1. Linear Programming Models: Graphical and Computer Methods2
Linear Programming Models: Graphical and Computer Methods

2. LEARNING OBJECTIVES
Understand the basic assumptions and properties of linear programming (LP).
Use graphical procedures to solve LP problems with only two variables to understand how LP problems are solved.
Understand special situations such as redundancy, infeasibility, unboundedness, and alternate optimal solutions in LP problems.
Understand how to set up LP problems on a spreadsheet and solve them using Excel’s Solver.

3. Introduction
Management decisions involve the most effective use of resources
Most widely used modeling technique is linear programming (LP)
Deterministic models

4. Developing a LP Model
All LP models can be viewed in terms of the three distinct steps
Formulation of simple mathematical expressions
Solution to identify an optimal (or best) solution to the model
Interpretation of the results and answer “what if?” questions

5. Properties of a LP Model
Seek to maximize or minimize some quantity (profit or cost)
Restrictions or constraints
Alternative courses of action
Linear equations or inequalities (=, ≤, ≥)

6. LP Characteristics
Feasible Region – The set of points that satisfies all constraints
Corner Point Property – An optimal solution must lie at one or more corner points
Optimal Solution – The corner point with the best objective function value is optimal

7. Formulating a LP Model A product mix problem Flair Furniture
Decide how much to make of two or more products
Objective is to maximize profit
Limited resources
Flair Furniture
Best combination of tables and chairs

8. Flair Furniture Produces tables and chairs
Each table takes 3 hrs of carpentry and 2 hrs of painting work
Each chair 4 hrs and 1 hr, respectively
2,400 hrs of carpentry time, 1,000 hrs of painting time
No more than 450 chairs
At least 100 tables
$7 and $5 profit for table and chair

9. Decision Variables What we are solving for
Two variables in the Flair problem
Number of tables (T, Tables or X1)
Number of chairs (C, Chairs or X2)
Decision variables can be in different units of measurement

10. The Objective Function
States the goal of a problem
A single objective function
Objective is often to maximize profit or minimize cost

11. The Objective Function
For Flair Furniture
Profit = ($7 profit per table) x (number of tables produced)
+ ($5 profit per chairs) x (numbers of chairs produced)
Using decision variables T and C
Maximize $7T + $5C

12. Constraints Restrictions or limits on our decisions
As many as necessary
Can be independent
Flair has four constraints
Carpentry time
Painting time
Number of chairs to make
Number of tables to make

13. Constraints For carpentry time There are 2,400 hours of time available
(3 hours per table) x (number of tables produced) +
(4 hours per chair) x (number of chairs produced)
There are 2,400 hours of time available
3T + 4C ≤ 2,400

14. Constraints All four constraints Carpentry time: 3T + 4C ≤ 2,400
Painting time: 2T + 1C ≤ 1,000
Chairs made: C ≤ 450
Tables made: T ≥ 100

15. Nonnegativity and Integers
Decision variables must be ≥ 0, so
Decision variables may have to be integers
T ≥ 0, and
C ≥ 0

16. Flair Model Matrix TABLES (T) CHAIRS (C) LIMIT
Profit Contribution $7 $5
Carpentry 3 hrs 4 hrs 2,400
Painting 2 hrs 1 hr 1,000
Chairs 0 unit 1 unit 450
Tables 1 unit 0 unit 100

17. Graphical Solution Complete model Maximize profit = $7T + $5C
Subject to
3T + 4C ≤ 2,400 (carpentry time)
2T + 1C ≤ 1,000 (painting time)
C ≤ 450 (maximum chairs allowed)
T ≥ 100 (minimum tables required)
T, C ≥ 0 (nonnegativity)

18. Graphical Representation
Number of Chairs (C)
Number of Tables(T)
1,000 – –
800 –
600 –
400 –
200 –
0 –
| | | | | | | | | | | |
,000
(T = 0, C = 600)
Carpentry Constraint Line
(T = 400, C = 300)
(T = 800, C = 0)
Figure 2.1

19. Graphical Representation
Number of Chairs (C)
Number of Tables(T)
1,000 – –
800 –
600 –
400 –
200 –
0 –
| | | | | | | | | | | |
,000
Region Satisfying
3T + 4C ≤ 2,400
(T = 300, C = 200)
(T = 600, C = 400)
Figure 2.2

20. Graphical Representation
Number of Chairs (C)
Number of Tables(T)
1,000 – –
800 –
600 –
400 –
200 –
0 –
| | | | | | | | | | | |
,000
(T = 0, C = 1,000)
(T = 100, C = 700)
(T = 0, C = 600)
Painting Constraint
(T = 300, C = 200)
Carpentry Constraint
(T = 500, C = 200)
(T = 500, C = 0)
(T = 800, C = 0)
Figure 2.3

21. Graphical Representation
Painting Constraint
Number of Chairs (C)
Number of Tables(T)
1,000 – –
800 –
600 –
400 –
200 –
0 –
| | | | | | | | | | | |
,000
Infeasible Solution (T = 50, C = 500)
Minimum Tables Required Constraint
Maximum Chairs Allowed Constraint
Feasible Region
(T = 300, C = 200)
Carpentry Constraint
Infeasible Solution
(T = 500, C = 200)
Figure 2.4

22. Using Level Lines 800 – – (T = 0, C = 560) 600 – 400 –
| | | | | | | | | |
,000
800 – –
600 –
400 –
200 –
0 –
Number of Chairs (C)
Number of Tables(T)
(T = 0, C = 560)
(T = 0, C = 420)
$7T + $5C = $2,800
Feasible Region
$7T + $5C = $2,100
(T = 300, C = 0)
(T = 400, C = 0)
Figure 2.5

23. Using Level Lines 0 200 400 600 800 1,000 800 – – 600 – 400 – 200 –
| | | | | | | | | |
,000
800 – –
600 –
400 –
200 –
0 –
Number of Chairs (C)
Number of Tables(T)
Optimal Level Profit Line
Carpentry Constraint
$7T + $5C = $2,800 1 2 3 4 5
Optimal Corner Point Solution
$7T + $5C = $2,100
Level Profit Line with No Feasible Points ($7T + $5C = $4,200)
Painting Constraint
Figure 2.6

24. Calculating a Solution
Optimal point 4 is the intersection of two constraints, carpentry and painting
Solving simultaneously
6T + 8C = 4,800
– (6T + 3C = 3,000)
5C = 1,800
implies C = 360
and T = 320

25. Using All Corner Points
Point 1 (T = 100, C = 0) Profit = $7 x $5 x 0 = $700 Point 2 (T = 100, C = 450) Profit = $7 x $5 x 450 = $2,950 Point 3 (T = 200, C = 450) Profit = $7 x $5 x 450 = $3,650 Point 4 (T = 320, C = 360) Profit = $7 x $5 x 360 = $4,040 Point 5 (T = 500, C = 0) Profit = $7 x $5 x 0 = $3,500

26. Extension to the Model 800 – –
| | | | | | | | | |
,000
800 – –
600 –
400 –
200 –
0 –
Number of Chairs (C)
Number of Tables(T)
Optimal Level Profit Line for Revised Problem
(T = 300, C = 375) is the New Optimal Corner Point Solution
(T = 320, C = 360) is No Longer Feasible 1 5 2 3 4 6 7
Additional Constraint C – T ≥ 75 Has a Positive Slope
($7T + $5C = $2,800)
This Portion of the Original Feasible Region Is No Longer Feasible
Figure 2.7

27. Minimization Problem Minimize cost Holiday Meal Turkey Ranch
Two types of feed
Minimize cost = $0.10A + $0.15B
subject to
5A + 10B ≥ 45 (protein required)
4A + 3B ≥ 24 (vitamin required)
0.5A ≥ 1.5 (iron required)
A,B ≥ 0 (nonnegativity)

28. Minimization Problem Data for Holiday Meal Turkey Ranch
NUTRIENTS PER POUND OF FEED MINIMUM REQUIRED
PER TURKEY PER
NUTRIENT BRAND A FEED BRAND B FEED MONTH
Protein (units)
Vitamin (units)
Iron (units)
Cost Per Pound $ $0.15
Table 2.1

29. Minimization Problem 10 – 9 – Iron Constraint 8 – 7 – 6 – 5 – 4 –
Pounds of Brand B (B)
Pounds of Brand A (A)
10 –
9 –
8 –
7 –
6 –
5 –
4 –
3 –
2 –
1 –
0 –
| | | | | | | | | | | 1 2 3
Feasible Region is Unbounded
Iron Constraint
Vitamin Constraint
Protein Constraint
Figure 2.8

30. Graphical Solution 10 – 9 – Level Cost Line for Minimum Cost 8 – 7 –
Pounds of Brand B (B)
Pounds of Brand A (A)
10 –
9 –
8 –
7 –
6 –
5 –
4 –
3 –
2 –
1 –
0 –
| | | | | | | | | | |
Unbounded Feasible Region 1 2 3
Level Cost Line for Minimum Cost
$0.10A + $0.15B = $1
Direction of Decreasing Cost
Level Cost Line
Optimal Corner Point Solution
(A = 4.2, B = 2.4)
Figure 2.9

31. Calculating a Solution
Optimal point 2 is the intersection of two constraints, vitamin and protein
Solving simultaneously
4(5A + 10B = 45) implies 20A + 40B = 180
– 5(4A + 3B = 24) implies – (20A + 15B = 120)
25B = 60
implies B = 2.4
and A = 4.2

32. Special Situations Redundant Constraints
Do not affect the feasible region
Changed constraint in Flair Furniture problem
T ≥ becomes T ≤ 100

33. Special Situations C ≤ 450 Number of Chairs (C)
Number of Tables(T)
1,000 – –
800 –
600 –
400 –
200 –
0 –
| | | | | | | | | | | |
,000
C ≤ 450
Carpentry Constraint Is Redundant
Feasible Region
Constraint Changed to T ≤ 100
Painting Constraint Is Redundant
Figure 2.10

34. Special Situations Infeasibility
No one solution satisfies all the constraints
Changed constraint in Flair Furniture problem
T ≥ becomes T ≥ 600

35. Special Situations Constraint Changed to T ≥ 600 C ≤ 450
Number of Chairs (C)
Number of Tables(T)
1,000 – –
800 –
600 –
400 –
200 –
0 –
| | | | | | | | | | | |
,000
Region Satisfying Fourth Constraint
C ≤ 450
Two Regions Do Not Overlap
3T + 4C ≤ 2,400
Region Satisfying Three Constraints
2T + C ≤ 1,000
Figure 2.11

36. Special Situations Alternate Optimal Solutions
More than one solution satisfies all the constraints
Changed objective in Flair Furniture problem
$7T + $5C becomes $6T + $3C

37. Special Situations 0 200 400 600 800 1,000 800 – – 600 – 400 – 200 –
| | | | | | | | | |
,000
800 – –
600 –
400 –
200 –
0 –
Number of Chairs (C)
Number of Tables(T)
Level Profit Line for Maximum Profit Overlaps Painting Constraint
Level Profit Line Is Parallel to Painting Constraint 1 2 3 4 5
Feasible Region
$6T + $3C = $2,100
Optimal Solution Consists of All Points Between Corner Points 4 and 5
Figure 2.12

38. Special Situations Unbounded Solution
May or may not have a finite solution
Usually improper formulation
Changed objective in Holiday Meal problem
Minimize = $0.10A + $0.15B
becomes
Maximize = 8A + 12B

39. Special Situations Pounds of Brand B (B) Pounds of Brand A (A) 10 –
9 –
8 –
7 –
6 –
5 –
4 –
3 –
2 –
1 –
0 –
| | | | | | | | | | |
Unbounded
Feasible Region
Iron Constraint
Direction of Increasing Value
Value Can Be Increased to Infinity
8A + 12B = 100
8A + 12B = 80
Vitamin Constraint
Protein Constraint
Figure 2.13

40. Using Excel’s Solver Excel’s built-in LP solution tool for LP
Commonly available and easy access
Familiar software

41. Using Solver
Screenshot 2-1A

42. Using Solver
Screenshot 2-1B

43. Using Solver
Screenshot 2-1B

44. Using Solver
Screenshot 2-1C

45. Using Solver
Screenshot 2-1D

46. Using Solver
Screenshot 2-1E

47. Using Solver
Screenshot 2-1F

48. Using Solver
Screenshot 2-2

49. Using Solver
Screenshot 2-3A

50. Using Solver
Screenshot 2-3B