exam b answers

8/11/2019 Exam b Answers

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Quantitative Analysis BA 452 Exam B

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BA 452 Print name: ________________

Exam B

This is a 100-minute exam (1hr. 40 min.). There are 4 questions (25

minutes per question). To avoid the temptation to cheat, you must abide by

these rules then sign below after you understand the rules and agree to

them:

Turn off your cell phones. You cannot leave the room during the exam, not even to use the

restroom. The only things you can have in your possession are pens or pencils

and a simple non-graphing, non-programmable, non-text calculator. All other possessions (including phones, computers, or papers) are

prohibited and must be placed in the designated corner of the room. Possession of any prohibited item (including phones, computers, or

papers) during the exam (even if you dont use them but keep themin your pocket) earns you a zero on this exam, and you will bereported to the Academic Integrity Committee for further action.

Sign here:__________________________________________________

Part 1. You may only use blank or graph paper,pencils, a ruler, and a calculator. You may notuse a computer or notes. After you finish all

questions in Part 1, turn in your answers, thenyou may use the school computer to run theManagement Scientist to complete Part 2 of theexam.


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Rounding Off

Question 1: Exxon Mobil Corporation seeks to

maximize profit by making two grades of gasoline

from crude oil and additives. It just received this

day's allocation of 1.5 thousand gallons of crude oil,

and 0.8 thousand gallons of additives.

It takes 0.6 gallons of crude oil to make a gallon of Premium gasoline, and

0.3 gallons of crude oil to make a gallon of Regular gasoline. It also takes

0.2 gallons of additives to make a gallon of Premium gasoline, and 0.3

gallons of additives to make a gallon of Regular gasoline. Premium

gasoline has unit profit 3 of dollars, and Regular gasoline has 4 dollars.

a. Develop a linear programming model for this problem to determine

how much should be produced.

b. Graphically solve the linear-programming problem from Part a if you

require that production units be integers.

c. Graphically solve the linear-programming problem from Part a if you

do not require that production units be integers (instead, production

units are continuous variables).

d. Compare your solutions in Parts b and c.

Tip: Your written answer should define the decision variables, formulate the

objective and constraints, and solve for the optimum. --- You will not earn

full credit if you just solve for the optimum; you must also define the

decision variables, and formulate the objective and constraints.

Answer to Question 1:


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Answer to Quest ion:

Part a:

Let X = thousands of units of Premium gasoline produced.

Let Y = thousands of units of Regular gasoline produced.

Max 3X + 4Y

s.t. 0.6X + 0.3Y < 1.5 (crude oil)

0.2X + 0.3Y < 0.8 (additives)

X, Y 0

Part c:

A graph of the feasible set and isovalue lines (dashed lines above) reveals

the isovalue lines are steeper than the second constraint, and the optimum

occurs where the firstand the secondconstraint bind. Solving the bindingform of those two constraints yields the optimal solution: X = 1.75, Y = 1.50

0

1

2

3

4

5

0 1 2 3 4 5


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Part b: You would earn full credit if you noted that since (X,Y) are in

thousands of units, X = 1.75 thousand and Y = 1.50 thousand are integer

solutions. You would also earn full credit by solving the problem if you

require that X and Y are integers.

Max 3X + 4Y

s.t. 0.6X + 0.3Y < 1.5 (crude oil)

0.2X + 0.3Y < 0.8 (additives)

X, Y 0

A graph of the feasible set with dots to indicate all feasible integer solutions

and isovalue lines reveals the optimum occurs at X = 1, Y = 2

Part d: The integer solution in Part b is not the result of rounding off thecontinuous solution in Part c.

0

1

2

3

4

5

0 1 2 3 4 5


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Assignment with Supply and Demand

Question 2: The Goodyear Tire and Rubber

Company uses rubber in its American, Asian, and

European tire manufacturing plants. Goodyear can

buy rubber from either India, or Indonesia, or

Malaysia, or Thailand. The number of tons of rubber needed daily by each

tire plant and the price per ton charged by each supplier are as follows:

The cost (dollars per ton) for shipping from each supplier to each

manufacturing plant are as follows:

American Asian European

India 3 6 4

Indonesia 5 9 8

Malaysia 2 6 4

Thailand 7 1 2

To reduce fixed costs, Goodyear wants each manufacturing plants

demand to be assigned to exactly one rubber supplier. And because of

capacity constraints, each rubber supplier can supply at most one tire plant.

Formulate the optimal assignment of rubber suppliers to manufacturing

plant s as a linear-programming problem. Formulate the problem, but you

need not solve the problem.

Answer to Quest ion 2:

Demand (tons) Price (dollars per ton)

American 5 India 3

Asian 4 Indonesia 4

European 3 Malaysia 2

Thailand 5


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Linear programming formulation (supply inequality, demand equality).

Variables: Xij = 1 if Supplier i is assigned to Plant j, else 0

Assignment Costs:

The total cost is the sum of the purchase cost and the transportation

cost.

Supplier 1 (India) assigned to Plant 1 (American) (cost in dollars):

o Purchase cost: (5 x $3) = $15

o Transportation Cost: (5 x $3) = $15

o Total Cost: $30

Assignment Costs: Cij = Cost of assigning Supplier i to Plant j

Cij Plant 1 Plant 2 Plant 3

Sup 1 5(3+3)=30 4(3+6)=36 3(3+4)=21

Sup 2 5(4+5)=45 4(4+9)=52 3(4+8)=36

Sup 3 5(2+2)=20 4(2+6)=32 3(2+4)=18

Sup 4 5(5+7)=60 4(5+1)=24 3(5+2)=21


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Make or Buy with Fixed Costs

Question 3: Danchuk Manufacturing produces a

variety of classic automobiles, including a 1955 Chevy

and a 1955 Thunderbird. Each car consists of three

components that can be manufactured by Danchuk: a

body, an interior, and an engine. Both cars use the same engine, but

different bodies and different interiors.

Danchuks sales forecast indicates that 300 Chevys and 500 Thunderbirds

will be needed to satisfy demand during the next year. Because only 2000

hours of in-house manufacturing time is available, Danchuk is considering

purchasing some, or all, of the components from outside suppliers. If

Denchuk manufactures a component in-house, it incurs a fixed setup costas well as a variable manufacturing cost. The following table shows the

setup cost, the manufacturing time per component, the manufacturing cost

per component, and the cost to purchase each of the components from an

outside supplier:

Component

Setup Cost(thousandsof dollars)

ManufacturingTime per Unit

(hours)

ManufacturingCost per Unit(thousands of

dollars)

PurchaseCost per Unit(thousandsof dollars)

Chevy Body 100 3 4 5Thunderbird

Body90 2 2 3

ChevyInterior

10 1 2 3

ThunderbirdInterior

20 2 3 4

Engine 20 2 4 5

Formulate a linear program for Denchuk to minimize total cost to meet thesales forecasts. But you need not compute an optimum.

Tip: Your written answer should define the decision variables, carefully

state which variables are continuous and which are binary, and formulate

the objective and constraints.



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Continuous (or integer) variables are used for the number of units made:

Let CB = the number of Chevy Bodies to make

Let TB = the number of Thunderbird Bodies to make

Let CI = the number of Chevy Interiors to makeLet TI = the number of Thunderbird Interiors to make

Let E = the number of Engines to make

Continuous (or integer) variables are also used for the number of units

purchased:

Let CBP = the number of Chevy Bodies to purchase

Let TBP = the number of Thunderbird Bodies to purchase

Let CIP = the number of Chevy Interiors to purchaseLet TIP = the number of Thunderbird Interiors to purchase

Let EP = the number of Engines to purchase

Binary variables are used to indicate whether production is positive, and

setup costs are incurred:

Let CBS = 1 if Chevy Bodies are produced; 0, if not.

Let TBS = if Thunderbird Bodies are produced; 0, if not.

Let CIS = 1 if Chevy Interiors are produced; 0, if not.

Let TIS = if Thunderbird Interiors are produced; 0, if not.

Let ES = if Engines are produced; 0, if not.

Objective: Minimize Total Cost

4CB + 2TB + 2CI + 3TI + 4E (production costs)

+5CBP + 3TBP + 3CIP + 4TIP + 5EP (purchase costs)

+100CBS + 90TBS + 10CIS + 20TIS + 20ES (setup costs)

Input Constraints:

3CB + 2TB + 1CI + 2TI + 2E < 2000 (In-house hours)


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Sales Constraints:

CB + CBP = 300 (manufactured or purchased bodies used for Chevys)

TB + TBP = 500 (manufactured or purchased bodies used for

Thunderbirds)

CI + CIP = 300 (manufactured or purchased interiors used for Chevys)TI + TIP = 500 (manufactured or purchased interiors used for

Thunderbirds)

E + EP = 800 (manufactured or purchased engines used for both Chevys

and Thunderbirds)

Setup Constraints (given sales constraints imply CB < 300, TB < 500, CI 500

Walmart demand constraint is X36 + X46 > 600

Transshipment constraints are

X35 + X36 < X13 + X23, through Nevada

X45 + X46 < X14 + X24, through New York

And setup indicator constraints are

X35 + X36 < 1350Y3, through Nevada

X45 + X46 < 1350Y4, through New York


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That solution is the same if, instead of Xij variables being integer, they were

continuous.

In that solution, Nevada transshipment node is used, but New York is not

used.

Finally, the minimum cost is $10,050 per day.

exam b answers

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