İŞLETME KARAR MODELLERİ - Deneme Sınavı - 10

A slack variable represents the remainder of an expendable resource and applies to less than or equal constraints. On the other hand, a surplus represents the excess amount of a restricted resource and applies to greater than or equal constraints.

Thus, the solution space is presented by m linear equations and n variables. If the numbers of the linear equations and the variables are equal, the system has only one solution. However, the majority of linear programs have a greater number of variables compared equations n>m.

A basic solution whether it is feasible or not is a corner point of the solution space. The corner points of system of equations are obtained by setting n – m number of variables equal to zero and solving the equations for the remaining m number of variables. The number of basic solutions can be calculated as below:

Here, the system has two equations (m = 3) and four variables. (n = 6) Thus, there are 20 basic solutions:

The phases of algebraic method are these:

Iteration is a repetition of a mathematical procedure applied to the result of a previous application.

The simplex algorithm iteratively switches to the next basic feasible solution that is adjacent to the previous basic feasible solution until it reaches the optimum Z.

The coefficients of the variables in the objective function refer to the effect of one-unit increase of respective variable on the function output Z. Hence, these coefficients correspond to the rate of improvements of the respective variables in Z. According to the objective function of the program, increasing x₂ from zero yields more than increasing other variables from zero. Thus, the entering variable is x₂.

If the solution is optimal, then we stop the process.

Identity matrix of n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. An identity matrix is also called a unity matrix.

The pivot number in the tableau below is 2:

The pivot number in the tableau below is 6:

In transportation models, the total supply equals the total demand. Some transportation problems may be less restrictive; the total supply may exceed aggregate demand. In such a case, a dummy destination must be defined to fictionally absorb an excessive amount of the supply. Thus, the model becomes balanced and fits the general form of the transportation model.

The solution of a transportation model follows the steps that the simplex method offers:

Strong duality property refers that the objective value of Z and the objective value of W are equal only if the basic feasible solution is the optimal for the primal as well as the dual model

A balanced transportation problem has m + n – 1 basic variables in its basic feasible solution, whereas the number of its dual variables is m + n.

There are various methods for selecting an arbitrary variable; the most prominent ones are:

The initial basic feasible solution obtained in the table below has the basic variables given below.

x₁₁ = 10, x₂₁ = 1, x₂₂ = 3, x₃₂ = 4, x₄₂ = 4, x₄₃ = 5,

The objective value for this solution, which is the total transportation cost is 168 TL.

Z = 5× 10 + 1 × 7 + 3 × 5 + 4× 6+ 4 × 6+ 5 × 6 = 168 TL.

The initial basic feasible solution obtained in the table below has the basic variables given below.

x₁₂ = 9, x₁₃ = 1, x₂₃ = 4, x₃₁ = 4, x₄₁ = 7, x₄₂ = 5,

The objective value for this solution, which is the total transportation cost is 149 TL.

Z = 8× 9 + 3 × 1 + 2 × 4 + 2× 4+ 4 ×7+ 6 × 8 = 149 TL.

The term “Assignment” refers to the case of job allocations to a group of workers.

The assignment model has a tailored solution method such as the transportation model does. The solution method of the assignment model is called the Hungarian Method, named by the nationality of its developers. The steps of Hungarian Method are given below.

The optimum basic feasible solution has basic variables given below

x₁₃ = 1, x₂₅ = 1, x₃₄ = 1, x₄₂ = 1, x₅₁ = 1

The lowest cost of the assignments is 110 TL.

Z = 1 × 30 + 1 × 50 + 1 × 15 + 1 × 15+ 1 × 0= 110 TL.