Chapter 6—Integer Linear Programming

Chapter 6—Integer Linear Programming

Chapter 6—Integer 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

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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.

X4£ M4Y4,
X4³ 100

b.

X4£ M4Y4,
X4 = 100 Y4

c.

X4£ M4Y4,
X4³ 100 Y4

d.

X4£ M4Y4,
X4£ 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.

X31£ M3Y3,
X32³ 50Y3

b.

X31£ M3Y3,
X31³ 50

c.

X32³ (1/50)X31,
X31£ 50

d.

X32£ M3Y3,
X31³ 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.

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

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.

X1³ 3, X1³ 4

b.

X1 = 4

c.

3£ X1, X1
£ 4

d.

X1£ 3, X1³ 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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