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Exam 19: Linear Programming

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Question 21

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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 objective function?

A) Z = $150K + $300Q
B) Z = $500K + $300Q
C) Z = $300K + $150Q
D) Z = $300K + $500Q
E) Z = $100K + $90Q

F) All of the above
G) B) and E)

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Question 22

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The production planner for a private label soft drink maker is planning the production of two soft drinks: root beer (R) and sassafras soda (S) . Two resources are constrained: production time (T) , of which she has at most 12 hours per day; and carbonated water (W) , of which she can get at most 1,500 gallons per day. A case of root beer requires 2 minutes of time and 5 gallons of water to produce, while a case of sassafras soda requires 3 minutes of time and 5 gallons of water. Profits for the root beer are $6.00 per case, and profits for the sassafras soda are $4.00 per case. What is the objective function?

A) $4R + $6S = Z
B) $2R + $3S = Z
C) $6R + $4S = Z
D) $3R + $2S = Z
E) $5R + $5S = Z

F) B) and D)
G) All of the above

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Question 23

Multiple Choice

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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 maximum profit?

The linear optimization technique for allocating constrained resources among different products is:

The production planner for a private label soft drink maker is planning the production of two soft drinks: root beer (R) and sassafras soda (S) . Two resources are constrained: production time (T) , of which she has at most 12 hours per day; and carbonated water (W) , of which she can get at most 1,500 gallons per day. A case of root beer requires 2 minutes of time and 5 gallons of water to produce, while a case of sassafras soda requires 3 minutes of time and 5 gallons of water. Profits for the root beer are $6.00 per case, and profits for the sassafras soda are $4.00 per case. For the production combination of 180 root beer and 0 sassafras soda, which resource is slack (not fully used) ?

When a change in the value of an objective function coefficient remains within the range of optimality, the optimal solution also remains the same.

The operations manager for the Blue Moon Brewing Co. produces two beers: Lite (L) and Dark (D) . Two of his resources are constrained: production time, which is limited to 8 hours (480 minutes) per day; and malt extract (one of his ingredients) , of which he can get only 675 gallons each day. To produce a keg of Lite beer requires 2 minutes of time and 5 gallons of malt extract, while each keg of Dark beer needs 4 minutes of time and 3 gallons of malt extract. Profits for Lite beer are $3.00 per keg, and profits for Dark beer are $2.00 per keg. What is the objective function?

LP problems must have a single goal or objective specified.

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. What is the constraint for sugar?

Wood Specialties Company produces wall shelves, bookends, and shadow boxes. It is necessary to plan the production schedule for next week. The wall shelves, bookends, and shadow boxes are made of oak, of which the company has 600 board feet. A wall shelf requires 4 board feet, bookends require 2 board feet, and a shadow box requires 3 board feet. The company has a power saw for cutting the oak boards into the appropriate pieces; a wall shelf requires 30 minutes, bookends require 15 minutes, and a shadow box requires 15 minutes. The power saw is expected to be available for 36 hours next week. After cutting, the pieces of work in process are hand finished in the finishing department, which consists of 4 skilled and experienced craftsmen, each of whom can complete any of the products. A wall shelf requires 60 minutes of finishing, bookends require 30 minutes, and a shadow box requires 90 minutes. The finishing department is expected to operate for 40 hours next week. Wall shelves sell for $29.95 and have a unit variable cost of $17.95; bookends sell for $11.95 and have a unit variable cost of $4.95; a shadow box sells for $16.95 and has a unit variable cost of $8.95. (A) Is this a problem in maximization or minimization? (B) What are the decision variables? Suggest symbols for them. (C) What is the objective function? (D) What are the constraints?

In a linear programming problem involving minimization, at least one constraint must be of the __________ type.

For the following constraints, which point is in the feasible solution space of this maximization problem? $\begin{aligned}14 x + 6 y & \leq 42 \\x - y & \leq 3\end{aligned}$

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. What are optimal weekly profits?

In the range of feasibility, the value of the shadow price remains constant.

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 slack (unused amount) for each resource for the optimum production combination?

The equation 5x + 7y = 10 is linear.

Consider the following linear programming problem: Maximize $Z = \$ 15 X + \$ 20 y$ $\begin{array} { c } \text{Subject to:}&8 x + 5 y \leq 40 \\&4 x + y \geq 4\end{array}$ Solve the values of x and y that will maximize revenue. What revenue will result?

When we use less of a resource than was available, in linear programming that resource would be called non-__________.

Nonzero slack or surplus is associated with a binding constraint.

The value of an objective function decreases as it is moved away from the origin.