A generalized linear production model: A unifying model

doi:10.1007/BF01580585

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A generalized linear production model: A unifying model

Journal Article•DOI•

A generalized linear production model: A unifying model

Daniel Granot¹•Institutions (1)

University of British Columbia¹

01 Mar 1986-Mathematical Programming (Springer-Verlag New York, Inc.)-Vol. 34, Iss: 2, pp 212-222

TL;DR: It appears that the generalized linear production model is a unifying model which can be used to explain the non-emptiness of the core of cooperative games generated by various, seemingly different, optimization models.

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Abstract: We introduce a generalized linear production model whose attractive feature being that the resources held by any subset of producersS is not restricted to be the vector sum of the resources held by the members ofS. We provide sufficient conditions for the non-emptiness of the core of the associated generalized linear production game, and show that if the core of the game is not empty then a solution in it can be produced from a dual optimal solution to the associated linear programming problem. Our generalized linear production model is a proper generalization of the linear production model introduced by Owen, and it can be used to analyze cooperative games which cannot be studied in the ordinary linear production model framework. We use the generalized model to show that the cooperative game induced by a network optimization problem in which players are the nodes of the network has a non-empty core. We further employ our model to prove the non-emptiness of the core of two other classes of cooperative games, which were not previously studied in the literature, and we also use our generalized model to provide an alternative proof for the non-emptiness of the core of the class of minimum cost spanning tree games. Thus, it appears that the generalized linear production model is a unifying model which can be used to explain the non-emptiness of the core of cooperative games generated by various, seemingly different, optimization models.

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Journal Article•DOI•

Cost Allocation in Collaborative Forest Transportation

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Mikael Frisk¹, Maud Göthe-Lundgren², Kurt Jörnsten³, Mikael Rönnqvist³•Institutions (3)

Forestry Research Institute of Sweden¹, Linköping University², Norwegian School of Economics³

01 Nov 2006-Social Science Research Network

TL;DR: In this article, the authors investigate a number of possibilities based on economic models including Shapley value, the nucleolus, separable and non-separable costs, shadow prices and volume weights.

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Abstract: Transportation planning is an important part of the wood flow chain in forestry. There are often several forest companies operating in the same region and co-ordination between two or more companies is rare. However, there is an increasing interest in collaborative planning as the potential savings are large, often in the range 5-15%. A key question is how savings should be distributed among the participants. In this paper we investigate a number of possibilities based on economic models including Shapley value, the nucleolus, separable and non-separable costs, shadow prices and volume weights. We also propose a new allocation method based on finding as equal relative profits as possible among the participants. A case study including eight forest companies is described and analyzed.

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335 citations

Journal Article•DOI•

Cost allocation in collaborative forest transportation

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Mikael Frisk¹, Maud Göthe-Lundgren², Kurt Jörnsten³, Mikael Rönnqvist³, Mikael Rönnqvist¹ - Show less +1 more•Institutions (3)

Forestry Research Institute of Sweden¹, Transport Research Institute², Norwegian School of Economics³

01 Sep 2010-European Journal of Operational Research

TL;DR: In this paper, the authors investigated a number of sharing mechanisms based on economic models including Shapley value, the nucleolus, separable and non-separable costs, shadow prices and volume weights, with the aim that the participants relative profits are as equal as possible.

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332 citations

Journal Article•DOI•

Operations Research Games: a Survey

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Peter Borm¹, Herbert Hamers¹, Ruud Hendrickx¹•Institutions (1)

Tilburg University¹

01 Dec 2001-Top

TL;DR: This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers (players) are involved on the basis of a distinction between the nature of the underlying optimisation problem: connection, routing, scheduling, production and inventory.

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Abstract: This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers (players) are involved. Cooperating players not only face a joint optimisation problem in trying, e.g., to minimise total joint costs, but also face an additional allocation problem in how to distribute these joint costs back to the individual players. This interplay between optimisation and allocation is the main subject of the area of operations research games. It is surveyed on the basis of a distinction between the nature of the underlying optimisation problem: connection, routing, scheduling, production and inventory.

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211 citations

Cites background or methods from "A generalized linear production mod..."

...Granot (1986) and Curiel et al. (1989) consider LP processes where (simple) control games on (bundles of) resources determine the resource bundle available to each coalition....
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...4 some results are stated from Hamers (1997), Granot et al. (1999) and Granot and Hamers (2000)....
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...Consistency of the home-down painting solution is studied in Granot et al. (1996), Granot and Maschler (1998) and Van Gellekom and Potters (1999)....
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...and Granot and Hamers (2000) have characterised concavity by the structure of the available edges....
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...The following theorem comes from Granot and Huberman (1981)....
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Journal Article•DOI•

Algorithmic Aspects of the Core of Combinatorial Optimization Games

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Xiaotie Deng¹, Toshihide Ibaraki², Hiroshi Nagamochi²•Institutions (2)

City University of Hong Kong¹, Kyoto University²

01 Aug 1999-Mathematics of Operations Research

TL;DR: The computational complexity and algorithms of the core are studied to answer important questions about the cores of various games on graphs, such as maximum flow, connectivity, maximum matching, minimum vertex cover, minimum edge cover, maximum independent set, and minimum coloring.

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Abstract: We discuss an integer programming formulation for a class of cooperative games. We focus on algorithmic aspects of the core, one of the most important solution concepts in cooperative game theory. Central to our study is a simple but very useful observation that the core for this class is nonempty if and only if an associated linear program has an integer optimal solution. Based on this, we study the computational complexity and algorithms to answer important questions about the cores of various games on graphs, such as maximum flow, connectivity, maximum matching, minimum vertex cover, minimum edge cover, maximum independent set, and minimum coloring.

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197 citations

Posted Content•

A Stochastic Programming Duality Approach to Inventory Centralization Games

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Xin Chen¹, Jiawei Zhang¹•Institutions (1)

New York University¹

01 Sep 2007-Social Science Research Network

TL;DR: It is proved that the newsvendor game with concave ordering cost has a nonempty core and it is NP-hard to determine whether a given allocation is in the core of the inventory games even in a very simple setting.

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Abstract: In this paper, we present a unified approach to study a class of cooperative games arisingfrom inventory centralization. The optimization problems corresponding to the inventory games are formulated as stochastic programs. We observe that the strong duality of stochastic linear programming not only directly leads to a series of recent results concerning the non-emptiness of the cores of such games, but also suggests a way to find an element in the core. The proposed approach is also applied to inventory games with concave ordering cost. In particular, we showthat the newsvendor game with concave ordering cost has a non-empty core. Finally, we prove that it is NP-hard to determine whether a given allocation is in the core for the inventory games even in a very simple setting.

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92 citations

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Journal Article•DOI•

College Admissions and the Stability of Marriage

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David Gale¹, Lloyd S. Shapley¹•Institutions (1)

RAND Corporation¹

01 Jan 1962-American Mathematical Monthly

TL;DR: In this article, the authors studied the relationship between college admission and the stability of marriage in the United States, and found that college admission is correlated with the number of stable marriages.

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Abstract: (2013). College Admissions and the Stability of Marriage. The American Mathematical Monthly: Vol. 120, No. 5, pp. 386-391.

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5,655 citations

Journal Article•DOI•

The assignment game I: The core

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Lloyd S. Shapley¹, Martin Shubik²•Institutions (2)

RAND Corporation¹, Yale University²

30 Sep 1971-International Journal of Game Theory

TL;DR: In this article, it was shown that the optimal assignment game is a dual problem of a linear programming problem dual to optimal assignment, and that these outcomes correspond exactly to the price lists that competitively balance supply and demand.

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Abstract: The assignment game is a model for a two-sided market in which a product that comes in large, indivisible units (e.g., houses, cars, etc.) is exchanged for money, and in which each participant either supplies or demands exactly one unit. The units need not be alike, and the same unit may have different values to different participants. It is shown here that the outcomes in thecore of such a game — i.e., those that cannot be improved upon by any subset of players — are the solutions of a certain linear programming problem dual to the optimal assignment problem, and that these outcomes correspond exactly to the price-lists that competitively balance supply and demand. The geometric structure of the core is then described and interpreted in economic terms, with explicit attention given to the special case (familiar in the classic literature) in which there is no product differentiation — i.e., in which the units are interchangeable. Finally, a critique of the core solution reveals an insensitivity to some of the bargaining possibilities inherent in the situation, and indicates that further analysis would be desirable using other game-theoretic solution concepts.

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1,751 citations

"A generalized linear production mod..." refers background in this paper

...2 The assignment game introduced by Shapley and Shubik [17] can be cast as a network flow problem in which the players are the nodes of a (bipartite) graph and the resources are the arcs in the graph, see also Granot and Granot [13]....
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Journal Article•DOI•

Maximum matching and a polyhedron with 0,1-vertices

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Jack Edmonds

01 Jan 1965-Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics

TL;DR: The emphasis in this paper is on relating the matching problem to the theory of continuous linear programming, and the algorithm described does not involve any "blind-alley programming" -which, essentially, amounts to testing a great many combinations.

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Abstract: An algorithm is described for optimally pamng a finit e set of objects. That is, given a real numerical weight for each unordered pair of objects in a se t Y, to selec t a family of mutually di sjoint pairs th e sum of whose wei ghts is maximum . The well-known optimum assignment proble m [5)2 is the sp ecial case where Y partitions into two se ts A and B suc h that pairs contained in A and pairs contain ed in Bare not positively weighted and therefo re are superfluous to the problem. For this \"bipartite\" case the algorithm becomes a variant of the Hungarian method [3]. The problem is treated in terms of a graph G whose nodes (vertices) are the objects Y and whose edges are pairs of objects, including at leas t all of th e positively weighted pairs. A matching in G is a subse t of its edges such that no two mee t the same node in in G. The proble m is to find a maximum-weight-sum matching in C. Th e special case where all th e positive weights are one is treated in detail in [2] and [6]. The description here of the more general algorithm uses the terminology set up in [2]. Paper [2] (especially sec . 5) helps also to motivate thi s paper, though it is not r eally a prerequisate till section 7 here . The incr ease in difficulty of the maximum weightsum matc hing algorithm relative to the s ize of the graph is not expone ntial, and only moderately algebraic. The algorithm does not involve any \"blind-alley programming\" -which, essentially, amounts to testing a great many combinations . The emphasis in this paper is on relating the matching problem to the theory of continuous linear

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1,712 citations

"A generalized linear production mod..." refers background or methods in this paper

...By Edmonds [7], the feasible region (7) has the 'integrality property' ....
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...Next, we will use results of Edmonds [7] to formulate the m.c.s.t, game as a cost generalized linear production game....
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...Indeed, following Edmonds [7], the linear programming problem whose optimal solution is the characteristic vector of an m.c.d.s.t, iffN = ( N w {0},/~N) is essentially (6), (7) 4 and the addit ional set o f constraints - ~ ( x ~ s : i c N ) > ~ - l , j ~ N ....
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...By Edmonds [6], the weighted matching problem can be formulated as an ordinary linear programming problem....
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...As expected, there is more than one way to use Edmonds [6] in order to formulate, for a subset S, a linear programming problem whose optimal value is the maximum weighted matching value ~(S)....
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Journal Article•DOI•

On balanced sets and cores

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Lloyd S. Shapley¹•Institutions (1)

RAND Corporation¹

01 Jan 1967-Naval Research Logistics Quarterly

TL;DR: In this paper, the authors established a direct correspondence between the balanced sets of coalitions of a multi-person game and the conditions that determine whether the game has a core, and showed that such a correspondence can be found for any game.

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Abstract: This Memorandum establishes a direct correspondence between the balanced sets of coalitions of a multi-person game and the conditions that determine whether the game has a core.

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1,046 citations

"A generalized linear production mod..." refers background in this paper

...The weighted matching game (N; ~) is the side payment cooperative game associated with the room-mates problem which was introduced by Gale and Shapley [8]....
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...2 The assignment game introduced by Shapley and Shubik [17] can be cast as a network flow problem in which the players are the nodes of a (bipartite) graph and the resources are the arcs in the graph, see also Granot and Granot [13]....
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...Further, a network game, derived from a general transhipment problem, with players as nodes of a network is formulated as a controlled programming problem by Dubey and Shapley [5], and is shown therein to have a non-empty core....
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...It is shown by Bondareva [3] and Shapley [16], see also Charnes and Kortanek [4], that (N ; v) has a non-empty core iff it is balanced....
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Journal Article•DOI•

On the core of linear production games

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Guillermo Owen¹•Institutions (1)

Rice University¹

01 Dec 1975-Mathematical Programming

TL;DR: An economic production game is treated, in which players pool resources to produce finished goods which can be sold at a given market price anduality theory of linear programming is used to obtain equilibrium price vectors and to prove the non-emptiness of the core.

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Abstract: An economic production game is treated, in which players pool resources to produce finished goods which can be sold at a given market price. The production process is linear, so that the characteristic function can be obtained by solving linear programs.

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468 citations

Additional excerpts

...Our generalized linear production model is a proper generalization of the linear production model introduced by Owen, and it can be used to analyze cooperative games which cannot be studied in the ordinary linear production model framework....
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...Our generalized linear product ion model is a proper generalization o f Owen's (ordinary) linear product ion model....
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...In the linear p roduc t ion game model in t roduced by Owen [15] each of n players is in possession of a resource vector b ~= (b~l, b ~ , ....
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...The core of a resource game (N; b~), in the ordinary linear production model, contains the unique imputation (bl, b ~ , . . . , bT), and as it was shown by Owen, the imputations (b~, b 2 , . . . , b~) (i = 1 , . . . , m) can be used to generate efficiently a vector in the core of the l inear p roduc t ion game....
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...Perhaps the more important feature of the generalized linear production model, aside from being a proper generalization of Owen's model, is that it is a unifying model....
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