# Linear Programming and Network Modeling

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Quiz Business Modeling

1. Consider the constraint

X3 + X4 + X5 + X6 + X7 > 27

representing Air Express’ Monday worker requirement. Why was a “>” used versus an “=”?

a. The “>” is needed to accommodate workers held over from Sunday.

b. Solver only accepts “>” constraints.

c. The “>” is less restrictive.

d. The “=” will always produce an infeasible constraint.

2. A company wants to select no more than 2 projects from a set of 4 possible projects. Which of the following constraints

ensures that no more than 2 will be selected.

a. X1 + X2 + X3 + X4 = 2

b. X1 + X2 + X3 + X4 £ 2

c. X1 + X2 + X3 + X4 ³ 2

d. X1 + X2 + X3 + X4 ³ 0

3. A company wants to select 1 project from a set of 4 possible projects. Which of the following constraints ensures that only

1 will be selected.

a. X1 + X2 + X3 + X4 = 1

b. X1 + X2 + X3 + X4 £ 1

c. X1 + X2 + X3 + X4 ³ 1

d. X1 + X2 + X3 + X4 ³ 0

4. If a company produces Product 1, then it must produce at least 150 units of Product 1. Which of the following constraints

enforce this condition?

a. X1 < 150Y1

b. X1 – 150Y1 > 0

c. X1Y1 < 150

d. X1 > 150 + Y1

5. A production company wants to ensure that if Product 1 is produced, production of Product 1 not exceed production of

Product 2. Which of the following constraints enforce this condition?

a. X1 > M2Y2

b. X1 < M2X2

c. X1 < M1Y1 , X1 < Y1X2

d. X1 < X2

6. A company must invest in project 1 in order to invest in project 2. Which of the following constraints ensures that project

1 will be chosen if project 2 is invested in?

a. X1 + X2 = 0

b. X1 + X2 = 1

c. X1 – X2 ³ 0

d. X1 – X2 £ 0

7. If a company selects Project 1 then it must also select either Project 2 or Project 3. Which of the following constraints

enforce this condition?

a. X1 – X2 – X3 > 0

b. X1 + (X2 – X3) < 0

c. X1 + X2 + X3 < 2

d. X1 – X2 – X3 < 0

8. If a company selects either of Project 1 or Project 2 (or both), then either Project 3 or Project 4 (or both) must also be

selected. Which of the following constraints enforce this condition?

a. X1 + X2 < 2(X3 + X4)

b. X1 + X2 < X3 + X4

c. X1 – X3 = X2 – X4

2

d. X1 + X2 + X3 + X4 < 2

9. A company is developing its weekly production plan. The company produces two products, A and B, which are processed

in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of

A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The

data for the problem are summarized below.

Hours required by

Operation A B Hours

Cutting 3 4 48

Welding 2 1 36

The decision variables are defined as

Xi = the amount of product i produced

Yi = 1 if Xi > 0 and 0 if Xi = 0

What is the objective function for this problem?

a. MAX: 17 X1 + 21 X2

b. MAX: 17 X1 + 21 X2 – 60 Y1 – 80 Y2

c. MIN: 17 X1 + 21 X2 – 60 Y1 – 80 Y2

d. MIN: 60 Y1 + 80 Y2

10. A company is developing its weekly production plan. The company produces two products, A and B, which are processed

in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of

A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The

data for the problem are summarized below.

Hours required by

Operation A B Hours

Cutting 3 4 48

Welding 2 1 36

The decision variables are defined as

Xi = the amount of product i produced

Yi = 1 if Xi > 0 and 0 if Xi = 0

Which of the following constraints creates the link between setting up to produce A’s and making some A’s for this

problem?

a. X1 £ 16Y1

b. X1 – Y1 = 0

c. X1 – 18Y1 > 0

d. =if(X1 > 0, Y1 = 1, Y1 = 0)

11. A company is developing its weekly production plan. The company produces two products, A and B, which are processed

in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of

A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The

data for the problem are summarized below.

Hours required by

Operation A B Hours

Cutting 3 4 48

Welding 2 1 36

The decision variables are defined as

3

Xi = the amount of product i produced

Yi = 1 if Xi > 0 and 0 if Xi = 0

What is the appropriate value for M1 in the linking constraint for product A?

a. 2

b. 3

c. 16

d. 18

12. A company is developing its weekly production plan. The company produces two products, A and B, which are processed

in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of

A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The

data for the problem are summarized below.

Hours required by

Operation A B Hours

Cutting 3 4 48

Welding 2 1 36

What is the appropriate formula to use in cell E8 of the following Excel implementation of the ILP model for this

problem?

A B C D E

1 Fixed charge problem

3

4 Product A Product

B

5 Number to produce

6

7 Unit profit 17 21 Total

profit:

8 Fixed cost 60 80

9

10 Resources Hours required Used Available

11 Cutting 3 4 48

12 Welding 2 1 36

13

14 Binary variables

15 Linking constraints

a. =SUMPRODUCT(B5:C5,B7:C7) – SUMPRODUCT(B8:C8,B14:C14)

b. =SUMPRODUCT(B8:C8,B14:C14) – SUMPRODUCT(B5:C5,B7:C7)

c. =SUMPRODUCT(B5:C5,B7:C7) – B8:C8

d. =SUMPRODUCT(B5:C5,B7:C7) – SUMPRODUCT(B8:C8,B15:C15)

13. A company is developing its weekly production plan. The company produces two products, A and B, which are processed

in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of

A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The

data for the problem are summarized below.

Hours required by

Operation A B Hours

Cutting 3 4 48

Welding 2 1 36

What is the appropriate formula to use in cell B15 of the following Excel implementation of the ILP model for this

problem?

A B C D E

1 Fixed charge problem

3

4 Product A Product

B

5 Number to produce

6

7 Unit profit 17 21 Total

profit:

8 Fixed cost 60 80

9

10 Resources Hours required Used Available

11 Cutting 3 4 48

12 Welding 2 1 36

13

14 Binary variables

15 Linking constraints

a. = B5 – MIN($E$11/B11, $E$11/C11)*B14

b. =B5 – MIN($E$11/B11, $E$12/B12)

c. =B5 – $E$12/B12*B14

d. =B5 – MIN($E$11/B11, $E$12/B12)*B14

14. A company is planning next month’s production. It has to pay a setup cost to produce a batch of X4’s so if it does produce

a batch it wants to produce at least 100 units. Which of the following pair of constraints shows the relationship(s) between

the setup variable Y4 and the production quantity variable X4?

a. X4 £ M4Y4, X4 ³ 100

b. X4 £ M4Y4, X4 = 100 Y4

c. X4 £ M4Y4, X4 ³ 100 Y4

d. X4 £ M4Y4, X4 £ 100 Y4

15. A company will be able to obtain a quantity discount on component parts for its three products, X1, X2 and X3 if it

produces beyond certain limits. To get the X1 discount it must produce more than 50 X1’s. It must produce more than 60

5

X2’s for the X2 discount and 70 X3’s for the X3 discount. How many binary variables are required in the formulation of

this problem?

a. 3

b. 6

c. 9

d. 12

16. A company will be able to obtain a quantity discount on component parts for its three products, X1, X2 and X3 if it

produces beyond certain limits. To get the X1 discount it must produce more than 50 X1’s. It must produce more than 60

X2’s for the X2 discount and 70 X3’s for the X3 discount. How many decision variables are required in the formulation of

this problem?

a. 3

b. 6

c. 9

d. 12

17. A company will be able to obtain a quantity discount on component parts for its three products, X1, X2 and X3 if it

produces beyond certain limits. To get the X1 discount it must produce more than 50 X1’s. It must produce more than 60

X2’s for the X2 discount and 70 X3’s for the X3 discount. Which of the following pair of constraints enforces the quantity

discount relationship on X3?

a. X31 < M3Y3 , X32 > 50Y3

b. X31 < M3Y3 , X31 > 50

c. X32 > (1/50)X31 , X31 < 50

d. X32 < M3Y3 , X31 > 50Y3

18. A wedding caterer has several wine shops from which it can order champagne. The caterer needs 100 bottles of

champagne on a particular weekend for 2 weddings. The first supplier can supply either 40 bottles or 90 bottles.

The relevant decision variable is defined as

X1 = the number of bottles supplied by supplier 1

Which set of constraints reflects the fact that supplier 1 can supply only 40 or 90 bottles?

a. X1 £ 40 Y11, X1 £ 90(1 – Y11)

b. X1 = 40Y11 + 90Y12 , Y11 + Y12 < 1

c. X1 = 40Y1 + 90(1 – Y1) , Y1 = 0 OR 1

d. X1 = 40Y11 + 90Y12 , Y11 + Y12 = 1

The Questions 19 to 24 are based on the problem below.

UAM installs high pressurized instruments for small aircrafts. UAM is developing a 3-period production and inventory plan

model based on the relevant information in the table below.

Period Production

Cost

($)

Demand

(Uni

ts)

Inventory

Cost

($)

Regular

Production

Capacity

1 3.8 500 1 250

2 3.9 300 1.1 300

3 4.1 400 1.5 300

The beginning inventory for period 1 is 100 units. In addition, UAM would like to have at least 50 units in ending inventory

for period 3. The ending inventory (t) = the beginning inventory (t) + production (t) – demand (t) in the same period (t).

4.1 Develop a linear programming model for UAM to minimize its total production and inventory cost:

19. Define Decision variables clearly:

a. P1, P2 and P3

6

b. P1, P2, P3, I1, I2 and I3 for periods 1, 2 and 3, respectively.

c. P1, P2, P3, I1, I2 and I3 for periods 1, 2 and 3, respectively.

d. P1, P2, and P3 = the number of units to make in the periods 1, 2, and 3, respectively.

20. Write the objective function (with its goal):

a. MAX: 3.8 P1+3.9P2 + 4.1 P3 +1 (100+I1)/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2

b. MIN: P1+P2 + P3 +1 (100+I1)/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2

c. MIN: 3.8 P1+3.9P2 + 4.1 P3 +I1/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2

d. MIN: 3.8 P1+3.9P2 + 4.1 P3 +1 (100+I1)/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2

21. Identify which one of the following is NOT a constraint for the problem:

a. I3 = I2 + P3 – 400

b. 0 P3 300

c. P1, P2, P3, I1, I2, and I3 0

d. I3 = 50

22. The table below shows the selling prices per unit. UAM would like to maximize its total profit. Write out the

objective function (with its goal):

Period Selling Price ($)

1 15

2 16

3 15.5

a. MIN 3.8 P1+3.9P2 + 4.1 P3 +1 (100+I1)/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2

b. MAX $15 * 500 + $16 * 300 + $15.5 * 400

c. MAX $15 * 500 + $16 * 300 + $15.5 * 400 – [P1+P2 + P3 + I1/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2]
d. MAX $15 * 500 + $16 * 300 + $15.5 * 400 – [3.8 P1+3.9P2 + 4.1 P3 +1 (100+I1)/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2]
The Table below shows the overtime production capacity and cost of production in overtime. UAM would like to minimize

the total production and inventory cost based on the additional information given in the Problem above.

Period Overtime Production Capacity Overtime Production Cost ($)

1 100 6

2 100 6.4

3 125 6.8

23. Define additional decision variables used here?

a. O1, O2 and O3

b. O1, O2 and O3 for periods 1, 2 and 3

c. O1, O2 and O3 = the overtimes in the periods 1, 2 and 3

d. O1, O2 and O3 = the number of units made in overtimes in the periods 1, 2 and 3.

24. Write out the objective function with its goal:

a. MIN: 3.8 P1+3.9P2 + 4.1 P3 + 6O1 + 6.4O2 + 6.8 O3+ I1 + (I1 + I2) + (I2 + I3)

b. MIN: 3.8 P1+3.9P2 + 4.1 P3 + O1 + O2 + O3+ 1 (100+I1)/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2

c. MAX: P1+P2 + P3 + 6O1 + 6.4O2 + 6.8 O3+ 1 (100+I1)/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2

d. MIN: 3.8 P1+3.9P2 + 4.1 P3 + 6O1 + 6.4O2 + 6.8 O3+ 1 (100+I1)/2 + 1.1 (I1 + I2)/2 + 1.5 (I2 + I3)/2

25. Write out the constraints: Which one of the following is not a constraint here:

a. I3 = I2 + P3 + O3 – 400

b. 0 P2 300

c. 0 O3 125

d. 50 O1 100

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