İŞLETME KARAR MODELLERİ

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.

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

1. Determine a basic feasible solution to initiate,

2. Check the optimality of the basic feasible solution as the simplex method does. If it is not, optimal, then move to the next step. If optimal, stop.

3. Iterate to the next basic feasible solution by determining the entering and leaving variable using the rate of improvement on the objective value.

A linear program can be related to another one reflecting the opposite of itself. These two programs have the same conditions (cost and constraint coefficients), yet they are modeled from counter-viewpoints. Consequently, they reach the same optimal solution mutually. Here, the inversed pair of the original model is called the dual of the original (or primal) model. The dual model interchanges costs and constraints of the primal model, and converts the objective from maximization to minimization.

For a basic feasible solution of the primal model, the objective value, Z is equal or greater than W, the objective value of the corresponding dual solution. This is called weak duality property: the researcher begins at any point, then reaches the optimum at a level by lowering the costs and thus increasing the revenues simultaneously.

The simplex method concurrently identifies basic feasible solutions for a primal model and its dual. This is a property of the duality called complementary solutions property. This property allows checking for the optimality of the primal solution by inspecting the feasibility of its dual solution.

The dual of a balanced transportation problem is interpreted as a problem of a counterpart who provides an alternative to transportation. The decision variables of the dual model are the prices of the alternative service offered by the counterpart actor. The correct answer is A.

Least-cost Method: The Least-cost method considers the respective costs of the variables so as to find a basic feasible solution that approximates the optimum solution more. Unlike the Northwest Method, the Leastcost algorithm begins with selecting the route that has the smallest unit cost of transportation. If there is more than one variable that has the smallest cost, any variable among these can be selected arbitrarily. The value allocated to the selected route is the greater one of the values of the supply and demand. The satisfied column (corresponding demands) or row (corresponding supplies) is masked out with grey and ignored for the next allocations. Next, subtract the allocation value from the supply and demand and repeat the same allocation process with the remainders of the supply and demand. The process continues until the last row or column is satisfied. The correct answer is C.

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 is given below. 1. Identify the smallest value of each row for the cost matrix of the assignment problem. Subtract each row’s smallest value from all the costs in the respective row. 2. Identify the smallest value of each column for this altered matrix. Subtract each column’s smallest value from all the costs in the respective column. 3. Mask the columns and rows out that have a zero value. The number of masked out rows and columns must be at a minimum. 4. If the number of masked out rows and columns is equal to n, then the optimum can be obtained from the present matrix; move on to the next step. If not, skip to Step 6. 5. Identify the optimal solution by the coordinates of the zero-valued elements in the present matrix. 6. Identify the smallest value except for the ones in masked out rows and columns. This value is then subtracted from the values of unmasked rows and columns and, added to the intersections of masked out rows and columns. Return to Step 3. The correct answer is D.

Vogel’s Approximation Method: Vogel’s approximation method (VAM) can be regarded as an improved version of the least-cost method. Instead of using the direct cost of the transportation, VAM utilizes the concept of the penalty cost. Each row or column has its own penalty cost, which is used for determining which variables are the basic ones. A penalty cost is the difference between the smallest cost of a row (or column) and the cost that is smaller than the others except the smallest one for that row (or column).

The solution of a transportation model follows three steps to reach the optimum. First step is to determine a basis, which is the initial basic feasible solution. The next step is to check the optimality of the solution. The third step is conditional to the second: if the current solution is not optimal, then the process is to iterate to a new basic feasible solution that includes the entering variable determined in the previous step. The correct answer is A.

Also known as the method of simplex multipliers or the u-v method, MODI is a tailored version of the simplex method for the transportation model. MODI determines whether a basic feasible solution is optimal or not, and if not, identifies the entering variable by following these steps: 1. Denote the simplex multipliers, ui for each row and vi for each column alongside the transportation tableau 2. Write down the equations ui + vi = cij for each basic variable xij 3. Set one of the multipliers to zero and find the values of other multipliers on this system of equations 4. Calculate the values of cij – (ui + vi ) for each non-basic variable 5. If any of these values is greater than or equal zero, then the solution is optimal 6. If not, the non-basic variable that has the most negative value is the entering variable. The correct answer is C.