Linear Programming:
Modeling Examples
Chapter 4- Part2
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-1
Chapter Topics

A Marketing Example

A Blend Example

A Multiperiod Scheduling Example
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-2
A Marketing Example
Data and Problem Definition (1 of 6)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-3
A Marketing Example
Model Summary (2 of 6)
Maximize Z = 20,000x1 + 12,000x2 + 9,000x3
subject to:
15,000x1 + 6,000x 2+ 4,000x3  100,000
x1  4
x2  10
x3  7
x1 + x2 + x3  15
x1, x2, x3  0
where
x1 = number of television commercials
x2 = number of radio commercials
x3 = number of newspaper ads
4-4
A Marketing Example
Solution with Excel (3 of 6)
Exhibit 4.10
4-5
A Marketing Example
Solution with Excel Solver Window (4 of 6)
Exhibit 4.11
4-6
A Marketing Example
Integer Solution with Excel (5 of 6)
Exhibit 4.12
Exhibit 4.13
4-7
A Marketing Example
Integer Solution with Excel (6 of 6)
Exhibit 4.14
4-8
A Blend Example
Problem Definition and Data (1 of 8)
Component
Maximum Barrels
Available/day
Cost/barrel
1
4,500
$12
2
2,700
10
3
3,500
14
4-9
A Blend Example
Problem Definition and Data (2 of 8)
Grade
Component Specifications
Selling Price ($/bbl)
Super
At least 50% of 1
Not more than 30% of 2
$23
Premium
At least 40% of 1
Not more than 25% of 3
Extra
At least 60% of 1
At least 10% of 2
20
18
4-10
A Blend Example
Problem Definition and Data (3 of 8)
Component
Maximum Barrels
Available/day
Cost/barrel
1
4,500
$12
2
2,700
10
3
3,500
14
Grade
Component Specifications
Selling Price ($/bbl)
Super
At least 50% of 1
Not more than 30% of 2
$23
Premium
At least 40% of 1
Not more than 25% of 3
Extra
At least 60% of 1
At least 10% of 2
20
18
4-11
A Blend Example
Problem Statement and Variables (4 of 8)
■
Determine the optimal mix of the three components in each grade
of motor oil that will maximize profit. Company wants to produce
at least 3,000 barrels of each grade of motor oil.
■
Decision variables: The quantity of each of the three components
used in each grade of gasoline (9 decision variables);
xij = barrels of component i used in motor oil grade j per day,
where i = 1, 2, 3 and j = s (super), p (premium), and
e (extra).
4-12
A Blend Example
Model Summary (5 of 8)
Maximize Z = 11x1s + 13x2s + 9x3s + 8x1p + 10x2p + 6x3p + 6x1e
+ 8x2e + 4x3e
subject to:
x1s + x1p + x1e  4,500 bbl.
x2s + x2p + x2e  2,700 bbl.
x3s + x3p + x3e  3,500 bbl.
0.50x1s - 0.50x2s - 0.50x3s  0
0.70x2s - 0.30x1s - 0.30x3s  0
0.60x1p - 0.40x2p - 0.40x3p  0
0.75x3p - 0.25x1p - 0.25x2p  0
0.40x1e- 0.60x2e- - 0.60x3e  0
0.90x2e - 0.10x1e - 0.10x3e  0
x1s + x2s + x3s  3,000 bbl.
x1p+ x2p + x3p  3,000 bbl.
all xij  0
x1e+ x2e + x3e  3,000 bbl.
4-13
A Blend Example
Solution with Excel (6 of 8)
Exhibit 4.17
4-14
A Blend Example
Solution with Solver Window (7 of 8)
Exhibit 4.18
4-15
A Blend Example
Sensitivity Report (8 of 8)
Exhibit 4.19
4-16
A Multi-Period Scheduling Example
Problem Definition and Data (1 of 5)
4-17
A Multi-Period Scheduling Example
Decision Variables (2 of 5)
Decision Variables:
rj = regular production of computers in week j
(j = 1, 2, …, 6)
oj = overtime production of computers in week j
(j = 1, 2, …, 6)
ij = extra computers carried over as inventory in week j
(j = 1, 2, …, 5)
4-18
A Multi-Period Scheduling Example
Model Summary (3 of 5)
Model summary:
Minimize Z = $190(r1 + r2 + r3 + r4 + r5 + r6) + $260(o1+o2
+o3 +o4+o5+o6) + 10(i1 + i2 + i3 + i4 + i5)
subject to:
rj  160 computers in week j (j = 1, 2, 3, 4, 5, 6)
oj  50 computers in week j (j = 1, 2, 3, 4, 5, 6)
r1 + o1 - i1 = 105
week 1
r2 + o2 + i1 - i2 = 170 week 2
r3 + o3 + i2 - i3 = 230 week 3
r4 + o4 + i3 - i4 = 180 week 4
r5 + o5 + i4 - i5 = 150 week 5
r6 + o6 + i5 = 250
week 6
rj, oj, ij  0
4-19
A Multi-Period Scheduling Example
Solution with Excel (4 of 5)
Exhibit 4.20
4-20
A Multi-Period Scheduling Example
Solution with Solver Window (5 of 5)
Exhibit 4.21
4-21
4-22