İŞLETME KARAR MODELLERİ

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 dual problems are imaginary and thus, purchasing is not a real option for the Factory. The purchasing price concept in the text is sometimes called shadow price so as to stress the virtuality. The real alternative for this factory is “not to produce”, no purchasing.

In a degenerate basic feasible solution, a degenerate basic variable acts as a non-basic variable, however, it does not prevent reaching to the optimal.

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

Northwest corner method

Least-cost method

Vogel’s approximation method

A transportation problem is a linear optimization problem, which seeks to minimize the total costs of shipment of items from certain origins to a number of destinations while fulfilling the requirements of destinations and emptying the origins. The decision variables of this problem are the amounts to be shipped for each origin-destination pairs.

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. Note that a transportation problem has a feasible solution only if the total availability meets all the requirements.

According to these, the correct answer is given in the option C.

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. Return to step 2.

In this sense, the correct answer is A.

The dual model interchanges costs and constraints of the primal model, and converts the objective from maximization to minimization. Moreover, the values of the decision variables of the dual model reveal the prices for the constraints of the original problem
for a basic feasible solution. This interpretation provides an economic perspective for solving the primal problem. Combining this economic insight and the properties of duality allows for an easier method altering the regular version of the simplex algorithm for the solution of a balanced transportation model. According tı these, the correct answer is given in the option B.

Some of the properties of duality contribute solving of the primal problem by considering primal-dual model relationships. Before getting into these, recall that x’s are the components of a feasible solution for the primal model and, whereas the corresponding dual solution comprises of y’s. 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. In this sense the correct answer is E.

The transportation problem, as it was formulated at the beginning of this chapter, is a minimization problem just as the one exemplified for the economic interpretation of the dual model. However, the difference is that the constraints of the transportation model are in the form of equality. Thus, the duality relation here is asymmetric.

The dual model interchanges costs and constraints of the primal model, and converts the objective from maximization to minimization. Hence, there is a dual variable y for each constraint. In the case of the transportation model, the dual variable y is defined with a pair of variables; u, for the constraints of origins and v, for the constraints of destinations.

The dual of the transportation model can be interpreted as the dual model explained in the previous subsection. For the problem of the production decision, a counterpart actor was assumed as if it provided a purchasing alternative. A similar assumption can be made to interpret the dual of the transportation model. The original problem is to select the most economical distribution routes that meet the demands of the destinations. Total cost of the shipment depends on total costs of transportation for the selected routes. The decision variables for the primal model are the amounts to be carried from the origins (e.g. production locations) to the destinations (e.g. retailing locations).

According to these information, the correct answer is option E.

The solution of a transportation problem begins with determining a basic feasible solution to proceed to the next phase. A balanced transportation problem has m + n – 1 basic variables in its basic feasible solution, whereas the number of the constraint equations is m + n. The correct answer is given in option A.

Selecting a variable to be basic means allocating the greatest value possible to the respective variable, under the constraints of the total supply and demand for the origins and destinations, respectively. There are various methods for selecting an arbitrary variable; the most prominent ones are:
1. Northwest corner method
2. Least-cost method
3. Vogel’s approximation method

The correct answer is C.

(based on the model on page 114)

As a linear algebraic property of balanced transportation model, the number of independent equations is m + n – 1. Therefore, a feasible basic solution has m + n – 1 basic variables.

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. In this sense, the correct aswer is the option D.

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. Northwest corner method is the most convenient method to initialize the solution of a transportation model as compared to least-cost and Vogel's approximation methods. so the answer for this question is A.

As a special form of the linear programming model, the transportation model and its solution method have been introduced thus far. This section introduces the assignment model, a special case of the transportation model. The term “Assignment” refers to the case of job allocations to a group of workers. In general, a worker has a single job and a job is done by a single worker. Similarly, the assignment model has the origin locations from where a single unit can be transported to a dedicated destination, and vice versa—the model has the destinations to which a single unit is sent solely from one origin location. Simply put, the number of the origins (represented by the workers) and the destinations (represented by the jobs) are equal, and the amount to be sent and received is strictly 1.

(see the relevant section in your book for more details.)

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. So, the correct answer is option B.

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.

According to these, the correct answer is given in option C.