A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs

This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.

Low carbon joint strategy and coordination for a dyadic supply chain with Nash bargaining fairness

<p style='text-indent:20px;'>In the paper, fairness concern criterion is utilized to explore the coordination of a dyadic supply chain with a fairness-concerned retailer (acting as a newsvendor), who is committed to low carbon efforts. Two models are developed for stochastic demand disturbances in the forms of multiplicative case and additive case, respectively. Firstly, the optimal joint decision of the retailer and the supply chain are proposed in two scenarios, i.e., decentralized decision and the centralized decision. Secondly, in order to realize channel coordination, the contract of revenue sharing combined with the mechanism of low-carbon cost sharing is designed. Moreover, the influences of the retailer's fairness concern and bargaining power on the joint decision and the contract parameters are also investigated. Finally, numerical examples are given to illustrate the theoretical results and some suggestions to supply chain management are also provided. The results show that the revenue sharing contract can make the supply chain achieved coordination with the cost sharing mechanism of low-carbon efforts. Furthermore, the optimal low-carbon effort level and ordering quantity decrease in terms of fairness-concerned parameter and Nash bargaining power parameter, which increases in unit cost. However, the optimal pricing makes the opposite change.</p>

Changed Industrial Relationship by Offshoring Threat and Labor Share: An Empirical Game-Theoretic Analysis

Research indicates that the labor share of the aggregate income has decreased steadily since the mid-1970s, i.e. when the globalization process began. This paper discusses the ways in which qualitative changes in globalization, coupled with increased offshoring, have changed industrial relationships. In our analysis, we consider a simple Nash bargaining model between employers and employees. Our model proposes the hypotheses that employees gain the power to increase their wages when employers do not have the option of offshoring. However, employees typically lose this power when employers possess an offshoring threat, culminating in wage deduction. Leveraging a panel set of data obtained from 18 OECD countries during the period 1975–2017, we have empirically confirmed these hypotheses by comparing the first phase of globalization—not characterized by an offshoring threat—with the second phase, which entails an offshoring threat. Our findings reveal that workers’ bargaining power, positively affects labor share in the first phase; however, it loses its effect in the second phase when offshoring exerts its negative effects on labor share. We conclude that a qualitative change in globalization with increased offshoring radically changed industrial relationship through the threat effect.

Socio-legal systems and implementation of the Nash solution in Debreu–Hurwicz equilibrium

In this article we combine Debreu’s (Proc Natl Acad Sci 38(10):886–893, 1952) social system with Hurwicz’s (Econ Design 1(1):1–14, 1994; Am Econ Rev 98(3):577–585, 2008) ideas of embedding a “desired” game form into a “natural” game form that includes all feasible behavior, even if it is “illegal” according to the desired form. For the resulting socio-legal system we extend Debreu’s concepts of a social system and its social equilibria to a socio-legal system with its Debreu–Hurwicz equilibria. We build on a more general version of social equilibrium due to Shafer and Sonnenschein (J Math Econ 2(3):345–348, 1975) that also generalizes the dc-mechanism of Koray and Yildiz (J Econ Theory 176:479–502, 2018) which relates implementation via mechanisms with implementation via rights structures as introduced by Sertel (Designing rights: invisible hand theorems, covering and membership. Tech. rep. Mimeo, Bogazici University, 2001). In the second part we apply and illustrate these new concepts via an application in the narrow welfarist framework of two person cooperative bargaining. There we provide in a socio-legal system based on Nash’s demand game an implementation of the Nash bargaining solution in Debreu–Hurwicz equilibrium.

Green Supply Chain Management with Nash Bargaining Loss-Averse Reference Dependence

This paper investigates a two-echelon green supply chain (GSC) with a single loss-averse manufacturer and a single loss-averse retailer. Since the Nash bargaining solution exactly characterizes endogenous power and the contribution of the GSC members, it is introduced as the loss-averse reference point for the GSC members. Based on this, a decision model of the two-echelon GSC with loss aversion is formulated. The optimal strategies of price and product green degree are derived in four scenarios: (a) the centralized decision scenario with rational GSC members, namely the CD scenario; (b) the decentralized decision scenario with rational GSC members, namely the DD scenario; (c) the decentralized decision scenario with the GSC members loss-averse, where the manufacturer’s share is below its own loss-averse reference point, namely the DD(∆m ≥ πm) scenario; (d) the decentralized decision scenario with the GSC members loss-averse, where the retailer’s share is below its own loss-averse reference point, namely the DD(∆r ≥ πr) scenario. Then, a comparative analysis of the optimal strategies and profits in these four scenarios is conducted, and the impacts of loss aversion and green efficiency coefficient of products (GECP) on the GSC are also performed. The results show that (i) GECP has a critical influence on the retail price and the wholesale price; (ii) the GSC with loss aversion provide green products with the lowest green degree; (iii) the retail price, the wholesale price and product green degree are decreasing monotonically with the loss aversion level of the GSC member without incurring loss; (iv) furthermore, the effect of the loss aversion level of the GSC member with incurring loss on the optimal strategies is related to GECP and the gap between the GSC members’ loss aversion levels.

Efficiency decomposition in a three-stage network structure: Cooperative DEA, Nash bargaining game models and SOCP formulations

In this paper, for evaluating the efficiency in a three-stage DEA structure we use the additive and the multiplicative cooperative models that comply with the cooperation paradigm in the organizations, where for improving efficiency of system, stages cooperate together. Since the overall efficiency from the cooperative models may not be unique and consequently the stages’ efficiencies, then we combine them with the Nash bargaining game approach that besides maximizing efficiency scores for stages and the whole system, provides a unique and fair efficiency decomposition. Second order programming relaxation of the proposed nonlinear models are given in contrast to the parametric linear models in the literature. Finally, the effectiveness of the proposed models are illustrated with two numerical examples.