Question 1

Suppose an LP problem was subject to constraints of
2X + Y> 10
X + 3Y> 20
Suppose that a new constraint is added, of the form 3X + A ∗ Y> 90. What is the largest value that A can have so that this new constraint is redundant?

Accepted Answer

The answer of Suppose an LP problem was subject to...

Question 2

A manager must decide on the mix of products to produce for the coming week. Product A requires three minutes per unit for molding, two minutes per unit for painting, and one minute for packing. Product B requires two minutes per unit for molding, four minutes for painting, and three minutes per unit for packing. There will be 600 minutes available for molding, 600 minutes for painting, and 420 minutes for packing. Both products have contributions of $1.50 per unit. Answer the following questions; base your work on the solution panel provided.
 $$\begin{array} { | l | r | r | r | r | r | } 
\hline & \mathrm { A } & \mathrm { B } & & \text { RHS } & \text { Dual } \
\hline \text { Maximize } & 1.5 & 1.5 & & & \
\hline \text { Molding } & 3 & 2 . & \square & 600 & 0.375 \
\hline \text { Painting } & 2 & 4 &\square & 600 & 0.1875 \
\hline \text { Packing } & 1 . & 3 & \square& 420 & 0 . \
\hline \text { Solution --- } & 150 & 75 & & 337.5 & \
\hline
\end{array}$$ a. What combination of A and B will maximize contribution?
b. What is the maximum possible contribution?
c. Are any resources not fully used up? Explain.

Accepted Answer

The answer of A manager must decide on the mix...

Question 3

A manager must decide on the mix of products to produce for the coming week. Product A requires three minutes per unit for molding, two minutes per unit for painting, and one minute for packing. Product B requires two minutes per unit for molding, four minutes for painting, and three minutes per unit for packing. There will be 600 minutes available for molding, 600 minutes for painting, and 420 minutes for packing. Both products have contributions of $1.50 per unit.
a. Algebraically state the objective and constraints of this problem.
b. Plot the constraints on the grid below and identify the feasible region.

Accepted Answer

The answer of A manager must decide on the mix...

Question 4

Tom is a habitual shopper at garage sales. Last Saturday he stopped at one where there were several types of used building materials for sale. At the low prices being asked, Tom knew that he could resell the items in another town for a substantial gain. Four things stood in his way: he could only make one round trip to resell the goods; his pickup truck bed would hold only 1000 pounds; the pickup truck bed could hold at most 70 cubic feet of merchandise; and he had only $200 cash with him. He wants to load his truck with the mix of materials that will yield the greatest profit when he resells them.
 $\begin{array}{|c|c|c|c|c|}
\hline \text { Item } & \text { Cubic feet per } & \text { Price per unit } &\text { Weight per } & \text { Can resell for } \
&\text { Item } &&\text { Item } \
\hline 2 \times 4 \text { studs } & 1 & \$ 0.10 & 5 \text { pounds } & \$ 0.80\
\hline 4 \times 8 \text { plywood } & 3 & \$ 0.50 & 20 \text { pounds } & \$ 3.00 \
\hline \text { Concrete blocks } & 0.5 & \$ 0.25 & 10 \text { pounds } & \$ 0.75 \\hline
\end{array}$ State the decision variables (give them labels). State the objective function. Formulate the constraints of this problem. Do not solve, but speculate on what might be a good solution for Tom. You must supply a set of quantities for the decision variables. Provide a sentence or two of support for your choice.

Accepted Answer

The answer of Tom is a habitual shopper at garage...

Question 5

The property manager of a city government issues chairs, desks, and other office furniture to city buildings from a centralized distribution centre. Like most government agencies, it operates to minimize its costs of operations. In this distribution centre, there are two types of standard office chairs, Model A and Model B. Model A is considerably heavier than Model B, and costs $20 per chair to transport to any city building; each model B costs $14 to transport. The distribution centre has on hand 400 chairs-200 each of A and B.
The requirements for shipments to each of the city's buildings are as follows:
Building 1 needs at least 100 of A
Building 2 needs at least 150 of B.
Building 3 needs at least 100 chairs, but they can be of either type, mixed.
Building 4 needs 40 chairs, but at least as many B as A.
Write out the objective function and the constraints for this problem. (Hint: there are eight variables-chairs for building 1 cannot be used to satisfy the demands for another building).

Accepted Answer

The answer of The property manager of a city government...

Question 6

A stereo mail order centre has 8,000 cubic feet available for storage of its private label loudspeakers. The ZAR3 speakers cost $295 each and require 4 cubic feet of space; the ZAR2ax speakers cost $110 each and require 3 cubic feet of space; and the ZAR4 model costs $58 and requires 1 cubic foot of space. The demand for the ZAR3 is at most 20 units per month. The wholesaler has $100,000 to spend on loudspeakers this month. Each ZAR3 contributes $105, each ZAR2ax contributes $50, and each ZAR4 contributes $28. The objective is to maximize total contribution. Write out the objective and the constraints.

Accepted Answer

The answer of A stereo mail order centre has 8,000...

Question 7

The objective of a linear programming problem is to maximize 1.50A + 1.50B, subject to 3A + 2B ≤ 600, 2A + 4B ≤ 600, and 1A + 3B ≤ 420.
a. Plot the constraints on the grid below
b. Identify the feasible region and its corner points. Show your work.
c. What is the optimal product mix for this problem?

Accepted Answer

The answer of The objective of a linear programming problem...

Question 8

A financial advisor is about to build an investment portfolio for a client who has $100,000 to invest. The four investments available are A, B, C, and D. Investment A will earn 4% and has a risk of two "points" per $1,000 invested. B earns 6% with 3 risk points; C earns 9% with 7 risk points; and D earns 11% with a risk of 8. The client has put the following conditions on the investments: A is to be no more than one-half of the total invested. A cannot be less than 20% of the total investment. D cannot be less than C. Total risk points must be at or below 1,000.Identify the decision variables of this problem. Write out the objective function and constraints. Do not solve.

Accepted Answer

The answer of A financial advisor is about to build...

Question 9

A craftsman builds two kinds of birdhouses, one for wrens (X1), and one for bluebirds (X2). Each wren birdhouse takes four hours of labour and four units of lumber. Each bluebird house requires two hours of labour and twelve units of lumber. The craftsman has available 60 hours of labour and 120 units of lumber. Wren houses profit $6 each and bluebird houses profit $15 each.
Use the software output that follows to interpret the problem solution. Include a statement of the solution quantities (how many of which product), a statement of the maximum profit achieved by your product mix, and a statement of "resources unused" and "shadow prices." $$\begin{array} { | l | r | r | r | r | r | } 
\hline  { \text { Bird Houses Solution } } \
\hline \text { Variable } & \text { Value } & \text { Reduced } & \text { Original Val } & \text { Lower Bound } & \text {Upper Bound } \
\hline X 1 & 12 . & 0 . & 6 . & 5 . & 30 . \
\hline X 2 & 6 . & 0 . & 15 . & 3 . & 18 . \
\hline \text { Constraint } & \text { Dual Value } & \text { sack/Surplus } & \text { Original Val} & \text { Lower Bound } & \text {Upper Bound } \
\hline \text { Constraint 1 } & 0.3 & 0 & 60 . & 20 . & 120 . \
\hline \text { Constraint 2 } & 1.2 & 0 . & 120 . & 60 . & 360 . \
\hline
\end{array}$$

Accepted Answer

The answer of A craftsman builds two kinds of birdhouses,...

Question 10

Suppose that a chemical manufacturer is deciding how to mix two chemicals, A and B. A costs $5/gram and B costs $4/gram if they are ordered above the current supply level. There are currently 40 grams of A and 30 grams of B that must be used in the mix or they will expire. If a customer wants 1 kg of the mix with at least 40% A but no more than 55% A, how many grams of each chemical should be included in the mix?

Accepted Answer

The answer of Suppose that a chemical manufacturer is deciding...

Question 11

Phil Bert's Nuthouse is preparing a new product, a blend of mixed nuts. The product must be at most 50% peanuts, must have more almonds than cashews, and must be at least 10% pecans. The blend will be sold in one-pound bags. Phil's goal is to mix the nuts in such a manner that all conditions are satisfied and the cost per bag is minimized. Peanuts cost $1 per pound. Cashews cost $3 per pound. Almonds cost $5 per pound and pecans cost $6 per pound. Identify the decision variables of this problem. Write out the objective and the set of constraints for the problem. Do not solve.

Accepted Answer

The answer of Phil Bert's Nuthouse is preparing a new...

Question 12

Rienzi Farms grows sugar cane and soybeans on its 500 acres of land. An acre of soybeans brings a $1000 contribution to overhead and profit; an acre of sugar cane has a contribution of $2000. Because of a government program no more than 200 acres may be planted in soybeans. During the planting season 1200 hours of planting time will be available. Each acre of soybeans requires 2 hours, while each acre of sugar cane requires 5 hours. The company seeks maximum contribution (profit) from its planting decision.
a. Algebraically state the decision variables, objective and constraints.
b. Plot the constraints
c. Solve graphically, using the corner-point method.

Accepted Answer

The answer of Rienzi Farms grows sugar cane and soybeans...

Question 13

Lost Maples Winery makes three varieties of contemporary Texas Hill Country wines: Austin Formation (a fine red), Ste. Genevieve (a table white), and Los Alamos (a hearty pink Zinfandel). The raw materials, labour, and contribution per case of each of these wines is summarized below.
 $$\begin{array} { | l | c | c | c | c | c | } 
\hline & \begin{array} { c } 
\text { Grapes } \
\text { Variety A } \
\text { bushels }
\end{array} & \begin{array} { c } 
\text { Grapes } \
\text { Variety B } \
\text { bushels }
\end{array} & \begin{array} { c } 
\text { Sugar } \
\text { pounds }
\end{array} & \begin{array} { c } 
\text { Labour (man- } \
\text { hours) }
\end{array} & \begin{array} { c } 
\text { Contrib. } \
\text { per case }
\end{array} \
\hline \text { Austin Formation } & 4 & 0 & 1 & 3 & \$ 24 \
\hline \text { Ste. Genevieve } & 0 & 4 & 0 & 1 & \$ 28 \
\hline \text { Los Alamos } & 2 & 2 & 2 & 2 & \$ 20 \
\hline
\end{array}$$ The winery has 2800 bushels of Variety A grapes, 2040 bushels of Variety B grapes, 800 pounds of sugar, and 1060 man-hours of labour available during the next week. The firm operates to achieve maximum contribution. Refer to the POM for Windows panels showing the solution to this problem.  Answer the following questions.
a. For maximum contribution, how much of each wine should be produced?
b. How much contribution will be made by selling the output?
c. Is there any sugar left over? If so, how much? If not, what is its shadow price (dual value)? Explain what this value means to Lost Maples' management.
d. Interpret the meaning of the lower bound to Labour in the Ranging analysis. That is, explain how the solution would change if the amount of labour fell below that lower value.
e. Interpret the meaning of the upper bound to Los Alamos wine in the Ranging analysis.

Accepted Answer

The answer of Lost Maples Winery makes three varieties of...

Question 14

A linear programming problem contains a restriction that reads "the quantity of S must be no more than one-fourth as large as T and U combined." Formulate this as a constraint ready for use in problem solving software.

Accepted Answer

The answer of A linear programming problem contains a restriction...

Question 15

A feedlot is trying to decide on the lowest cost mix that will still provide adequate nutrition for its cattle. Suppose that the numbers in the chart represent the number of grams of ingredient per 100 grams of feed and that the cost of Feed X is $5/100 grams, Feed Y is $3/100 grams, and Feed X is $8/100 grams. Each cow will need 50 grams of A per day, 20 grams of B, 30 grams of C, and 10 grams of D. If the feedlot can get no more than 200 grams per day per cow of any of the feed types determine the constraints governing the problem.
 $$\begin{array} { | l | l | l | l | } 
\hline \text { Ingredient } & \text { X } & \text { Y } & Z \
\hline \mathrm { A } & 10 & 15 & 5 \
\hline \mathrm { B } & 30 & 10 & 20 \
\hline \mathrm { C } & 40 & 0 & 20 \
\hline \mathrm { D } & 0 & 20 & 30 \
\hline
\end{array}$$

Accepted Answer

The answer of A feedlot is trying to decide on...

Question 16

South Coast Papers wants to mix two lubricating oils (A and
B) for its machines in order to minimize cost. It needs no less than 3,000 gallons in order to run its machines during the next month. It has a maximum oil storage capacity of 4,000 gallons. There are 2,000 gallons of Oil A and 4,000 of Oil B available. The mixed fuel must have a viscosity rating of no less than 40.
When mixing fuels, the amount of oil obtained is exactly equal to the sum of the amounts put in. The viscosity rating is the weighted average of the individual viscosities, weighted in proportion to their volumes. The following is known: Oil A has a viscosity of 45 and costs 60 cents per gallon; Oil B has a viscosity of 37.5 and costs 40 cents per gallon.
State the objective and the constraints of this problem. Plot all constraints and highlight the feasible region. Use your (by now, well-developed) intuition to suggest a feasible (but not necessarily optimal) solution. Be certain to show that your solution meets all constraints.

Accepted Answer

The answer of South Coast Papers wants to mix two...

Question 17

The Queen City Nursery manufactures bags of potting soil from compost and topsoil. Each cubic foot of compost costs 12 cents and contains 4 pounds of sand, 3 pounds of clay, and 5 pounds of humus. Each cubic foot of topsoil costs 20 cents and contains 3 pounds of sand, 6 pounds of clay, and 12 pounds of humus. Each bag of potting soil must contain at least 12 pounds of sand, 12 pounds of clay, and 10 pounds of humus. Explain how this problem meets the conditions of a linear programming problem. Plot the constraints and identify the feasible region. Graphically or with corner points find the best combination of compost and topsoil that meets the stated conditions at the lowest cost per bag. Identify the lowest cost possible.

Accepted Answer

The answer of The Queen City Nursery manufactures bags of...

Question 18

Suppose that a constraint for assembly time has a shadow price of $50/hour for 15 hours in either direction and that all available assembly time is currently used (would require overtime to do more). If the salary of workers is $30 and they receive 50% extra pay for overtime what should management do?

Accepted Answer

The answer of Suppose that a constraint for assembly time...

Question 19

Suppose that a constraint is given by X + Y≤10. If another constraint is given to be 3X + 2Y≥15 determine the corners of the feasible solution. If the profit from X is 5 and the profit from Y is 10, determine the maximum profit.

Accepted Answer

The answer of Suppose that a constraint is given by...

Question 20

John's Locomotive Works manufactures a model locomotive. It comes in two versions-a standard (X1), and a deluxe (X2). The standard version generates $250 per locomotive for the standard version, and $350 per locomotive for the deluxe version. One constraint on John's production is labour hours. He only has 40 hours per week for assembly. The standard version requires 250 minutes each, while the deluxe requires 350 minutes. John's milling machine is also a limitation. There are only 20 hours a week available for the milling machine. The standard unit requires 60 minutes, while the deluxe requires 120. Formulate as a linear programming problem, and solve using either the graphical or corner points solution method.

Accepted Answer

The answer of John's Locomotive Works manufactures a model locomotive....

Quiz 19: Linear Programming

Suppose an LP problem was subject to constraints of 2X + Y> 10 X + 3Y> 20 Suppose that a new constraint is added, of the form 3X + A ∗ Y> 90. What is the largest value that A can have so that this new constraint is redundant?

A linear programming problem contains a restriction that reads "the quantity of S must be no more than one-fourth as large as T and U combined." Formulate this as a constraint ready for use in problem solving software.

Suppose that a constraint is given by X + Y≤10. If another constraint is given to be 3X + 2Y≥15 determine the corners of the feasible solution. If the profit from X is 5 and the profit from Y is 10, determine the maximum profit.