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Exam 19: Linear Programming

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Question 1

True/False

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Linear programming techniques will always produce an optimal solution to an LP problem.

A) True
B) False

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Question 2

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Coordinates of all corner points are substituted into the objective function when we use the approach called:

A) least squares.
B) regression.
C) enumeration.
D) graphical linear programming.
E) constraint assignment.

F) A) and B)
G) A) and E)

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Question 3

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The production planner for Fine Coffees, Inc., produces two coffee blends: American (A) and British (B) . Two of his resources are constrained: Columbia beans, of which he can get at most 300 pounds (4,800 ounces) per week; and Dominican beans, of which he can get at most 200 pounds (3,200 ounces) per week. Each pound of American blend coffee requires 12 ounces of Colombian beans and 4 ounces of Dominican beans, while a pound of British blend coffee uses 8 ounces of each type of bean. Profits for the American blend are $2.00 per pound, and profits for the British blend are $1.00 per pound. What is the Columbia bean constraint?

What combination of x and y will yield the optimum for this problem? Maximize $Z=\$ 3 x+\$ 15 y$ $\begin{array}{l}\text{Subject to:}&2 x+4 y \leq 12 \\&5 x+2 y \leq 10\end{array}$

An electronics firm produces two models of pocket calculators: the A-100 (A) , which is an inexpensive four-function calculator, and the B-200 (B) , which also features square root and percent functions. Each model uses one (the same) circuit board, of which there are only 2,500 available for this week's production. Also, the company has allocated a maximum of 800 hours of assembly time this week for producing these calculators, of which the A-100 requires 15 minutes (.25 hours) each, and the B-200 requires 30 minutes (.5 hours) each to produce. The firm forecasts that it could sell a maximum of 4,000 A-100s this week and a maximum of 1,000 B-200s. Profits for the A-100 are $1.00 each, and profits for the B-200 are $4.00 each. For the production combination of 1,400 A-100s and 900 B-200s, which resource is slack (not fully used) ?

The theoretical limit on the number of constraints that can be handled by the simplex method in a single problem is:

The logistics/operations manager of a mail order house purchases two products for resale: king beds (K) and queen beds (Q) . Each king bed costs $500 and requires 100 cubic feet of storage space, and each queen bed costs $300 and requires 90 cubic feet of storage space. The manager has $75,000 to invest in beds this week, and her warehouse has 18,000 cubic feet available for storage. Profit for each king bed is $300 and for each queen bed is $150. For the purchase combination 0 king beds and 200 queen beds, which resource is slack (not fully used) ?

The feasible solution space only contains points that satisfy all constraints.

A company produces two products (A and B) using three resources (I, II, and III). Each product A requires 1 unit of resource I and 3 units of resource II and has a profit of $1. Each product B requires 2 units of resource I, 3 units of resource II, and 4 units of resource III and has a profit of $3. Resource I is constrained to 40 units maximum per day; resource II, 90 units; and resource III, 60 units. What is the objective function?

The production planner for Fine Coffees, Inc., produces two coffee blends: American (A) and British (B) . Two of his resources are constrained: Columbia beans, of which he can get at most 300 pounds (4,800 ounces) per week; and Dominican beans, of which he can get at most 200 pounds (3,200 ounces) per week. Each pound of American blend coffee requires 12 ounces of Colombian beans and 4 ounces of Dominican beans, while a pound of British blend coffee uses 8 ounces of each type of bean. Profits for the American blend are $2.00 per pound, and profits for the British blend are $1.00 per pound. Which of the following is not a feasible production combination?

A small firm makes three products, which all follow the same three-step process, which consists of milling, inspection, and drilling. Product A requires 6 minutes of milling, 5 minutes of inspection, and 4 minutes of drilling; product B requires 2.5 minutes of milling, 2 minutes of inspection, and 2 minutes of drilling; and product C requires 5 minutes of milling, 4 minutes of inspection, and 8 minutes of drilling. The department has 20 hours available during the next period for milling, 15 hours for inspection, and 24 hours for drilling. Product A contributes $6.00 per unit to profit, product B contributes $4.00 per unit, and product C contributes $10.00 per unit. Use the following computer output to find the optimum mix of products in terms of maximizing contributions to profits for the next period. PROBLEM TITLE: LINEAR PROGRAMMING PROBLEM IS A MAX WITH 3 VARIABLES AND 3 CONSTRAINTS. $\begin{array} { | l | l | l | l | l | } \hline \text { ROW } & \text { X1 } & \text { X2 } & \text { X3 } & \text { RHS } \\\hline\end{array}$ $\begin{array} { | l | l | l | l | } \hline \text { COST } & 6.00 & 4.00 & 10.00 \\\hline\end{array}$ $\begin{array} { | l | l | l | l | l | r | } \hline 1 - & 6.00 & 2.50 & 5.00 & \leq & 1,200.00 \\\hline 2 - & 5.00 & 2.00 & 4.00 & \leq & 900.00 \\\hline 3 - & 4.00 & 2.00 & 8.00 & \leq & 1,440.00 \\\hline\end{array}$ NUMBER OF ITERATIONS: 2 OPTIMAL SOLUTION: OBJECTIVE FUNCTION VALUE = 2,070 DECISION VARIABLE SECTION: $\begin{array} { | c | c | c | c | } \hline \text { VARIABLE } & \text { STATUS } & \text { VALUE } & \text { REDUCED COST } \\\hline \mathrm { X } 1 & \text { Non-basic } & 0 & 3.5 \\\hline \mathrm { X } 2 & \text { Basic } & 180 & 0 \\\hline \mathrm { X } 3 & \text { Basic } & 135 & 0 \\\hline\end{array}$ SLACK VARIABLES SECTION: $\begin{array} { | c | c | c | c | } \hline \text { SLACK } & \text { STATUS } & \text { VALUE } & \text { SHADOW PRICE } \\\hline \mathrm { X } 4 & \text { Basic } & 75 & 0 \\\hline \mathrm { X5 } & \text { Non-basic } & 0 & 1.5 \\\hline \mathrm { X6 } & \text { Non-basic } & 0 & .5 \\\hline\end{array}$

An objective function represents a family of parallel lines.

Graphical linear programming can handle problems that involve any number of decision variables.

The equation 3xy = 9 is linear.

A change in the value of an objective function coefficient does not change the optimal solution.

A company produces two products (A and B) using three resources (I, II, and III). Each product A requires 1 unit of resource I and 3 units of resource II and has a profit of $1. Each product B requires 2 units of resource I, 3 units of resource II, and 4 units of resource III and has a profit of $3. Resource I is constrained to 40 units maximum per day; resource II, 90 units; and resource III, 60 units. What are the corner points of the feasible solution space?

The term range of optimality refers to a constraint's right-hand-side quantity.

A company produces two products (A and B) using three resources (I, II, and III). Each product A requires 1 unit of resource I and 3 units of resource II and has a profit of $1. Each product B requires 2 units of resource I, 3 units of resource II, and 4 units of resource III and has a profit of $3. Resource I is constrained to 40 units maximum per day; resource II, 90 units; and resource III, 60 units. What is the constraint for resource I?

The owner of Crackers, Inc., produces two kinds of crackers: Deluxe (D) and Classic (C) . She has a limited amount of the three ingredients used to produce these crackers available for her next production run: 4,800 ounces of sugar; 9,600 ounces of flour, and 2,000 ounces of salt. A box of Deluxe crackers requires 2 ounces of sugar, 6 ounces of flour, and 1 ounce of salt to produce; while a box of Classic crackers requires 3 ounces of sugar, 8 ounces of flour, and 2 ounces of salt. Profits for a box of Deluxe crackers are $.40; and for a box of Classic crackers, $.50. For the production combination of 800 boxes of Deluxe and 600 boxes of Classic, which resource is slack (not fully used) ?

The logistics/operations manager of a mail order house purchases two products for resale: king beds (K) and queen beds (Q) . Each king bed costs $500 and requires 100 cubic feet of storage space, and each queen bed costs $300 and requires 90 cubic feet of storage space. The manager has $75,000 to invest in beds this week, and her warehouse has 18,000 cubic feet available for storage. Profit for each king bed is $300 and for each queen bed is $150. What is the storage space constraint?