Please respond to the following 4 open-ended prompts by demonstrating the fulles

ID: 379161 • Letter: P

Question

Please respond to the following 4 open-ended prompts by demonstrating the fullest extent of your knowledge.

(a) What are the 4 assumptions of classic linear programming problems? Under what circumstances can you relax one or more of these assumptions?

(b) What are some similarities and differences between transportation and assignment problems?

(d) What situations in linear programming would it be beneficial to use the duality theory of the Simplex method?

Explanation / Answer

(a)

1. Proportionality: The basic assumption in the linear programming is that any change in the constraint inequalities will bring the proportional change in the objective function. That is, if the value of a variable is doubled, the contribution of that variable to the objective function and each constraint is also doubled.

2. Additivity: This assumption states that total value of the objective function and each constraint function is calculated by adding each variables individual contribution.

3. Continuity: This assumption states that the decision variables are continuous. This means both fractional and integer values can be used.

5. Certainty: This assumption states that the parameters of objective function coefficients and the coefficients of constraint inequalities are known.

6. The values of the regressors, the X's, are fixed in repeated sampling.

7. There is no exact linear relationship (i.e., multicollinear-ity) in the regressors.

8. Assumption: Population model

This assumption specifies endogenous variable and its functional form. It also states that the model is linear on the parameters.

Perfect multicollinearity is usually not seen in practice, unless the model is wrongly designed. In a situation like this, the estimators obtained will generally not be very accurate. The relationship between regressors makes it difficult to quantify accurately the effect each one has on the regressand and this is because of the fact that the variances of the estimators are high. So, when there is a linear relationship between the regressors, then no perfect multicollinearity exists.

(b)

Similarities:

Both are an extended form of LPP simplex problems and can be solved by simplex method.

Both the models focus on reducing cost or maximizing profits.

The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines

Assignment Problem

Transportation Problem

There are facilities, and jobs which have to be assigned to those facilities.

sources and destinations are present

Number of facilities (sources) is equal to number of jobs (destinations)

This is not the case here.

Each supply and demand value is 1

Total supply must equal to total demand

It is concerned with the allocation of different jobs to the different people within the organization or firm. There are 2 methods of solving this type of problem 1) Hungarian method 2)Completer enumeration method.

It is concerned with the type of transportation method along with transportation routes for the distribution of a product. There are 3 method to solve this type of problem.

(C.)

A heuristic is a guide that aids decision-making. These in a way are rules used by people to make judgements and make decisions.

(D)

The Duality in Linear Programming implies that every linear programming problem has another linear programming problem related to it. So, in this case the original linear programming problem is called “Primal,” while the derived linear problem is called “Dual.”

The primal or original linear programming problem is of the maximization type while the dual problem is of minimization type.

The first column in the constraint inequality of primal problem becomes the first row in a dual problem and similarly the second column of constraint has become the second row in the dual problem.

Dual problem is used for sensitivity analysis.

Situations where specialized algorithms are required, this can be used.

In situations where it is possible to solve the related linear program with the shadow prices.