Deck 7: Linear Programming Applications

Question 1

The fixed per-unit profit contribution coefficients of the objective function in a linear programming problem imply the following economic assumptions except:

A) selling prices per unit of the products (outputs) are constant
B) constant returns to scale in the production process
C) buying prices per unit of the resources (inputs) are proportional to the amount purchased
D) both b and c
E) both a and c

Answer 1

buying prices per unit of the resources (inputs) are proportional to the amount purchased

Question 2

Which of the following statements concerning dual variables is (are) true?

A) Dual variables are obtained automatically in an algebraic solution of a linear programming problem.
B) Dual variables are similar to the artificial variables used in the LaGrange Multiplier technique.
C) A dual variable indicates how much the objective function will change if one additional unit of a given resource is made available, provided that the increase in the resource does not shift the optimal solution to another corner of the feasible solution space.
D) a and c
E) a, b, and c

Answer 2

a, b, and c

Question 3

If the primal linear programming problem has two variables and four constraints (excluding the non-negativity constraints), the corresponding dual linear programming problem will have ____.

A) two variables and four constraints
B) four variables and two constraints
C) two variables and two constraints
D) four variables and four constraints
E) none of the above

Answer 3

four variables and two constraints

Question 4

A dual variable equal to zero in the optimal solution to a profit-maximization linear programming problem indicates that the objective function will not increase if an additional unit of the given resources is made available.

Question 5

An optimal solution of a linear programming problem always lies on the boundary of the feasible solution space.

Question 6

A primal linear programming problem has multiple optimal solutions if it contains two or more variables.

Question 7

Slack variables are given coefficients of ____ in the objective function.

A) +1
B) 0
C) -1
D) +.00001
E) none of the above

Question 8

In a maximization linear programming problem, the ____ variables represent the difference between the right-hand sides and left-hand sides of less than or equal to ( $\le$ ) inequality constraints.

A) dual
B) slack
C) primal
D) both a and b
E) none of the above

Question 9

In a minimization linear programming problem, the ____ variables are subtracted from the greater than or equal to ( $\ge$ ) inequality constraints in order to convert these constraints to equalities.

A) dual
B) primal
C) surplus
D) both a and b
E) both a and c

Question 10

A computer solution of large-scale linear programming problems typically employs a procedure (or variation of the procedure) known as the ____ method.

A) least squares
B) analysis of variance
C) simplex
D) primal/dual
E) none of the above

Question 11

____ variables are given coefficients of zero in the objective function of a minimization linear programming problem.

A) Primal
B) Dual
C) Surplus
D) Dummy
E) none of the above

Question 12

The Value-Pack Canning Company buys fresh fruits and vegetables from farmers, processes and cans them, and sells the output to various supermarket chains. The company is trying to determine the optimal mix of peas and green beans to process during the forthcoming period at its Fresno plant. Output is limited by the capacity of the plant and the firm's financial resources. The plant can process 4,000,000 pounds of peas or green beans (or any linear combination thereof) during the forthcoming period. The company buys peas and green beans for $.10 and $.20 per pound respectively. Purchases must be paid for at the time of delivery and the firm's current cash balance limits purchases to $600,000 during the forthcoming period. As part of long-term contracts with several supermarket chains, the firm is required to process at least 800,000 pounds of peas and 1,200,000 pounds of green beans during the period. The profit contribution of peas and green beans are $.015 and $.025 respectively. The firm desires to find the mix of peas and green beans to produce in order to maximize the total profit contribution of the plant during the forthcoming period. Let X₁ be the number of pounds of peas processed and X₂ be the number of pounds of green beans processed.
(a)Formulate the problem algebraically in the linear programming framework.
(b)Using graphical methods, determine the optimal mix of vegetables to process.