An Effective Generalization of the Direct Support Method

Abstract

The main objective of our paper is to solve a problem which was encountered in an industrial firm. It concerns the conception of a weekly production planning with the aim to optimize the quantities to be launched. Indeed, one of the problems raised in that company could be modeled as a linear multiobjective program where the decision variables are of two kinds: the first ones are upper and lower bounded, and the second ones are nonnegative. During the resolution process of the multiobjective case, we were faced with the necessity of developing an effective method to solve the mono-objective case without any increase in the linear program size, since the industrial case to solve is already very large. So, we propose an extension of the direct support method presented in this paper. Its particularity is that it avoids the preliminary transformation of the decision variables. It handles the bounds as they are initially formulated. The method is really effective, simple to use, and permits speeding up the resolution process.

1. Introduction

The company Ifri is one of the largest and most important Algerian companies in the agroalimentary field. Ifri products mainly mineral water and various drinks.

From January to October of the year 2003, the company production was about 175 million bottles. Expressed in liters, the production in this last period has exceeded the 203 million liters of finishedproducts (all products included). Having covered the national market demand, Ifri left to the acquisition of new international markets.

The main objective of our application [1] is to conceive a data-processing application to carry out an optimal weekly production planning which will replace the planning based primarily on the good management and the experiment of the decision makers.

This problem relates to the optimization problem of the quantities to launch in production. It is modeled as a linear multi-objective program where the objective functions involved are linear, the constraints are linear and the decision variables are of two kinds: the first ones are upper and lower bounded, and the second ones are nonnegative.

Multicriteria optimization problems are a class of difficult optimization problems in which several different objective functions have to be considered at the same time. It is seldom the case that one single point will optimize all the several objective functions. Therefore, we search the so-called efficient points, that is, feasible points having the property that no other feasible point improves all the criteria without deteriorating at least one.

In [2], we developed a method to solve the multi-objective linear programming problem described above. To avoid the preliminary transformation of the constraints, hence the augmentation of the problem dimension, we propose to extend the direct support method of Gabasov et al. [3] known in single-objective programming.

In [2], we proposed a procedure for finding an initial efficient extreme point, a procedure to test the efficiency of a nonbasic variable, and a method to compute all the efficient extreme points, the weakly efficient extreme points, and the 𝜖-weakly efficient extreme points of the problem.

A multiobjective linear program with the coexistence of the two types of the decision variables can be presented in the following canonical form: 𝐶𝑥𝐴𝑑+𝑥𝑄+𝑦𝐻⟶𝑦=m𝑏,₋ax,≤𝑦≥𝑥≤0,𝑑⁺,(1.1) where 𝐶 and 𝑄 are 𝑘×𝑛_𝑥 and 𝑘×𝑛_𝑦 matrices, respectively, 𝐴 and 𝐻 are matrices of order 𝑚×𝑛_𝑥 and 𝑚×𝑛_𝑦, respectively, with rang(𝐴∣𝐻)=𝑚<𝑛_𝑥+𝑛_𝑦, 𝑏∈ℝ^𝑚, 𝑑⁻∈ℝ^𝑛𝑥, and 𝑑⁺∈ℝ^𝑛𝑥.

We denote by 𝑆 the set of feasible decisions: 𝑆=(𝑥,𝑦)∈ℝ^{𝑛𝑥+𝑛𝑦},𝐴𝑥+𝐻𝑦=𝑏,𝑑⁻≤𝑥≤𝑑⁺,𝑦≥0.(1.2)

Definition 1.1. A feasible decision (𝑥⁰,𝑦⁰)∈ℝ^{𝑛𝑥+𝑛𝑦} is said to be efficient (or Pareto optimal) for the problem (1.1), if there is no other feasible solution (𝑥,𝑦)∈𝑆 such that 𝐶𝑥+𝑄𝑦≥𝐶𝑥⁰+𝑄𝑦⁰ and 𝐶𝑥+𝑄𝑦≠𝐶𝑥⁰+𝑄𝑦⁰.

Definition 1.2. A feasible decision (𝑥⁰,𝑦⁰)∈ℝ^{𝑛𝑥+𝑛𝑦} is said to be weakly efficient (or Slater optimal) for the problem (1.1), if there is no other feasible solution (𝑥,𝑦)∈𝑆 such that 𝐶𝑥+𝑄𝑦>𝐶𝑥⁰+𝑄𝑦⁰.

Definition 1.3. Let 𝜖∈ℝ^𝑘, 𝜖≥0. A feasible decision (𝑥^𝜖,𝑦^𝜖)∈𝑆 is said to be 𝜖-weakly efficient for the problem (1.1), if there is no other feasible solution (𝑥,𝑦)∈𝑆 such that 𝐶𝑥+𝑄𝑦−𝐶𝑥^𝜖−𝑄𝑦^𝜖>𝜖.

The multiobjective linear programming consists of determining the whole set of all the efficient decisions, all weakly efficient decisions and all 𝜖-weakly efficient decisions of problem (1.1) for given 𝐶,𝑄,𝐴,𝐻,𝑏,𝑑⁻,and𝑑⁺.

During the resolution process, we need to use an efficiency test of nonbasic variables. This problem can be formulated as a single-objective linear program where the decision variables are of two types: upper and lower bounded variables and nonnegative variables. We propose in this paper to solve this latter problem by an adapted direct support method. Our approach is based on the principle of the methods developed by Gabasov et al.[3], which permit to solve a single-objective linear program with nonnegative decision variables or a single-objective linear program with bounded decision variables. Our work aims to propose a generalization for the single-objective linear program with the two types of decision variables: the upper and lower bounded variables and the nonnegative variables.

This work is devoted to present this method. Its particularity is that it avoids the preliminary transformation of the decision variables. It handles the constraints of the problems such as they are initially formulated. The method is really effective, simple to use, and direct. It allows us to treat problems in a natural way and permits speeding up the whole resolution process. It generates an important gain in memory space and CPU time. Furthermore, the method integrates a suboptimal criterion which permits to stop the algorithm with a desired accuracy. To the best of our knowledge, no other linear programming method uses this criterion which could be useful in practical applications.

The principle of this iterative method is simple: starting with an initial feasible solution and an initial support, each iteration consists of finding an ascent direction and a step along this direction to improve the value of the objective function without leaving the problem’s feasible space. The initial feasible solution and the initial support could be computed independently. In addition to this, the initial feasible point need not to be an extreme point such as in the simplex method. The details of our multiobjective method will be presented in our future works.

2. Statement of the Problem and Definitions

The canonical form of the program is as follows: 𝑧(𝑥,𝑦)=𝐴𝑑𝑥𝑐^𝑡+𝑥+𝐻𝑦𝑘^𝑡=𝑦⟶𝑏,max,((22..12))₋≤𝑦𝑥≥≤0,𝑑⁺,((22..34)) where 𝑐 and 𝑥 are 𝑛_𝑥-vectors, 𝑘 and 𝑦 are 𝑛_𝑦-vectors, 𝑏 an 𝑚-vector, 𝐴=𝐴(𝐼,𝐽_𝑥) an 𝑚×𝑛_𝑥-matrix, 𝐻=𝐻(𝐼,𝐽_𝑦) an 𝑚×𝑛_𝑦-matrix, with rank(𝐴∣H)=𝑚<𝑛_𝑥+𝑛_𝑦; 𝐼={1,2,…,𝑚}, 𝐽_𝑥={1,2,…,𝑛_𝑥}, 𝐽_𝑦={𝑛_𝑥+1,𝑛_𝑥+2,…,𝑛_𝑥+𝑛_𝑦}.

Let us set 𝐽=𝐽_𝑥∪𝐽_𝑦, such that 𝐽_𝑥=𝐽_𝑥𝐵∪𝐽_𝑥𝑁, 𝐽_𝑦=𝐽_𝑦𝐵∪𝐽_𝑦𝑁, with 𝐽_𝑥𝐵∩𝐽_𝑥𝑁=𝜙, 𝐽_𝑦𝐵∩𝐽_𝑦𝑁=𝜙 and |𝐽_𝑥𝐵|+|𝐽_𝑦𝐵|=𝑚.

We set 𝐽_𝐵=𝐽_𝑥𝐵∪𝐽_𝑦𝐵, 𝐽_𝑁=𝐽⧵𝐽_𝐵=𝐽_𝑥𝑁∪𝐽_𝑦𝑁, and we note by 𝐴_𝐻 the 𝑚×(𝑛_𝑥+𝑛_𝑦)-matrix (𝐴∣𝐻).

Let the vectors and the matrices be partitioned in the following way: 𝑥=𝑥_𝐵𝑥=𝑥𝐽_𝑥=𝑥_𝑗,𝑗∈𝐽_𝑥,𝑦=𝑦𝐽_𝑦=𝑦_𝑗,𝑗∈𝐽_𝑦,𝑥_𝑁,𝑥^𝐵𝑦=𝑦_𝐵=𝑥𝐽_𝑥𝐵=𝑥_𝑗,𝑗∈𝐽_𝑥𝐵,𝑥_𝑁=𝑥𝐽_𝑥𝑁=𝑥_𝑗,𝑗∈𝐽_𝑥𝑁,𝑦_𝑁,𝑦^𝐵𝑐=𝑐_𝐵=𝑦𝐽_𝑦𝐵=𝑦_𝑗,𝑗∈𝐽_𝑦𝐵,𝑦_𝑁=𝑦𝐽_𝑦𝑁=𝑦_𝑗,𝑗∈𝐽_𝑦𝑁,𝑐_𝑁,𝑐^𝐵𝑘=𝑘_𝐵=𝑐𝐽_𝑥𝐵=𝑐_𝑗,𝑗∈𝐽_𝑥𝐵,𝑐_𝑁=𝑐𝐽_𝑥𝑁=𝑐_𝑗,𝑗∈𝐽_𝑥𝑁,𝑘_𝑁,𝑘^𝐵𝐴=𝐴𝐼,𝐽_𝑥=𝑘𝐽_𝑦𝐵=𝑘_𝑗,𝑗∈𝐽_𝑦𝐵,𝑘_𝑁=𝑘𝐽_𝑦𝑁=𝑘_𝑗,𝑗∈𝐽_𝑦𝑁,𝐴_𝐵=𝑎_𝑖𝑗,1≤𝑖≤𝑚,1≤𝑗≤𝑛_𝑥=𝑎_𝑗,𝑗∈𝐽_𝑥=𝐴_𝐵∣𝐴_𝑁,𝐻=𝐻𝐼,𝐽_𝑦=𝐴𝐼,𝐽_𝑥𝐵,𝐴_𝑁=𝐴𝐼,𝐽_𝑥𝑁,𝑎_𝑗isthe𝑗thcolumnof𝐴,𝐻_𝐵=ℎ_𝑖𝑗,1≤𝑖≤𝑚,𝑛_𝑥+1≤𝑗≤𝑛_𝑥+𝑛_𝑦=ℎ_𝑗,𝑗∈𝐽_𝑦=𝐻_𝐵∣𝐻_𝑁,𝐴=𝐻𝐼,𝐽_𝑦𝐵,𝐻_𝑁=𝐻𝐼,𝐽_𝑦𝑁,ℎ_𝑗isthe𝑗thcolumnof𝐻,_𝐻𝐴=𝐴_𝐻𝐻𝐵(𝐼,𝐽)=𝑎_𝐻𝑖𝑗,1≤𝑖≤𝑚,1≤𝑗≤𝑛_𝑥+𝑛_𝑦=𝑎^𝐻_𝑗,𝑗∈𝐽_𝑥∪𝐽_𝑦=𝐴^𝐻_𝐵∣𝐴_𝐻𝑁,=𝐴_𝐻𝐼,𝐽_𝑥𝐵∪𝐽_𝑦𝐵=𝐴_𝐵∣𝐻_𝐵,𝐴_𝐻𝑁=𝐴_𝐻𝐼,𝐽_𝑥𝑁∪𝐽_𝑦𝑁=𝐴_𝑁∣𝐻_𝑁.(2.5)

Definition 2.1. (i) A vector (𝑥,𝑦), satisfying the constraints (2.2)–(2.4), is called a feasible solution of the problem (2.1)–(2.4).
(ii) A feasible solution (𝑥⁰,𝑦⁰) is said to be optimal if 𝑧(𝑥⁰,𝑦⁰)=𝑐^𝑡𝑥⁰+𝑘^𝑡𝑦⁰=max(𝑐^𝑡𝑥+𝑘^𝑡𝑦), where (𝑥,𝑦) is taken among all the feasible solutions of the problem (2.1)–(2.4).
(iii) On the other hand, a feasible solution (𝑥^𝜖,𝑦^𝜖) is called 𝜖-optimal or suboptimal if 𝑧𝑥⁰,𝑦⁰−𝑧(𝑥^𝜖,𝑦^𝜖)=𝑐^𝑡𝑥⁰−𝑐^𝑡𝑥^𝜖+𝑘^𝑡𝑦⁰−𝑘^𝑡𝑦^𝜖≤𝜖,(2.6) where (𝑥⁰,𝑦⁰) is an optimal solution of the problem (2.1)–(2.4), and 𝜖 is a nonnegative number, fixed in advance.
(iv) The set 𝐽_𝐵=𝐽_𝑥𝐵∪𝐽_𝑦𝐵⊂𝐽,|𝐽_𝐵|=𝑚 is called a support if det𝐴_𝐻𝐵=det(𝐴_𝐵,𝐻_𝐵)≠0.
(v) A pair {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)}, formed by a feasible solution (𝑥,𝑦) and a support (𝐽_𝑥𝐵,𝐽_𝑦𝐵), is called a support feasible solution.
(vi) The support feasible solution is said to be nondegenerate, if 𝑑^−𝑗<𝑥_𝑗<𝑑^+𝑗,forany𝑗∈𝐽_𝑥𝐵,𝑦_𝑗>0,forany𝑗∈𝐽_𝑦𝐵.(2.7)

3. Increment Formula of the Objective Function

Let {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} be a support feasible solution for the problem (2.1)–(2.4), and let us consider any other feasible solution (𝑥,𝑦)=(𝑥+Δ𝑥,𝑦+Δ𝑦).

We define two subsets 𝐽_𝑦𝑁+ and 𝐽_𝑦𝑁0 of 𝐽_𝑦𝑁 as follows: 𝐽_𝑦𝑁+=𝑗∈𝐽_𝑦𝑁,𝑦_𝑗>0,𝐽_𝑦𝑁0=𝑗∈𝐽_𝑦𝑁,𝑦_𝑗=0.(3.1) The increment of the objective function is as follows: Δ𝑧=−𝑐^𝑡𝐵,𝑘^𝑡𝐵𝐴^𝐻−1_𝐵𝐴_𝑁−𝑐^𝑡𝑁Δ𝑥^𝑁−𝑐^𝑡𝐵,𝑘^𝑡𝐵𝐴^𝐻−1_𝐵𝐻_𝑁−𝑘^𝑡𝑁Δ𝑦^𝑁.(3.2)

The potential vector 𝑢 and the estimations vector 𝐸 are defined by 𝐸_𝑡𝑢^𝑡=𝑐^𝑡𝐵,𝑘^𝑡𝐵𝐴^𝐻−1_𝐵,𝐸=𝐸^𝑡𝐵,𝐸^𝑡𝑁,𝐸^𝑡𝐵=𝐸^𝑡𝑥_𝐵,𝐸^𝑡𝑦_𝐵=(0,0),_𝑡𝑁=𝐸^𝑡𝑥_𝑁,𝐸^𝑡𝑦_𝑁,𝐸^𝑡𝑥_𝑁=𝑢^𝑡𝐴_𝑁−𝑐^𝑡𝑁,𝐸^𝑡𝑦_𝑁=𝑢^𝑡𝐻_𝑁−𝑘^𝑡𝑁.(3.3) Then, the increment formula presents the following final form: Δ𝑧=−𝐸^𝑡𝑥_𝑁Δ𝑥_𝑁−𝐸^𝑡𝑦_𝑁Δ𝑦_𝑁.(3.4)

4. Optimality Criterion

Theorem 4.1. Let {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} be a support feasible solution for the problem (2.1)–(2.4). Then, the following relations 𝐸𝐸_{𝑥𝑗𝑥𝑗}≥0,if𝑥_𝑗=𝑑^−𝑗,𝑗∈𝐽_𝑥𝑁,𝐸_𝑥𝑗≤0,if𝑥_𝑗=𝑑^+𝑗,𝑗∈𝐽_𝑥𝑁,𝐸=_𝑦𝑗0,if𝑑^−𝑗<𝑥_𝑗<𝑑^+𝑗,𝑗∈𝐽_𝑥𝑁,𝐸_𝑦𝑗≥0,if𝑦_𝑗=0,𝑗∈𝐽_𝑦𝑁,=0,if𝑦_𝑗>0,𝑗∈𝐽_𝑦𝑁,(4.1) are sufficient for the optimality of the feasible solution (𝑥,𝑦). They are also necessary if the support feasible solution is nondegenerate.

Proof. Sufficiency
Let {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} be a support feasible solution satisfying the relations (4.1). For any feasible solution (𝑥,𝑦) of the problem (2.1)–(2.4), the increment formula (3.4) gives the following: Δ𝑧=−_{𝑗∈𝐽𝑥𝑁}𝐸_𝑥𝑗𝑥_𝑗−𝑥_𝑗−_{𝑗∈𝐽𝑦𝑁+}𝐸_𝑦𝑗𝑦_𝑗−𝑦_𝑗−_{𝑗∈𝐽𝑦𝑁0}𝐸_𝑦𝑗𝑦_𝑗−𝑦_𝑗.(4.2)
Since 𝑑^−𝑗≤𝑥_𝑗≤𝑑^+𝑗, for all 𝑗∈𝐽_𝑥, and from the relations (4.1), we have −_{𝑗∈𝐽𝑥𝑁}𝐸_𝑥𝑗𝑥_𝑗−𝑥_𝑗=−_{𝑗∈𝐽𝑥𝑁,𝐸𝑥𝑗>0}𝐸_𝑥𝑗𝑥_𝑗−𝑑^−𝑗−_{𝑗∈𝐽𝑥𝑁,𝐸𝑥𝑗<0}𝐸_𝑥𝑗𝑥_𝑗−𝑑^+𝑗≤0.(4.3)
On the other hand, the condition 𝑦_𝑗≥0,forall𝑗∈𝐽_𝑦, implies that −_{𝑗∈𝐽𝑦𝑁+}𝐸_𝑦𝑗𝑦_𝑗−𝑦_𝑗−_{𝑗∈𝐽𝑦𝑁0}𝐸_𝑦𝑗𝑦_𝑗−𝑦_𝑗=−_{𝑗∈𝐽𝑦𝑁0}𝐸_𝑦𝑗𝑦_𝑗≤0.(4.4) Hence, Δ𝑧=𝑧𝑥,𝑦−𝑧(𝑥,𝑦)≤0,𝑧𝑥,𝑦≤𝑧(𝑥,𝑦).(4.5) The vector (𝑥,𝑦) is, consequently, an optimal solution of the problem (2.1)–(2.4).
Necessity
Let {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} be a nondegenerate optimal support feasible solution of the problem (2.1)–(2.4) and assume that the relations (4.1) are not satisfied, that is, there exists at least one index 𝑗₀∈𝐽_𝑁=𝐽_𝑥𝑁∪𝐽_𝑦𝑁 such that 𝐸𝐸_𝑥𝑗0>0,for𝑥_𝑗0>𝑑^−𝑗₀,𝑗₀∈𝐽_𝑥𝑁,or,_𝑥𝑗0<𝐸0,for𝑥_𝑗0<𝑑^+𝑗₀,𝑗₀∈𝐽_𝑥𝑁,or,_𝑦𝑗0𝐸<0,for𝑗₀∈𝐽_𝑦𝑁0,or,_𝑦𝑗0≠0,for𝑗₀∈J_𝑦𝑁+.(4.6) We construct another feasible solution (𝑥,𝑦)=(𝑥+𝜃𝑙_𝑥,𝑦+𝜃𝑙_𝑦), where 𝜃 is a positive real number, and ^𝑙𝑥𝑙_𝑦=^{𝑙(𝐽𝑥)𝑙(𝐽}_𝑦⁾=𝑙(𝐽)=𝑙 is a direction vector, constructed as follows.
For this, two cases can arise: (i)if 𝑗₀∈𝐽_𝑥𝑁, we set 𝑙_𝑥𝑗𝑙_𝑥𝑗0=−sign𝐸_𝑥𝑗0,=𝑙_𝑦𝑗0,𝑗≠𝑗₀,𝑗∈𝐽_𝑥𝑁,𝑙_𝐵=0,𝑗∈𝐽_𝑦𝑁,=𝑙^𝑥𝐵𝑙_𝑦𝐵=𝐴^𝐻−1𝐵𝑎_𝑗0sign𝐸_𝑥𝑗0,(4.7) where 𝑎_𝑗0 is the 𝑗₀th column of the matrix 𝐴; (ii)if 𝑗₀∈𝐽_𝑦𝑁, we set 𝑙_𝑦𝑗𝑙_𝑦𝑗0=−sign𝐸_𝑦𝑗0,=𝑙_𝑥𝑗0,𝑗≠𝑗₀,𝑗∈𝐽_𝑦𝑁,𝑙_𝐵=0,𝑗∈𝐽_𝑥𝑁,=𝑙^𝑥𝐵𝑙_𝑦𝐵=𝐴^𝐻−1𝐵ℎ_𝑗0sign𝐸_𝑦𝑗0,(4.8)where ℎ_𝑗0 is the 𝑗₀th column of the matrix 𝐻. From the construction of the direction 𝑙, the vector (𝑥,𝑦) satisfies the principal constraint 𝐴𝑥+𝐻𝑦=𝑏.
In order to be a feasible solution of the problem (2.1)–(2.4), the vector (𝑥,𝑦) must in addition satisfy the inequalities 𝑑⁻≤𝑥≤𝑑⁺ and 𝑦≥0, or in its developed form 𝑑𝑑^−𝑗_−𝑗−𝑥_𝑗≤𝜃𝑙_𝑥𝑗≤𝑑^+𝑗−𝑥_𝑗,𝑗∈𝐽_𝑥𝐵,𝜃𝑙_𝑦𝑗−𝑥_𝑗≤𝜃𝑙_𝑥𝑗≤𝑑^+𝑗−𝑥_𝑗,𝑗∈𝐽_𝑥𝑁,≥−𝑦_𝑗,𝑗∈𝐽_𝑦𝐵,𝜃𝑙_𝑦𝑗≥−𝑦_𝑗,𝑗∈𝐽_𝑦𝑁.(4.9) As the support feasible solution {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} is nondegenerate, we can always find a small positive number 𝜃 such that the relations (4.9) are satisfied. Thus, for a small positive number 𝜃, we can state that the vector (𝑥,𝑦) is a feasible solution for the problem (2.1)–(2.4). The increment formula gives in both cases 𝑧𝑥,𝑦−𝑧(𝑥,𝑦)=𝜃𝐸_𝑗0sign𝐸_𝑗0=𝜃||𝐸_𝑗0||>0,(4.10) where 𝐸_𝑗0=𝐸_𝑥𝑗0if𝑗₀∈𝐽_𝑥𝑁or𝐸_𝑗0=𝐸_𝑦𝑗0if𝑗₀∈𝐽_𝑦𝑁.(4.11) Therefore, we have found another feasible solution (𝑥,𝑦)≠(𝑥,𝑦) with the inequality 𝑧(𝑥,𝑦)>𝑧(𝑥,𝑦) which contradicts the optimality of the feasible solution (𝑥,𝑦). Hence the relations (4.1) are satisfied.

5. The Suboptimality Condition

In order to evaluate the difference between the optimal value 𝑧(𝑥⁰,𝑦⁰) and another value 𝑧(𝑥,𝑦) for any support feasible solution {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)}, when 𝐸_𝑦≥0, we use the following formula: 𝛽(𝑥,𝑦),𝐽_𝑥𝐵,𝐽_𝑦𝐵=_{𝑗∈𝐽𝑥𝑁𝐸𝑥𝑗>0}𝐸_𝑥𝑗𝑥_𝑗−𝑑^−𝑗+_{𝑗∈𝐽𝑥𝑁𝐸𝑥𝑗<0}𝐸_𝑥𝑗𝑥^𝑗−𝑑^+𝑗+_{𝑗∈𝐽𝑦𝑁}𝐸_𝑦𝑗𝑦_𝑗,(5.1) which is called the suboptimality condition.

Theorem 5.1 (the suboptimality condition). Let {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} be a support feasible solution of the problem (2.1)–(2.4) and 𝜖 an arbitrary nonnegative number.
If 𝐸_𝑦≥0 and _𝑗∈𝐽_{𝑥𝑁𝐸𝑥𝑗>0}𝐸_𝑥𝑗𝑥_𝑗−𝑑^−𝑗+_{𝑗∈𝐽𝑥𝑁𝐸𝑥𝑗<0}𝐸_𝑥𝑗𝑥^𝑗−𝑑^+𝑗+_{𝑗∈𝐽𝑦𝑁}𝐸_𝑦𝑗𝑦_𝑗≤𝜖,(5.2) then the feasible solution (𝑥,𝑦) is 𝜖-optimal.

Proof. We have 𝑧𝑥⁰,𝑦⁰−𝑧(𝑥,𝑦)≤𝛽(𝑥,𝑦),𝐽_𝑥𝐵,𝐽_𝑦𝐵.(5.3) Then, if 𝛽(𝑥,𝑦),𝐽_𝑥𝐵,𝐽_𝑦𝐵≤𝜖,(5.4) we will have 𝑧𝑥⁰,𝑦⁰−𝑧(𝑥,𝑦)≤𝜖,(5.5) therefore, (𝑥,𝑦) is 𝜖-optimal.
In the particular case where 𝜖=0, the feasible solution (𝑥,𝑦) is consequently optimal.

6. Construction of the Algorithm

Given any nonnegative real number 𝜖 and an initial support feasible solution {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)}, the aim of the algorithm is to construct an 𝜖-optimal solution (𝑥^𝜖,𝑦^𝜖) or an optimal solution (𝑥⁰,𝑦⁰). An iteration of the algorithm consists of moving from {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} to another support feasible solution {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} such that 𝑧(𝑥,𝑦)≥𝑧(𝑥,𝑦). For this purpose, we construct the new feasible solution (𝑥,𝑦) as follows: (𝑥,𝑦)=(𝑥,𝑦)+𝜃(𝑙_𝑥,𝑙_𝑦), where 𝑙=(𝑙_𝑥,𝑙_𝑦) is the appropriate direction, and 𝜃 is the step along this direction.

In this algorithm, the simplex metric is chosen. We will thus vary only one component among those which do not satisfy the relations (4.1).

In order to obtain a maximal increment, we must take 𝜃 as great as possible and choose the subscript 𝑗₀ such that ||𝐸_𝑗0||=max|||𝐸_𝑥𝑗0|||,|||𝐸_𝑦𝑗0|||,(6.1) with |||𝐸_𝑥𝑗0|||=max|||𝐸_𝑥𝑗|||,𝑗∈𝐽_{𝑥𝑁𝑁𝑂},|||𝐸^𝑦_𝑗0|||=max|||𝐸_𝑦𝑗|||,𝑗∈𝐽_{𝑦𝑁𝑁𝑂},(6.2) where 𝐽_{𝑥𝑁𝑁𝑂} and 𝐽_{𝑦𝑁𝑁𝑂} are the subsets, respectively, of 𝐽_𝑥𝑁 and 𝐽_𝑦𝑁, whose the subscripts do not satisfy the relations of optimality (4.1).

6.1. Computation of the Direction 𝑙

We have two cases.(i)If |𝐸_𝑗0|=|𝐸_𝑥𝑗0|, we set: 𝑙_𝑥𝑗𝑙_𝑥𝑗0=−sign𝐸_𝑥𝑗0,=𝑙_𝑦𝑗0,𝑗≠𝑗₀,𝑗∈𝐽_𝑥𝑁,𝑙_𝐵=0,𝑗∈𝐽_𝑦𝑁,=𝑙^𝑥𝐵𝑙_𝑦𝐵=𝐴^𝐻−1𝐵𝑎_𝑗0sign𝐸_𝑥𝑗0.(6.3)(ii)If |𝐸_𝑗0|=|𝐸_𝑦𝑗0|, we will set 𝑙_𝑦𝑗𝑙_𝑦𝑗0=−sign𝐸_𝑦𝑗0,=𝑙_𝑥𝑗0,𝑗≠𝑗₀,𝑗∈𝐽_𝑦𝑁,𝑙_𝐵=0,𝑗∈𝐽_𝑥𝑁,=𝑙^𝑥𝐵𝑙_𝑦𝐵=𝐴^𝐻−1𝐵ℎ_𝑗0sign𝐸_𝑦𝑗0.(6.4)

6.2. Computation of the Step 𝜃

The step 𝜃⁰ must be taken as follows: 𝜃⁰=min𝜃_𝑥,𝜃_𝑦.(6.5)(i)If |𝐸_𝑗0|=|𝐸_𝑥𝑗0|, then 𝜃_𝑥=min(𝜃_𝑥𝑗0,𝜃_𝑥𝑗1), where 𝜃_𝑥𝑗0=𝑑^+𝑗0𝑥−𝑥_𝑗0,if𝐸_𝑥𝑗0<0,_𝑗0𝜃−𝑑^−𝑗₀,if𝐸_𝑥𝑗0>0,_𝑥𝑗1=min𝜃_𝑥𝑗,𝑗∈𝐽_𝑥𝐵,(6.6) with 𝜃_𝑥𝑗=⎧⎪⎪⎨⎪⎪⎩𝑑^+𝑗−𝑥_𝑗𝑙_𝑥𝑗𝑑,if𝑙_𝑥𝑗>0,_−𝑗−𝑥_𝑗𝑙_𝑥𝑗∞,,iiff𝑙𝑙_{𝑥𝑗𝑥𝑗}<0,=0.(6.7) The number 𝜃_𝑦 will be computed in the following way: 𝜃_𝑦=𝜃_𝑦𝑗1=min𝜃_𝑦𝑗,𝑗∈𝐽_𝑦𝐵,(6.8) where 𝜃_𝑦𝑗=⎧⎪⎨⎪⎩−𝑙𝑦^𝑗_𝑦𝑗∞,,iiff𝑙𝑙_{𝑦𝑗𝑦𝑗}<0,≥0.(6.9)(ii)If |𝐸_𝑗0|=|𝐸_𝑦𝑗0|, then 𝜃_𝑥=𝜃_𝑥𝑗1=min𝜃_𝑥𝑗,𝑗∈𝐽_𝑥𝐵,(6.10)

where 𝜃_𝑥𝑗=⎧⎪⎪⎨⎪⎪⎩𝑑^+𝑗−𝑥_𝑗𝑙_𝑥𝑗𝑑,if𝑙_𝑥𝑗>0,_−𝑗−𝑥_𝑗𝑙_𝑥𝑗∞,,ifif𝑙𝑙_{𝑥𝑗𝑥𝑗}<0,𝜃=0,_𝑦=min𝜃_𝑦𝑗0,𝜃_𝑦𝑗1,(6.11) with 𝜃_𝑦𝑗0=𝑦^𝑗0∞,,iiff𝐸𝐸_{𝑦𝑗0𝑦𝑗0}>0,𝜃<0,_𝑦𝑗1=min𝜃_𝑦𝑗,𝑗∈𝐽_𝑦𝐵,(6.12) where 𝜃_𝑦𝑗=⎧⎪⎨⎪⎩−𝑙𝑦^𝑗_𝑦𝑗∞,,iiff𝑙𝑙_{𝑦𝑗𝑦𝑗}<0,≥0.(6.13)

The new feasible solution is 𝑥,𝑦=𝑥+𝜃⁰𝑙_𝑥,𝑦+𝜃⁰𝑙_𝑦.(6.14)

6.3. The New Suboptimality Condition

Let us calculate the suboptimality condition of the new support feasible solution in the case of 𝐸_𝑦≥0. We have 𝛽𝑥,𝑦,𝐽_𝑥𝐵,𝐽_𝑦𝐵=_{𝑗∈𝐽𝑥𝑁𝐸𝑥𝑗>0}𝐸_𝑥𝑗𝑥_𝑗−𝑑^−𝑗+_{𝑗∈𝐽𝑥𝑁𝐸𝑥𝑗<0}𝐸_𝑥𝑗𝑥_𝑗−𝑑^+𝑗+_{𝑗∈𝐽𝑦𝑁}𝐸_𝑦𝑗𝑦_𝑗.(6.15)(i)If 𝑗₀∈𝐽_𝑥𝑁, then the components 𝑥_𝑗, for 𝑗∈𝐽_𝑥𝑁, are equal to 𝑥_𝑗=⎧⎪⎨⎪⎩𝑥^𝑗𝑥_𝑗0,for𝑗≠𝑗₀,𝑥−𝜃⁰,if𝐸_𝑥𝑗0>0,_𝑗0+𝜃⁰,if𝐸_𝑥𝑗0<0,(6.16) and the components 𝑦_𝑗 are 𝑦_𝑗=𝑦_𝑗,∀𝑗∈𝐽_𝑦𝑁.(6.17) Hence, 𝛽𝑥,𝑦,𝐽_𝑥𝐵,𝐽_𝑦𝐵=𝛽(𝑥,𝑦),𝐽_𝑥𝐵,𝐽_𝑦𝐵−𝜃⁰|||𝐸_𝑥𝑗0|||.(6.18)(ii)If 𝑗₀∈𝐽_𝑦𝑁, then the components 𝑥_𝑗, for 𝑗∈𝐽_𝑥𝑁, are equal to 𝑥_𝑗=𝑥_𝑗,∀𝑗∈𝐽_𝑥𝑁,(6.19)

and the components 𝑦_𝑗, for 𝑗∈𝐽_𝑦𝑁, are 𝑦_𝑗=𝑦^𝑗𝑦_𝑗0,for𝑗≠𝑗₀,−𝜃⁰,for𝑗=𝑗₀.(6.20)

Hence, 𝛽𝑥,𝑦,𝐽_𝑥𝐵,𝐽_𝑦𝐵=𝛽(𝑥,𝑦),𝐽_𝑥𝐵,𝐽_𝑦𝐵−𝜃⁰𝐸_𝑦𝑗0.(6.21) In both cases, we will have 𝛽𝑥,𝑦,𝐽_𝑥𝐵,𝐽_𝑦𝐵=𝛽(𝑥,𝑦),𝐽_𝑥𝐵,𝐽_𝑦𝐵−𝜃⁰||𝐸_𝑗0||,(6.22) with |𝐸_𝑗0|=|𝐸_𝑥𝑗0|∨𝐸_𝑦𝑗0.

6.4. Changing the Support

If 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵))≤𝜖, then the feasible solution (𝑥,𝑦) is 𝜖-optimal and we can stop the algorithm; otherwise, we will change 𝐽_𝐵 as follows: (i)if 𝜃⁰=𝜃_𝑥𝑗0∨𝜃_𝑦𝑗0, then 𝐽_𝐵=𝐽_𝐵, 𝑥=𝑥+𝜃⁰𝑙_𝑥, 𝑦=𝑦+𝜃⁰𝑙_𝑦, (ii)if 𝜃⁰=𝜃_𝑥𝑗1∨𝜃_𝑦𝑗1, then 𝐽_𝐵=(𝐽_𝐵⧵𝑗₁)∪𝑗₀, 𝑥=𝑥+𝜃⁰𝑙_𝑥, 𝑦=𝑦+𝜃⁰𝑙_𝑦.

Then we start a new iteration with the new support feasible solution {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)}, where the support 𝐽_𝐵 satisfies the algebraic condition det𝐴_𝐻𝐵=det𝐴_𝐻𝐵𝐼,𝐽_𝐵≠0.(6.23)

Remark 6.1. The step 𝜃⁰=∞ may happen only if 𝐽_𝑥𝐵=∅, |𝐸_𝑗0|=|𝐸_𝑦𝑗0| and 𝜃_𝑦=∞. In such a case, the objective function is unbounded with respect to 𝑦.

7. Algorithm

Let 𝜖 be any nonnegative real number and {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} an initial support feasible solution. The steps of the algorithm are as follows. (1)Compute the estimations vector: 𝐸^𝑡𝑁=𝐸^𝑡𝐽_𝑁=𝐸𝑢^𝑡𝑥_𝑁𝑡,𝐸^𝑡𝑦_𝑁=𝑢^𝑡𝐴_𝑁−𝑐^𝑡𝑁,𝑢^𝑡𝐻_𝑁−𝑘^𝑡𝑁,=𝑐^𝑡𝐵,𝑘^𝑡𝐵𝐴^𝐻−1_𝐵.(7.1)(2)Optimality test of the support feasible solution {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)}. (i)If 𝐸_𝑦≥0, then (a)calculate the value of suboptimality 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)), (b)if 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵))=0, the process is stopped with {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} as an optimal support solution, (c) if 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵))≤𝜖, the process is stopped with {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} as an 𝜖-optimal support solution, (d)if 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵))>𝜖, go to (3), (ii) if 𝐸_𝑦≱0, go directly to (3). (3)Change the feasible solution (𝑥,𝑦) by (𝑥,𝑦): 𝑥=𝑥+𝜃⁰𝑙_𝑥 and 𝑦=𝑦+𝜃⁰𝑙_𝑦. (i)Choose a subscript 𝑗₀. (ii)Compute the appropriate direction 𝑙=^𝑙𝑥_𝑙𝑦. (iii)Compute the step 𝜃⁰. (a)If 𝜃⁰=∞ then the objective function is unbounded with respect to 𝑦 and the process is stopped. (b)Otherwise, compute (𝑥,𝑦)=(𝑥+𝜃⁰𝑙_𝑥,𝑦+𝜃⁰𝑙_𝑦). (4)Optimality test of the new feasible solution (𝑥,𝑦). (i)If 𝐸_𝑦≥0, then (a)calculate 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵))=𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵))−𝜃⁰|𝐸_𝑗0|, (b)if 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵))=0, the process is stopped with {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} as an optimal support solution, (c) if 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵))≤𝜖, the process is stopped with {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)} as an 𝜖-optimal support solution,(d)otherwise, go to (5). (ii)If 𝐸_𝑦≱0, then go to (5). (5)Change the support 𝐽_𝐵 by 𝐽_𝐵.(i)If 𝜃⁰=𝜃_𝑥𝑗0∨𝜃_𝑦𝑗0, then 𝐽_𝑥𝐵=𝐽_𝑥𝐵,𝐽_𝑥𝑁=𝐽_𝑥𝑁,𝐽_𝑦𝐵=𝐽_𝑦𝐵,𝐽_𝑦𝑁=𝐽_𝑦𝑁,𝑥=𝑥+𝜃⁰𝑙_𝑥,𝑦=𝑦+𝜃⁰𝑙_𝑦.(7.2)(ii)If 𝜃⁰=𝜃_𝑥𝑗1∨𝜃_𝑦𝑗1, then two cases can arise: (a)case where 𝐸_𝑗0=|𝐸_𝑥𝑗0|∶*if 𝜃⁰=𝜃_𝑥𝑗1, then 𝐽_𝑥𝐵=𝐽_𝑥𝐵⧵𝑗₁∪𝑗₀,𝐽_𝑥𝑁=𝐽_𝑥𝑁⧵𝑗₀∪𝑗₁,𝐽_𝑦𝐵=𝐽_𝑦𝐵,𝐽_𝑦𝑁=𝐽_𝑦𝑁,(7.3)*if 𝜃⁰=𝜃_𝑦𝑗1, then 𝐽_𝑥𝐵=𝐽_𝑥𝐵∪𝑗₀,𝐽_𝑥𝑁=𝐽_𝑥𝑁⧵𝑗₀,𝐽_𝑦𝐵=𝐽_𝑦𝐵⧵𝑗₁,𝐽_𝑦𝑁=𝐽_𝑦𝑁∪𝑗₁.(7.4)(b)Case where 𝐸_𝑗0=|𝐸_𝑦𝑗0|: *if 𝜃⁰=𝜃_𝑦𝑗1, then 𝐽_𝑦𝐵=𝐽_𝑦𝐵⧵𝑗₁∪𝑗₀,𝐽_𝑦𝑁=𝐽_𝑦𝑁⧵𝑗₀∪𝑗₁,𝐽_𝑥𝐵=𝐽_𝑥𝐵,𝐽_𝑥𝑁=𝐽_𝑥𝑁,(7.5)*if 𝜃⁰=𝜃_𝑥𝑗1, then 𝐽_𝑥𝐵=𝐽_𝑥𝐵⧵𝑗₁,𝐽_𝑥𝑁=𝐽_𝑥𝑁∪𝑗₁,𝐽_𝑦𝐵=𝐽_𝑦𝐵∪𝑗₀,𝐽_𝑦𝑁=𝐽_𝑦𝑁⧵𝑗₀.(7.6)(iii)Go to (1) with the new support feasible solution {(𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)}, where 𝑥=𝑥+𝜃⁰𝑙_𝑥 and 𝑦=𝑦+𝜃⁰𝑙_𝑦.

8. Numerical Example

For the sake of clarity, let us illustrate the theoretical development of the method by considering the following linear program: 𝑧(𝑥,𝑦)=𝑥2𝑥₁−3𝑥₂−𝑦₃+𝑦₄⟶max,₁−7𝑥₁−𝑥₂+3𝑦₃+2𝑦₄=2,+−𝑥2₂≤+2𝑥𝑦₃₁+3𝑦₄=2,−4≤𝑥₂≤2,𝑦≤4,₃𝑦≥0,₄≥0,(8.1) where 𝑥=(𝑥₁,𝑥₂) and 𝑦=(𝑦₃,𝑦₄).

We define 𝐴=(−171−1), 𝐻=(3223), 𝑐^𝑡=(2−3), and 𝑘^𝑡=(−11).

Let (𝑥,𝑦)=(1302) be an initial feasible solution of the problem. We set 𝐽_𝐵=𝐽_𝑥𝐵,𝐽_𝑦𝐵={1,3},𝐽_𝑁={2,4}.(8.2) Let 𝜖=0.

Thus, we have an initial support feasible solution {(𝑥,𝑦),𝐽_𝐵} with 𝐴_𝐻𝐵=−17𝑧32(𝑥,,𝑦)=𝐴−5_𝐻𝑁.=−1123,(8.3)

First Iteration
Let us calculate 𝑢^𝑡=𝑐^𝑡𝐵,𝑘^𝑡𝐵𝐴^𝐻−1_𝐵=−3𝐸23−273_𝑡𝑁=𝑢^𝑡𝐴_𝐻𝑁−𝑐^𝑡𝑁,𝑘^𝑡𝑁=6523−2530.(8.4)Choice of 𝑗₀:
Among the nonoptimal indices 𝐽_{𝑥𝑁𝑁𝑂}∪𝐽_{𝑦𝑁𝑁𝑂}={2,4}, 𝑗₀ is chosen such that |𝐸_𝑗0| is maximal; we then have 𝑗₀=2.
Computation of 𝑙: 𝑙𝑙_𝑥2=−1,𝑙_𝑦4=0,_𝐵=𝑙^𝑥1𝑙_𝑦3=𝐴^𝐻−1𝐵𝑎₂=⎛⎜⎜⎜⎝−−256323⎞⎟⎟⎟⎠.(8.5) Hence 𝑙_𝑥=−−2513,𝑙_𝑦=−0263.(8.6)Computation of 𝜃⁰: 𝜃𝜃_𝑥𝑗0=𝜃_𝑥2=𝑥₂−𝑑⁻²=7,_𝑥1=𝑑⁻¹−𝑥₁𝑙_𝑥1=69𝜃_𝑦35,=−𝑦³𝑙_𝑦3=0.(8.7) The maximal step is then 𝜃⁰=𝜃_𝑦𝑗1=𝜃_𝑦3=0.(8.8)Computation of (𝑥,𝑦): 𝑥=𝑥+𝜃⁰𝑙_𝑥=13,𝑦=𝑦+𝜃⁰𝑙_𝑦=02.(8.9)Change the support: 𝐽_𝑥𝐵={1,2},𝐽_𝑥𝑁=∅,𝐽_𝑦𝐵=∅,𝐽_𝑦𝑁={3,4}.(8.10)

Second Iteration
We have 𝐽(𝑥,𝑦)=1302_𝑥𝐵,𝐽={1,2},𝐽_𝑥𝑁=∅,_𝑦𝐵𝐴=∅,𝐽_𝑦𝑁={3,4},_𝐻𝐵=−17−11,𝐴_𝐻𝑁=3223.(8.11) We compute 𝑢^𝑡=𝑐^𝑡𝐵,𝑘^𝑡𝐵𝐴^𝐻−1_𝐵=19𝐸_𝑡𝑁616,=𝑢^𝑡𝐴_𝐻𝑁−𝑐^𝑡𝑁,𝑘^𝑡𝑁=656365.(8.12) Computation of 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)): 𝛽(𝑥,𝑦),𝐽_𝑥𝐵,𝐽_𝑦𝐵=𝐸_𝑦3𝑦₃+𝐸_𝑦4𝑦₄=353.(8.13) Then, (𝑥,𝑦) is not optimal.
Choice of 𝑗₀: As the set of nonoptimal indices is 𝐽_{𝑥𝑁𝑁0}∪𝐽_{𝑦𝑁𝑁0}={4}, we have 𝑗₀=4.
Computation of 𝑙: 𝑙_𝑦4=−1,𝑙_𝑦3=0,𝑙_𝐵=𝑙^𝑥1𝑙_𝑥2=𝐴^𝐻−1𝐵ℎ₄=⎛⎜⎜⎜⎝−−15676⎞⎟⎟⎟⎠.(8.14) Hence, 𝑙_𝑥=(^{−5/6−17/6}),𝑙_𝑦=(⁰⁻¹).
Computation of 𝜃⁰: 𝜃_𝑥𝑗1==mmiinn𝜃𝑑_𝑥1,₋₁𝜃_𝑥2−𝑥₁𝑙_𝑥1,𝑑⁻²−𝑥₂=min18𝑙_𝑥25,4172=4127=𝜃^𝑥2𝜃_𝑦3,=∞,𝜃_𝑦𝑗0=𝜃_𝑦4=2.(8.15) The maximal step is thus 𝜃⁰=𝜃_𝑦4=2.
Consequently, the support remains unchanged: 𝐽_𝐵=𝐽_𝐵={1,2},𝐽_𝑁=𝐽_𝑁={3,4}.(8.16)Computation of (𝑥,𝑦): 𝑥=𝑥+𝜃⁰𝑙_𝑥=−23−38,𝑦=𝑦+𝜃⁰𝑙_𝑦=00.(8.17)Computation of 𝛽((𝑥,𝑦),(𝐽_𝑥𝐵,𝐽_𝑦𝐵)): 𝛽𝑥,𝑦,𝐽_𝑥𝐵,𝐽_𝑦𝐵=0.(8.18) Then, the vector (𝑥,𝑦)=(−2/3,−8/3,0,0) is an optimal solution and the maximal value of the objective function is 𝑧=20/3.

9. Conclusion

The necessity of developing the method presented in this paper occurred during a more complex optimization scheme involving the resolution of a multicriteria decision problem [2]. Indeed, an efficiency test of nonbasic variables is necessary to be executed several times along the resolution process. This efficiency test yields to solve a monocriteria program with two kinds of variables: upper and lower bounded variables and nonnegative ones. This kind of linear models can be found as subproblems in quadratic programming [4] and optimal control for example. In these cases, the use of the simplex method is not suitable since the transformed problems are often degenerated. An other particularity of our method is that it uses a suboptimal criterion which can stop the algorithm with a desired precision. It is effective, fast, simple, and permits a time reduction in the whole optimization process.