İŞLETME KARAR MODELLERİ - Deneme Sınavı - 6

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.

A transportation problem is a linear optimization problem, which seeks to minimize the total transportation cost from the origins to the destinations. The total cost is the sum of the total cost of each route, which is the multiplication of the unit cost with the amount of the units to be transported on the respective route. The decision variables of this problem are the amounts of shipment, as the transportation costs are given data. A balanced transportation model is a linear model with equality constraints. The inverse of a linear problem can be modeled as the dual of the original model. 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. At the optimum point, the objective value of the primal and the dual models are the same, and the optimum solution is the only solution that meets the restrictions for both models. Hence, the optimality of a basic feasible solution for the transportation model can be tested by checking the feasibility of the dual solution. The correct answer is B.

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. In the equation below, the asterisk denotes that x and y are the optimum feasible solutions. cx* = Z = W = y* b. The correct answer is D.

When the primal-dual pairs’ objective values are equal, the solutions are optimal if the solutions are both feasible—neither of the solutions does violate their respective constraints. Put differently, a feasible solution of the primal solution (x) is not optimal if the dual solution (y) violates a constraint of the dual model. 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.

Northwest Corner Method: The first basic variable is the one that is at the northwest on the transportation tableau. The value allocated to this variable is the value of the smaller one of the supply and demand. If the respective origin has any supply remaining after the allocation, then move one column to the right and allocate the value in the same manner as the first one. Otherwise, move one row down and repeat the allocation process with the remaining demand. The process is finalized when there is nothing left to allocate at the last row. The correct answer is A.

The concept of linear programming and its solution method “simplex” was first introduced by George Dantzig, the so-called father of linear programming. His research on the planning methods for the US Army Air Force was a game changer, as the method had great potential for dealing with difficult decision making problems by utilizing technological/computational advances.

The correct answer is E.

The variables that are set to zero are called non-basic variables. These have zero coefficients and are not related to the basic solution, as they have no effect on the objective function. On the contrary, the remaining are the only variables that determine the basic solution.

The correct answer is E.

The value at the intersection of the pivotal column and row is called the pivotal number. It is alternatively called as pivot number or pivot element 

The correct answer is D.

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 correct answer is D.

Here, the system has two equations (m = 2) and four variables. (n = 4) Thus, there are 6 basic solutions.

C2-4 = 4!/ 2!(4 − 2)! = 6

The correct answer is C.

For this system of equations, basic solutions can be determined by setting two (n – m = 2) of the variables to zero and solving the equations for the remaining two (m = 2). 

The correct answer is C.

If two corner-point feasible solutions are connected by a line segment, these cornerpoint feasible solutions are adjacent. Two corner-point feasible solutions are adjacent if all but one of their respective variables have the same value.

The correct answer is B.

A slack or surplus is utilized to reconstruct a constraint to obtain equation format. 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.

The correct answer is B.

A slack or surplus is utilized to reconstruct a constraint to obtain equation format. A slack variable represents the remainder of an expendable resource and applies to less than or equal constraints. 

The correct answer is A.

In the simplex tableau, this is implied by a negative coefficient in the pivotal column. Before moving forward, the steps of determining the pivotal row are given below:

1. Take each positive coefficient in the pivotal column.

2. Divide each of these into the right-hand side constant for the corresponding row.

3. Identify the row that has the minimum ratio.

4. Pivotal row is the row that is identified in the previous step.

The correct answer is A.

All of them are true for simplex method.

The value at the intersection of the pivotal column and row is called the pivotal number.