Posted: February 18th, 2023

The shaded area in the following graph represents the feasible region of a linear programming problem whose objective function is to be maximized, where x1 and x2 represent the level of the two activities.

Label each of the following statements as True or False, and then justify your answer based on the graphical method. In each case, give an example of an objective function that illustrates your answer.

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a. If (3, 3) produces a larger value of the objective function than (0, 2) and (6, 3), then (3, 3) must be an optimal solution.

b. If (3, 3) is an optimal solution and multiple optimal solutions exist, then either (0, 2) or (6, 3) must also be an optimal solution.

c. The point (0, 0) cannot be an optimal solution.

SOLUTION

Linear programming is a mathematical optimization technique used to determine the best outcome given a set of constraints and a linear objective function. The objective function is a linear equation that is to be maximized or minimized, subject to constraints that are also linear equations. The variables in the objective function and constraints are continuous, meaning they can take on any real value.

A typical linear programming problem can be formulated as follows:

Maximize (or minimize) Z = c1x1 + c2x2 + … + cnxn

Subject to:

a11x1 + a12x2 + … + a1nxn <= b1

a21x1 + a22x2 + … + a2nxn <= b2

…

am1x1 + am2x2 + … + amnxn <= bm

where Z is the objective function, c1, c2, …, cn are the coefficients of the variables x1, x2, …, xn in the objective function, and aij is the coefficient of the variable xi in the jth constraint. The values b1, b2, …, bm represent the right-hand side of the constraints.

The variables x1, x2, …, xn are non-negative and represent the decision variables, which can take on any non-negative real value. The objective is to find the values of x1, x2, …, xn that maximize (or minimize) the objective function Z while satisfying all of the constraints.

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