1. Nonlinear Programming Prepared by Lee Revere and John Large
Chapter 11
Integer, Goal,
and
Nonlinear Programming
Prepared by Lee Revere and John Large
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11-1

2. Learning Objectives Students will be able to:
Understand the difference between LP and integer programming.
Understand and solve the three types of integer programming problems.
Apply the branch and bound method to solve integer programming problems.
Solve goal programming problems graphically and using a modified simplex technique.
Formulate nonlinear programming problems and solve using Excel.
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3. Chapter Outline 11.1 Introduction 11.2 Integer Programming
11.3 Modeling with 0-1 (Binary) Variables
11.4 Goal Programming
11.5 Nonlinear Programming
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4. Introduction
Integer programming is the extension of LP that solves problems requiring integer solutions.
Goal programming is the extension of LP that permits more than one objective to be stated.
Nonlinear programming is the case in which objectives or constraints are nonlinear.
All three above mathematical programming models are used when some of the basic assumptions of LP are made more or less restrictive.
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5. Summary: Linear Programming Extensions
Integer Programming
Linear, integer solutions
Goal Programming
Linear, multiple objectives
Nonlinear Programming
Nonlinear objective and/or constraints
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6. Integer Programming
Solution values must be whole numbers in integer programming .
There are three types of integer programs:
pure integer programming;
mixed-integer programming; and
0–1 integer programming.
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7. Integer Programming (continued)
The Pure Integer Programming problems are cases in which all variables are required to have integer values.
The Mixed-Integer Programming problems are cases in which some, but not all, of the decision variables are required to have integer values.
The Zero–One Integer Programming problems are special cases in which all the decision variables must have integer solution values of 0 or 1.
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11-7

8. Integer Programming Example: Harrison Electric Company
The Company produces two products popular with home renovators: old-fashioned chandeliers and ceiling fans.
Both the chandeliers and fans require a two-step 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 only 12 hours of wiring time and 30 hours of assembly time are available.
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11-8

9. Integer Programming: Example (continued)
If each chandelier produced nets the firm $7 and each fan $6, Harrison’s production mix decision can be formulated using LP as follows:
maximize profit = $7X1 + $6X2
subject to: 2X1 + 3X2 ≤ 12 (wiring hours)
6X1 + 5X2 ≤ 30 (assembly hours)
X1, X2 ≥ 0 (nonnegative)
X1 = number of chandeliers produced
X2 = number of ceiling fans produced
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11-9

10. Integer Programming: Example (continued)
With only two variables and two constraints, the graphical LP approach to generate the optimal solution is given below:
6X1 + 5X2 ≤ 30
+ = Possible Integer Solution
Optimal LP Solution
(X1 = 33/4, X2 = 11/2,
Profit = $35.25
2X1 + 3X2 ≤ 12
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11. Integer Solution to Harrison Electric Co.
Optimal solution
Solution if rounding off
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12. Integer Solution to Harrison Electric Co. (continued)
Rounding off is one way to reach integer solution values, but it often does not yield the best solution.
An important concept to understand is that an integer programming solution can never be better than the solution to the same LP problem.
The integer problem is usually worse in terms of higher cost or lower profit.
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11-12

13. Branch and Bound Method
Branch and Bound break the feasible solution region into sub-problems until an optimal solution is found.
There are Six Steps in Solving Integer Programming Maximization Problems by Branch and Bound.
The steps are given over the next several slides.
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14. Branch and Bound Method: The Six Steps
Solve the original problem using LP.
If the answer satisfies the integer constraints, it is done.
If not, this value provides an initial upper bound.
Find any feasible solution that meets the integer constraints for use as a lower bound.
Usually, rounding down each variable will accomplish this.
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15. Branch and Bound Method Steps: (continued)
Branch on one variable from Step 1 that does not have an integer value.
Split the problem into two sub-problems based on integer values that are immediately above and below the non-integer value.
For example, if X2 = 3.75 was in the final LP solution, introduce the constraint X2 ≥ 4 in the first sub-problem and X2 ≤ 3 in the second sub-problem.
Create nodes at the top of these new branches by solving the new problems.
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16. Branch and Bound Method Steps: (continued)5.
If a branch yields a solution to the LP problem that is not feasible, terminate the branch.
If a branch yields a solution to the LP problem that is feasible, but not an integer solution, go to step 6.
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17. Branch and Bound Method Steps: (continued)
If the branch yields a feasible integer solution, examine the value of the objective function.
If this value equals the upper bound, an optimal solution has been reached.
If it is not equal to the upper bound, but exceeds the lower bound, set it as the new lower bound and go to step 6.
Finally, if it is less than the lower bound, terminate this branch.
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18. Branch and Bound Method Steps: (continued)
Examine both branches again and set the upper bound equal to the maximum value of the objective function at all final nodes.
If the upper bound equals the lower bound, stop.
If not, go back to step 3.
Minimization problems involve reversing the roles of the upper and lower bounds.
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11-18

19. Harrison Electric Co: Revisited
Figure 11.1 shows graphically that the optimal, non-integer solution is
X1 = 3.75 chandeliers
X2 = 1.5 ceiling fans
profit = $35.25
Since X1 and X2 are not integers, this solution is not valid.
The profit value of $35.25 will serve as an initial upper bound.
Note that rounding down gives X1 = 3, X2 = 1, profit = $27, which is feasible and can be used as a lower bound.
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11-19

20. Integer Solution: Creating Sub-problems
The problem is now divided into two sub-problems: A and B.
Consider branching on either variable that does not have an integer solution; pick X1 this time.
Subproblem A
maximize profit = $7X1 + $6X2
Subject to: X1 + 3X2 ≤ 12
6X1 + 5X2 ≤ 30
X ≥ 4
Subproblem B
maximize profit = $7X1 + $6X2
Subject to: X1 + 3X2 ≤ 12
6X1 + 5X2 ≤ 30
X ≤ 3
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21. Optimal Solution for Sub-problems
Optimal solutions are:
Sub-problem A: X1 = 4; X2 = 1.2, profit=$35.20
Sub-problem B: X1=3, X2=2,
profit=$33.00
(see figure on next slide)
Stop searching on the Subproblem B branch because it has an all-integer feasible solution.
The $33 profit becomes the lower bound.
Subproblem A’s branch is searched further since it has a non-integer solution.
The second upper bound becomes $35.20, replacing $35.25 from the first node.
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22. Optimal Solution for Sub-problem
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23. Sub-problems C and D
Subproblem A’s branching yields Subproblems C and D.
Subproblem C
maximize profit = $7X1 + $6X2
Subject to: X1 + 3X2 ≤ 12
6X1 + 5X2 ≤ 30
X ≥ 4
X2 ≥ 2
Subproblem D
maximize profit = $7X1 + $6X2
Subject to: X1 + 3X2 ≤ 12
6X1 + 5X2 ≤ 30
X ≥ 4
X2 ≤ 1
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11-23

24. Sub-problems C and D (continued)
Subproblem C has no feasible solution at all because the first two constraints are violated if the X1 ≥ 4 and X2 ≥ 2 constraints are observed.
Terminate this branch and do not consider its solution.
Subproblem D’s optimal solution is
X1 = 4 , X2 = 1, profit = $35.16.
This non-integer solution yields a new upper bound of $35.16, replacing the original $35.20.
Subproblems C and D, as well as the final branches for the problem, are shown in the figure on the next slide.
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11-24

25. Harrison Electric’s Full Branch and Bound Solution
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26. Subproblems E and F Finally, create subproblems E and F and
solve for X1 and X2 with the added
constraints X1 ≤ 4 and X1 ≥ 5. The
subproblems and their solutions are:
Subproblem E
maximize profit = $7X1 + $6X2
Subject to: X1 + 3X2 ≤ 12
6X1 + 5X2 ≤ 30
X ≥ 4
X ≤ 4
X2 ≤ 1
Optimal solution for E:
X1 = 4, X2 = 1, profit = $34
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11-26

27. Subproblems E and F (continued)
Subproblem F
maximize profit = $7X1 + $6X2
Subject to: X1 + 3X2 ≤ 12
6X1 + 5X2 ≤ 30
X ≥ 4
X ≥ 5
X2 ≤ 1
Optimal solution for F:
X1 = 5, X2 = 0, profit = $35
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11-27

28. Using Software to Solve Harrison Electric Co. Problem
POM-QM for Windows Analysis of Harrison Electric’s Problem Using Integer programming: Input Screen.
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29. Using Software to Solve Harrison Electric Co. Problem (continued)
Output Screen Using POM-QM for Windows on Harrison Electric’s Integer Programming Problem
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30. Goal Programming maximizing total profit, maximizing market share,
Firms usually have more than one goal. For example,
maximizing total profit,
maximizing market share,
maintaining full employment,
providing quality ecological management,
minimizing noise level in the neighborhood, and
meeting numerous other non-economic goals.
It is not possible for LP to have multiple goals unless they are all measured in the same units (such as dollars),
a highly unusual situation.
An important technique that has been developed to supplement LP is called goal programming.
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11-30

31. Goal Programming (continued)
Goal programming “satisfices,”
as opposed to LP, which tries to “optimize.”
Satisfice means coming as close as possible to reaching goals.
The objective function is the main difference between goal programming and LP.
In goal programming, the purpose is to minimize deviational variables,
which are the only terms in the objective function.
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11-31

32. Example of Goal Programming Harrison Electric Revisited
Goals Harrison’s management wants to
achieve, each equal in priority:
Goal 1: to produce as much profit above $30 as possible during the production period.
Goal 2: to fully utilize the available wiring department hours.
Goal 3: to avoid overtime in the assembly department.
Goal 4: to meet a contract requirement to produce at least seven ceiling fans.
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11-32

33. Example of Goal Programming Harrison Electric Revisited
Need a clear definition of deviational variables, such as :
d1– = underachievement of the profit target
d1+ = overachievement of the profit target
d2– = idle time in the wiring dept. (underused)
d2+ = overtime in the wiring dept. (overused)
d3– = idle time in the assembly dept. (underused)
d3+ = overtime in the wiring dept. (overused)
d4– = underachievement of the ceiling fan goal
d4+ = overachievement of the ceiling fan goal
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34. Ranking Goals with Priority Levels
A key idea in goal programming is that one goal is more important than another. Priorities are assigned to each deviational variable.
Priority 1 is infinitely more important than Priority 2, which is infinitely more important than the next goal, and so on.
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35. Analysis of First Goal 11-35
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36. Analysis of First and Second Goals
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37. Analysis of All Four Priority Goals
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38. Goal Programming Versus Linear Programming
Multiple goals (instead of one goal)
Deviational variables minimized (instead of maximizing profit or minimizing cost of LP)
“Satisficing” (instead of optimizing)
Deviational variables are real (and replace slack variables)
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39. Initial Goal Programming TableauCj
Quantity
Solution
Mix P1 P2 P4 P3 x1 x2
d1-
d2-
d3-
d4-
d1+
d2+
d3+
d4+ Zj
Cj - Zj { 7 2 6 3 5 1 -1 30 12 -2 -3 -7 -6
Pivot Column
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40. Second Goal Programming TableauCj
Quantity
Solution
Mix P1 P2 P4 P3 x1 x2
d1-
d2-
d3-
d4-
d1+
d2+
d3+
d4+ Zj
Cj - Zj { 1
6/7
9/7
-1/7
1/7
-2/7
-6/7
+2/7 -1
30/7
24/7 7 +1
-9/7
2/7
Pivot Column
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41. Final Solution to Harrison Electric’s Goal ProgrammingCj
Quantity
Solution
Mix P1 P2 P4 P3 x1 x2
d1-
d2-
d3-
d4-
d1+
d2+
d3+
d4+ Zj
Cj - Zj {
8/5
6/5
1/5
-6/5 1 -1
3/5
-1/5
-3/5 6
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42. Harrison Electric’s Goal Programming Using POM-QM for Windows
Final Tableau for Harrison Electric Using POM-QM for Windows.
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43. Harrison Electric’s Goal Programming Using POM-QM for Windows
Summary Solution Screen for Harrison Electric’s Goal Programming Problem Using POM-QM for Windows.
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44. Nonlinear Programming
Nonlinear objective function, linear constraints
Nonlinear objective function and nonlinear constraints
Linear objective function and nonlinear constraints
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45. Nonlinear Programming
Nonlinear objective function, linear constraints
Max: 28X X X22
Subject to:
X X2 ≤ 1000
0.5X X2 ≤ 500
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46. Nonlinear Programming
An Excel Formulation of Great Western Appliance’s Nonlinear Programming Problem.
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47. Nonlinear Programming
Nonlinear objective function and nonlinear constraints.
Max: 13X1 + 6X1X2 + 5X2 + X2–1
Subject to:
2X12+ 4X22 ≤ 90
X X23 ≤ 75
8X1 – 2X2 ≤ 61
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48. Nonlinear Programming
The Problem has both Nonlinear Objective Function and Nonlinear Constraints.
The solution to Great Western Appliance’s NLP Problem using Excel Solver:
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11-48

49. Nonlinear Programming
The problem has both Nonlinear Objective Function and Nonlinear Constraints.
An Excel Formulation of Hospicare Corp.’s NLP Problem:
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11-49

50. Nonlinear Programming
The problem has both Nonlinear Objective Function and Nonlinear Constraints.
Excel Solution to the Hospicare Corp.’s NLP Problem using Solver:
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11-50

51. Nonlinear Programming
Linear objective function and nonlinear constraints
Max: X1 + 7X2
Subject to:
3X X12 + 4X X22 ≥ 125
13X1 + X13 ≥ 80
0.7X1 + X2 ≥ 17
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52. Nonlinear Programming
The problem has both Linear Objective Function with Nonlinear Constraints.
Excel Formulation of Thermlock’s NLP Problem:
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53. Nonlinear Programming
The problem has both Linear Objective Function with Nonlinear Constraints.
The solution to Thermlock’s NLP Problem Using the Excel Solver:
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54. Computational Procedures -Nonlinear Programming
Gradient method (steepest descent)
Separable programming - linear representation of nonlinear problem
Separable programming deals with a class of problems in which the objective and constraints are approximated by linear functions. In this way, the powerful simplex algorithm may again be applied.
In general, work in the area of NLP is the least charted and most difficult of all the quantitative analysis models.
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11-54