## Chapter 6âInteger Linear Programming

## Chapter 6âInteger Linear Programming

31. 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 appropriate value for M1 in the linking

constraint for product A?

a.

2

b.

3

c.

16

d.

12

32. 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

2

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)

33. 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

2

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

34. 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 pairs of

constraints show the relationship(s) between the setup variable Y4

and the production quantity variable X4?

a.

X4Â£ M4Y4,

X4Â³ 100

b.

X4Â£ M4Y4,

X4 = 100 Y4

c.

X4Â£ M4Y4,

X4Â³ 100 Y4

d.

X4Â£ M4Y4,

X4Â£ 100 Y4

35. 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 binary variables are required in the formulation of this

problem?

a.

3

b.

6

c.

9

d.

12

36. 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

37. 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.

X31Â£ M3Y3,

X32Â³ 50Y3

b.

X31Â£ M3Y3,

X31Â³ 50

c.

X32Â³ (1/50)X31,

X31Â£ 50

d.

X32Â£ M3Y3,

X31Â³ 50Y3

38. 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.

X1Â£ 40 Y11,

X1Â£ 90(1- Y11)

b.

X1 = 40Y11 + 90Y12,

Y11 + Y12Â£ 1

c.

X1 = 40Y1 + 90(1

– Y1), Y1 = 0 OR 1

d.

X1 = 40Y11 + 90Y12,

Y11 + Y12 = 1

39. The branch-and-bound algorithm starts by

a.

relaxing all the integrality

conditions in an ILP and solving the resulting LP problem.

b.

relaxing all the RHS values in an ILP

and solving the resulting LP problem.

c.

solving two LP problems in which X1

is set at 0 and 1 respectively.

d.

determining the most likely RHS values

and solving for them.

40. Any integer variable in an ILP that assumes a fractional value in

the optimal solution to the relaxed LP problem can be designated

a.

a diverging variable.

b.

a branching variable.

c.

a bifurcating variable.

d.

a splitting variable.

41. The optimal relaxed solution for an ILP has X1 = 3.6

and X2 = 2.9. If we branch on X1, what constraints must

be added to the two resulting LP problems?

a.

X1Â³ 3, X1Â³ 4

b.

X1 = 4

c.

3Â£ X1, X1

Â£ 4

d.

X1Â£ 3, X1Â³ 4

42. A sub-problem in a B & B is solved and found infeasible.

Should the B & B algorithm continue further analysis on this candidate

problem?

a.

Yes, a feasible solution may be found

when additional constraints are added.

b.

Yes, removing a constraint in further

analysis may restore feasibility.

c.

No, adding more constraints will not

restore problem feasibility.

d.

No, the result cannot occur so

re-examine the formulation and start over.

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