Integer Program Examples
1)
You have $14,000 to invest in any of 4 investments. The following table
indicates how much is needed in order to purchase the investment and the expected value
of the investment after one year. You cannot partially invest in the investments (must
fully purchase or not purchase). How do you maximize your investment?
Investment
1
2
3
4
Purchase Price
$5
$7
$4
$3
Expected Value at year end
$16
$22
$12
$8
Integer Program Examples
1)
You have $14,000 to invest in any of 4 investments. The following table
indicates how much is needed in order to purchase the investment and the expected value
of the investment after one year. You cannot partially invest in the investments (must
fully purchase or not purchase). How do you maximize your investment?
Investment
1
2
3
4
Purchase Price(000s)
$5
$7
$4
$3
Let xi = 1 if investment i is made
0 if not
Max Z = 16x1 + 22x2 + 12x3 + 8x4
s.t
5x1 + 7x2 + 4x3 + 3x4 <= 14
x1, x2, x3, x4 <= 1
x1, x2, x3, x4 >= 0
LP OPTIMUM FOUND AT STEP
3
OBJECTIVE FUNCTION VALUE
1)
44.00000
VARIABLE
VALUE
X1
1.000000
X2
1.000000
X3
0.500000
X4
0.000000
ROW
2)
3)
4)
5)
6)
REDUCED COST
0.000000
0.000000
0.000000
1.000000
SLACK OR SURPLUS DUAL PRICES
0.000000
3.000000
0.000000
1.000000
0.000000
1.000000
0.500000
0.000000
1.000000
0.000000
NO. ITERATIONS=
3
Expected Value at year end (000s)
$16
$22
$12
$8
Integer Program Examples
Max Z = 16x1 + 22x2 + 12x3 + 8x4
s.t
5x1 + 7x2 + 4x3 + 3x4 <= 14
x1, x2, x3, x4 = 0 or 1
LP OPTIMUM FOUND AT STEP
0
OBJECTIVE VALUE = 44.0000000
NEW INTEGER SOLUTION OF 42.0000000
RE-INSTALLING BEST SOLUTION...
AT BRANCH
OBJECTIVE FUNCTION VALUE
1)
42.00000
VARIABLE
VALUE
X1
0.000000
X2
1.000000
X3
1.000000
X4
1.000000
ROW
2)
3)
4)
5)
6)
REDUCED COST
-16.000000
-22.000000
-12.000000
-8.000000
SLACK OR SURPLUS DUAL PRICES
0.000000
0.000000
1.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
NO. ITERATIONS=
0
BRANCHES= 0 DETERM.= 1.000E 0
0 PIVOT
0
Integer Program Examples
2)
3-job, 4-machine Job Shop scheduling problem. Three products, A,B, and C are to be
produced using four machines. The technological sequence and the processing time on the
machines for the three products are shown in the figure below. For instance, product A is
processed on machine 1 first for a1 hours, then on machine 3 for a3 hours, and finally on machine
4 for a4 hours. Each machine can work on only one product at a time. Moreover, each product
requires a different set of tools for machining, which requires each machine to complete the
processing of one product before taking up the next one (no preemption).
In addition, it is required to complete product B in no more than d hours from the starting
time. The problem is to determine the sequence in which the various products are processed on
the machines so as to complete all the products in the least possible time.
machine / process time
Product A:
1 / a1 ------------- 3 / a3 ----------------- 4 / a4
Product B:
1 / b1 ------------- 2 / b2 ----------------- 4 / b4
Product C:
2 / c2 ------------- 3 / c3
Integer Program Examples
2)
3-job, 4-machine Job Shop scheduling problem. Three products, A,B, and C are to be
produced using four machines. The technological sequence and the processing time on the
machines for the three products are shown in the figure below. For instance, product A is
processed on machine 1 first for a1 hours, then on machine 3 for a3 hours, and finally on machine
4 for a4 hours. Each machine can work on only one product at a time. Moreover, each product
requires a different set of tools for machining, which requires each machine to complete the
processing of one product before taking up the next one (no preemption).
In addition, it is required to complete product B in no more than d hours from the starting
time. The problem is to determine the sequence in which the various products are processed on
the machines so as to complete all the products in the least possible time.
machine / process time
Product A:
1 / a1 ------------- 3 / a3 ----------------- 4 / a4
Product B:
1 / b1 ------------- 2 / b2 ----------------- 4 / b4
Product C:
2 / c2 ------------- 3 / c3
xA1 – start time of product A on machine 1
.
xC3 – start time of product C on machine 3
Technological
Route
A) xA1 + a1 <= xA3
xA3 + a3 <= xA4
B) xB1 + b1 <= xB2
xB2 + b2 <= xB4
C) xC2 + c2 <= xC3
One product
Machine at
a time
1)
2)
3)
4)
xA1 + a1 <= xB1 or xB1 + b1 <= xA1
xA1 + a1 – xB1 <= My1
where y1 = 0 or 1
xB1 + b1 – xA1 <= M(1-y1)
xB2 + b2 – xC2 <= My2
xC2 + c2 – xB2 <= M(1-y2)
xA3 + a3 – xC3 <= My3
xC3 + c3 – xA3 <= M(1-y3)
xA4 + a4 – xB4 <= My4
xB4 + b4 – xA4 <= M(1-y4)
B finished by d
xB4 + b4 <= d
All jobs completed
z >= xA4 + a4
z >= xB4 + b4
z >= xC3 + c3
Objective:
Min z
Integer Program Examples
! Job Shop Scheduling – Integer Programming Example 2
!
! Product A) 1/3 -> 3/4 -> 4/2
! Product B) 1/2 -> 2/4 -> 4/2 d = 12
! Product C) 2/3 -> 3/4
!
Min z
ST xA1 - xA3 <= -3
xA3 - xA4 <= -4
xA4 - z <= -2
xB1 - xB2 <= -2
xB2 - xB4 <= -4
xB4 <= 9
xB4 - z <= -2
xC2 - xC3 <= -3
xC3 - z <= -4
xA1 - xB1 - 99 y1 <= -3
xB1 - xA1 + 99y1 <= 97
xB2 - xC2 - 99 y2 <= -4
xC2 - xB2 + 99 y2 <= 96
xA3 - xC3 - 99 y3 <= -4
xC3 - xA3 + 99 y3 <= 95
xA4 - xB4 - 99 y4 <= -2
xB4 - xA4 + 99 y4 <= 96
y1 <= 1
y2 <= 1
y3 <= 1
y4 <= 1
End
INT y1
INT y2
INT y3
INT y4
Integer Program Examples
LAST INTEGER SOLUTION IS THE BEST FOUND
RE-INSTALLING BEST SOLUTION...
OBJECTIVE FUNCTION VALUE
1)
11.00000
VARIABLE
VALUE
REDUCED COST
Y1
0.000000
-99.000000
Y2
1.000000
0.000000
Y3
0.000000
0.000000
Y4
0.000000
0.000000
Z
11.000000
0.000000
XA1
0.000000
1.000000
XA3
3.000000
0.000000
XA4
7.000000
0.000000
XB1
3.000000
0.000000
XB2
5.000000
0.000000
XB4
9.000000
0.000000
XC2
0.000000
0.000000
XC3
7.000000
0.000000
ROW SLACK OR SURPLUS DUAL PRICES
2)
0.000000
0.000000
3)
0.000000
0.000000
4)
2.000000
0.000000
5)
0.000000
1.000000
6)
0.000000
1.000000
7)
0.000000
0.000000
8)
0.000000
1.000000
9)
4.000000
0.000000
10)
0.000000
0.000000
11)
0.000000
1.000000
12)
94.000000
0.000000
13)
90.000000
0.000000
14)
2.000000
0.000000
15)
0.000000
0.000000
16)
91.000000
0.000000
17)
0.000000
0.000000
18)
94.000000
0.000000
19)
1.000000
0.000000
20)
0.000000
0.000000
21)
1.000000
0.000000
22)
1.000000
0.000000
NO. ITERATIONS= 107
BRANCHES= 2 DETERM.= 1.000E 0
Integer Program Examples
** Solution without integer constraint **
LP OPTIMUM FOUND AT STEP
11
OBJECTIVE FUNCTION VALUE
1)
9.000000
VARIABLE
VALUE
Z
9.000000
XA1
0.000000
XA3
3.000000
XA4
7.000000
XB1
0.000000
XB2
2.000000
XB4
7.000000
XC2
0.000000
XC3
5.000000
Y1
0.030303
Y2
0.060606
Y3
0.020202
Y4
0.020202
REDUCED COST
0.000000
1.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
ROW SLACK OR SURPLUS DUAL PRICES
2)
0.000000
1.000000
3)
0.000000
1.000000
4)
0.000000
1.000000
5)
0.000000
0.000000
6)
1.000000
0.000000
7)
2.000000
0.000000
8)
0.000000
0.000000
9)
2.000000
0.000000
10)
0.000000
0.000000
11)
0.000000
0.000000
12)
94.000000
0.000000
13)
0.000000
0.000000
14)
92.000000
0.000000
15)
0.000000
0.000000
16)
91.000000
0.000000
17)
0.000000
0.000000
18)
94.000000
0.000000
19)
0.969697
0.000000
20)
0.939394
0.000000
21)
0.979798
0.000000
22)
0.979798
0.000000
NO. ITERATIONS=
11
Integer Program Examples
3) There are six cities in a county. The county must determine where to build fire stations. The
county wants to build the minimum number of fire stations needed to ensure that at least one fire
station is with 15 minutes (driving time) of each city. The times (in minutes) required to drive
between the cities are shown below. A fire station must be built within a city. Determine where
the fire stations are to be built.
To
From City 1 City 2 City 3 City 4 City 5 City 6
City 1
0
10
20
30
30
20
City 2
10
0
25
35
20
10
City 3
20
25
0
15
30
20
City 4
30
35
15
0
15
25
City 5
30
20
30
15
0
14
City 6
20
10
20
25
14
0
Integer Program Examples
3) There are six cities in a county. The county must determine where to build fire stations. The
county wants to build the minimum number of fire stations needed to ensure that at least one fire
station is with 15 minutes (driving time) of each city. The times (in minutes) required to drive
between the cities are shown below. A fire station must be built within a city. Determine where
the fire stations are to be built.
To
From City 1 City 2 City 3 City 4 City 5 City 6
City 1
0
10
20
30
30
20
City 2
10
0
25
35
20
10
City 3
20
25
0
15
30
20
City 4
30
35
15
0
15
25
City 5
30
20
30
15
0
14
City 6
20
10
20
25
14
0
x1 – 1 if built in city 1, 0 otherwise
Min x1 + x2 + x3 + x4 + x5 + x6
ST x1 + x2
>= 1
x1 + x2
+ x6 >= 1
x3 + x4
>= 1
x3 + x4 + x5
>= 1
x4 + x5 + x6 >= 1
x2
+ x5 + x6 >= 1
xi = 0 or 1 for all i