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Payoff Allocation Methods for Several Operational Research Games

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Payoff Allocation Methods for Several Operational Research Games UNIVERSITY OF SOUTHAMPTON FACULTY OF BUSINESS, LAW AND ART Southampton Business School Payoff Allocation Methods for Several Operational Research Games by Phuoc Hoang Le Thesis for the degree of Doctor of Philosophy May 2017 University of Southampton Doctoral Thesis Payoff Allocation Methods for Several Operational Research Games Author: Phuoc Hoang Le Supervisor: Dr. Tri-Dung Nguyen Prof. Tolga Bektas¸ Examiners: Dr. Patrick Beullens Dr. Katerina Papadaki Southampton 2016 ii UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF BUSINESS, LAW AND ART Southampton Business School Doctor of Philosophy in Management Science PAYOFF ALLOCATION METHODS FOR SEVERAL OPERATIONAL RESEARCH GAMES by Phuoc Hoang Le From the start of this century, enforcing and maintaining collaborations became one of the trends in business and management, ranging from production and inventory to transportation and network communication. Because of the economies of scale, many enterprises might collaborate to generate more value and save costs instead of working on their own. The rapid change in science and information technologies enables users to communicate and share data more efficiently through different digital platforms. These emerging technologies open several opportunities for companies and institutions to allow potential coordination and collaboration. This development has created the sharing economy which enables individuals, households, businesses, and organisations to engage in collaborative production, distribution, and consumption of goods and services. The interactions among players willing to cooperate are the underlying theme of Cooper- ative Game Theory. In this field, payoff (yield/cost) sharing problems aim at producing solutions with two main criteria: fairness and stability. These principles lead to some popular solution concepts such as the Shapley value and the core, which concern dis- tributing the payoff to members of a coalition. The main contributions of this thesis are the developments of some computationally efficient methods to calculate these solution concepts in the generalised minimum spanning tree, linear production, and multidimen- sional integer knapsack games. These are the subclasses of operational research games where the characteristic functions of these cooperative games are described by linear or integer linear programming formulations. The minimum-cost spanning tree game is a particular class of cooperative games defined on a graph, where each player owns a vertex. Solutions of the game represent ways to distribute the total cost of a minimum-cost spanning tree among all the players. When we partition the graph into clusters, the generalised minimum spanning tree problem is to determine a minimum-cost tree including exactly one vertex from each cluster. iii The first chapter introduces and studies the generalised minimum spanning tree game and some of its properties. We propose a constraint generation algorithm to calculate a stable payoff distribution and present some computational results obtained using the proposed algorithm. Another class of operational research games, called linear production game, is concerned with allocating the total payoff of an enterprise among the owners of the resources in a fair way. With cooperative game theory provides a mathematical framework for sharing the benefit of the cooperation, the Shapley value is one of the widely used solution concepts as a fair measurement in this area. Finding the exact Shapley values for linear production games is, however, challenging when the number of players exceeds 30. This chapter describes the deploy of linear programming sensitivity analysis for more efficient computation. We also propose the stratified sampling technique to estimate the Shapley value for large-scale linear production games. Computational results show the effectiveness of these compared to other existing methods. The integer optimisation game, where the characteristic function is generated by solving an integer optimisation problem, forms an important class in cooperative game theory. In this chapter, we introduce a new subclass of integer optimisation games namely the multidimensional integer knapsack game. Finding the Shapley value is challenging for most integer optimisation games with a large number of players, particularly for the multidimensional integer knapsack game due to the structure of its characteristic function. We describe an algebraic approach using Gr¨obnerbases to be able to compute the Shapley value efficiently. Some computational experiments are presented to show the potential of the proposed method compared to CPLEX, the state-of-the-art commercial solver on some randomly generated instances. iv Tóm t­t khóa luªn PHƯƠNG PHÁP PHÂN PHÈI KẾT QUẢ CHO MËT SÈ TRÁ CHƠI VẬN TRÙ Tø b­t đầu cõa th¸ kỷ 21, vi»c t«ng cường và duy tr¼ sự cëng t¡c đã trở thành mët xu hướng m¤nh trong kinh doanh và qu£n lý, đi·u đó được thº hi»n trong nhi·u lĩnh vực tø s£n xu§t và lưu trú hàng ho¡ cho đến vªn chuyºn và m¤ng lưới thông tin. Do lñi th¸ kinh t¸ nhờ quy mô, c¡c c¡ thº thường muèn hñp t¡c để t¤o ra nhi·u gi¡ trị hơn và ti¸t ki»m chi ph½ thay v¼ ho¤t động ri¶ng l´. Sự thay đổi nhanh chóng trong khoa học và công ngh» thông tin ngày nay cho ph²p người dùng có thº giao ti¸p và chia s´ dú li»u hi»u qu£ thông qua nhi·u n·n t£ng kỹ thuªt sè kh¡c nhau. Nhúng công ngh» mới đó đã mở ra nhi·u cơ hëi hñp t¡c cho c¡c công ty và tªp đoàn kinh doanh. Sự ph¡t triºn này t¤o ra n·n kinh t¸ chia s´ để cho ph²p c¡c c¡ nh¥n, hë gia đình, và công ty có kh£ n«ng cùng s£n xu§t, ph¥n phèi và ti¶u thụ c¡c hàng hóa và dich vụ. Sự tương t¡c giúa nhúng người chơi muèn cùng nhau làm vi»c là mët trong nhúng chõ đề ch½nh cõa lý thuy¸t trá chơi hñp t¡c. Trong ngành nghi¶n cùu này, sự chia s´ k¸t qu£ (chi ph½) nh¬m mục đích t¼m ra c¡ch ph¥n phèi tho£ m¢n c¡c thành vi¶n tham gia theo hai ti¶u ch½ ch½nh: t½nh công b¬ng và t½nh ên định. Nhúng nguy¶n lý này d¨n đến mët sè kh¡i ni»m nêi ti¸ng trong lý thuy¸t trá chơi như tªp lãi (h¤t nh¥n) và gi¡ trị Shapley, nh¬m ph¥n phèi k¸t qu£ đầu ra cho c¡c thành vi¶n trong nhóm cëng t¡c. Đóng góp ch½nh cõa khóa luªn là sự ph¡t triºn nhúng phương ph¡p kh¡c nhau để t½nh to¡n nhúng kh¡i ni»m này trong bài to¡n nèi c¥y tèi thiºu têng qu¡t, bài to¡n s£n xu§t tuy¸n t½nh và bài to¡n balô nhi·u chi·u. Đây là nhúng lớp con cõa trá chơi vªn trù, trong đó hàm đặc trưng cõa nhúng trá chơi hñp t¡c này được mô t£ bởi h» qui ho¤ch tuy¸n t½nh hoặc h» qui ho¤ch nguy¶n. Trá chơi nèi c¥y tèi thiºu là mët lớp con cõa trá chơi hñp t¡c được định nghĩa tr¶n mët đồ thị với tªp đỉnh và tªp c¤nh x¡c định, trong đó méi người chơi sở húu mët đỉnh. Mët lời gi£i cõa trá chơi này đại di»n cho c¡ch ph¥n phèi têng chi ph½ cõa bài to¡n c¥y tèi thiºu cho nhúng thành vi¶n tham gia. Khi chúng ta chia đồ thị này thành nhúng cụm nhỏ, bài to¡n c¥y nèi tèi thiºu têng qu¡t nh¬m mục đích x¡c định mët c¥y tèi thiºu sao cho nó chùa ch½nh x¡c mët đỉnh trong méi cụm. Chương đầu ti¶n này giới thi»u và t¼m hiºu trá chơi nèi c¥y tèi thiºu và nhúng đặc t½nh cõa nó. Chúng tôi đề xu§t phương ph¡p gi£i thuªt t¤o lớp c­t nh¬m ph¥n phèi chi ph½ mët c¡ch ên định và đưa ra nhúng k¸t qu£ t½nh to¡n cho phương ph¡p này. Trá chơi s£n xu§t tuy¸n t½nh nh¬m ph¥n phèi mët c¡ch công b¬ng têng gi¡ trị cõa k¸t qu£ đầu ra cõa mët nhóm đầu tư cho c¡c thành vi¶n sở húu tài nguy¶n. Gi¡ trị Shapley v là mët trong nhúng kh¡i ni»m được dùng r§t phê bi¸n trong lý thuy¸t trá chơi hñp t¡c, nh¬m cung c§p mët khuôn khê t½nh to¡n cho vi»c chia s´ k¸t qu£ cõa sự cëng t¡c mët c¡ch công b¬ng. Tuy nhi¶n, vi»c t¼m ch½nh x¡c gi¡ trị Shapley cho trá chơi s£n xu§t tuy¸n t½nh là r§t khó kh«n khi trá chơi có hơn 30 thành vi¶n. Chương này mô t£ c¡ch dùng sự k¸t hñp giúa phương ph¡p ph¥n t½ch độ nh¤y qui ho¤ch tuy¸n t½nh và phương ph¡p l§y m¨u thû x¸p t¦ng để x§p x¿ gi¡ trị Shapley cho trá chơi s£n xu§t tuy¸n t½nh với nhi·u thành vi¶n. K¸t qu£ t½nh to¡n s³ ch¿ ra sự hi»u qu£ cõa nhúng phương ph¡p này khi chúng tôi so s¡nh với c¡c c¡ch thùc chia s´ đã có cho cùng bài to¡n.
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