1. Introduction to Linear Programming
Chapter 3
Introduction to Linear Programming

2. Introduction Linear programming An application of linear programming
Programming means planning
Model contains linear mathematical functions
An application of linear programming
Allocating limited resources among competing activities in the best possible way
Applies to wide variety of situations

3. 3.1 Prototype Example Wyndor Glass Co.
Produces windows and glass doors
Plant 1 makes aluminum frames and hardware
Plant 2 makes wood frames
Plant 3 produces glass and assembles products

4. Prototype Example Company introducing two new products
Product 1: 8 ft. glass door with aluminum frame
Product 2: 4 x 6 ft. double-hung, wood-framed window
Define the problem: What mix of products would be most profitable (determine the production rate of each product to maximize the profit)?
Assuming company could sell as much of either product as could be produced

5. Prototype Example Products produced in batches of 20
Data needed (data gathering process)
Number of hours of production time available per week in each plant for new products
Production time used in each plant for each batch of each new product
Profit per batch of each new product

6. Prototype Example

7. Prototype Example From bottom row of Table 3.1
Formulating the model (decision variables, objective function and constraints)
x1 = number of batches of product 1 produced per week
x2 = number of batches of product 2 produced per week
Z = total profit per week (thousands of dollars) from producing these two products
From bottom row of Table 3.1
𝑍 = 3 𝑥1+5𝑥2

8. Prototype Example Constraints (see Table 3.1)
𝑥1≤4
2𝑥2≤12
3𝑥1+2𝑥2≤18
𝑥1≥0
𝑥2≥0
Classic example of resource-allocation problem
Most common type of linear programming problem

9. Prototype Example Problem can be solved graphically
Two dimensional graph with x1 and x2 as the axes
First step: sketch the feasible region (values of x1 and x2 in this region satisfy the constraint restrictions)
See Figures 3.1 and Figure 3.2
Next step: find out a point in the feasible region that maximizes value of Z = 3x1 + 5x2
See Figure 3.3

10. Prototype Example

11. Prototype Example

12. Prototype Example
Rewrite z = 3x1 + 5x2 as x2 = -3x1/5 + z/5, the graph of this is a line with slope -3/5 and y-interception z/5. To maximize z is equivalent to maximizing the y-intercept z/5 (among all parallel line segments in the feasible region having slope -3/5). Figure 3.3 illustrates three parallel lines of different y-intercepts. Note that since |-3/5| < |-3/2|
(-3/2 is the slope of the third constraint boundary), the lines of the objective function is less steep. The objective function line on the feasible region that maximizes the y-intercept is the line having the corner point (2, 6) on the line (the only point from the feasible region is on the line).

13. 3.2 The Linear Programming Model
General problem terminology and examples
Resources: money, particular types of machines, vehicles, or personnel
Activities: investing in particular projects, advertising in particular media, or shipping from a particular source
Problem involves choosing levels of activities to maximize overall measure of performance

14. The Linear Programming Model

15. The Linear Programming Model

16. The Linear Programming Model
Standard form
The first m
constraints
are called
the
functional
constraints.
The last n
constraints
are
nonnegativity
constraints.

17. The Linear Programming Model
Other legitimate forms
Minimizing (rather than maximizing) the objective function
Functional constraints with greater-than-or- equal-to inequality
Some functional constraints in equation form
Some decision variables may be negative

18. The Linear Programming Model

19. The Linear Programming Model
Feasible solution
Solution for which all constraints are satisfied
Might not exist for a given problem
Infeasible solution
Solution for which at least one constraint is violated
Optimal solution
Has most favorable value of objective function (the largest value if the objective function is to be maximized, or the smallest value if the objective function is to be minimized)
Might not exist for a given problem (no feasible solutions or unbounded Z)
Might have multiple optimal solutions

20. The Linear Programming Model
Corner-point feasible (CPF) solution
Solution that lies at the corner of the feasible region (note that there are corner point solutions that do not lie in the feasible region, so they are corner-point infeasible solutions).

21. The Linear Programming Model
Linear programming problem with feasible solution and bounded feasible region
Must have CPF solutions and optimal solution(s)
Best CPF solution must be an optimal solution

22. The Linear Programming Model
Example: a problem having no feasible solutions

23. The Linear Programming Model
Example: a problem having multiple optimal solutions

24. The Linear Programming Model
Example: a problem having feasible solutions but no optimal solution (unbounded Z)

25. The Linear Programming Model
More examples (sketch the constraint boundaries and see):
No feasible solutions: x1 ≤ 1, x2 ≤ 2, x1 + x2 ≥ 5, x1 ≥ 0, x2 ≥ 0.
2. No optimal solutions (unbounded Z):
maximize Z = 3x1 + 5x2 s.t. x1 + x2 ≥ 4, x1 ≥ 0, x2 ≥ 0
3. Multiple optimal solutions:
maximize Z = 2x1 + 2x2 s.t. x1 + x2 ≤ 2, x1 ≥ 0, x2 ≥ 0
All points on the line segment between (2,0) and (0, 2) on the feasible region are optimal solutions and Z = 4.

26. 3.3 Assumptions of Linear Programming
A Linear Programming model must simultaneously satisfy all four assumptions below
1. Proportionality assumption
The contribution of each activity to the value of the objective function (or left-hand side of a functional constraint) is proportional to the level of the activity (cjxj terms or aijxj terms)
If assumption does not hold, one must use nonlinear programming (Chapter 13)

27. Assumptions of Linear Programming
2. Additivity
Every function in a linear programming model is the sum of the individual contributions of the activities (∑cjxj or ∑aijxj)
3. Divisibility
Decision variables in a linear programming model may have any real number values
Including noninteger values
Assumes activities can be run at fractional values

28. Assumptions of Linear Programming
4. Certainty
Value assigned to each parameter (cj and aij and bi) of a linear programming model is assumed to be a known constant
Seldom satisfied precisely in real applications
Sensitivity analysis used

29. 3.4 Additional Examples
The optimal solutions of the examples in this section can be obtained by using Excel Solver (introduced in Section 3.5) and are available on the course website.
Example 1: Design of radiation therapy for Mary’s cancer treatment
Goal: select best combination of beams and their intensities to generate best possible dose distribution
Dose is measured in kilorads
This is an example of a cost-benefit-trade-off problem (it seeks the best trade-off between some cost and benefits).

30. Example 1: Radiation Therapy Design

31. Example 1: Radiation Therapy Design
Linear programming model
Using data from Table 3.7

32. Example 1: Radiation Therapy Design
A type of cost- benefit tradeoff problem

33. Example 2: Regional Planning
It’s like the prototype example, it’s another resource-allocation problem).

34. Example 2: Regional Planning

35. Example 2: Regional Planning

36. Example 2: Regional Planning

37. Example 2: Regional Planning

38. Example 3: Personnel Scheduling
It’s a cost-benefit-trade-off problem.

39. Example 3: Personnel Scheduling
Without the integer constraints, due to the special structure of the model, the optimal solution
turns out to have integer values anyway.

40. Example 4: Distributing Goods through a Distribution Network
It’s a minimum cost flow problem.

41. Example 4: Distributing Goods through a Distribution Network

42. Example 4: Distributing Goods through a Distribution Network

43. Example 5: Reclaiming Solid Wastes
SAVE-IT company collects and treats four types of solid waste materials
Materials amalgamated into salable products
Three different grades of product possible
Fixed treatment cost covered by grants
Objective: maximize the net weekly profit
Determine amount of each product grade
Determine mix of materials to be used for each grade

44. Example 5: Reclaiming Solid Wastes
It’s an example of a blending problem (find the best blend of ingredients into final products to meet certain specifications).

45. Example 5: Reclaiming Solid Wastes

46. Example 5: Reclaiming Solid Wastes
Decision variables
𝑥𝑖𝑗 (for i = A, B, C; j = 1,2,3,4) =
number of pounds of material j allocated to product grade i per week

47. Example 5: Reclaiming Solid Wastes

48. Example 5: Reclaiming Solid Wastes

49. Example 5: Reclaiming Solid Wastes

50. 3.5 Formulating and Solving Linear Programming Models on a Spreadsheet
Excel and its Solver add-in
Popular tools for solving small linear programming problems

51. Formulating and Solving Linear Programming Models on a Spreadsheet
The Wyndor example
Data entered into a spreadsheet

52. Formulating and Solving Linear Programming Models on a Spreadsheet
Changing cells
Cells containing the decisions to be made
C12 and D12 in the Wyndor example below

53. Formulating and Solving Linear Programming Models on a Spreadsheet

54. Formulating and Solving Linear Programming Models on a Spreadsheet

55. 3.6 Formulating Very Large Linear Programming Models
Actual linear programming models
Can have hundreds or thousands of functional constraints
Number of decision variables may also be very large
Modeling language
Used to formulate very large models in practice
Expedites model management tasks

56. Formulating Very Large Linear Programming Models
Modeling language examples
AMPL, MPL, OPL, GAMS, and LINGO
Example problem with a huge model

57. An Example Problem of Huge Model

58. An Example Problem of Huge Model
Functional constraints:
1,000 production capacity constraints
1,000 production balance constraints
100 maximum inventory constraints (total inventory ≤ inventory capacity)
1,000 maximum sales constraints (sales ≤ demand)

59. An Application Vignette

60. 3.7 Conclusions Linear programming technique applications
Resource-allocation problems
Cost-benefit tradeoffs
Not all problems can be formulated to fit a linear programming model
Alternatives: integer programming or nonlinear programming models