , then (28) is satisfied. Finally, if   a > 2C1 and Q b 6 2k0 , then w ¼ C b and K ¼ k0 is an optimal b

solution. (ii) When k1 > 0; k2 > 0, based on (26) and (27), w ¼ C b and
 According to (24) and (25), K ¼ max k0 ; 12 Q b .     wC b b and k should be ¼ c ð K  k Þ  Q k1 ¼ aw  wþC 2 0 b 4w 4K  
 positive. if a > 2C1 and K ¼ max k0 ; 12 Q b ¼ 12 Q b ; k1 and k2 b

 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi   h i 16aC b þ1þ1 1 > b > ; if a 6 2C1 ; Q b > 2k0 and c > QwC ; Q > b 2 2k0 8a > b b > > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    h i 16aC b þ1þ1 cQ b ðwC b Þþc2 k20 þak0 wC b 1 ðw; KÞ ¼ ; if a 6 ; > 2k and c 6 ; Q 0 b 2C b 2c Q b 2k0 8a > > > >   > > :
C ; maxk ; 1 Q ; 1 if a > 2C 0 2 b b

ð21Þ

b

Proof. According to concavity of profit function (please see Appendix B) and linearity of constraints in symmetric reliable suppliers’ problem, we can elicit optimal price and capacity from first-order derivatives of profit function with respect to the variables that are presented in (22) and (23). Then, by adding Lagrangian Multipliers into (22) and (23) based on Kuhn-Tucker conditions and solving Eqs. (24)–(27), optimal policies of competitive reliable suppliers are obtained. It is mentioned that the necessary conditions for existence of optimal point are represented in B.4 and B.5 (please see the Appendix B). So, for each case if these conditions are not satisfied, the obtained point can not be an optimal solution.






@ pBi Q m ; Q b ; wbi ; K i ; wbj ; K j ¼ @wbi

K 2i Q b w2bj þ K i K j wbj C bi Q b
2 K i wbj þ K j wbi

!

 ai Q b wbi ð22Þ

is positive. Consequently, when a >



1 2C b



and Q b > 2k0 , then

w ¼ C b and K ¼ is an optimal solution.
 (iii) If k1 ¼ 0; k2 > 0, then w P C b and K ¼ max k0 ; 12 Q b according to (26) and (27) respectively. So, from (24), we get pffiffiffiffiffiffiffiffiffiffiffiffiffiffi   16aC b þ1þ1 b w¼ . According to (25), k2 ¼ cðK  k0 Þ  Q b wC 4K 8a 1 Qb 2

should be positive. It is obvious that if K ¼
 b , then k2 > 0. Finally, if max k0 ; 12 Q b ¼ 12 Q b and c > QwC b 2k0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i   16aC b þ1þ1 wC b 1 < and Q c ; a 6 > 2k , then, w ¼ ; 0 b Q 2k0 2C 8a b

b

K ¼ 12 Q b is an optimal solution. (iv) In cases where k1 ¼ 0; k2 ¼ 0, based on (26) and (27) w P C b
 and K P max k0 ; 12 Q b . According to (24) and (25) we get pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 h i 16aC b þ1þ1 cQ b ðwC b Þþc2 k0 þak0 b ; ;K ¼ . Finally, if c 6 QwC w¼ 2c 2k0 8a b

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Q b > 2k0 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

cQ b ðwC b Þþc2 k0 þak0 2c

a6



1 2C b



,

w¼

then

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

16aC b þ1þ1 ;K 8a

¼

is an optimal solution.

For more resolutions about this proposition we present a numerical example in Appendix D. From Proposition 2 the following viewpoints are concluded assuming fixed quantity of manufacturer’s order:
It is seen that each reliable supplier’s optimal price does not depend on their production capacities and the total orders made by manufacturer. Consequently, in case of symmetric reliable suppliers disruption has no effect on optimal their prices.
In some cases, production capacity of each reliable supplier depends on the amount of manufacturer’s order of them.
It is mentioned that the optimal polices depend on the constant coefficients of pricing cost and capacity increasing cost of reliable supplier.
When the constant coefficient of capacity increasing cost is larger than the determined limit, optimal production capacity of each of reliable suppliers is equal to the manufacturer’s order; it means that when capacity increasing cost is high for reliable suppliers, they increase their primary capacity only up to the manufacturer’s order.
When the constant coefficient of pricing cost is larger than the obtained limit, optimal selling price of each reliable supplier is equal to the production cost; it means that when pricing cost is high, reliable suppliers have to reduce their selling price as much as possible. In fact, each supplier should determine the constant coefficient of its pricing cost accurately based on other cost factors. If a supplier considers a large value for the constant coefficient of its pricing cost, he has to reduce its price and in some cases it is possible to incur some losses. Actually, in this situation the supplier cannot make any deals with the manufacturer. So, in real cases if a manufacturer and a supplier make deal to work with each other, the third case in Proposition 2 does not happened; because in this situation the supplier will suffer from the deal. 3.3.2. Cooperative reliable suppliers When both reliable suppliers make a coalition to cooperate with each other, the optimal policies should be determined to maximize the sum of their profit functions. So, the problem of cooperative reliable suppliers is achieved by substituting the following Equation instead of Eq. (17). 2 X






pBi Q m ; Q b ; wbi ; K i ; wbj ; K j ; i – j

ð30Þ

i¼1

Eqs. (34) and (35) and (26) and (27), the optimal policies are derived. It is mentioned that the necessary conditions for existence of optimal point are defined in C.4 and C.5 (please see the Appendix C). So, for each case if these conditions are not satisfied, the obtained point can not be an optimal solution.

@

P2 i¼1

¼

@

P2 i¼1

¼




pBi Q m ; Q b ; wbi ; K i ; wbj ; K j



@wbi

 ! K i Q b w2bj K i þ K j þ K i K j wbj C bi  C bj Q b  ai Q b wbi
2 K i wbj þ K j wbi


pBi Q m ; Q b ; wbi ; K i ; wbj ; K j

ð32Þ



@K i
! Q b K j wbi wbj wbi  wbj  C bi þ C bj  ci ðK i  K 0i Þ
2 K i wbj þ K j wbi

ð33Þ

1=2  aw þ k1 ¼ 0

ð34Þ

cðK  k0 Þ þ k2 ¼ 0

ð35Þ

ðwb  C b Þk1 ¼ 0    1 k2 ¼ 0 K  max k0 ; Q b 2 wb P C b   1 K P max k0 ; Q b 2 k1 ; k2 P 0 The optimal policies of cooperative reliable suppliers will be achieved by solving above problem. After solving the system of Equations, we should analyze four possible cases: (i) If k1 > 0; k2 ¼ 0, then w ¼ C b and K ¼ k0 according to (26) and (35) respectively. Based on (34), k1 ¼ aw  ð1=2Þ should   be positive. So, if a > 2C1 ; k1 is positive. Also, if b
 max k0 ; 12 Q b ¼ k0 , then constraint (18) will be satisfied.   Finally, if a > 2C1 and 2k0 P Q b then w ¼ C b and K ¼ k0 is b

an optimal solution. (ii) In cases where k1 ¼ 0; k2 ¼ 0, based on (26) and (27), w P C b  
 and K P max k0 ; 12 Q b . Therefore, if a 6 2C1 according to b

and (35) we get w ¼ 1 ; K ¼ k0 . So, if   2a
1  1 1 K ¼ max k0 ; 2 Q b ¼ k0 and a 6 2C ; w ¼ 2a ; K ¼ k0 is an (34)

b

Proposition 3. The optimal price and capacity of symmetric cooperative reliable suppliers is:

8
    > < C b ; max k0 ; 12 Q b ; if a > 2C1 b   ðw; K Þ ¼
> 1 ; max k ; 1 Q ; if a 6 1 : 0 2 b 2a 2C

ð31Þ

b

optimal solution.
 (iii) If k1 > 0; k2 > 0, then w ¼ C b and K ¼ max k0 ; 12 Q b according to (26) and (27) respectively. Based on (34) and (35) k1 ¼ aw  ð1=2Þ and k2 ¼ cðK  k0 Þ that should be positive.  
 It is clear when a > 2C1 and K ¼ max k0 ; 12 Q b ¼ 12 Q b ; k1 b   and and k2 are positive. Consequently, if a > 2C1 b

Proof. In Appendix C, we proved that sum of profit functions of reliable suppliers is concave according to some conditions. Due to the concave profit function and linear constraints, we can elicit optimal price and capacity from first-order derivatives of profit function with respect to the variables that are presented in (32) and (33). By adding Lagrangian Multipliers based on KuhnTucker conditions into (32) and (33) and solving the system of

Q b > 2k0 ; w ¼ C b and K ¼ 12 Q b is an optimal solution. (iv) When k1 ¼ 0; k2 > 0, based on (26) and (27), w P C b and
 1 . According K ¼ max k0 ; 12 Q b . So, from (34), we get w ¼ 2a to (35), k2 ¼ cðK  k0 Þ should be positive. Consequently, if  
 1 and K ¼ 12 Q b K ¼ max k0 ; 12 Q b ¼ 12 Q b and a 6 2C1 ; w ¼ 2a b

can be an optimal solution.

N. Mohammadzadeh, S.H. Zegordi / Computers & Industrial Engineering 101 (2016) 194–215

For more resolutions about this proposition we present a numerical example in Appendix D. From Proposition 3 the following viewpoints are concluded assuming fixed quantity of manufacturer’s order:
It is seen that each reliable supplier’s optimal price only depends on the constant coefficient of pricing cost. When this coefficient is larger than the obtained limit, optimal selling price of each reliable supplier is equal to the production cost; it means that when pricing cost is high, reliable suppliers have to reduce their selling price as much as possible. As explained in Proposition 2 for the third case, the first case in Proposition 3 also does not happened in real cases; because of some losses that the supplier incurs.
When the constant coefficient of pricing cost is smaller than the obtained limit, optimal selling price of each reliable supplier does not depend on their production costs.
Production capacity of each reliable supplier does not depend on the value of constant coefficient of capacity increasing cost.
It is mentioned that in cooperative situations, reliable suppliers increase their production capacities only up to the manufacturer’s order if the sum of primary capacities is smaller than the sum of manufacturer’s order from reliable suppliers.
Due to Proposition 2 and 3, selling price equilibrium of reliable suppliers in cases of competition and cooperation depends on the constant coefficient of pricing cost. It is observed that if the constant coefficient of pricing cost is larger than the calculated limit, the optimal policies of reliable suppliers in cooperative situation is the same as competitive situation. When this coefficient is smaller than the calculated limit, the selling price of competitive reliable suppliers is smaller than the cooperative reliable suppliers. 4. ‘‘CmCm Model: bi-level competitive model with competitive suppliers In this model, supply chain is considered in decentralized decision making conditions in which the manufacturer is leader and reliable suppliers are followers. Reliable suppliers compete to achieve their optimal prices and capacities based on Nash equilibrium. Competitive games include two groups of Nash and Stackelberg Games based on equilibrium point. In Nash equilibrium, power of all actors is same; although in Stackelberg equilibrium, power of one or more actors to determine optimal strategy is more than others (the actor(s) are known as leader(s) and other are followers). It means that first, leader(s) of game predicts optimal strategy of followers and then, followers determine their optimal strategies. Therefore, in this equilibrium, decision making would occur consequentially. However, in Nash equilibrium, determining optimal strategies would be occurred simultaneously because of their same powers. (For more information about applying game theory in supply chain please see (Konstantin & Charles, 2007)). According to the defined allocation function, the supplier who proposes lower price and higher capacity receives a greater share of manufacturer’s orders. In developing this model, the following steps have been taken:

3. When prices and capacities are determined, according to the allocation function, the optimal size of order is specified for each supplier. According to the above explanations, the ‘‘CmCm” model is a bilevel model in competitive conditions. Manufacturer makes decisions as the leader of the game in the first step. Reliable suppliers, who play the role of followers in this game, make their decisions after being informed of manufacturer’s decision. Thus, the competitive sub-game among reliable suppliers in the second level (which are presented in (17)–(20)) as limitations of the first level (that are presented in (2) and (3)) will be studied. At the end, the competitive model with competitive suppliers is presented as follows:

Max




pM Q m ; Q b ; wbi ; K i ; wbj ; K j



S:T Q m; Q b P 0 Max






pBi Q m ; Q b ; wbi ; K i ; wbj ; K j i 2 f1; 2g 8i – j

S:T wbi P C bi K i P k0i Ki P

i 2 f1; 2g 8i – j i 2 f1; 2g 8i – j

K i wbj K j wbi þK i wbj

Qb

i 2 f1; 2g 8i – j

In general, if values of ai ; ci are exactly determined (similar to symmetric case in subsection 3–2), according to concavity of the objective function and linearity of constraints in reliable supplier’s problem (the second level is a convex optimization problem), the bi-level competitive model between the manufacturer and reliable suppliers turn into a single-level model while having Kuhn-Tucker conditions, so by adding Lagrangian multipliers can be solved using an optimization software. Therefore, the bi-level competitive model with competitive reliable suppliers turns into a single level model by inserting the following Equations and conditions into the constraints of first level instead of the profit function of second level:

@ pB1 ðQ m ; Q b ; wb1 ; K 1 ; wb2 ; K 2 Þ þ k1 þ K 2 k5 ¼ 0 @wb1

ð36Þ

@ pB1 ðQ m ; Q b ; wb1 ; K 1 ; wb2 ; K 2 Þ þ k2 þ wb2 k5 ¼ 0 @K 1

ð37Þ

@ pB2 ðQ m ; Q b ; wb1 ; K 1 ; wb2 ; K 2 Þ þ k3 þ K 1 k6 ¼ 0 @wb2

ð38Þ

@ pB2 ðQ m ; Q b ; wb1 ; K 1 ; wb2 ; K 2 Þ þ k4 þ wb1 k6 ¼ 0 @K 2

ð39Þ

ðwb1  C b1 Þk1 ¼ 0

ð40Þ

ðK 1  K 01 Þk2 ¼ 0

ð41Þ

ðwb2  C b2 Þk3 ¼ 0

ð42Þ

ðK 2  K 02 Þk4 ¼ 0 1. At the first stage, manufacturer decides on the optimal quantity of orders from unreliable supplier and the sum of orders from reliable suppliers. Manufacturer leads the game, so has enough power and information to predict optimal strategies of reliable suppliers and make its decisions according to their optimal strategies and then informs them of its total order size. 2. Reliable suppliers participate in a Nash competitive game, having been informed of the total size of order and each decides on its own optimal price and capacity.

203

 K1   K2 

K 1 wb2 K 1 wb2 þ K 2 wb1

ð43Þ  Q b k5 ¼ 0

 K 2 wb1 Q b k6 ¼ 0 K 1 wb2 þ K 2 wb1

k1 ; k2 ; k3 ; k4 ; k5 ; k6 P 0

ð44Þ

ð45Þ ð46Þ

204

N. Mohammadzadeh, S.H. Zegordi / Computers & Industrial Engineering 101 (2016) 194–215

To achieve analytical solutions of the competitive bi-level model in recursive method, first we should solve the above equations to compute the optimal prices and capacities in sub- game. After inserting optimal policies obtained from sub- game into the profit function of manufacturer, according to Kuhn-Tucker conditions, optimal orders can be obtained using first-order derivatives of manufacturer’s profit with respect to the order variables. In cases where both reliable suppliers are symmetric, variable K is omitted from the manufacturer’s profit function. Also, variable w is independent from the order variables. Consequently, optimal manufacturer’s orders from suppliers in the ‘‘CmCm” model by assuming symmetric reliable suppliers are obtained by inserting Eq. (21) into Eq. (4). 5. ‘‘CmCo Model: bi-level competitive model with cooperative suppliers The difference between ‘‘CmCo” and ‘‘CmCm” model is the subgame between reliable suppliers. In cases where supply disruption occurrence is possible for the manufacturer, both reliable suppliers might make a coalition against the manufacturer and cooperate to decide on their prices and capacities with the aim of taking advantage of the situation and gain more profit. In such situation, objective function of the sub-game is obtained by sum of objective functions of both reliable suppliers. In cases where both suppliers join a coalition they are making a decision making institution, so they must decide so as to maximize their total profit function. Therefore, in the second model, in order to gain the optimal policies of reliable suppliers in sub- game, concavity of the sum of reliable suppliers function should be proved with respect to the variables of price and capacity. In this model as well, like the previous one, concavity of sum of reliable suppliers’ profit functions has been presented in Appendix C in case of symmetric reliable suppliers due to simplifying computations. The only difference between this model and previous one is seen in the objective function of their sub- games. So, like as previous model by inserting Eqs. (47)–(50), (40)–(46) into the constrains of first level, the bi- level competitive model with cooperative reliable suppliers turns into the single level model and can be solved using an optimization software.

@ @

@

P2 i¼1




pBi Q m ; Q b ; wbi ; K i ; wbj ; K j

 þ k1 þ K 2 k5 ¼ 0

@wb1
 i¼1 pBi Q m ; Q b ; wbi ; K i ; wbj ; K j þ k2 þ wb2 k5 ¼ 0 @K 1

P2

P2 i¼1




pBi Q m ; Q b ; wbi ; K i ; wbj ; K j

P2




i¼1 pBi Q m ; Q b ; wbi ; K i ; wbj ; K j

@K 2

ð48Þ

 þ k3 þ K 1 k6 ¼ 0

@wb2 @

ð47Þ

ð49Þ

 þ k4 þ wb1 k6 ¼ 0

ðwb1  C b1 Þk1 ¼ 0

ð50Þ

k1 ; k2 ; k3 ; k4 ; k5 ; k6 P 0 To achieve analytical solutions of the ‘‘CmCo” model, we should perform computations based on explanations that are mentioned for the ‘‘CmCm” model in previous Section. According to the explanations, the optimal manufacturer’s orders in cases of symmetric cooperative reliable suppliers are obtained by inserting Eq. (31) into Eq. (4). After obtaining the optimal values of second level variables of the model with the aim of maximizing total profit of reliable suppliers, the approach of sharing maximum profit between two reliable suppliers is considered as an incentive for accepting cooperation among them. In order to obtain the optimal share of profit for each reliable supplier Nash bargaining game has been used (For more explanations about Nash Bargaining Solution please see (Compte & Jehiel, 2010)). This method was also applied by Li et al. (2010). Based on the Nash bargaining solution, two reliable suppliers bargain over the share of total profit pCB obtained from their cooperation. The set of feasible agreements is A1 ¼ 
  pCB1 ; pCB2 pCB1 þ pCB2 ¼ pCB where pCB1 and pCB2 are shares of total profit to the first and second reliable supplier, respectively. According to bargaining method, it is necessary to determine a disagreement point. We assume that the optimal profit obtained from competitive game (‘‘CmCm” model) for each reliable supplier is the disagreement point for the bargaining. It means that if the share of total profit to each reliable supplier obtained from bargaining solution in cooperative game is less than the competitive profit obtained in ‘‘CmCm” model, so the agreement will not be accepted.
 1 The disagreement point is denoted by d ¼ pB1 ; pB2 . Consequently, the Nash bargaining solution of the cooperative game described between two reliable suppliers in ‘‘CmCo” model is obtained from solving the following maximization problem:

max where




h





pCB1  pB1  pCB2  pB2 








pCB1 ; pCB2 2 A1 ; pCB1 ; pCB2 P d1

i

ð51Þ

It is obvious that the above problem will reach the maximum value if Eq. (52) is true.









pCB1  pB1 ¼ pCB2  pB2



ð52Þ

According to the equation p þ p ¼ p and Eq. (52), the optimal profit for each reliable supplier in cooperation is obtained from Eqs. (53) and (54).  CB1

1
2



1
2



pCB1 ¼  pCB þ pB1  pB2 pCB2 ¼  pCB þ pB2  pB1

 CB2

 CB

ð53Þ ð54Þ

If optimal calculated profit shares, for each reliable supplier is more than the optimal profit obtained from competitive game (‘‘CmCm” model), accepting cooperation game has propriety over competition for both reliable supplier. If the optimal profit share is less than competitive game for both or either of suppliers, the cooperative contract will not be accepted.

ðK 1  K 01 Þk2 ¼ 0 6. ‘‘CCo Model: coordination with cooperative approach

ðwb2  C b2 Þk3 ¼ 0 ðK 2  K 02 Þk4 ¼ 0   K 1 wb2 Q b k5 ¼ 0 K1  K 1 wb2 þ K 2 wb1   K 2 wb1 Q b k6 ¼ 0 K2  K 1 wb2 þ K 2 wb1

The third model has been developed in order to improving coordination in relations among manufacturer and reliable suppliers in disruption conditions. To achieve this, a cooperative contract has been developed among manufacturer and reliable suppliers. So, in this model, the manufacturer will cooperate with both reliable suppliers. When the manufacturer and reliable suppliers agree upon the contract, the optimal values of the amount of orders, selling prices of component and production capacity of each reliable

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supplier can be obtained by maximizing the total profit of them. Thus, in the ‘‘CCo” model, there is actually a decision maker for deciding on the optimal values of all variables. The constraints of this model are same as those mentioned for the previous models. In order to obtain the optimal parametric solutions, it is necessary to prove the concavity of the sum of manufacturer’s objective functions and reliable suppliers with respect to the variables. With the calculation of first-order condition of the sum of manufacturer’s objective functions and reliable suppliers to all variables, it can simply be shown that this objective function is neither concave nor convex. Thus, the optimal values of variables cannot be obtained as parametric solutions. In order to analyze this model we only mention the local optimal answers using GAMS optimization software which solves LINDO solver. Thus, the ‘‘CCo” model can be explained in cooperative conditions among manufacturer and both reliable suppliers as shown below:

pC ðQ m ; Q b ; wb1 ; K 1 ; wb2 ; K 2 Þ ¼ pM þ pB1 þ pB2

ð55Þ

S:T Q m; Q b P 0 i 2 f1; 2g 8i – j

wbi P C bi

i 2 f1; 2g 8i – j K i wbj Ki P Q i 2 f1; 2g 8i – j K j wbi þ K i wbj b K i P k0i

Having obtained the optimal values of the variables aiming at maximizing the sum profit of manufacturer and reliable suppliers, sharing approach of maximum profit among manufacturer and reliable suppliers is regarded as an incentive for accepting cooperation. As in the ‘‘CmCo” model, here, the optimal share of profit of each member for participation in cooperation contract is obtained through Nash bargaining game. In this model, the manufacturer and reliable suppliers’ profit in cooperative game are respectively,   p CB1 ; pCB2 ; pCM and the sum of optimal profit obtained from cooperative game is pC . Thus, feasible set of profit sharing among manufacturer and each of reliable suppliers will be presented as follows:

         A2 ¼ p CB1 ; pCB2 ; pCM pCB1 þ pCB2 þ pCM ¼ pC


ð56Þ

In this model, the point of disagreement is defined as the anticipated profit for manufacturer and each of reliable suppliers in
 2 competitive game which equals to d ¼ pB1 ; pB2 ; pM . So, when the profit share for each member in the cooperative game is less than the competitive game (‘‘CmCm” model), cooperative contract is not accepted. On this account, the optimal answer of bargaining game is obtained through the solution of optimization problem in (57).

max where


h








     p CB1  pB1  pCB2  pB2  pCM  pM










2 2      p CB1 ; pCB2 ; pCM 2 A ; pCB1 ; pCB2 ; pCM P d

i

ð57Þ

It is clear that the above problem produces the maximum value when condition (58) is true:














     p CM  pM ¼ pCB1  pB1 ¼ pCB2  pB2



ð58Þ

   According to the equation p CB1 þ pCB2 þ pCM ¼ pC and Eq. (58), the optimal share of profit for manufacturer and each of reliable suppliers are calculated and presented in Eqs. (59)–(61).

1
3



1
3



1
3



 pC þ 2pM  pB1  pB2 p CM ¼

p  pC þ 2pB1  pM  pB2 CB1 ¼  pC þ 2pB2  pM  pB1 p CB2 ¼

ð59Þ ð60Þ ð61Þ

7. Computational study In this section, some numerical problems based on 3 cases with different parameters’ values have been designed and solved for each developed model in this paper. Since optimal parametric solutions related to the models were not computed in the cases of asymmetric reliable suppliers, we used numerical problems to investigate and analyze features of models. Models have been solved in cases of asymmetric reliable suppliers using LINDO and CONOPT solvers in version 24.1.3 of GAMS software in a personal computer with 32-bit OS, Intel(R) Core(TM) 2 DUO CPU, 2.40 GHZ and 2 GB RAM. LINDO and CONOPT solvers are appropriate for nonlinear problems (For more information please see (McCarl et al., 2014)). The obtained results have been presented in computational results subsection. Values of parameters in developed numerical problems also appear in Table 1. Then in sensitivity analysis subsection, the impact of supply disruption on the profit functions and some decision variables in each model is investigated and illustrated using appropriate statistical tests and also some graphs. Finally, in limitations and future directions subsection we have explained some of assumptions which make some limitations for the study and proposed some directions to release the assumptions in future researches. 7.1. Computational results In this Subsection, all developed models are solved using numerical problems for several disruption probabilities from 0.1 to 1. Results obtained from the solution of three models in this paper are explained in Tables 2–4, respectively. In the tables, optimal values of manufacturer’s orders, optimal selling prices and production’ capacities of each reliable supplier and profit functions of each supply chain’s member are observed. The profit of unreliable supplier and whole the supply chain are denoted by pS and pt in the tables respectively. Also, the fraction of order’s allocation to each reliable supplier is observed in the tables. Designed examples are related to conditions which two reliable suppliers are not symmetric. Moreover, reliable suppliers are different from unreliable supplier in terms of quality, capacity and technology of production. Unreliable supplier has unlimited capacity of production and has also higher level of technology and production quality than two reliable suppliers. Hence, production cost for unreliable supplier is lower than reliable suppliers. Moreover, technological level of two reliable suppliers is different and hence, production costs of the two reliable suppliers are also different. In examples, it has been assumed that second reliable supplier with higher level of technology for production and lower cost has also more initial capacity. Moreover, the second reliable supplier would tolerate lower cost than the first reliable supplier to increase its

Table 1 Parameters’ values for numerical problems. Problems

P

r

l

wm

Cq

h

Cl

Cm

Cp

a2

a1

C b2

C b1

c2

c1

k02

k01

I II III

70 120 45

20 35 50

100 200 500

16 32 9

10 20 2

17 35 8

30 50 15

4 10 2

15 30 7

0.03 0.006 0.03

0.01 0.012 0.02

6 17 3

10 14 5

0.3 0.5 0.2

0.6 0.2 0.3

40 70 200

30 90 170

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N. Mohammadzadeh, S.H. Zegordi / Computers & Industrial Engineering 101 (2016) 194–215

Table 2 Optimal values of decision variables in ‘‘CmCm” model with respect to disruption probability.

a

Q b

Q

wb1

wb2

K 1

K 2

h1

h2

pM

pB1

pB2

pS

pt

Problem I 0.1 109.15 0.2 42.13 0.3 27.5 0.4 27.5 0.5 27.5 0.6 27.5 0.7 26.13 0.8 22.83 0.9 20.35 1 0

0 67.02 81.65 81.65 81.65 81.65 83.02 86.32 88.8 90.74

109.15 109.15 109.15 109.15 109.15 109.15 109.15 109.15 109.15 90.74

– 22.76 22.77 22.77 22.77 22.77 22.94 23.32 23.6 23.79

– 16.58 16.58 16.58 16.58 16.58 16.1 18.01 18.76 19.34

– 38.5 39.9 39.9 39.9 39.9 40.2 40.9 41.3 41.7

– 50.7 52.5 52.5 52.5 52.5 53.2 54.7 55.9 56.8

0 0.36 0.36 0.36 0.36 0.36 0.36 0.37 0.37 0.37

0 0.64 0.64 0.64 0.64 0.64 0.64 0.63 0.63 0.63

2550.43 1993.42 1903.85 1830.75 1757.65 1684.55 1612.60 1554.47 1508.92 1472.1

0 109.39 130.08 130.08 130.08 130.08 135.94 150.3 161.14 169.68

0 163.06 195.67 195.67 195.67 195.67 199.16 204.96 206.76 206.6

1178.9 404.42 230.94 197.95 164.96 131.97 94.08 54.79 24.43 0

3729.33 2670.29 2460.54 2354.45 2248.36 2142.3 2041.78 1964.52 1901.25 1848.38

Problem II 0.1 210.43 0.2 210.43 0.3 56.83 0.4 37.03 0.5 28.56 0.6 23.47 0.7 20 0.8 17.46 0.9 15.52 1 0

0 0 153.6 173.39 181.87 186.96 190.43 192.96 194.91 196.46

210.43 210.43 210.43 210.43 210.43 210.43 210.43 210.43 210.43 196.46

– – 37.908 37.955 37.974 37.985 37.992 37.997 38.001 38.005

– – 42.033 41.971 41.946 41.931 41.922 41.915 41.909 41.905

– – 128.56 133.13 130.02 130.88 131.47 131.89 132.21 132.47

– – 92.4 98 93.2 93.7 94.1 94.3 94.5 94.7

0 0 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61

0 0 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39

7336.3 5410.1 4151.83 3957.71 3855.12 3789.47 3743.36 3709.05 3682.45 3661.20

0 0 776.39 870.24 910.18 934.06 950.31 962.19 971.3 978.53

0 0 599.2 663.81 690.97 707.11 718.06 726.03 732.15 736.99

4166.5 3703.5 875.21 488.86 314.10 206.52 132.02 76.85 34.14 0

11502.8 9113.62 6402.63 5980.62 5770.37 5637.16 5543.75 5474.12 5420.04 5376.72

Problem III 0.1 529.34 0.2 139.31 0.3 139.31 0.4 139.31 0.5 139.31 0.6 139.31 0.7 132.69 0.8 115.81 0.9 115.81 1 0

0 390.02 390.02 390.02 390.02 390.02 396.65 413.52 413.52 413.52

529.34 529.34 529.34 529.34 529.34 529.34 529.34 529.34 529.34 413.52

– 13.839 13.839 13.839 13.839 13.839 14.041 14.529 14.529 14.529

– 12.636 12.636 12.636 12.636 12.636 13.169 14.529 14.529 14.529

– 185.24 185.24 185.24 185.24 185.24 185.84 187.37 187.37 187.37

– 220.9 220.9 220.9 220.9 220.9 222.3 226.2 226.2 226.2

0 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43

0 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57

11378.07 10308.16 9925.20 9542.24 9159.28 8776.33 8395.59 8096.16 7812.92 7529.68

0 713.16 713.16 713.16 713.16 713.16 756.2 867.29 867.29 867.29

0 1150.7 1150.7 1150.7 1150.7 1150.7 1179.3 1229.5 1229.5 1229.5

3334.8 780.16 682.64 585.12 487.60 390.08 278.64 162.14 81.07 0

14712.9 12952.2 12471.7 11991.3 11510.8 11030.3 10609.7 10355.1 9990.84 9626.53

Q m

Table 3 Optimal values of decision variables in ‘‘CmCo” model with respect to disruption probability.

a

Q m

Q b

Q

wb1

wb2

K 1

K 2

h1

h2

pCM

pCB

pCB1

pCB2

pS

pt

Problem I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

109.15 109.15 78.36 45.26 45.26 45.26 45.26 45.26 45.26 0

0 0 30.79 63.89 63.89 63.85 63.89 63.89 63.89 63.89

109.15 109.15 109.15 109.15 109.15 109.15 109.15 109.15 109.15 63.89

– – 32.4 33.1 33.1 33.1 33.1 33.1 33.1 33.1

– – 23.1 22.6 22.6 22.6 22.6 22.6 22.6 22.6

– – 32 34.43 34.43 34.43 34.43 34.43 34.43 34.43

– – 40 40 40 40 40 40 40 40

0 0 0.36 0.37 0.37 0.37 0.37 0.37 0.37 0.37

0 0 0.64 0.63 0.63 0.63 0.63 0.63 0.63 0.63

2550.4 1933.7 1324.9 1127.6 948.92 770.21 591.49 412.78 234.07 55.36

0 0 176.54 369.17 369.17 369.17 369.17 369.17 369.17 369.17

0 0 55.47 151.79 151.79 151.79 152.98 157.26 161.77 166.13

0 0 121.07 217.38 217.38 217.38 216.19 211.92 207.4 203.04

1178.9 1047.9 658.26 325.88 271.57 217.26 162.94 108.63 54.31 0

3729.33 2981.61 2159.66 1822.69 1589.66 1356.64 1123.6 890.58 657.55 424.53

Problem II 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

210.43 210.43 210.43 210.43 56.47 56.47 56.47 56.47 56.47 0

0 0 0 0 153.9 153.9 153.9 153.9 153.9 153.9

210.43 210.43 210.43 210.43 210.43 210.43 210.43 210.43 210.43 153.9

– – – – 52.1 52.1 52.1 52.1 52.1 52.1

– – – – 63.7 63.7 63.7 63.7 63.7 63.7

– – – – 90 90 90 90 90 90

– – – – 78.3 78.3 78.3 78.3 78.3 78.3

0 0 0 0 0.58 0.58 0.58 0.58 0.58 0.58

0 0 0 0 0.42 0.42 0.42 0.42 0.42 0.42

7336.3 5410.1 3483.8 1557.5 893.92 603.4 312.88 22.356 268.16 558.68

0 0 0 0 2017.4 2017.4 2017.4 2017.4 2017.4 2017.4

0 0 0 0 1118.3 1122.2 1124.8 1126.8 1128.3 1129.5

0 0 0 0 899.11 895.24 892.59 890.64 889.14 887.94

4166.5 3703.5 3240.6 2777.7 621.21 496.97 372.73 248.49 124.24 0

11502.8 9113.62 6724.42 4335.21 3532.56 3117.79 2703.03 2288.27 1873.51 1458.74

Problem III 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

529.34 529.34 177.32 177.32 177.32 177.32 177.32 177.32 177.32 0

0 0 352.0 352.0 352.0 352.0 352.0 352.0 352.0 352.0

529.32 529.32 529.32 529.32 529.32 529.32 529.32 529.32 529.32 352.0

– – 21.4 21.4 21.4 21.4 21.4 21.4 21.4 21.4

– – 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1

– – 170.5 170.5 170.5 170.5 170.5 170.5 170.5 170.5

– – 200 200 200 200 200 200 200 200

0 0 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43

0 0 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57

11378 9280.5 7223.6 6674.8 6126 5577.1 5028.3 4479.5 3930.6 3381.8

0 0 2174.7 2174.7 2174.7 2174.7 2174.7 2174.7 2174.7 2174.7

0 0 868.55 868.55 868.55 868.55 875.79 906.20 906.20 906.20

0 0 1306.1 1306.1 1306.1 1306.1 1298.9 1268.5 1268.5 1268.5

3334.8 2964.3 868.88 744.76 620.63 496.50 372.38 248.25 124.13 0

14712.9 12244.8 10267.2 9594.24 8921.28 8248.32 7575.36 6902.40 6229.44 5556.49

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a

Q b

Q

wb1

wb2

K 1

K 2

h1

h2

pC

p CM

p CB1

p CB2

pS

pt

Problem I 0.1 34.59 0.2 22.99 0.3 16.99 0.4 13.35 0.5 11.04 0.6 9.43 0.7 8.25 0.8 7.33 0.9 6.60 1 0

74.56 86.16 92.16 95.80 98.11 99.72 100.90 101.82 102.55 103.15

109.15 109.15 109.15 109.15 109.15 109.15 109.15 109.15 109.15 103.15

10 10 10 10 10 10 10 10 10 10

6 6 6.32 6.79 7.06 7.25 7.4 7.5 7.6 7.65

30.82 33.77 35.23 36.10 36.67 37.07 37.37 37.61 37.8 37.95

56.07 65.90 69.88 71.33 72.22 72.84 73.29 73.63 73.90 74.12

0.25 0.24 0.24 0.25 0.26 0.27 0.27 0.28 0.27 0.28

0.75 0.76 0.76 0.75 0.74 0.73 0.73 0.72 0.72 0.72

2864.6 2788.1 2749.4 2727.1 2712.8 2702.7 2695.2 2689.4 2684.7 2680

2655.1 2167.5 2077.1 2020 1967.4 1915.3 1861.8 1814.3 1778.2 1749.6

104.72 283.47 303.33 320.29 339.87 360.87 385.10 410.18 430.44 447.21

104.72 337.15 368.93 385.89 405.47 426.47 448.32 464.84 476.07 484.12

373.57 220.71 142.75 96.13 66.25 45.29 29.69 17.59 7.92 0

3228.17 3008.82 2892.12 2823.28 2779.04 2747.98 2724.87 2706.97 2692.66 2680.96

Problem II 0.1 210.4 0.2 43.38 0.3 30.55 0.4 23.80 0.5 19.59 0.6 16.68 0.7 14.54 0.8 12.9 0.9 11.59 1 0

0 167.04 179.88 186.62 190.84 193.75 195.89 197.53 198.84 199.9

210.43 210.43 210.43 210.43 210.43 210.43 210.43 210.43 210.43 199.9

– 14 14 14 14 14 14 14 14 14

– 17 17 17 17 17 17 17 17 17

– 108.5 118.5 123.9 127.2 129.5 131.1 132.4 133.5 134.3

– 71.14 74.47 76.21 77.31 78.06 78.61 79.04 79.38 79.65

0 0.65 0.66 0.66 0.67 0.67 0.67 0.67 0.67 0.67

0 0.35 0.34 0.34 0.33 0.33 0.33 0.33 0.33 0.33

7336.3 6715.3 6587.8 6516.6 6470.8 6438.7 6414.9 6396.6 6381.9 6370.0

7336.3 5845.2 4505.3 4299.3 4193.3 4125.5 4077.8 4042.2 4014.5 3992.3

0 435.09 1129.8 1211.8 1248.3 1270.1 1284.7 1295.3 1303.3 1309.6

0 435.09 952.65 1005.4 1029.1 1043.1 1052.5 1059.1 1064.2 1068.1

4166.5 763.58 470.42 314.23 215.47 146.77 95.964 56.747 25.507 0

11502.8 7478.91 7058.18 6830.82 6686.27 6585.49 6510.90 6453.32 6407.46 6370.03

Problem III 0.1 104.6 0.2 80.24 0.3 64.56 0.4 54.18 0.5 46.84 0.6 41.37 0.7 37.12 0.8 33.71 0.9 30.91 1 0

424.76 449.1 464.77 475.16 482.49 487.96 492.22 495.63 498.43 500.78

529.34 529.34 529.34 529.34 529.34 529.34 529.34 529.34 529.34 500.78

5 5 5 5 5 5 5 5 5 5

5 5 5 5 5 5 5 5 5 5

187.9 197.6 203.9 208.1 211 213.2 214.9 216.2 217.4 218.3

236.9 251.5 260.9 267.1 271.5 274.8 277.3 279.4 281.1 282.5

0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44

0.55 0.55 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56

14112 13924.5 13806.1 13725.7 13667.5 13623.2 13588.3 13560.1 13536.6 13516.9

12289.4 10892.3 10597 10315 10041 9770.7 9481.3 9218.5 9021.9 8826.5

911.31 1297.3 1385.49 1486.34 1594.59 1707.49 1841.95 1989.64 2076.25 2164.09

911.31 1734.9 1823.07 1923.92 2032.17 2145.07 2265.04 2351.90 2438.51 2526.35

658.83 449.33 316.36 227.55 163.95 115.84 77.9541 47.194 21.6342 0

14770.8 14373.8 14122.4 13953.2 13831.4 13739.1 13666.3 13607.3 13558.3 13516.9

Q m

capacity. First reliable supplier (with higher production cost and lower capacity than the second one) would tolerate higher costs to enhance capacity. Also, the supplier costs lower for pricing because of high production and capacity increasing costs. In order to determine values of constant coefficients of pricing cost and capacity increasing cost, obtained values for the two parameters presented in Appendices B and C have been considered. Cost of quality and also holding cost is generally calculated as a percent of value of product. As can be observed in tables, both reliable suppliers in the ‘‘CCo” model have the lowest proposed prices, but in the second they have the highest proposed prices in 3 numerical problems. The price of both reliable suppliers in the ‘‘CmCo” model with any disruption probability is more than the ‘‘CmCm” model and in the ‘‘CmCm” model is more than the ‘‘CCo” model. With the review of results, it can be understood that the selling price of both reliable suppliers are not much sensitive to the disruption probability. It must also be noted that the probability of supply disruption has an indirect effect on the optimal selling price of reliable suppliers. In fact, a minor change in prices with respect to the rise of probability results from changes in production’s capacity which is depends on changes in quantity of manufacturer’s order due to disruption probability. According to the tables, optimal production’ capacities of cooperative reliable suppliers are lower than the competitive ones in each disruption’s probability. Other features of the models are presented in next Subsection using appropriate statistical tests and figures.

ply chain in each model and each numerical problem. We present the results of sensitivity analysis for each member and finally whole supply chain separately in specified sub-sections as follows:

7.2. Sensitivity analysis

7.2.1. Manufacturer As shown in Fig. 2, manufacturer’s profit in 3 problems reduces when the disruption probability rises under three different models. In all possible likelihoods of disruption, manufacturer’s profit in the ‘‘CCo” model is more than or equal to the ‘‘CmCm” model and in the ‘‘CmCm” model is more than or equal to the ‘‘CmCo” model. It means that cooperation among reliable suppliers makes increase in their prices; so the manufacturer’s order from reliable suppliers decrease and profit of manufacturer also decreases relatively. Also it can be said that the amount of orders from unreliable supplier remains constant or reduces with the increase in disruption probability in 3 cases and three different models. The amount of orders from unreliable supplier in the ‘‘CmCo” model is more than other models due to the higher prices proposed by reliable suppliers in the ‘‘CmCo” model as compared with the others. According to Fig. 3, manufacturer’s orders from reliable suppliers increase with the rise of disruption probability in the ‘‘CCo” model. In the ‘‘CmCm” and ‘‘CmCo” models, the trend of changing in the amount of manufacturer’s orders from both reliable suppliers, with the rise of probability of disruption occurrence, either remains constant or increases. From Figs. 2 and 3 it seems that manufacturer’s profit and orders are affected by 3 different situations of modeling and also disruption probabilities. We describe 4 hypotheses about manufacturer’s profit and orders in the following:

In this subsection, we analyze and investigate the impact of supply disruption on the profit of each member and the whole sup-

1. Different relationships between manufacturer and reliable suppliers have some effects on manufacturer’s profits.

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Problem II Problem I

8000

3000 7000

Profit of Manufacturer

Profit of Manufacturer

2500 2000 1500 1000

6000 5000 4000 3000 2000 1000

500 0 0

-1000 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.

Models

CmCm

CmCo

CCo

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.

Models

CmCm

CmCo

CCo

Problem III 12000

Profit of Manufacturer

11000 10000 9000 8000 7000 6000 5000 4000 3000 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.

Models

CmCm

CmCo

CCo

Fig. 2. The effects of disruption probability on manufacturer’s profit under different models.

2. Disruption probabilities affect manufacturer’s profit. 3. Different relationships between manufacturer and reliable suppliers have some effects on manufacturer’s order from reliable suppliers. 4. Disruption probabilities have some effects on manufacturer’s order from reliable suppliers. To prove these hypotheses, it is reasonable to perform some statistical test. In cases where the equality of variances has not been rejected, we have used two way- ANOVA tests and in other cases we have performed nonparametric test named Friedman test instead of two way- ANOVA test because of inequality of variances. Also we have considered disruption probability and type of models as two factors in statistical tests. To prove hypotheses 1 and 3, 3 different models are considered as 3 different groups and to prove hypotheses 2 and 4, 10 different disruption probabilities (from 0.1 to 1) are considered as 10 different groups. The results obtained from statistical tests are presented in Table 5. The significant values of comparing the results of 3 models in hypotheses 1 and 3 have been achieved less than 0.05(significance level). So, the null hypotheses (indifferences among manufacturer’s profits and orders from reliable suppliers in 3 models) have been rejected in 3 numerical problems. Also, the significant values of comparing the results under various disruption probabilities in hypotheses 2 and 4 have been achieved less than 0.05 (significance level). So, the null hypotheses (indifferences among manufacturer’s profits and orders from reliable suppliers under various disruption probabilities) have been rejected in 3 numerical problems. Consequently,

the 4 described hypotheses have been proved in 3 numerical problems. So we present the obtained results about manufacturer as follows:
Increasing of disruption probabilities makes reduction in manufacturer’s profit in 3 models. But the reduction of profit in the cooperative model is less than the both competitive models.
Cooperation among reliable suppliers makes more reduction in manufacturer’s profit and also orders in comparison with the case of competitive reliable suppliers.
Cooperation among manufacturer and reliable suppliers in any disruption probability is dominant and profitable strategy.
Manufacturer’s orders from reliable suppliers increase with the rise of disruption probabilities in 3 models.
Manufacturer’s orders from reliable suppliers in cooperative model are more than the both competitive models.
As we have declared in Proposition 1, optimal ordering policy of manufacturer in each case is related to the suppliers’ prices and quality cost. In numerical problems presented in the computational study section, optimal ordering policy is single sourcing with unreliable supplier for very low disruption probability in ‘‘CmCm” model and also for low or medium disruption probabilities in ‘‘CmCo” model.
It is clear that when the disruption probability equals to 1, optimal policy of manufacturer is dual sourcing with two reliable sources, but with different order’s quantity in 3 different models.

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Problem II 200

100

Orders from Reliable Suppliers

Orders from Reliable Suppliers

Problem I

80 60 40 20 0

150

100

50

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.

Models

CmCm

CmCo

CCo

Models

CmCm

CmCo

CCo

Problem III

Orders from Reliable Suppliers

500

400

300

200

100

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.

Models

CmCm

CmCo

CCo

Fig. 3. The effects of disruption probability on manufacturer’s order from reliable suppliers under different models.

7.2.2. Reliable suppliers Profit of both reliable suppliers is rising when the probability of disruption occurrence rises in the ‘‘CCo” and ‘‘CmCm” models. The highest amounts of their optimal profits occur in the ‘‘CCo” model for both reliable suppliers as occurred for manufacturer. In low probabilities of disruption occurrence both reliable suppliers’ profit in competitive situation is more than cooperative situation. It seems that the profits of reliable suppliers are sensitive to variation of disruption probabilities and also the relationships between them and manufacturer. Also, it seems that optimal selling prices of reliable supplier are not much sensitive to disruption probabilities; but their optimal prices are sensitive to relationships between reliable suppliers and manufacturer. So, the following hypotheses are described: 1. Different relationships between reliable suppliers and manufacturer have some effects on reliable suppliers’ profits. 2. Disruption probabilities affect reliable suppliers’ profits. 3. Different relationships between reliable suppliers and manufacturer have some effects on optimal selling prices of reliable suppliers. 4. Disruption probabilities have not some meaningful effects on optimal selling prices of reliable suppliers. To prove the above hypotheses like as previous we use appropriate tests and the obtained results for reliable supplier 1 and 2 are presented in Table 5 for 3 numerical problems. The significant values of comparing reliable suppliers’ profits in 3 different models (hypothesis 1) are less than 0.05. Also, the significant values of

comparing reliable suppliers’ profits under various disruption probabilities (hypothesis 2) are less than 0.05 also for reliable supplier 1 and 2 in 3 problems. So, the null hypotheses (indifferences among profits of reliable suppliers in 3 models and under various disruption probabilities) have been rejected. The results of statistical tests to analyze the impact of various disruption probabilities on optimal selling prices of reliable suppliers show that the null hypotheses cannot be rejected; because the significant values are greater than 0.05 for reliable supplier 1 and 2 in 3 problems. So, it is concluded that variation of disruption probabilities has not meaningful effect on optimal selling prices of reliable suppliers. In contrast, effect of different relationships between reliable suppliers and manufacturer on their optimal selling prices has not been rejected; because the significant values are less than 0.05 for both reliable suppliers in 3 problems. Consequently, 4 described hypotheses have been proved. Figs. 4 and 5 display reaction curves of prices for both reliable suppliers comparatively in competitive and cooperative games (‘‘CmCm” and ‘‘CmCo” models) with probability 0.5 for disruption occurrence in numerical problem I. Reaction curve of first (second) reliable supplier to price of the second reliable supplier (first) is in fact optimal price of first reliable supplier in response to (second) changes in price of second (first) reliable supplier. Also, reaction curve of production capacity of first (second) reliable supplier to production capacity of second (first) reliable supplier (first) is in fact optimal production capacity of first (second) reliable supplier in response to changes in production capacity of second (first) reliable supplier. Therefore, intersection points of reaction curves of reliable suppliers against each other are in fact optimum responses

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Table 5 Results of statistical tests. Statistical results Chi-square or F

Df

Asymp. Sig.

10 3 10 3 10 3 10 3 10 3 10 3 10 3

19.538 27.000 31.51 6.52 116.71 10.13 16.000 9.196 92.07 11.72 16.000 9.258 12.359 27.000

2 9 2 9 2 9 2 9 2 9 2 9 2 9

.000 .001 .000 .000 .000 .000 .000 .239 .000 .000 .000 .235 .002 .001

Problem II Manufacturer’s profit in 3 models Manufacturer’s profit in different disruption probabilities Manufacturer’s order from reliable suppliers in 3 models Manufacturer’s order from reliable suppliers in different disruption probabilities Reliable supplier 1’s profit in 3 models Reliable supplier 1’s profits in different disruption probabilities Selling price of reliable supplier 1 in 3 models Selling price of reliable supplier 1 in different disruption probabilities Reliable supplier 2’s profit in 3 models Reliable supplier 2’s profit in different disruption probabilities Selling price of reliable supplier 2 in 3 models Selling prices of reliable supplier 2 in different disruption probabilities Supply chain’s profit in 3 models Supply chain’s profit in different disruption probabilities

10 3 10 3 10 3 10 3 10 3 10 3 10 3

25.23 8.68 8.86 6.83 6.55 9.34 20.000 9.000 8.51 8.41 20.000 1.43 13.14 11.45

2 9 2 9 2 9 2 9 2 9 2 9 2 9

.000 .000 .002 .000 .007 .000 .000 .437 .003 .000 .000 .998 .000 .000

Problem III Manufacturer’s profit in 3 models Manufacturer’s profit in different disruption probabilities Manufacturer’s order from reliable suppliers in 3 models Manufacturer’s order from reliable suppliers in different disruption probabilities Reliable supplier 1’s profit in 3 models Reliable supplier 1’s profits in different disruption probabilities Selling price of reliable supplier 1 in 3 models Selling price of reliable supplier 1 in different disruption probabilities Reliable supplier 2’s profit in 3 models Reliable supplier 2’s profit in different disruption probabilities Selling price of reliable supplier 2 in 3 models Selling prices of reliable supplier 2 in different disruption probabilities Supply chain’s profit in 3 models Supply chain’s profit in different disruption probabilities

10 3 10 3 10 3 10 3 10 3 10 3 10 3

20.000 27.000 14.38 3.73 92.02 8.97 20.000 12.495 55.39 9.64 20.000 4.113 19.538 27.000

2 9 2 9 2 9 2 9 2 9 2 9 2 9

.000 .001 .000 .008 .000 .000 .000 .187 .000 .000 .000 .904 .000 .001

50 45 40 35 30 25 20 15 10 5 0

Selling Price of Reliable Supplier 1

N

Problem I Manufacturer’s profit in 3 models Manufacturer’s profit in different disruption probabilities Manufacturer’s order from reliable suppliers in 3 models Manufacturer’s order from reliable suppliers in different disruption probabilities Reliable supplier 1’s profit in 3 models Reliable supplier 1’s profits in different disruption probabilities Selling price of reliable supplier 1 in 3 models Selling price of reliable supplier 1 in different disruption probabilities Reliable supplier 2’s profit in 3 models Reliable supplier 2’s profit in different disruption probabilities Selling price of reliable supplier 2 in 3 models Selling prices of reliable supplier 2 in different disruption probabilities Supply chain’s profit in 3 models Supply chain’s profit in different disruption probabilities

Selling Price of Reliable Supplier 1

Comparison between

wb2 = R2(wb1)

wb1 = R1(wb2)

(wb1 , wb2) 0

10

20

30

40

50

Selling Price of Reliable Supplier 2 Fig. 4. Reaction curve of prices for both reliable suppliers in ‘‘CmCm” model, Problem I (denoted by Ri for reliable supplier i).

or equilibriums of games described between them, which has been evaluated under competitive and cooperative situations.

50 45

wb2 = R2 (wb1)

40

wb1 = R1 (wb2)

35 30 25

(wb1 , wb2 )

20 15 10 5 0 0

10

20

30

40

50

Selling Price of Reliable Supplier 2 Fig. 5. Reaction curve of prices for both reliable suppliers in ‘‘CmCo” model, Problem I (denoted by R0i for reliable supplier i).

It is shown that reaction curves have the same behaviors in competition and cooperation. It is mentioned that the increasing

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slope of reactions curves in competitive situation is slower than cooperative situation. As it is shown in the figures, Nash equilibrium point of prices in the ‘‘CmCm” model takes place prior to the ‘‘CmCo” model, in the sense that equilibrium prices in competitive situation are smaller than the cooperative situation. Contrary to this behavior is also true for equilibrium capacities, in the sense that Nash equilibrium point for capacity in the ‘‘CmCm” model takes place after the ‘‘CmCo” model. Finally according to the hypotheses and Figures we present the obtained results about reliable suppliers as follows:
Cooperation between reliable suppliers against the manufacturer increases their selling prices. In addition, cooperation among reliable suppliers and manufacturer decrease the optimal selling prices approximately to their production costs.
Cooperation between reliable suppliers against the manufacturer increases their profits in medium or high disruption probabilities in comparison to competition.
Disruption probability has significant effect on the profit of reliable suppliers in 3 models.
Cooperation among reliable suppliers and manufacturer is dominant strategy for both reliable suppliers like as the manufacturer.
Disruption probability has no significant effect on optimal selling prices of reliable suppliers in 3 models. It can be an important result of considering two reliable suppliers in addition to unreliable source. In fact, selection of two reliable sup-

pliers under supply disruption and the relationship between them can control indiscriminate increase in price of reliable source in supply disruption situation. 7.2.3. Supply chain It is seen in Fig. 6 that the total profit of supply chain as a whole, with low, medium or high probabilities in the ‘‘CCo” model is more than those of the other two models. Only in very low disruption probability total profit of supply chain in the ‘‘CCo” model can be equal (problem 2), greater (problem 3) or less (problem 1) than those of the other two models. Also, for medium or high probabilities, the total profit in the ‘‘CmCm” model is more than the ‘‘CmCo” model. With the rise of occurrence probability, the total profit of supply chain reduces in all three models. This reduction in the ‘‘CmCm” and ‘‘CCo” model occurs with a roughly constant slope, but the reduction slopein the ‘‘CmCo” model is more than the other models. We investigate the effectiveness of disruption probability and relationships between reliable suppliers and manufacturer on whole supply chain using appropriate statistical tests. The obtained results are presented in Table 5. As can be seen in Table 5, the significant values for both tests are less than 0.05 in 3 numerical problems. So, the impacts of supply disruption and also relationship between reliable suppliers and manufacturer on profit of supply chain have been proved. According to the obtained results, the following viewpoints have been concluded about the whole supply chain:

Problem I

Problem II 12000

Total supply chain's profit

Total supply chain's profit

4000

3000

2000

1000

10000 8000 6000 4000 2000

0

0 1 2 3 4 5 6 7 8 9 0 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0. 1 .

CmCm

1 2 3 4 5 6 7 8 9 0 0 . 0 . 0. 0. 0. 0 . 0 . 0. 0. 1 .

CmCo

CCo

1 2 3 4 5 6 7 8 9 0 0. 0. 0. 0. 0 . 0 . 0 . 0 . 0. 1.

Models

CmCm

1 2 3 4 5 6 7 8 9 0 0. 0 . 0. 0. 0 . 0 . 0. 0. 0. 1 .

CmCo

Problem III 15000

Total supply chain's profit

Models

1 2 3 4 5 6 7 8 9 0 0 . 0 . 0 . 0. 0. 0. 0. 0 . 0 . 1.

12500

10000

7500

5000 1 2 3 4 5 6 7 8 9 0 0 . 0. 0. 0 . 0. 0 . 0 . 0 . 0 . 1.

Models

CmCm

1 2 3 4 5 6 7 8 9 0 0 . 0. 0. 0. 0. 0 . 0 . 0. 0. 1.

CmCo

1 2 3 4 5 6 7 8 9 0 0. 0. 0 . 0. 0 . 0 . 0. 0 . 0. 1.

CCo

Fig. 6. The effects of disruption probability on Total Supply Chain’s profit under different models.

1 2 3 4 5 6 7 8 9 0 0. 0 . 0. 0 . 0 . 0. 0 . 0. 0. 1.

CCo

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Disruption probability has negative impact on profit of whole supply chain.
In cases where there is medium or high possibility of supply disruption for manufacturer and it leads the market, cooperation of followers (reliable suppliers) against manufacturer (as leader), not only fails to improve the critical situation, but also, it might intense the negative impacts resulted from disruption on the performance of supply chain more than ever.
Cooperation among manufacturer and reliable suppliers under disruption is dominant strategy for supply chain as a whole in all probabilities of disruption with the exception of very low probabilities in some cases. 7.3. Limitations and future directions There are some limitation associated with the assumptions of this paper which can be released and investigate in future researches. The first constraint is associated with normal distribution of demand random variable. Obtained results from models have been derived by assuming normal distribution of demand. Under conditions that demand follows a discrete distribution such as Poisson or other distributions, no fundamental changes would be created in models. However, it could not be mentioned that results are same, since behavior of demand variable can affect obtained results certainly. Therefore, it is necessary to evaluate and solve models again based on new demand distribution function. The second limitation is related to unreliable supplier. In this study, it has been assumed that offered price of unreliable supplier is fixed and hence, the supplier has no role in the game. In some cases, the assumption may be rejected. For example, unreliable supplier may be informed of manufacturer’s decision to supply a part of orders from reliable suppliers and so make decision to reduce its offered price. In fact, unreliable supplier may enter to competition with reliable suppliers and propose some incentive options to manufacturer. This issue can be evaluated in furthers studies. In this study, it has been assumed that production capacity of unreliable supplier is unlimited. It is possible for unreliable to determine a low level for manufacturer’s order and the level may be effective in regard with determining optimal policy of manufacturer. In this study, it is assumed that reliability of two reliable suppliers and also quality of their products are same. Each of the two assumptions can be released. Under such conditions, certainly quality and reliability as two other factors, along with price and capacity, should consider in models; so that can affect decision making of manufacturer. Therefore, the two parameters should be considered in allocation function suitably. Even it may be necessary to change allocation function from linear form to other forms like exponential or nonlinear functions, which can be evaluated in further researches. Also, if the disruption probability is considered for all suppliers we can define reliability level for them and use allocation function for allocating orders to suppliers according to reliability level, price, quality, etc. Another assumption is that manufacturer has more power at the market and so is leader in games. In real world, such conditions may be existed that suppliers or one of them and even unreliable supplier has more power in market and consequently is leader of the game. In these cases, obtained results would be certainly different from the results of this study. Evaluation of this problem and comparison of obtained results with the results of this study is important and valuable. Models of the study have been developed for such conditions that a part of final product is ordered by the manufacturer. Models can be developed for conditions that several parts or even several products are ordered from suppliers. However, in this case, it is important to find that which supplier(s) supplies which part(s). If

a specific part is produced only by a supplier, there would be no way for manufacturer to supply all orders from the supplier. But if a part is supplied by several suppliers, whether reliable or unreliable, other factors can also affect decision making of manufacturer. Under such conditions, decision variables from one index to two or even more indices would be changed (depending on number of suppliers, who produce specific part); although allocation function should be defined for each part separately. Moreover, if several unreliable suppliers exist, competition among them can affect manufacturer’s orders which would be investigated in further studies. In this study, selling price of final product has been considered as constant and the assumption can be released; because the cost of purchasing parts can affect the price of final product. Under such conditions, price of final product can be considered as a variable in the models and its optimal value can be derived based on other costs, such as cost of purchasing parts. The last limitation is about reliability of two suppliers. In some cases it is probable that all suppliers are prone to disruption. In this case the rate of reliability of each supplier is an important parameter in determining optimal policies. So, this issue can be investigated in future researches.

8. Conclusion In this paper the optimal ordering policy of the manufacturer and pricing/production capacity policy of two reliable suppliers was investigated by considering demand uncertainty and probability of supply disruption from the main supplier in different situations. The main supplier had been considered as an unreliable source for manufacturer, thus manufacturer proceeded to get two reliable suppliers in order to reduce possible losses from any probable disruption. Two reliable suppliers had a primary capacity. Manufacturer had used an allocation function to assign its orders among them. Effective factors on this function were proposed price and capacity of each reliable supplier. In this paper, at first, we investigated optimal policy of each member separately by considering fixed policies of the others. It is mentioned that for fixed policy of reliable suppliers, the optimal ordering policy of manufacturer depends on the selling price of unreliable source and quality cost. In this situation, we achieved 2 important bounds for selling price of unreliable supplier which can change the optimal policy of manufacturer. It means that if the price is smaller than a specified bound, optimal ordering manufacturer is single sourcing with the unreliable source. When the price gives a value between two calculated bounds, optimal policy of manufacturer is triple sourcing with unreliable and reliable sources. Finally, when the price is greater than a specified bound, optimal policy of manufacturer is dual sourcing with two reliable sources. Also, we proved that optimal policies of reliable suppliers by considering fixed policy of manufacturer are influenced by two important parameters ‘‘constant coefficient of pricing cost” and ‘‘constant coefficient of capacity increasing cost”. In the ‘‘CmCm” and ‘‘CmCo” models which were bi-level models and developed based on game theory (Stackelberg type), it was assumed that the manufacturer is leader of game and reliable suppliers are followers. In the ‘‘CmCm” model, a competitive sub-game and in the ‘‘CmCo” model a cooperative sub-game was considered among both reliable suppliers. The ‘‘CCo” model was developed to improve the performance of members of the chain and reduce losses resulted from any disruption. In this model, manufacturer and both reliable suppliers joined a coalition and all decision variables were determined cooperatively. After analyzing the results, it was understood that the acceptance of agreement based on cooperation among manufacturer and reliable suppliers (‘‘CCo” model) with any probability of dis-

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ruption’s occurrence results in promoting the profit for each of them in comparison with the previous competitive states (‘‘CmCm” and ‘‘CmCo” models). Making coalition and cooperation of the two reliable suppliers is not beneficial for the manufacturer. When both reliable suppliers make a coalition against manufacturer to cooperate, they will have higher proposed prices as compared with competitive situation. This causes the reduction of manufacturer’s orders from reliable suppliers (even more than case of competitive situation) and also brings about an intense drop in gaining profit for the manufacturer. Although cooperation between two reliable suppliers leads to the rise of their proposed prices and reduction of their receiving orders, the profit of both reliable suppliers is more than competitive conditions if the probability of disruption occurrence is medium or high. Since any probability of disruption will affect the chain as a whole, it is important to mentioned that accepting the agreement of cooperation among manufacturer and two reliable suppliers results in the improvement of the profit for the supply chain as compared with other two competitive states, if the disruption probability to be not very low. From obtained results of statistical tests, it is found that in case of working with two reliable suppliers along with unreliable supplier, disruption probability has no significant effect on optimal selling prices of reliable suppliers when manufacturer is leader in market. Also, disruption probability has significant effect on profit of manufacturer and suppliers and also manufacturer’s orders. It is important to mention that the effectiveness of different relationships between manufacturer and reliable suppliers on their profits, orders and prices has been proved by statistical tests. Recommended issues for future studies on disruption and coordination in supply chain include: the unreliable supplier can also enter to competition with reliable suppliers and propose some incentive contracts to manufacturer for gaining more orders. The models can be developed for situations in which n reliable suppliers and m unreliable suppliers exist and the manufacturer needs to order n parts from the suppliers. Other allocation functions like as nonlinear or exponential ones is also significant with a view to other competitive factors such as quality, orders delivery time, and reliability level. Optimal answers for these models in situations where reliable suppliers are leaders and manufacturer is the follower with a comparison to the existing results of this paper could be highly beneficial. Also, it is valuable to investigate the situation in which all suppliers are prone to disruption. Appendix A In order to prove the concavity of the manufacturer’s profit function to the quantity of order variables, first and second- order conditions of this function is studied and then the hessian matrix is formed.


 @ 2 pM Q m ; Q b ; wbi ; K i ; wbj ; K j @Q 2m

¼ ð1  aÞ
  cp  p  h  cl f ðQ m þ Q b Þ <0 ðA:1Þ



 @ 2 pM Q m ; Q b ; wbi ; K i ; wbj ; K j @ 2 pM Q m ; Q b ; wbi ; K i ; wbj ; K j ¼ @Q m @Q b @Q b @Q m ¼ ð1  aÞ
  cp  p  h  cl f ðQ m þ Q b Þ <0 ðA:2Þ

@ 2 pM ðQ m ; Q b ; wb Þ


@Q 2b

 ¼ cp  p  h  cl ða  f ðQ b Þ þ ð1  aÞ  f ðQ m þ Q b ÞÞ < 0  2  @ pM ðQ m ;Q b Þ  @Q 2 jHðQ b ; Q m Þj ¼  @ 2 p ðQm ;Q Þ m b M  @Q m @Q b

ðA:3Þ

 @ 2 pM ðQ m ;Q b Þ   @Q b @Q m  @ 2 pM ðQ m ;Q b Þ   2 @Q b


2 ¼ að1  aÞ  cp  p  h  cl f ðQ b Þf ðQ m þ Q b Þ >0

ðA:4Þ

According to the negativity of the first element of the hessian matrix for manufacturer’s profit function and the positivity of the matrix determinant assuming a – 0; 1, concavity of profit function with respect to variables of the amount of orders will be proved. When a ¼ 1, based on Eq. (2) Y ¼ 0. So, Q m variable is omitted from the manufacturer’s profit function and this function turns into a single variable function. In this case, the profit function is concave based on negativity of (A.3). Appendix B In order to prove the concavity of profit function of each reliable supplier, second-order condition for this function with respect to variables of price and capacity has been calculated using the first-order derivatives presented in (22) and (23):


 @ 2 pBi Q m ; Q b ; wbi ; K i ; wbj ; K j ¼ @w2bi

K j K 2i Q b w2bj  K i K 2j wbj C bi Q b
3 K i wbj þ K j wbi

!

 ai Q b <0 ðB:1Þ
 @ 2 pBi Q m ; Q b ; wbi ; K i ; wbj ; K j @K 2i

!

¼

Q b K j wbi w2bj ðwbi  C bi Þ  ci < 0
3 K i wbj þ K j wbi ðB:2Þ

With the constant capacity, it can be seen that profit function for each reliable supplier with respect to price variable is concave. It was also found that, with the constant price, the profit function for each reliable supplier is concave with respect to capacity. The proof of concavity of profit function for each reliable supplier with respect to the variables of price and capacity simultaneously need to perform a set of complex computations to achieve the range of two parameters a; c. So, concavity of profit function for each reliable supplier with respect to the variables of price and capacity can be proved assuming symmetric reliable suppliers. On this account, Parameter C indicates the cost of component production for suppliers; parameter a shows the constant coefficient of pricing cost, parameter K 0 stand for primary capacity, c for constant coefficient of capacity increasing cost, w for price and K shows the increased capacity for each supplier.   2  @ pB ðw;K Þ @ 2 pB ðw;K Þ   @w2 @K@w   jHðw;K Þj ¼  2 2  B ðw;K Þ @ pB ðw;K Þ   @ p@w@K @K 2  

  Qb   Q wþC þ a   b 4K 8w2       ¼  Qb wC Q b 8K 2  c   4K 2  3  Q 2b w2  C 2 þ 8aQ 2b w2 ðw  C Þ þ 8cQ b K 2 ðw þ C Þ 4 5=64K 2 w2 ¼ þ64acQ b K 2 w2  4Q 2b w2 ðB:3Þ

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According to the determinant of the hessian matrix, we can say that concavity of the objective functions for each reliable supplier depends on the values of two parameters a; c. So, we should determine domain of these parameters in such a way that the sign of (B.3) become positive. On one hand, If (B.3) is positive, (B.4) should be satisfied. On the other hand based on assumptions, 0 < c < 1.

4Q b w2 Q b ðw2 C 2 Þ8aQ b w2 ðwC Þ So, < 1 should be satisfied that is equiv8K 2 ðw þ C Þ þ 64aK 2 w2 alent to (B.5). We could determine the lower bound for both parameters a; c in such a way that the profit function of each of reliable supplier is concave with respect to the variables of price and capacity. Thus, if the values of these two parameters are truly determined, the profit function for each reliable supplier will be concave with respect to the variables of price and capacity.

  2 39 4Q b w2  Q b w2  C 2  8aQ b w2 ðw  C Þ = 4 5
  2 39 4Q b w2  Q b w2  C 2  8K 2 ðw þ C Þ = 5
ðB:4Þ




i¼1 pBi Q m ; Q b ; wbi ; K i ; wbj ; K j

According to the determinant of the above matrix, it can be said that concavity of the sum of objective functions of reliable suppliers depends on the values of two parameters a; c such as competitive situation. So, we should determine domain of these parameters in such a way that the sign of (C.3) become positive. On one hand, If (C.3) is positive (C.4) should be satisfied. On the other hand based on assumptions, 0 < c < 1. So,

In order to clarify conclusions which are extracted from Propositions 1–3, some numerical examples have been presented.
Numerical example from proposition 1: Here we use a numerical example to better demonstrate the function of proposition 1. The parameters’ values are presented below:

a ¼ 0:5; K 1 ¼ 52:5; K 2 ¼ 39:9; wb1 ¼ 16:58; wb2 ¼ 22:77; P ¼ 70; l ¼ 100; r ¼ 20; C q ¼ 10; h ¼ 17;



@w2bi h

 i1 2K j Q b K i w2bj K i þ K j þ K i K j wbj C bi  C bj A  ai Q b < 0 ¼@
3 K i wbj þ K j wbi


i¼1

pBi Q m ; Q b ; wbi ; K i ; wbj ; K j

E¼

ðC:2Þ

Assuming the constancy of capacity, it was understood that the sum of profit functions of reliable suppliers is concave with respect to the price variable. It was also observed that, if the prices are constant, the sum of profit function for reliable suppliers is concave with respect to the capacity variable. The concavity of such function with respect to both variables of price and capacity is so complicated to prove contemporarily. So, as in competitive situation, concavity of the sum of profit functions of reliable suppliers is proved with respect to both variables of price and capacity by assuming symmetric reliable suppliers in cooperative situation.

    2
1    @ pB ðw;K Þ @ 2 pB ðw;K Þ   þ a Q4Kb   @w2  Q b 2w @K@w  jHðw; K Þj ¼  @2 p ðw;K Þ @ 2 p ðw;K Þ  ¼  Q b  B   B  c  4K @w@K @K 2 " #   1 Q 2b ¼ cQ b þa  2w 16K 2

c>

16K

Qb
2 1 2w

  wb1 wb2 ðK 1 þ K 2 Þ þ cq ¼ 28:785 K 2 wb1 þ K 1 wb2

According to Proposition 1, if wm > 28:785 so 
opt opt   1 Q m ; Q b ¼ 0; F ð0:551Þ ¼ ð0; 102:564Þ. In this example D gets a negative value. So, when wm 6 28:785 for example equal
opt opt   1 16, Q m ; Q b ¼ F ð0:676Þ F 1 ð0:112Þ; F 1 ð0:676ÞÞ ¼

to

ð27:5; 81:65Þ.

 ðK 1 þK 2 Þ It is important to say that when cq P a p  cp þ cl  wKb12 wwb2þK , 1w b1

ðC:3Þ

ðC:4Þ

b2

so the value of D is positive. In this example if cq P 23:715, so D gets a positive value. For more clarification we assume that cq ¼ 30. In this situation D = 12.57. It means that if wm < 12:57 
  1 opt ¼ F ð0:735Þ; 0 ¼ for example equal to 10, so Q opt m ; Qb ð112:578; 0Þ.
Numerical example from Proposition 2: We use the following numerical example to more clarify properties of Proposition 2.

C b ¼ 10;

!  þa





1 wb1 wb2 ðK 1 þ K 2 Þ a  þ cq  p  cp þ cl ð1  aÞ K 2 wb1 þ K 1 wb2 ð1  aÞ

¼ 27:43

@K 2i

¼

C p ¼ 15

 D¼




! 2Q b K j wbi w2bj wbi  wbj  C bi þ C bj  ci < 0
3 K i wbj þ K j wbi

C l ¼ 30;

Now we calculate D and E:

ðC:1Þ P2

<1

should be satisfied that is equivalent to (C.5). We could determine the lower bound for both parameters a; c in such a way that the sum of profit functions of reliable suppliers is concave with respect to the variables of price and capacity. Thus, if the values of these two parameters are truly determined, the sum of profit functions of reliable suppliers will be concave with respect to the variables of price and capacity.

0

@2

Qb 1 þa 16K 2 ð2w Þ

Appendix D

In order to prove the concavity of the function obtained from the sum of the profit functions of two reliable suppliers, secondorder condition of this function are studied with respect to the variables of price and capacity using first-order derivatives presented in (32) and (33).

P2

ðC:5Þ

ðB:5Þ

Appendix C

@2

   Qb 1 a > max 0;  16K 2 2w

K 0 ¼ 30;

Q b ¼ 90;

It is clear that a ¼ 0:01 <



1 2C b



a ¼ 0:01;

c ¼ 0:6

¼ 0:05 and c ¼ 0:6 <

h

wC b Q b 2k0

i

¼

0:75. So, ðw ; K  Þ ¼ ð32:656; 47:781Þ. If, c ¼ 0:8, so a ¼ 0:01 <   h i 1 b ¼ 0:05 and c ¼ 0:8 > QwC ¼ 0:75. Therefore, ðw ; K  Þ ¼ 2C b b 2k0   ð32:656; 45Þ. Finally if a ¼ 0:06 > 2C1 ¼ 0:05, so ðw ; K  Þ ¼ b

ð10; 45Þ.

N. Mohammadzadeh, S.H. Zegordi / Computers & Industrial Engineering 101 (2016) 194–215


Numerical example from Proposition 3: We calculate the optimal strategies based on Proposition 3 by using the same parameters’ values applied to solve Proposition 2.   It is clear that a ¼ 0:01 < 2C1 ¼ 0:05. So, ðw ; K  Þ ¼ ð50; 45Þ. If, b   a ¼ 0:06 > 2C1 ¼ 0:05, therefore ðw ; K  Þ ¼ ð10; 45Þ. By comb

paring optimal policies of reliable suppliers in two competitive   and cooperative situations, it is found that if a 6 2C1 , cooperb

ative prices are greater than competitive prices. When   a > 2C1 , cooperative prices and competitive prices get the b

same value which is equal to production cost of reliable suppliers.

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