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Math - College, section 1, Fall 2019
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Anderson, et al - Intro to Management Science 15e (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 85 / 101

Due : Sunday, January 27, 2030 00:00 EST

Last Saved : n/a Saving... ()

Ask Your Teacher Print Assignment

Question

Points

1	2	3	4	5	6	7	8	9	10
0/3	–/0	2/2	1/1	16/16	20/20	2/10	16/16	22/24	6/8

Total

85/101 (84.2%)

Instructions

Anderson/Sweeney/Williams/Camm/Cochran/Fry/Ohlmann's An Introduction to Management Science: Quantitiatve Approaches to Decision Making, 15th edition, published by Cengage Learning, equips students with a sound conceptual understanding of the role that management science plays in the decision-making process. This edition reflects the latest developments in Microsoft Office Excel 2016. This market leader provides unwavering accuracy with emphasis on applications and timely examples. A hallmark problem-scenario approach introduces each quantitative technique within an applications setting. Students apply the management science model to generate solutions and recommendations for management. An all new WebAssign online course management system is available with this powerful managerial science solution.

Question 1 is a randomized multipart question that utilizes the graphing tool, which allows the student to actively sketch a line and corresponding region.

Question 2 is a randomized question that demonstrates how images can be randomized in addition to the standard practice of randomized numbers.

Question 3 lets a student enter coordinate points in a natural (x, y) form.

Question 4 is representative of a test bank question offered.

Question 5 contains a randomized computer output for students to interpret.

Question 6 features grading for an objective function as well as constraint inequalities.

Question 7 accepts alternative optima as an ordered list that satisfies constraints developed by the student.

Question 8 is a randomized multimode question that allows students to determine numerical values and are then guided through the interpretation of those values.

Question 9 contains an image with randomized values for a seamless experience for the student.

Question 10 accepts alternative optima as a matrix that satisfies constraints developed by the student. This demo assignment allows many submissions and allows you to try another version of the same question for practice wherever the problem has randomized values.

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

Assignment Scoring

Your last submission is used for your score.

1. 0/3 points | Previous Answers ASWMSci15 2.E.002. My Notes

2. –/0 points ASWMSci15 2.E.007. My Notes

Identify the feasible region for the following set of constraints.

0.5A + 0.25B	≥	35
1A + 5B	≥	255
0.25A + 0.5B	≤	50
A, B ≥ 0

The A B-coordinate plane is given. There are 3 lines and a shaded region on the graph.
The first line enters the window at B = 140 on the positive B-axis, goes down and right, passes through the approximate point (27, 87) crossing the third line, passes through the approximate point (49, 41) crossing the second line, and exits the window at A = 70 on the positive A-axis.
The second line enters the window at B = 51 on the positive B-axis, goes down and right, passes through the approximate point (49, 41) crossing the first line, passes through the approximate point (163, 18) crossing the third line, and exits the window at A = 255 on the positive A-axis.
The third line enters the window at B = 100 on the positive B-axis, goes down and right, passes through the approximate point (27, 87) crossing the first line, passes through the approximate point (163, 18) crossing the second line, and exits the window at A = 200 on the positive A-axis.
The region is above the first line, below the second line, and below the third line.

Viewing Saved Work Revert to Last Response

3. 2/2 points | Previous Answers ASWMSci15 2.E.010. My Notes

For the linear program

Max

8A + 9B

s.t.

1A	+	2B ≤ 8
7A	+	5B ≤ 35
	A,	B ≥ 0

find the optimal solution using the graphical solution procedure. What is the value of the objective function at the optimal solution?

1433

$webMathematica generated answer key$ at (A, B) =

103, 73

$webMathematica generated answer key$

Solution or Explanation

The A B-coordinate plane is given. There are 2 lines and a shaded region on the graph.
The first line labeled A + 2B = 8 enters the window at B = 4 on the positive B-axis, goes down and right, passes through the approximate point (3.33, 2.33) crossing the second line labeled 7A + 5B = 35, and exits the window at A = 8 on the positive A-axis.
The second line labeled 7A + 5B = 35 enters the window at B = 7 on the positive B-axis, goes down and right, passes through the approximate point (3.33, 2.33) crossing the first line labeled A + 2B = 8, and exits the window at A = 5 on the positive A-axis.
The region is below the first line labeled A + 2B = 8 and below the second line labeled 7A + 5B = 35.
The intersection of the lines is labeled Optimal Solution.
The value of the objective function is labeled 8(10⁄3) + 9(7⁄3) = 143⁄3.

(1)

(2)

(1) ✕ 7

14B

(3)

(2) − (3)

−

−21

From (1),

A = 8 − 2

= 8 −

Viewing Saved Work Revert to Last Response

4. 1/1 points | Previous Answers ASWMSci15 2.TB.010. My Notes

The point

(5, 3)

is feasible for the constraint

2x₁ + 6x₂ ≤ 30.

True False

Viewing Saved Work Revert to Last Response

5. 16/16 points | Previous Answers ASWMSci15 3.E.009. My Notes

Tom's, Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom's, Inc., makes two salsa products: Western Foods Salsa and Mexico City Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20% tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs 10 ounces.

For the current production period, Tom's, Inc., can purchase up to 285 pounds of whole tomatoes, 140 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound for these ingredients is $0.96, $0.64, and $0.56, respectively. The cost of the spices and the other ingredients is approximately $0.10 per jar. Tom's, Inc., buys empty glass jars for $0.02 each, and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Tom's contract with Western Foods results in sales revenue of $1.64 for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa. Letting

W = jars of Western Foods Salsa
M = jars of Mexico City Salsa

leads to the formulation (units for constraints are ounces):

Max	1W	+	1.25M
s.t.
	5W	+	7M	≤	4,560	oz of whole tomatoes
	3W	+	1M	≤	2,240	oz of tomato sauce
	2W	+	2M	≤	1,600	oz of tomato paste
	W,	M	≥ 0

The computer solution is shown below.

Optimal Objective Value = 870.00000

Variable	Value	Reduced Cost
W	520.00000	0.00000
M	280.00000	0.00000

Constraint	Slack/Surplus	Dual Value
1	0.00000	0.12500
2	400.00000	0.00000
3	0.00000	0.18750

Variable	Objective Coefficient	Allowable Increase	Allowable Decrease
W	1.00000	0.25000	0.10714
M	1.25000	0.15000	0.25000

Constraint	RHS Value	Allowable Increase	Allowable Decrease
1	4560.00000	1040.00000	400.00000
2	2240.00000	Infinite	400.00000
3	1600.00000	100.00000	297.14286

(a)

What is the optimal solution, and what are the optimal production quantities?

520

jars M

280

jars profit $

870

(b)

Specify the objective function ranges. (Round your answers to five decimal places.)

Western Foods Salsa

0.89286

1.25000

Mexico City Salsa

1.00000

1.40000

(d)

Identify each of the right-hand-side ranges. (Round your answers to two decimal places. If there is no upper or lower limit, enter NO LIMIT.)

constraint 1

4160

$webMathematica generated answer key$ to

5600

$webMathematica generated answer key$ constraint 2

1840

$webMathematica generated answer key$ to

NO LIMIT

$webMathematica generated answer key$ constraint 3

1302.86

$webMathematica generated answer key$ to

1700

$webMathematica generated answer key$

Solution or Explanation

(a)

Optimal solution calls for the production of 520 jars of Western Foods Salsa and 280 jars of Mexico City Salsa; profit is $870.

(b)

Variable	Objective Coefficient Range
Western Foods Salsa	0.893 to 1.250
Mexico City Salsa	1.000 to 1.400

(c)

Constraint	Dual Value	Interpretation
1	0.125	One more ounce of whole tomatoes will increase profits by $0.125.
2	0.000	Additional ounces of tomato sauce will not improve profits; slack of 400 ounces.
3	0.187	One more ounce of tomato paste will increase profits by $0.187.

(d)

Constraint	Right-Hand-Side Range
1	4,160 to 5,600
2	1,840 to No Upper Limit
3	1,302.86 to 1,700

Viewing Saved Work Revert to Last Response

6. 20/20 points | Previous Answers ASWMSci15 3.E.029. My Notes

Question Part

Points

Submissions Used

1/1

2/100

3/100

2/100

5/100

2/100

3/100

2/100

1/100

2/100

1/100

Total

20/20

La Jolla Beverage Products is considering producing a wine cooler that would be a blend of a white wine (W), a rosé wine (R), and fruit juice (F). To meet taste specifications, the wine cooler must consist of at least 50% white wine, at least 20% and no more than 30% rosé, and exactly 20% fruit juice. La Jolla purchases the wine from local wineries and the fruit juice from a processing plant in San Francisco. For the current production period, 10,000 gallons of white wine and 8,000 gallons of rosé wine can be purchased; an unlimited amount of fruit juice can be ordered. The costs for the wine are $1.00 per gallon for the white and $1.40 per gallon for the rosé; the fruit juice can be purchased for $0.50 per gallon. La Jolla Beverage Products can sell all of the wine cooler they can produce for $2.80 per gallon.

(a)

Is the cost of the wine and fruit juice a sunk cost or a relevant cost in this situation? Explain.

The cost of the wine and fruit juice is a sunk cost as it does not depend on the production decision made. The cost of the wine and fruit juice is a sunk cost as it depends on the production decision made. correct

The cost of the wine and fruit juice is a relevant cost as it depends on the production decision made. The cost of the wine and fruit juice is a relevant cost as it does not depend on the production decision made.

(b)

Formulate a linear program to determine the blend of the three ingredients that will maximize the total profit contribution. (Let W be the gallons of white wine, R be the gallons of rosé wine, and F be the gallons of fruit juice.)

Max

2.3F+1.4R+1.8W

$webMathematica generated answer key$

s.t. white minimum %

−0.5F−0.5R+0.5W≥0

$webMathematica generated answer key$ rosé minimum %

−0.2F+0.8R−0.2W≥0

$webMathematica generated answer key$ rosé maximum %

−0.3F+0.7R−0.3W≤0

$webMathematica generated answer key$ fruit juice %

0.8F−0.2R−0.2W=0

$webMathematica generated answer key$ white available

W≤10000

$webMathematica generated answer key$ rosé available

R≤8000

$webMathematica generated answer key$

W, R, F ≥ 0

Solve the linear program to determine the number of gallons of each ingredient La Jolla should purchase and the total profit contribution (in dollars) they will realize from this blend.

10,000

gallons R

6,000

gallons F

4,000

gallons profit $

35,600

(c)

If La Jolla could obtain additional amounts of the white wine, should they do so?

Yes No

If so, how much (in dollars) should they be willing to pay for each additional gallon? (If not so, enter 0.)

4.56

dollars

If so, how many additional gallons would they want to purchase at the current price of $1.00? (If not so, enter 0.)

14,000

gallons

(d)

If La Jolla Beverage Products could obtain additional amounts of the rosé wine, should they do so?

Yes

If so, how much (in dollars) should they be willing to pay for each additional gallon? (If not so, enter 0.)

If so, how many additional gallons would they want to purchase at the current price of $1.40? (If not so, enter 0.)

gallons

(e)

Interpret the dual value for the constraint corresponding to the requirement that the wine cooler must contain at least 50% white wine. What is your advice to management given this dual value?

The dual value indicates that additional white wine

should not

be added.

(f)

Interpret the dual value for the constraint corresponding to the requirement that the wine cooler must contain exactly 20% fruit juice. What is your advice to management given this dual value?

The dual value indicates additional fruit juice

should

be added.

Solution or Explanation

(a)

Relevant cost since La Jolla Beverage Products can purchase wine and fruit juice on an as-needed basis.

(b)

Let

W = gallons of white wine
R = gallons of rosé wine
F = gallons of fruit juice

We want to maximize profits, so we need to first find the margins.

W margin	=	$2.80 − $1.00 = $1.80
R margin	=	$2.80 − $1.40 = $1.40
F margin	=	$2.80 − $0.50 = $2.30

Thus, we formulate:

Max	1.8W	+	1.4R	+	2.3F
s.t.
	0.5W	−	0.5R	−	0.5F	≥	0	% white minimum
	−0.2W	+	0.8R	−	0.2F	≥	0	% rosé minimum
	−0.3W	+	0.7R	−	0.3F	≤	0	% rosé maximum
	−0.2W	−	0.2R	+	0.8F	=	0	% fruit juice required
	W					≤	10,000	available white
			R			≤	8,000	available rosé
					W, R, F	≥	0

Optimal Solution:

W = 10,000, R = 6,000, F = 4,000

profit contribution = $35,600.

(c)

Since the cost of the white wine is a relevant cost, the dual value of $3.56 is the maximum premium (over the normal price of $1.00) that La Jolla Beverage Products should be willing to pay to obtain one additional gallon of white wine. In other words, at a price of

$4.56 = $3.56 + $1.00,

the additional cost is exactly equal to the additional revenue.

(d)

No; only 6,000 gallons of the rosé are currently being used.

(e)

Requiring 50% plus one gallon of white wine would reduce profit by $3.16.

(f)

Allowing the amount of fruit juice to exceed 20% by one gallon will increase profit by $0.90.

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7. 2/10 points | Previous Answers ASWMSci15 4.E.009. My Notes

A linear programming computer package is needed.

Epsilon Airlines services predominately the eastern and southeastern United States. A vast majority of Epsilon's customers make reservations through Epsilon's website, but a small percentage of customers make reservations via phone. Epsilon employs call-center personnel to handle these reservations along with any problems with the website reservation system and for the rebooking of flights for customers if their plans change or their travel is disrupted. Staffing the call center appropriately is a challenge for Epsilon's management team. Having too many employees on hand is a waste of money, but having too few results in very poor customer service and the potential loss of customers.

Epsilon analysts have estimated the minimum number of call-center employees needed by day of week for the upcoming vacation season (June, July, and the first two weeks of August). These estimates are given in the following table.

Day	Minimum Number of Employees Needed
Monday	70
Tuesday	45
Wednesday	40
Thursday	45
Friday	80
Saturday	65
Sunday	35

The call-center employees work five consecutive days and then have two consecutive days off. An employee may start work any day of the week. Each call-center employee receives the same salary. Assume that the schedule cycles and ignore start-up and stopping of the schedule. Develop a model that will minimize the total number of call-center employees needed to meet the minimum requirements. (Let

X_i

= the number of call-center employees who start work on day i where

i = 1 = Monday,

i = 2 = Tuesday,

etc).

Min

x1+x2+x3+x4+x5+x6+x7

$webMathematica generated answer key$

s.t. Monday

$webMathematica generated answer key$ Tuesday

$webMathematica generated answer key$ Wednesday

$webMathematica generated answer key$ Thursday

$webMathematica generated answer key$ Friday

$webMathematica generated answer key$ Saturday

$webMathematica generated answer key$ Sunday

$webMathematica generated answer key$

X₁, X₂, X₃, X₄, X₅, X₆, X₇ ≥ 0

Find the optimal solution.

(X₁, X₂, X₃, X₄, X₅, X₆, X₇) =

20,15,0,40,5,5,0

$webMathematica generated answer key$

Give the number of call-center employees that exceed the minimum required.

(M, Tu, W, Th, F, Sa, Su) =

0,0,0,30,0,0,15

$webMathematica generated answer key$

Solution or Explanation

Let X_i = the number of call center employees who start work on day i.

(i = 1 = Monday, i = 2 = Tuesday,

)

Min

X₁

X₂

X₃

X₄

X₅

X₆

X₇

s.t.

X₁

X₄

X₅

X₆

X₇

≥

X₁

X₂

X₅

X₆

X₇

≥

X₁

X₂

X₃

X₆

X₇

≥

X₁

X₂

X₃

X₄

X₇

≥

X₁

X₂

X₃

X₄

X₅

≥

X₂

X₃

X₄

X₅

X₆

≥

X₃

X₄

X₅

X₆

X₇

≥

X₁, X₂, X₃, X₄, X₅, X₆, X₇ ≥ 0

Solution: X₁ = 20, X₂ = 15, X₃ = 0, X₄ = 40, X₅ = 5, X₆ = 5, X₇ = 0.

Total Number of Employees = 85.

Excess employees: Thursday = 30, Sunday = 15, all others = 0.

Note: There are alternative optima to this problem (Number of employees may differ from above, but will have objective function value = 85).

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8. 16/16 points | Previous Answers ASWMSci15 5.E.015. My Notes

In a gambling game, Player A and Player B both have a $5 and a $10 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B's bill. If the bills match, Player B wins Player A's bill.

(a)

Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A.

		Player B
		$5	$10
Player A	$5	-5	10
Player A	$10	5	-10

(b)

Is there a pure strategy? Why or why not?

. Since the maximum of the row minimums is

-5

and the minimum of the column maximums is

there is no pure strategy

(c)

Determine the optimal strategies and the value of this game.

probability Player A selects $5 =

1/2

probability Player A selects $10 =

1/2

probability Player B selects $5 =

2/3

probability Player B selects $10 =

1/3

Does the game favor one player over the other?

Yes

(d)

Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A's winnings?

If Player B begins playing each bill 50% of the time, Player A should instead select $5 with probability

and select $10 with probability

Comment on why it is important to follow an optimal game theory strategy.

Following the optimal strategy

prevents

other players from taking advantage of the strategy you're playing, since they cannot improve their expected payout by not playing the optimal strategy.

Solution or Explanation

(a)

		Player B
		$5	$10
Player A	$5	−5	10
Player A	$10	5	−10

(b)

		Player B
		$5	$10	Minimum
Player A	$5	−5	10	−5
	$10	5	−10	−10
	Maximum	5	10

Since the maximum of the row minimums is −5 and the minimum of the column maximums is 5, this game has no pure strategy.

(c)

For Player A, let p = probability of $5 and

(1 − p) = probability

of $10

b₁ = $5, EV = −5p + 5(1 − p)

b₂ = $10, EV = 10p − 10(1 − p)

−5p + 5(1 − p)

10p − 10(1 − p)

−5p + 5 − 5p

10p − 10 + 10p

30p

(1 − p) = 1 −

For Player B, let q = probability of $5 and

(1 − q) = probability

of $10

a₁ = $5, EV = −5q + 10(1 − q)

a₂ = $10, EV = 5q − 10(1 − q)

−5q + 10(1 − q)

5q − 10(1 − q)

−5q + 10 − 10q

5q − 10 + 10q

30q

(1 − q) = 1 −

or q =

(1 − q) =

Value of game using Player A

EV = −5p + 5(1 − p)

−5

+ 5

This is a fair game. Neither player is favored.

(d)

If Player A realizes Player B is using a 50/50 strategy, we can use an expected value with these probabilities to show:

EV(a₁ = $5)

−5

+ 10

= 2.50

EV(a₂ = $10)

− 10

= −2.50

Player A should see that the expected value of a₁ is now larger than the expected value of a₂.

If Player A believes Player B will continue with a 50/50 strategy, then Player A should always play strategy a₁: reveal $5. But, if Player A does this, Player B will catch on and begin revealing a $5 bill all the time. The only way for a player to protect against the opponent taking advantage is to play the optimal strategy all the time.

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9. 22/24 points | Previous Answers ASWMSci15 6.E.002. My Notes

Question Part

Points

Submissions Used

1/1

0/1

1/1

0/1

1/1

2/100

3/100

2/100

3/100

2/100

Total

22/24

Consider the following network representation of a transportation problem.

A network diagram between five locations is shown. The left-hand side of the graph is labeled "Supplies" and lists Jefferson City and Omaha. The right-hand side of the graph is labeled "Demands" and lists Des Moines, Kansas City, and St. Louis. Lines are shown between various locations. The following list contains the numbers placed on the graph.

Jefferson City: 45

Omaha: 20

Des Moines: 30

Kansas City: 10

St. Louis: 25

Jefferson City–Des Moines: 14

Jefferson City–Kansas City: 9

Jefferson City–St. Louis: 8

Omaha–Des Moines: 8

Omaha–Kansas City: 11

Omaha–St. Louis: 24

The supplies, demands, and transportation costs per unit are shown on the network.

(a)

Develop a linear programming model for this problem; be sure to define the variables in your model.

Let

x₁₁ = amount shipped from Jefferson City to Des Moines
x₁₂ = amount shipped from Jefferson City to Kansas City
x₁₃ = amount shipped from Jefferson City to St. Louis
x₂₁ = amount shipped from Omaha to Des Moines
x₂₂ = amount shipped from Omaha to Kansas City
x₂₃ = amount shipped from Omaha to St. Louis

Min

x₁₁

x₁₂

x₁₃

x₂₁

x₂₂

x₂₃

s.t.

x₁₁

x₁₂

x₁₃

≤

x₂₁

x₂₂

x₂₃

≤

x₁₁

x₂₁

x₁₂

x₂₂

x₁₃

x₂₃

x₁₁, x₁₂, x₁₃, x₂₁, x₂₂, x₂₃ ≥ 0

(b)

Solve the linear program to determine the optimal solution.

	Amount	Cost
Jefferson City–Des Moines	10	140
Jefferson City–Kansas City	10	90
Jefferson City–St. Louis	25	200
Omaha–Des Moines	20	160
Omaha–Kansas City	0	0
Omaha–St. Louis	0	0
Total		590

Solution or Explanation

(a)

Let

x₁₁ = amount shipped from Jefferson City to Des Moines
x₁₂ = amount shipped from Jefferson City to Kansas City
x₁₃ = amount shipped from Jefferson City to St. Louis
x₂₁ = amount shipped from Omaha to Des Moines
x₂₂ = amount shipped from Omaha to Kansas City
x₂₃ = amount shipped from Omaha to St. Louis

Min

14x₁₁

9x₁₂

8x₁₃

8x₂₁

11x₂₂

24x₂₃

s.t.

x₁₁

x₁₂

x₁₃

≤

x₂₁

x₂₂

x₂₃

≤

x₁₁

x₂₁

x₁₂

x₂₂

x₁₃

x₂₃

x₁₁, x₁₂, x₁₃, x₂₁, x₂₂, x₂₃ ≥ 0

(b)

Optimal Solution

	Amount	Cost
Jefferson City–Des Moines	10	140
Jefferson City–Kansas City	10	90
Jefferson City–St. Louis	25	200
Omaha–Des Moines	20	160
Omaha–Kansas City	0	0
Omaha–St. Louis	0	0
Total		590

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10. 6/8 points | Previous Answers ASWMSci15 7.E.013. My Notes

The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas. The following is a linear program used to determine which cities Martin-Beck should construct a plant in.

Let

y₁ = 1 if a plant is constructed in Detroit; 0 if not

y₂ = 1 if a plant is constructed in Toledo; 0 if not

y₃ = 1 if a plant is constructed in Denver; 0 if not

y₄ = 1 if a plant is constructed in Kansas City; 0 if not.

The variables representing the amount shipped from each plant site to each distribution center are defined just as for a transportation problem.

x_ij = the units shipped in thousands from plant i to distribution center j

i = 1 (Detroit), 2 (Toledo), 3 (Denver), 4 (Kansas City), 5 (St.Louis) and

j = 1 (Boston), 2 (Atlanta), 3 (Houston)

The complete model for the Martin-Beck distribution system design problem is as follows.

Min 5x₁₁ + 2x₁₂ + 3x₁₃ + 4x₂₁ + 3x₂₂ + 4x₂₃ + 9x₃₁ + 7x₃₂ + 5x₃₃ + 10x₄₁ + 4x₄₂

+ 2x₄₃ + 8x₅₁ + 4x₅₂ + 3x₅₃ + 175y₁ + 300y₂ + 375y₃ + 500y₄

s.t.

x₁₁	+	x₁₂	+	x₁₃					≤ 10y₁	Detroit capacity

x₂₁	+	x₂₂	+	x₂₃					≤ 20y₂	Toledo capacity

x₃₁	+	x₃₂	+	x₃₃					≤ 30y₃	Denver capacity

x₄₁	+	x₄₂	+	x₄₃					≤ 40y₄	Kansas City capacity

x₅₁	+	x₅₂	+	x₅₃					≤ 30	St. Louis capacity

x₁₁	+	x₂₁	+	x₃₁	+	x₄₁	+	x₅₁	= 30	Boston demand

x₁₂	+	x₂₂	+	x₃₂	+	x₄₂	+	x₅₂	= 20	Atlanta demand

x₁₃	+	x₂₃	+	x₃₃	+	x₄₃	+	x₅₃	= 20	Houston demand

x_ij ≥ 0 for all i and j; y₁, y₂, y₃, y₄ = 0, 1

(a)

Modify the original formulation to account for the policy restriction that there must be one plant in either Detroit or Toledo, but not both.

new constraint

y1+y2=1

$webMathematica generated answer key$

(y₁, y₂, y₃, y₄)

1,0,1,0

$webMathematica generated answer key$

Place

x_ij

in row i column j.

[0, 10, 0; 0, 0, 0; 30, 0, 0; 0, 0, 0; 0, 10, 20]

value =

940

(b)

Modify the original formulation to account for the policy restriction that no more than two plants can be located in Denver, Kansas City, and St. Louis.

new constraint

y3+y4≤1

$webMathematica generated answer key$

(y₁, y₂, y₃, y₄)

0,0,0,1

$webMathematica generated answer key$

Place

x_ij

in row i column j.

[0, 0, 0; 0, 0, 0; 0, 0, 0; 0, 20, 20; 30, 0, 0]

value =

860

Solution or Explanation

(a)

One just needs to add the following multiple choice constraint to the problem.

y₁ + y₂ = 1

New Optimal Solution:

y₁ = 1, y₃ = 1, x₁₂ = 10, x₃₁ = 30, x₅₂ = 10, x₅₃ = 20

Value = 940

(b)

Since one plant is already located in St. Louis, it is only necessary to add the following constraint to the model

y₃ + y₄ ≤ 1

New Optimal Solution:

y₄ = 1, x₄₂ = 20, x₄₃ = 20, x₅₁ = 30

Value = 860

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Anderson, et al - Intro to Management Science 15e (Homework)

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