NETWORK SYNTHESIS PROBLEM: COST ALLOCATION AND ALGORITHMS

128 Pages·2010·7.37 MB·English

NETWORK SYNTHESIS PROBLEM: COST ALLOCATION AND
ALGORITHMS
By
MEHRAN HOJATI
B.Sc. (Economics), London School of Economics & Political Science, 1980
M.Sc. (Operational Research), London School of Economics & Political Science, 1981
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
FACULTY OF GRADUATE STUDIES
(COMMERCE AND BUSINESS ADMINISTRATION)
We accept this thesis as conforming to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA
© Mehran Hojati, 1987
October 1987
In presenting this thesis in partial fulfilment of the requirements for an advanced
degree at the University of British Columbia, I agree that the Library shall make it
freely available for reference and study. I further agree that permission for extensive
copying of this thesis for scholarly purposes may be granted by the head of my
department or by his or her representatives. It is understood that copying or
publication of this thesis for financial gain shall not be allowed without my written
permission.
Department
The University of British Columbia
1956 Main Mall
Vancouver, Canada
V6T 1Y3
nrJr 8, If89
Date
DE-6(3/81)
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ABSTRACT
This thesis is concerned with a network design problem which is referred to in the
literature as the network synthesis problem. The objective is to design an undirected
network, at a minimum cost, which satisfies known requirements, i.e., lower bounds on
the maximum flows, between every pair of nodes. If the requirements are to be satisfied
nonsimultaneously, i.e., one at a time, the problem is called the nonsimultaneous net
work synthesis problem, whereas if the requirements are to be satisfied simultaneously,
the problem is called the simultaneous network synthesis problem. The total construc
tion cost of the network is the sum of the construction cost of capacities on the edges,
where the construction cost of a unit capacity is fixed for any edge, independent of the
size of the capacity, but it may differ from edge to edge. The capacities are allowed
to assume noninteger nonnegative values. The simultaneous network synthesis problem
was efficiently solved by Gomory and Hu [60], whereas the nonsimultaneous network
synthesis problem can only be formulated and solved as a linear program with a large
number of constraints. However, the special equal-cost case, i.e., when the unit con
struction costs are equal across the edges, can be efficiently solved, see Gomory and Hu
[60], by some combinatorial method, other than linear programming. A cost allocation
problem which is associated with the network synthesis problem would naturally arise,
if we assume that the various nodes in the network represent different users or commu
nities. In this case, we need to find a fair method for allocating the construction cost of
the network among the different users. An interesting generalization of the nonsimul
taneous network synthesis problem, the Steiner network synthesis problem, is derived,
when only a proper subset of the nodes have positive requirements from each other.
The thesis is concerned with two issues. First, we will analyze the cost allocation
problems arising in the simultaneous and the equal cost nonsimultaneous network syn
thesis problem. Secondly, we will consider the Steiner network synthesis problem, with
particular emphasis on simplifying the computations in some special cases, not consid
ered before. We will employ cooperative game theory to formulate the cost allocation
problems, and we will prove that the derived games are 'concave', which implies the
existence of the core and the inclusion of the Shapley value and the nucleolus in the core,
and then we will present irredundant representations of the cores. For the equal cost
nonsimultaneous network synthesis game, we will use the irredundant representation of
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the core to provide an explicit closed form expression for the nucleolus of the game,
when the requirement structure is a spanning tree; then, we will develop, in a special
case, a decomposition of the game, which we will later use to efficiently compute the
Shapley value of the game when the requirement structure is a tree; the decomposition
will also be used for the core and the nucleolus of the game in the special case. For the
simultaneous network synthesis game, we will also use the irredundant representation
of the core to derive an explicit closed form expression for the nucleolus, and we will
also decompose the core of this game in the special case, and prove that the Shapley
value and the nucleolus coincide.
Secondly, for the Steiner network synthesis problem, two conceptually different con
tributions have been made, one being a simplifying transformation, and the other being
the case when the network has to be embedded in (i.e., restricted to) some special
graphs. Namely, when the requirement structure is sparse (because there are only a few
demand nodes and the rest are just intermediate nodes) and the positive requirements
are equal, we will employ a transformation procedure to simplify the computations.
This will enable us to efficiently solve the Steiner network synthesis problem with five
or less nodes which have equal positive requirements from each other. Finally, when
the solution network to the Steiner network synthesis problem is to be embedded in
(restricted to) some special graphs, namely trees, rings (circles), series-parallel graphs,
or M and M -free graphs, we will provide combinatorial algorithms which are expected
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to simplify the computations.
C on tents
ABSTRACT ii
Table of Contents iv
List of Tables vii
List of Figures viii
ACKNOWLEDGEMENT x
1 INTRODUCTION 1
1.1 The Network Synthesis Problem 1
1.2 Cost Allocation and Introduction to Cooperative Game Theory 4
1.3 The Steiner Network Synthesis Problem 12
2 COST ALLOCATION IN T HE N E T W O RK SYNTHESIS PROB
L EM 17
2.1 Introduction, Notation and Preliminaries 17
2.1.1 Introduction 17
2.1.2 Game Theory Notation and Preliminaries 18
2.1.3 Graph Theory Notation and Preliminaries 22
2.2 Concavity of the Network Synthesis Games 26
2.3 Irredundant Representation of the Cores of (iV;ci) and (N;c) 30
2
2.4 The Nucleolus of the Network Synthesis Games 36
2.5 Nearly Decomposable Network Synthesis Games 41
2.6 Decomposition of Cores of Nearly Decomposable Games (N; C\) and (N; c ) 43
2
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V
2.7 The Nucleolus of the Nearly Decomposable Game (N; c ) 47
2
2.8 Shapley Values of (iV;ci) and Nearly Decomposable (N;^) 51
2.9 Summary and Further Research for the Cost Allocation Problem . . .. 56
3 A SIMPLIFYING TRANSFORMATION FOR The STEINER NET
WORK SYNTHESIS PROBLEM 58
3.1 The Steiner Network Synthesis Problem 58
3.2 Simplifying the Equal-Requirement Case: Preliminaries 60
3.2.1 Linear Programming Formulations and the Transformation . .. 61
3.2.2 Dual Problems and the Laminarity of Dual Solutions to (P') . . 63
3.3 Optimality of the Subnetwork W 65
3.3.1 A Modified Dijkstra Algorithm and the Construction of a Solution
to (D) 65
3.3.2 Feasibility of the Derived Dual Solution 70
3.3.3 Implication of the Transformation 71
3.4 Summary and Further Research for the Transformation 72
4 T HE STEINER NETWORK SYNTHESIS PROBLEM IN SOME SPE
CIAL GRAPHS 73
4.1 Trees and Rings; Series-Parallel Graphs: Preliminaries and the Equal
Requirement Case 74
4.1.1 Trees and Rings 74
4.1.2 Series-Parallel Graphs: Preliminaries and the Equal Requirement
Case 75
4.2 Series-Parallel Graphs: Non-Equal Requirement Case 85
4.2.1 The Additional Complications 85
4.2.2 The Monotone Requirement Case 89
4.2.3 The Non-Monotone Requirement Case 93
4.3 M and M -Free Graphs 96
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vi
4.3.1 Preliminaries 98
4.3.2 Building Blocks and Composition rules 99
4.3.3 A Combinatorial Algorithm 101
4.3.4 A Fractional Solution 102
4.4 Summary and Further Research 104
4.5 A Final Comment 105
BIBLIOGRAPHY 107
List of Tables
4.1 The Series Composition Matrix 82
4.2 The Parallel Composition Matrix 83
4.3 The Computations for Example 4.1 85
4.4 The Computations used in Example 4.5 94
4.5 The Computations used in Example 4.6 97
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List of Figures
1.1 A Rectangular Warehouse, and Some Items (the thicklined nodes) to be
Picked up 15
2.1 A Requirement Structure (left), and a MRST in it (right) 23
2.2 The Equal-Requirement Subtrees 24
2.3 Optimal Solution Subnetworks for the Equal-Requirement Subtrees. . . 24
2.4 The Superimposed Optimal Solution Subnetwork 24
2.5 A Requirement Structure, and the Optimal Solution Networks 27
2.6 A Spanning Tree Requirement Structure Used in Theorem 2.3 34
2.7 A Spanning Tree Requirement Structure 42
2.8 The Requirement Structure (left) used in Example 2.4, and the Networks
Resulting from the Near-Decomposition (center and right) 54
3.1 Graph G, and point set Si 64
3.2 An Optimal Solution Subnetwork in G', and the Dijkstra Cuts 68
3.3 An Optimal Solution Subnetwork in G, and the Dijkstra Cuts 69
3.4 Mi (left), M (center), and M (right) 71
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4.1 Types i, ii, iii, iv, v, vi, and vii (from left to right) of Partial Extreme Solu
tion Subnetworks in a Simple Path Associated with the Steiner Network
Synthesis Problem, with Requirements Equal to 2, in a Series-Parallel
Graph 81
4.2 A Series-Parallel Graph (left), and its Reduced Graph (right) 83
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ix
4.3 A Binary Decomposition Tree for the Reduced Graph in Figure 4.2. . . 84
4.4 An Optimal Solution Subnetwork for Example 4.1 86
4.5 A Series-Parallel Graph (left), and an Optimal Solution Subnetwork in
it (right) 87
4.6 A MRST (left), Equal-Requirement Subtrees 1 (center) and 2 (right),
Derived in the Decomposition from the MRST 87
4.7 Optimal Solution Subnetworks 1 (left) and 2 (center), and the Superim
posed Subnetwork (right) 88
4.8 An Optimal Solution Subnetwork (left), and another Optimal Solution
Subnetwork (right) used in Example 4.3 89
4.9 A Series-Parallel Graph (left), and its Reduced Graph (right) 92
4.10 A MRST (left), and Equal-Requirement Subtrees 1 (center), and 2 (right),
Derived from the MRST by the Decomposition 92
4.11 Best Partial Composed Solution Subnetworks for (E,E,1C)-(E,E,1C) at
Stages 1 (left), 2 (center), and 3 (right) 93
4.12 The MRST used in Example 4.6 95
4.13 Best Partial Composed Solution Subnetworks of Type (E,E,lC)-(E,E,lC)
at Stages 1 (left), 2 (center), and 3 (right) 96
4.14 A Wheel (left), and a Propeller (right) 100
4.15 K (left), and its Cockade with Three Triangles (right) 101
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4.16 A Fractional Solution for the (Steiner) Network Synthesis Problem. . . . 103