The foundation of operations research is mostly cited to studies and applications in the U.S. Army right before The World War II. However, LeonidV. Kantorovich is recognized as the first to explore this area. In 1939, Kantorovich published a paper on a method for a production plan which he claimed to be the optimal way. This study included a linear program without a systematic solution. The systematic solution which is called “simplex algorithm” was later introduced by George B. Dantzig, who had planning experience in the U.S. Army Air Force. Tjalling C. Koopmans, who built a model for ship routes, was another contributor to the field. He and Kantorovich shared 1975 Nobel Prize in economics for their studies on the optimal allocation of resources. The correct answer is D.

A mixed strategy is a decision to choose a course of action for each play in accordance with some particular probability.

lowest in rows : -4 -5 0 -4 ; max : 0 . pg. 158. Correct answer is C.

Players can be individuals, organizations, teams or, in some cases, nature itself. The
number of players must be finite and must be known.

A state is called periodic, if it can only return to itself after a fixed number of transitions greater than 1. Otherwise, a state is said aperiodic.

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 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.

Max -Z = -x₁ + 3 x₂ - 2 x₃ - 0 s₁ - 0 s₂ ; x₂ : it has the maximum positive coefficient . pg. 95 . Correct answer is A.

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 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.

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.

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.

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.

then the following explanation is obtained 

and

Because of the

the result is 

The phases of algebraic method are these:

• Represent the problem with a system of equations, which has m equations and n non-negative variables.
• Determine the basic feasible solutions of the equations, of which one is expected to be the optimum.
• Use the objective function to determine the optimum basic feasible solution.

Decision Analysis generally deals with decision making in the face of uncertainty about one future event.

Historically, the game theory was first proposed by the French mathematician Emil Borel in 1921.

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.