B.3 Integer Programming
Review Questions
Lesson Topics
Rounding Off (5) solutions in continuous variables
to the nearest integer (like 2.67 rounded off to 3) is an
unreliable way to solve a linear programming problem
when decision variables should be integers.
Sensitivity Analysis with Integer Variables
is more important than with continuous variables
because a small change in a constraint coefficient can
cause a relatively large change in the optimal solution.
Assignment Problems with Valuable Time
(2) minimize the total time of assigning workers to
jobs. Minimizing total time is appropriate when each
worker has the same value of time.
Assignment Problems with Supply and
Demand (2) are Transportation Problems of
suppliers to demanders except that each demand is
assigned to exactly one supplier.
1
B.3 Integer Programming
Review Questions
Rounding Off
Question. Consider the following all-integer linear
program:
Max 1x1 + 1x2
s.t.
4x1 + 6x2 ≤ 22
1x1 + 5x2 ≤ 15
2x1 + 1x2 ≤ 9
x1, x2 ≥ 0 and integer
a. Graph the constraints for this problem. Use dots to indicate all
feasible integer solutions.
b. Find the optimal integer solution.
c. Solve the LP Relaxation of this problem (that is, the problem
without requiring decision variables to be integers).
2
B.3 Integer Programming
Review Questions
Answer to Question:
a.
b. The optimal solution to the LP Relaxation is shown on the above
graph to be x1 = 4, x2 = 1. Its value is 5.
c. The optimal integer solution is the same as the optimal solution to the
LP Relaxation. This is always the case whenever all the variables
take on integer values in the optimal solution to the LP Relaxation.
3
B.3 Integer Programming
Review Questions
Rounding Off
Question. Muir Manufacturing produces two
popular grades of commercial carpeting among its
many other products. In the coming production
period, Muir needs to decide how many rolls of each
grade should be produced in order to maximize profit. Each roll of Grade X
carpet uses 3 units of synthetic fiber, requires 1 hour of production time,
and needs 1 unit of foam backing. Each roll of Grade Y carpet uses 1 unit
of synthetic fiber, requires 3 hours of production time, and needs 1 units of
foam backing.
The profit per roll of Grade X carpet is $3 and the profit per roll of Grade Y
carpet is $2. In the coming production period, Muir has 9 units of synthetic
fiber available for use. Workers have been scheduled to provide up to 7
hours of production time. The company has 10 units of foam backing
available for use.
a. Develop a linear programming model for this problem to determine
how many rolls of each carpet should be produced.
b. Graphically solve the linear-programming problem from Part a if you
require that the number of rolls of each carpet be integers.
c. Graphically solve the linear-programming problem from Part a if you
do not require that the number of rolls of each carpet be integers
(instead, the number of rolls of each carpet 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.
4
B.3 Integer Programming
Review Questions
Answer to Question:
Part a:
Let X = the number of rolls of Grade X carpet to make
Let Y = the number of rolls of Grade Y carpet to make
Max 3X + 2Y
s.t.
3X + Y < 9
X + 3Y < 7
X + Y < 10
X,Y > 0
Part c:
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
A graph of the feasible set reveals that the third costraint is redundant --- it
does not affect the feasible set. Adding a graph of isovalue lines reveals
the optimum occurs where the first and second constraints bind (the third
constraint is redundant). Solving the binding form of those two constraints
yields the optimal solution: X = 2.5, Y = 1.5
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B.3 Integer Programming
Review Questions
Part b:
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
A graph of the feasible set reveals that the third costraint is redundant --- it
does not affect the feasible set. Use dots to indicate all feasible integer
solutions. Adding a graph of isovalue lines reveals the optimum occurs at
X = 3, Y = 0
Part d: Comparing solutions, the optimal integer solution in Part b is not the
result of rounding off the optimal continuous solution in Part c.
6
B.3 Integer Programming
Review Questions
Rounding Off
Question. The Iron Works, Inc. seeks to maximize
profit by making two products from steel and labor.
It just received this day's allocation of 9 pounds of
steel, and there is 8 hours of labor available.
It takes 3 pounds of steel to make a unit of Product 1, and 1 pound of steel
to make a unit of Product 2. It also takes 2 hours of labor to make a unit of
Product 1, and 5 hours of labor to make a unit of Product 2. The physical
plant has the capacity to make up to 5 units of total product (Product 1 plus
Product 2). Product 1 has unit profit 3 dollars, and Product 2 has 9 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 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.
7
B.3 Integer Programming
Review Questions
Answer to Question:
Part a:
Let X = units of Product 1 produced.
Let Y = units of Product 1 produced.
Max 3X + 9Y
s.t. 3X + Y < 9 (steel)
2X + 5Y < 8 (labor)
X+Y
< 5 (capacity)
X, Y  0
Part c:
X on horizontal axis
Y on vertical axis
First constraint, through (3,0) and (0,9)
Second constraint, through (4,0) and (0,1.6)
Third constraint, (5,0) and (0,5)
Feasible set of solutions is the region bounded
by first, second, and non-negativity constraints
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
A graph of the feasible set and isovalue lines reveals the optimum occurs
where the non-negativity of X and the second constraint binds. Solving the
binding form of those two constraints yields the optimal solution: X = 0, Y =
8/5 = 1.6
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B.3 Integer Programming
Review Questions
Part b:
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
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 = 1
Part d: Comparing solutions, the optimal integer solution in Part b is not the
result of rounding off the optimal continuous solution in Part c.
9
B.3 Integer Programming
Review Questions
Rounding Off
Question. 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 4 thousand gallons of crude oil,
and 2 thousand gallons of additives.
It takes 0.8 gallons of crude oil to make a gallon of Premium gasoline, and
0.4 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 0.6
gallons of additives to make a gallon of Regular gasoline. Premium
gasoline has unit profit 3 of dollars, and Regular gasoline has 1 dollar.
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.
10
B.3 Integer Programming
Review Questions
Answer to Question:
Part a:
Let X = thousands of units of Premium gasoline produced.
Let Y = thousands of units of Regular gasoline produced.
Max 3X + 1Y
s.t. 0.8X + 0.4Y < 4
0.2X + 0.6Y < 2
X, Y  0
(crude oil)
(additives)
Part c:
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
A graph of the feasible set and isovalue lines (dashed lines above) reveals
the optimum occurs where the non-negativity of Y and the first constraint
binds. Solving the binding form of those two constraints yields the optimal
solution: X = 5, Y = 0
Part b: Since the continuous solution is discrete, it is also the solution to
the discrete case. (Since (X,Y) are in thousands of units, (X,Y) would still
be an integer solution as long as 1000X and 1000Y were integers.
11
B.3 Integer Programming
Review Questions
Rounding Off
Question. 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.
12
B.3 Integer Programming
Review Questions
Answer to Question:
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
0.2X + 0.3Y < 0.8
X, Y  0
(crude oil)
(additives)
Part c:
5
4
3
2
1
0
0
1
2
3
4
5
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 first and the second constraint bind. Solving the binding
form of those two constraints yields the optimal solution: X = 1.75, Y = 1.50
13
B.3 Integer Programming
Review Questions
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
0.2X + 0.3Y < 0.8
X, Y  0
(crude oil)
(additives)
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
5
4
3
2
1
0
0
1
2
3
4
5
Part d: The integer solution in Part b is not the result of rounding off the
continuous solution in Part c.
14
B.3 Integer Programming
Review Questions
Rounding Off
Question. Jacuzzi produces two types of hot tubs:
Fuzion and Torino.
There are 5 pumps, 36 hours of labor, and 60 feet of
tubing available to make the tubs per week. Here are
the input requirements, and unit profits:
Pumps
Labor
Tubing
Unit Profit
Fuzion
1
9 hours
8 feet
$6
Torino
1
4 hours
15 feet
$9
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.
15
B.3 Integer Programming
Review Questions
Answer to Question:
Part a:
Let F = Fuzion units produced per week.
Let Y = Torino units produced per week.
Max 6 F + 9 T
s.t. F + T
< 5
9 F + 4 T < 36
8 F + 15 T < 60
F, T  0
(pumps)
(labor hours)
(tubing)
Part c:
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
A graph of the feasible set and isovalue lines (dashed lines above) reveals
the optimum occurs where the first and the third constraint bind. Solving
the binding form of those two constraints yields the optimal solution: F =
2.143, T = 2.857
16
B.3 Integer Programming
Review Questions
Part b:
Max 6 F + 9 T
s.t. F + T
< 5
9 F + 4 T < 36
8 F + 15 T < 60
F, T  0
(pumps)
(labor hours)
(tubing)
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
A graph of the feasible set dots and isovalue lines (dashed lines above)
reveals the optimum occurs at either (0,4) or (1,3) or (3,2). It turns out
there are two optima: (0,4) and (3,2).
Part d: The integer solutions in Part b are not the result of rounding off the
continuous solution in Part c.
17
B.3 Integer Programming
Review Questions
Assignment with Valuable Time
Question. Sibeau Custom Tailoring has five idle
tailors and two custom garments to make. The
estimated time (in hours) it would take each tailor to
complete a garment is shown in the next slide.
Tailor
Garment
1
2
3
4
5
Polyester Suit
9
11
6
11
8
Silk Suit
11
10
9
12
9
Formulate an appropriate binary-integer program for determining the tailorgarment assignments that minimize the total estimated time spent making
the two garments. No tailor is to be assigned more than one garment, and
each garment is to be worked on by exactly one tailor. Formulate the
problem, but you need not solve for the optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
18
B.3 Integer Programming
Review Questions
Answer to Question: Define the decision variables xij = 1 if garment i is
assigned to tailor j, and = 0 otherwise.
Minimize total time spent making garments:
Min 9x11 + 11x12 + 6x13 + 11x14 + 8x15
+ 11x21 + 10x22 + 9x23 + 12x24 + 9x25
Define the constraints of exactly one tailor per garment:
1) x11 + x12 + x13 + x14 + x15 = 1
2) x21 + x22 + x23 + x24 + x25 = 1
Define the constraints of no more than one garment per tailor:
3) x11 + x21 < 1
4) x12 + x22 < 1
5) x13 + x23 < 1
6) x14 + x24 < 1
7) x15 + x25 < 1
19
B.3 Integer Programming
Review Questions
Assignment with Valuable Time
Question. Sibeau Custom Tailoring has five idle
tailors and two custom garments to make. The
estimated time (in hours) it would take each tailor to
complete a garment is shown in the next slide. (An 'X'
in the table indicates an unacceptable tailor-garment assignment.)
Tailor
Garment
1
2
3
4
5
Polyester Suit
9
11
6
11
8
Silk Suit
11
10
X
12
9
Formulate and solve a binary-integer program for determining the tailorgarment assignments that minimize the total estimated time spent making
the two garments. No tailor is to be assigned more than one garment, and
each garment is to be worked on by exactly one tailor.
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.
20
B.3 Integer Programming
Review Questions
Answer to Question: Define the decision variables
xij = 1 if garment i is assigned to tailor j, and = 0 otherwise.
Minimize total time spent making garments:
Min 9x11 + 11x12 + 6x13 + 11x14 + 8x15
+ 11x21 + 10x22 + 12x24 + 9x25
Define the constraints of exactly one tailor per garment:
1) x11 + x12 + x13 + x14 + x15 = 1
2) x21 + x22 + x24 + x25 = 1
Define the constraints of no more than one garment per tailor:
3) x11 + x21 < 1
4) x12 + x22 < 1
5) x13 < 1
6) x14 + x24 < 1
7) x15 + x25 < 1
21
B.3 Integer Programming
Review Questions
Interpretation: Assign tailor 1 to garment 3 and tailor 2 to garment 5. Time
spent is 15 hours.
22
B.3 Integer Programming
Review Questions
Assignment with Supply and Demand
Question. Dow Chemical uses the chemical Rbase
in production operations at two divisions. Only three
suppliers of Rbase meet Dow’s quality standards.
The quantity of Rbase needed by each Dow division
and the price per gallon charged by each supplier are as follows:
Price per Gallon ($)
Demand (1000s
of gallons)
Div 1
40
Sup 1
12.60
Div 2
45
Sup 2
14.00
Sup 3
10.20
The cost per gallon ($) for shipping from each supplier to each division are
as follows:
Cij
Div 1
Div 2
Sup 1
1
0.80
Sup 2
2.50
0.20
Sup 3
3.15
5.40
Dow wants to diversify by spreading its business so that each division’s
demand is assigned to exactly one supplier.
Formulate the optimal assignment of suppliers to divisions as a linearprogramming problem. Formulate the problem, but you need not solve the
problem.
23
B.3 Integer Programming
Review Questions
Answer to Question:
Linear programming formulation (supply inequality, demand equality).
Variables: Xij = 1 if Supplier i is assigned to Division j, else 0
Assignment Costs:
 The total cost is the sum of the purchase cost and the transportation
cost.
 Supplier 1 assigned to Division 1 (cost in $1000s):
o Purchase cost: (40 x $12.60) = $504
o Transportation Cost: (40 x $1) = $40
o Total Cost: $544
 Assignment Costs: Cij = Cost of assigning Supplier i to Division j
Cij
Div 1
Div 2
Sup 1
544
603
Sup 2
660
639
Sup 3
534
702
24
B.3 Integer Programming
Review Questions
Linear programming formulation (supply inequality, demand equality).
Objective (minimize cost):
Min 544X11 + 603X12 + 660X21 + 639X22 + 534X31 + 702X32
Demand Constraints (since each division’s demand is assigned to exactly
one supplier):
X11 + X21 + X31 = 1
X12 + X22 + X32 = 1
Optional: There is no mention of supply constraints, which are common in
assignment problems. Here is what those common constraints would be in
this problem.
Supply Constraints (Each supplier can supply at most 1 Division):
X11 + X12 < 1
X21 + X22 < 1
X31 + X32 < 1
25
B.3 Integer Programming
Review Questions
Assignment with Supply and Demand
Question. 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:
Price (dollars per ton)
Demand (tons)
American
5
India
3
Asian
4
Indonesia
4
European
3
Malaysia
2
Thailand
5
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 plant’s
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.
26
B.3 Integer Programming
Review Questions
Answer to Question:
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
27
B.3 Integer Programming
Review Questions
Linear programming formulation (supply inequality, demand equality).
Objective (minimize cost):
Min 30X11 + 36X12 + 21X13
+ 45X21 + 52X22 + 36X23
+ 20X31 + 32X32 + 18X33
+ 60X41 + 24X42 + 21X43
Subject to:
Demand Constraints (each demand is assigned to exactly one supplier):
X11 + X21 + X31 + X41 = 1
X12 + X22 + X32 + X42 = 1
X13 + X23 + X33 + X43 = 1
Supply Constraints (each supplier can supply at most one demander):
X11 + X12 + X13 < 1
X21 + X22 + X23 < 1
X31 + X32 + X33 < 1
X41 + X42 + X43 < 1
28