# Linear Solution to Nonlinear Optimization Problems by Convex Cone Method and its Application to Multiple Objective Aggregate Production Planning

主査　教授　大里　有生
副査　教授　中村　和男
副査　教授　淺井　達雄
副査　助教授　山田　耕一
副査　助教授　植野　真臣

## 論文目次

Page
Title page　p.i
Acknowledgements　p.ii
Abstract　p.iii
List of figures　p.ix
List of tables　p.xi

Chapter1　INTRODUCTION
1.1　Introduction　p.3
1.2 Statement of the problem　p.6
1.3 Purpose of this research　p.6
1.4 Organization of the Dissertation　p.7
References　p.8

Chapter2　CONVEX CONE METHOD FOR LINEARIZATION OF CONVEX FUNCTIONS
2.1 Introduction　p.15
2.2 Convex set and cone concept　p.16
2.3 Convex functions　p.18
2.4 Convex cone method for linearization of convex functions　p.24
2.4.1 Convex cone method linearization of a triangular type function　p.25
2.4.2 Convex cone method linearization of convex polyhedral function　p.30
2.5 Conclusions　p.35
References　p.36

Chapter3　LINEAR CALCULATION OF A NONLINEAR OPITMIZATION PROBLEM WITH A CONVEX POLYHEDRAL OBJECTIVE FUNCTION AND LINEAR CONSTRAINTS BY CONVEX CONE METHOD
3.1 Introduction　p.41
3.2 Linear solution methods of nonlinear optimization problems　p.42
3.2.1　Linear solution methods of nonlinear optimization problems wiith a traingular objective function and linear constraints　p.43
3.2.2　Linear solution methods of nonlinear optimization problems wiith a convex polyhedral objective function and linear constraints　p.45
3.3 Linear calculational method for nonlinear optimization problems　p.48
3.3.1　Linear calculational method for a minimization with a traingular objective function and linear constraints　p.48
3.3.2 Relationship between GP and linear calsulation　p.49
3.3.3 Linear calculational method for nonlinear optimization wiith a convex polyhedral objective function and linear constraints　p.53
3.3.4 Numerical example
a） An example of the optimization wiith a traingular objective function and linear constraints　p.57
b）An example of the optimization wiith a convex polyhedral objective function and linear constraints　p.59
3.4 Conclusions　p.61
References　p.62

Chapter4　LINEAR CALCULATION FOR A MULTIPLE OBJECTIVE LINEAR PROGRAMMING WITH NONLINAER PREFERENCE FUNCTIONS AND LINAER CONSTRAINTS BY CONVEX CONE METHOD
4.1 Introduction　p.67
4.2 Multiple objective linear programming （MOLP）　p.68
4.3 Efficient linear coordination method for MOLP with nonlinear preference functions and linear constraints　p.70
4.4 A numerical example　p.79
4.5 Conclusions　p.83
References　p.83

Chapter5　LINEAR CALCULATION FOR A FUZZY MULTIPLE OBJECTIVE LINEAR PROGRAMMING WITH NONLINAER MEMBERSHIP FUNCTIONS AND LINEAR CONSTRAINTS BY CONVEX CONE METHOD

5.1 Introduction　p.89
5.2 Fuzzy goals　p.92
5.3 Satisfictiong linear coordination method for s fuzzy multiple objective linear programming （FMOLP） with nonlinear membership functions and linear constraints　p.96
5.4 A numerical example　p.97
5.5 Conclusions　p.101
References　p.101

Chapter6　MULTIPLE OBJECTIVE AGGREGATE PRODUCTION OLANNING BY LINEAR COORDINATION METHOD　p.105
6.1 Introduction　p.107
6.2 Decsion strategies　p.110
6.3 Aggregate production planning （APP） costs　p.110
6.4 An APP mathmatical formulation　p.111
6.4.1 Fuzzy goals and demands for an APP problem　p.113
6.4.2 Constrains　p.115
6.4.3 APP with fuzzy goals and demands formulation　117
6.5 A formulation of an APP problem of an APP problem by linear coordination method　p.118
6.5.1 A formulation of an APP problem　p.118
6.5.2 Computational algorithm　p.123
6.6　A numerical application　p.124
6.5 Conclusions　p.130
References　p.132

Chapter7　CONCLUSIONS
7.1 Summary from the reseach　p.137
7.2 Recommendation and future reseach　p.140
References　p.141

LIST OF AUTHOR'S PUBLICATIONS
Journal articles　p.143
Conference proceedings　p.143
REFERENCES　p.145

APPENDLIXES A : MATHMATICAL PRELIMINARIES
A1: linear programming　p.153
References
A2: Goal programming　p.154
References
A3:References point method　p.160
References
A4:References goal programming method　p.162
References
A5:Basic concept of fuzzy sets　p.165
References
A6:Min-max approach　p.167
References
A7:Augmented min-max approach　p.170
References

APPENDLIXES B : PROGRAMMING CONES
B1:Xpress programming code for fuzzy linear multiple objective aggregate production planning by min-max linear coordination　p.175
B2:Xpress programming code for fuzzy linear multiple objective aggregate production planning by efficient linear coordination　p.182
B3:Xpress programming code for fuzzy linear integer multiple objective aggregate production planning by min-max linear coordination　p.195
B4:Xpress programming code for fuzzy linear integer multiple objective aggregate production planning by efficient linear coordination　p.202

## 論文要旨

In this research, the linear solution method for nonlinear optimization problems with a convex polyhedral objective function by convex cone method, which makes it possible to solve the nonlinear optimization problems by use of effective linear calculation, is newly developed. With this method, it is not necessary to separate the objective function into linear segments, to prepare necessary preference information and to add upper bounded constraints as solving by the separable convex programming. This method is simple to formulate and easy to apply. Moreover, the number of additional constraints and new variables of this method are lower than solving by existing methods.
Reference goal programming （RGP）, the existing efficient multiple objective linear programming （MOLP） method, has been presented but this method considers the preference function as a certain target point, which is too rigid. Then, convex cone method is further applied to formulate MOLP problem, which has convex polyhedral preference functions. Reference point method （RPM） is integrated for ensuring the efficiency of the solutions. It is found that the proposed formulation, called a linear coordination method, can efficiently solve this MOLP problem with convex polyhedral preference functions. This method is a powerful and flexible method for finding the efficient solution
that is also close to the decision maker's requirements.
Moreover, it also can be applied to a fuzzy multiple objective linear programming （FMOLP） problem. A preference function shows the level of dissatisfaction. Conversely, in fuzzy set theory, a membership function is used to represent the satisfaction level of FMOLP problem. Linguistic variables are applied to state an aspiration level of each fuzzy goal, which is quantified by a concave membership function. This method has the better solution than the existing methods for concave membership functions in the sense of the M-Pareto optimality. The linguistic terms are
applied, which make the method more practical and easy to use. Furthermore, the flexibility in designing membership functions of satisficing method is also enhanced.
The linear coordination method is applied to Aggregate Production Planning （APP） problem, which involves multiple objectives and uncertainty of forecasted demands and production planning goals. In the objective functions of the APP problem, cost functions are included and found to be nonlinear in nature. The appropriate approach to handle an optimization problem with a nonlinear function is to formulate a linear approximation model. Existing methods are still insufficient to deal with such kind of realistic problems. Therefore, the linear coordination method is applied to increase the ability of existing multiple objective APP methods in dealing with uncertainty and nonlinearity of preference functions of production planning goals and forecasted demands. Flexible preference functions that can reflect exactly the decision maker's preferences can be designed by concave polyhedral membership functions. This newly developed APP method is a realistic method for multiple objective APP problems.