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Beyond Linear Programming: Mathematical Programming Extensions CHAPTER 7 and 17 (on CD). Integer Programming Goal Programming NonLinear Programming. Learning Objectives. Formulate integer programming (IP) models. Set up and solve IP models using Excel’s Solver.
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ADM2302  Rim Jaber
ADM2302  Rim Jaber
ADM2302  Rim Jaber
ADM2302  Rim Jaber
The Harison Electric Company, located in Chicago’s old town area, Produces two products popular with home renovators: oldfashion Chandelier and ceiling fans. Both the chandeliers and fans require a twostep production process involving wiring and assembly.
It takes about 2 hours to wire each chandelier, and 3 hours to wire a ceiling fan. Final assembly of the chandeliers and fans requires 6 and 5 hours, respectively. The production capability is such that 12 hours of wiring time and 30 hours of assembly time are available. If each chandelier produced nets the firm $600 and each fan $700. Harison’s production mix decision can be formulated using LP as follows:
ADM2302  Rim Jaber
Maximize Profit = $ 600X1 + 700X2
Subject to
2X1 + 3X2 <= 12 (wiring hours)
6X1 + 5X2 <= 30 (assembly hours)
X1, X2 >= 0
X1 = number of chandeliers produced
X2 = number of ceiling fans produced
ADM2302  Rim Jaber
X2
6
6X1 + 5X2£ 30
5
+ = possible Integer Solution
4
Optimal LP Solution
(X1 = 3 3/4, X2 = 1 1/2
Profit = $3,300)
+
3
+
+
+
2
2X1 + 3X2£ 12
+
+
+
+
1
X1
0
1
2
3
4
5
6
ADM2302  Rim Jaber
ADM2302  Rim Jaber
Chandeliers (X1)
Ceiling Fans(X2)
Profit: $600X1 + $700X2
0
1
2
3
4
5
0
1
2
3
4
0
1
2
3
0
1
0
0
0
0
0
0
0
1
1
1
1
1
2
2
2
2
3
3
4
$0
600
1,200
1,800
2,400
3,000
700
1,300
1,900
2,500
3,100
1,400
2,000
2,600
3,200
2.100
2,700
2,800
Solution if rounding off is used
Optimal Solution to integer Programming Problem
ADM2302  Rim Jaber
ADM2302  Rim Jaber
ADM2302  Rim Jaber
ADM2302  Rim Jaber
ADM2302  Rim Jaber
ADM2302  Rim Jaber
1 if an event takes place
Y =
0 if an event doesn’t take place
ADM2302  Rim Jaber
The Department head of a management science department at a major Midwestern university will be scheduling faculty to teach courses during the coming autumn term. Four core courses need to be covered. The four courses are at the UG, MBA, MS, and Ph.D. levels. Four professors will be assigned to the courses, with each professor receiving one of the courses. Student evaluations of professors are available from previous terms. Based on a rating scale of 4 (excellent), 3 (very good), 2 (average), 1(fair), and 0(poor), the average student evaluations for each professor are shown:
ADM2302  Rim Jaber
Professor D does not have a Ph.D. and cannot be assigned to teach the Ph.D.level course. If the department head makes teaching assignments based on maximizing the student evaluation ratings over all four courses, what staffing assignments should be made?
Course
UG
MBA
Professor
MS
Ph.D.
A
2.8
2.2
3.3
3.0
B
3.2
3.0
3.6
3.6
C
3.3
3.2
3.5
3.5
D
3.2
2.8
2.5

ADM2302  Rim Jaber
The Research and Development Division of the Progressive
Company has been developing four possible new product lines.
Management must now make a decision as to which of these
Four products actually will be produced and at what levels.
Therefore, a management science study has been requested to
Find the most profitable product mix.
A substantial cost is associated with beginning the production
Of any product, as given in the first row of the following table.
Management’s objective is to find the product mix that maximizes
The total profit (total net revenue minus start up costs).
4
1
2
3
$50,000
$40,000
$70,000
$60,000
Startupcost
Marginal revenue
$70
$60
$90
$80
1000
3000
2000
1000
Maximum Production
Let the Integer decisions variables x1, x2, x3, and
x4 be the total number of units produced of products 1,
2, 3 and 4, respectively. Management has imposed the
following policy constraints on these variables:
ADM2302  Rim Jaber
Or Either 4x1+6x2+3x3+5x4 <= 6,000
Use Auxiliary binary variables to formulate and solve the mixed BIP model.
ADM2302  Rim Jaber
0 if product i is not produced
ADM2302  Rim Jaber
Y 4 <= Y1 + Y2
Contingency Constraints: it involves binary variables that do not allow certain variables to equal 1 unless certain other variables are equal to 1
ADM2302  Rim Jaber
Or 1 if 4x1+6x2+3x3+5x4 <= 6,000 must hold
4x1+6x2+3x3+5x4 <= 6,000 + 1,000,000(1Y)
ADM2302  Rim Jaber
Max Z = $70x1 + 60x2 + 90x3 +80x4 
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0 if x1 is not produced
0 if x2 is not produced
0 if x3 is not produced
0 if x4 is not produced
$50,000Y1 + $40,000Y2 + $70,000Y3 + $60,000Y4
x1 <= 1000Y1
x2 <= 3000 Y2
x3 <= 2000 Y3
x4 <= 1000 Y4
ADM2302  Rim Jaber
Max Z = $70x1 + $60x2 + $90x3 +$80x4  $50,000Y1  $40,000Y2  $70,000Y3  $60,000Y4
Subject to
Y1 + Y2 + Y3 + Y4 <= 2
Y3 <= Y1 + Y2
Y 4 <= Y1 + Y2
5x1+3x2+6x3+4x4 <= 6,000 + 1,000,000Y
4x1+6x2+3x3+5x4 <= 6,000 + 1,000,00(1Y)
x1 <= 1000Y1
x2 <= 3000Y2
x3 <= 2000Y3
x4 <= 1000Y4
x1, x2, x3, x4 >= 0 and integer; Y,Y1,Y2,Y3,Y4 = 0, 1
Integer LP models have variety of applications including:
ADM2302  Rim Jaber