5.8 Numerical Example
5.8.4 Comparison among the results
The last few subsections have presented the results of the numerical applications using the proposed bargaining models. There are notable similarities in the results from different bar- gaining models. Both the Nash and Kalai Smorodinsky bargaining applications to the supply chain under consideration yields similar results. In both cases, the fixed price contract solu- tions are dominated for both the members of the supply chain. The project manager and the contractor both were found to have higher utility with the increase in the deciding contract
parameter b ∈ [0, 1]. Hence, the cost plus contract with b=1 was found to dominate any other solutions in both Nash and Kalai Smorodinsky bargaining cases. Apart from individual util- ities, Nash product in the case of Nash bargaining or the Kalai Smorodinsky value K in the case of Kalai Smorodinsky bargaining case were also found to dominate the solutions in the case of cost plus contracts with b=1.
In comparison to the Nash bargaining and Kalai Smorodinsky bargaining, the Utilitarian approach was found to have some distinctly different features. The sum utility function U was found to have a dominating solution for the cost plus contract with b=1 for all the distribution selected. This was mainly due to the existence of the dominating solutions of the utility function of the project manger at b = 1 (Cost plus contract) than in comparison to the solutions with b = 0.5 and b = 0.0. However, the utility of the contractor was found to be same for any values in the range b ∈ [0, 1]. In fact the utility of the contract was found to be independent of the nature of the probability distribution of the project cost. This can be explained from the equation (5.107). As evident from the equation, the Ucowas found to be independent of any
distribution specific parameters of the cost function. The utility of the contractor was found to be dependent only on the parameter η.
5.9
Chapter Summary
This chapter has investigated how to reach the optimal solution to coordinate supply chain with negotiation between the project manager and the contractor. This chapter used
• A cost based contract P= a+bX (Where X is a continuous cost function; a is the fixed component of the contract; and b is the variable part of the contract with b ∈ [0, 1]). • A time based contract with P=g-hT (where g is the fixed part and h is the penalty per
unit to entice the contractor for early completion)
• In either of the above contracts, if the variable part becomes zero (b or h), it becomes a fixed price contract
The bargaining approach used includes Nash bargaining, Kalai Smorodinsky bargaining, and Utilitarian approach. The models were prepared for two different situations: both the mem- bers are risk neutral, and the project manager is risk neutral and the contractor is risk averse.
For the case when both the supply chain members (the project manager and the contractor) are risk neutral, no clear dominance of solutions were observed by comparing the fixed price with either time based or cost based contracts. This, may entice the members of the supply chain under consideration to implement fixed price contract instead of implementing any complicated contracts.
On the contrary, if the project manager is risk neutral and the contractor is the risk averse, then the results are different and a dominance of solutions were observed. In comparison be- tween the time based and the fixed price contracts, the solutions from the fixed price contract were found to dominate the solutions from any time based contracts. In comparison between the cost based contract, and the fixed price contract, the results from the cost based contracts were found to dominate the results from the fixed price contracts. In fact, the cost plus con- tract with b = 1 was found to dominate any other cost sharing contracts with 0 < b < 1. This dominance was found to be strictly dominating for the case of Nash and Kalai Smorodinsky bargaining for all the members of the supply chain under consideration and the respective bargaining parameters (the Nash product N and the Kalai Smorodinsky bargaining value K). However, the contractor’s utility was found to be independent of the variable part of the of- fered contract (either b or h depending on the case) in the case of Utilitarian approach of bargaining. Hence, the utility of the contractor remained unchanged with respect to the fixed price contract in the utilitarian bargaining approach.
Chapter 6
Fairness and Fair Allocation in Project
Supply Chain
This research has addressed the issue of supply chain coordination in take it or leave it situ- ation in chapter 4 and extended it to bargaining situations in chapter 5. However, issues on the allocation of risk and benefits of supply chain coordination were not discussed. Chapter 2 highlighted the problems arise in absence of fair allocation of risk and benefits.
Justice or fairness has been conceptualized since the time of Aristotle and Plato (Liu et al. 2012). However, the authors concluded based on past research evidence how justice or fairness has been perceived differently in different contexts. The concept of fairness or justice has been studied for a long time in various economic and social exchange (Adams 1965, Lind & Tyler 1988, Greenberg & Cropanzano 1993).Various studies in economics and marketing have addressed the importance of fairness in the social exchange (Frazier 1983, Heide & John 1988, Corsten & Kumar 2005).
Liu et al. (2012) highlighted how four dimensions of justice have been developed in the literature over the last few decades. These are distributive, procedural, interpersonal and informational. The authors considered the first two dimensions as part of structural fairness and the last two as part of social fairness.
Despite its importance and long rooted existence in the economics and other social sci- ence related literature, the applications in supply chain related exchanges is relatively new. As per the knowledge of the author of the present research, Cui et al. (2007) is one of the pioneer authors who investigated the issues of distributive fairness in supply chain coordina-
tion. Following their study, authors including Loch & Wu (2008), Caliskan-Demirag et al. (2010), and Ho et al. (2014) proposed various models of supply chain coordination with fair- ness considerations. However, these studies were conducted with product supply chains with supply contracts and the quantity demanded as the decision variable. This chapter presents the models as an extension to the early studies in the project supply chain setting with the project contracts.
The question is what is the importance of fairness in the context of supply chain coordi- nation. The distributive and procedural fairness have been found to have a positive impact on long term relation between a firm and its distributor in a supply chain and ultimately on the overall performance (Griffith et al. 2006). Absence of fairness has been found as one of the factors leading to the failure of supply chain coordination relationship in some recent stud- ies. Katok & Pavlov (2013) explored the reasons leading to the termination of coordinated contractual relationships in a supply chain using behavioural laboratory experiments. The authors found lack of inequality aversion, incomplete information and bounded rationality as three reasons for this failure. This finding also supports the findings of Wu (2013b) where the authors found rejecting behaviours from retailers in a supply chain when experiencing unfair offers from the suppliers. Some of the examples from practice corroborate the importance of the need for the fair allocation of the risks and benefits in the coordinated relationship such as the termination of the contractual relationship between Walmart Canada and Lego group upon rejection by Lego group to reduce the price in the Canadian market. Lego group kept the price same as in the American market and reaped additional benefits due to the appre- ciation of Canadian dollar (Georgiades 2008). Similar was the case with the breakdown of contractual relationships between Chinese home appliance retailer Gome and air condition manufacturer Gree (Liu et al. 2012).
Traditionally, it used to be believed that the participants only care about the rational profit maximization as their objective in contractual agreements. However, some experimental studies have shown the existence of fairness considerations from the participants (Loch & Wu 2008, De Bruyn & Bolton 2008). More interestingly, this kind of caring behaviour has been observed not only in the take it or leave it environments, but also in the bargaining en- vironments as well (Camerer 2003). The classification of justice or fairness in buyer-seller relationship by Liu et al. (2012) provides the basic starting point. The interesting fact is the
consideration of distributive fairness as one of the forms of fairness consideration in supply chain literature. Authors including Fehr & Schmidt (1999), Bolton & Ockenfels (2000), and Charness & Rabin (2002) have defined fairness from somewhat different perspectives. Fehr & Schmidt (1999) defined fairness from an inequity aversion perspective, whereas the authors in two other studies defined fairness from reciprocity perspective. This has been supported in the literature of Falk & Fischbacher (2006). Some other notable extensions have been doc- umented in literature specific to the supply chain coordination such as peer-induced fairness (Ho et al. 2014). In a recent study by Du et al. (2014), the authors were critical of the models proposed by Fehr & Schmidt (1999) from applicability point of view. The authors used Nash bargaining solution as the fairness reference point solution. This research summarises this debate by the existence of context-specific nature of fairness consideration. This has been supported in the literature by Liu et al. (2012).
This research has used the definition of fairness proposed by Fehr & Schmidt (1999) as the reference for the take it or leave it situations and in some case for bargaining situations as well. One of the main reasons is its ability to best describe the context of the coordination problem considered for this research. Based on the definition by Fehr & Schmidt (1999), Cui et al. (2007) identified that simple wholesale price contracts can coordinate a manufacturer- retailer supply chain when members are fairness concerned. This model of Cui et al. (2007) has been extended in various different directions such as: models with non-linear demand (Caliskan-Demirag et al. 2010); and when the supplier’s fairness concerns are private infor- mation in a supplier-retailer supply chain (Katok et al. 2014). The authors found that under this situation the wholesale price contract can coordinate the supply chain as it did in the case of information symmetry in the models of Cui et al. (2007). Voigt & Inderfurth (2012) used the concepts of inequity aversion in a similar context with asymmetric holding cost informa- tion. There are other supply chain contexts where fairness in allocation has been considered such as cooperative advertising (Yang et al. 2013).
This research did not find any evidence of any supply chain coordination model including fairness consideration alongside profit maximizing objective in the project settings. Thus, this chapter addresses the third objective of this research
Objective 3. To investigate if the supply chain can be coordinated with fairly allocated risks and benefits in the scenarios mentioned in objectives 1 and 2.
The first part presents the analysis in take it or leave it situations with Stackelberg games. The approach proposed by Cui et al. (2007) has been used as the reference for this. The second part follows this up with some analysis of fairness considerations in bargaining situations.
6.1
Problem Description
As described earlier in chapter 4, the coordination problem is analysed with Stackelberg leader-follower games in the take it or leave it situation. The project manager is considered as the leader and the contractor is considered as the follower. In the supply chain literature, by Cui et al. (2007) and Caliskan-Demirag et al. (2010), the authors used a fixed wholesale price contract to see if it can coordinate the supply chain under consideration with the existence of the fairness concern. This research uses a fixed price contract to investigate if it can achieve the coordination requirements in the event of the presence of fairness concern.
Following the definition of the fairness in Fehr & Schmidt (1999) and the approach by Cui et al. (2007), the utility equation of the member i in a two member supply chain (with member i and member j) is
Ui(λ, P (T, C)) = πi+ Di(λ, P (T, C)); i ∈ {pm, co} (6.1)
The first part of the equation (6.1) corresponds to the monetary profit of the member i of the supply chain. The second part of the equation i.e. Di(λ, f ) is the member i’s disutility due
to inequity or unfairness. As per this model, the member would incur some disutility if (s)he earns more than or less than the profit (s)he believes is fair or equitable. This fair equitable profit perceived by the member is compared against the profit of the other member. Let γπpmand δπcoare the equitable profit as perceived by the contractor and the project manager
respectively in the supply chain under consideration (where γ > 0 and δ > 0). Based on the suggestion of Caliskan-Demirag et al. (2010), these factors γ and δ are exogenous to the members and are calculated based on the outside options available to the members of the supply chain under consideration. αi and βi (i ∈ {pm, co}) are the disutility to the member
per unit due to earning less (disadvantageous inequity) and more (advantageous inequity) in comparison to the other member. Authors including Fehr & Schmidt (1999), Cui et al. (2007) and Caliskan-Demirag et al. (2010), suggested based on previous research that the member is
more sensitive to disadvantageous inequity (earning less) than advantageous utility (earning more). Thus, it is assumed that αi ≥ βi. It is also assumed that 0 < βi < 1 in accordance with
the existing literature. Thus, based on the definition of Fehr & Schmidt (1999), the disutility due to inequity or unfairness is defined as below
Dpm(λ, P (T, C)) = −αpm[max{(δπco− πpm), 0}] − βpm[max{(πpm− δπco), 0}] (6.2)
and
Dco(λ, P (T, C)) = −αco[max{(γπpm− πco), 0}] − βco[max{(πco− γπpm), 0}] (6.3)
where αpm≥ βpm; 0 < βpm< 1; αco ≥ βco; and 0 < βco< 1.
As mentioned in the chapter 4, the coordination problem is solved using backward induction method from game theory. Given an offer of a contract price of P(T,C), the contractor would select a resource consumption rate λ that maximizes his profit if the contractor does not have any fairness concern. However, in the event of the presence of a fairness concern, the contractor will select a λ that maximizes his utility as below
Uco = πco− αco[max{(γπpm− πco), 0}] − βco[max{(πco− γπpm), 0}] (6.4)
The project manager would incorporate this requirement of λ in her take it or leave it offer and selects the value of P(T,C) that maximizes her profit (if she is not fairness concerned) given the constraint of λ. In the event, the project manager is fairness concerned, she selects a value of P(T,C) that maximizes her utility as below
Upm= πpm− αpm[max{(δπco− πpm), 0}] − βpm[max{(πpm− δπco), 0}] (6.5)
The next few subsections explore the coordination problems with fairness concerned mem- bers for different types of contracts used in chapter 4.