1. Computational Methods for Management and Economics Carla Gomes
Module 2
Introduction to LP
(Textbook – Hillier and Lieberman)

2. Mathematical Models
Need to formulate problems in a way that is convenient for analysis – typically using a mathematical model.
Mathematical models – idealized representations expressed in terms of mathematical symbols and expressions. Famous mathematical models
F = ma (Newton’s second law of motion) and
E=mc² (Einstein’s famous equation of conservation of energy into mass).

3. Optimization Problems
In an optimization problem we want to
maximize or minimize a specific quantity
(called the objective), which depends on a
finite number of decision variables and
parameters. The variables may be independent
of one another or may be related through one
or more constraints.

4. Mathematical model of a business problem: 
Decision variables - they represent quantifiable decisions to be made, under our control, say x1, x2, …, xn. The respective values are to be determined.
Objective function – expresses the appropriate measure of performance as a mathematical function of the decision variables. E.g., P = 2 x1 + 5 x2 + …+ xn
Constraints – any restriction on the values of the variables are also expressed mathematically, typically by means of equations (e.g. 2 x x2 <= 10)
Parameters of the model – constants (namely coefficients and right-end-sides) in the constraints and objective function.
Typical OR model
Choose the decision variables so as to maximize (or minimize) the objective function, subject to the specified constraints.

5. Example of Optimization Problem
Minimize: Z = x12 + x22
Subject to: 2x1 – 2x2 = 6 x
Z – objective;
x1,x2
- decision variables; two constraints;

6. Mathematical Program
Optimization problem in which the objective and constraints are given as mathematical functions and functional relationships.
Optimize: Z = f(x1, x2, …, xn)
Subject to:
g1(x1, x2, …, xn) = , , b1
g2(x1, x2, …, xn) = , , b2 …
gm(x1, x2, …, xn) = , , bm

7. Linear Programming (LP)
One of the most important scientific advances of the 20th century
A variety of applications:
   Financial planning, Marketing, E-business, Telecommunications, Manufacturing, Transportation Planning, System Design, Health Care
Remarkably efficient solution procedures to solve LP models – simplex method and interior point methods ---
Very fast LP solvers (CPLEX – from ,000,000X faster!)

8. Linear Programming (LP)
Linear – all the functions are linear (f and g functions are linear. Ex: f (x1, x2, …, xn)= c1x1 + c2 x2 + … cn xn
Programming – does not refer to computer programming but rather “planning” - planning of activities to obtain an optimal result i.e., it reaches the specified goal best (according to the mathematical model) among all feasible alternatives.

9. Wyndor Glass (Hillier and Liebermnan)
Prototype Example –
Wyndor Glass (Hillier and Liebermnan)

12. Wyndor Glass Co. Product Mix Problem
Wyndor has developed the following new products:
An 8-foot glass door with aluminum framing.
A 4-foot by 6-foot double-hung, wood-framed window.
The company has three plants
Plant 1 produces aluminum frames and hardware.
Plant 2 produces wood frames.
Plant 3 produces glass and assembles the windows and doors.
Questions:
Should they go ahead with launching these two new products?
If so, what should be the product mix?

13. Steps in setting up a LP Determine and label the decision variables.
Determine the objective and use the decision variables to write an expression for the objective function.
Determine the constraints - feasible region.
Determine the explicit constraints and write a functional expression for each of them.
Determine the implicit constraints (e.g., nonnegativity constraints).

14. The LP Mathematical model
Problem of determining the optimal product mix (maximizing profit) for Wyndor as an LP.
Step 1: Determine the decision variables (write a statement per variable). ?
Specify the unknowns that determine the profit; each variable has to be well defined in English; careful with units!!!
Put yourself in the shoes of the decision maker and then ask the question “What are the unknowns in this particular problem that impact the overall performance measure?”
(Often, the most difficult part of any modeling problem.
Once the decision variables are correctly identified then the remainder
of the modeling process usually goes smoothly.)

15. Objective Function Feasible Region
Step 2 Specify the problem objective (write an expression for the objective function)
Feasible Region
Step 3 Express the feasible region as the solution set of a finite collection of linear inequality and equality constraints.
3.1. Determine the functional constraints
3.1. Determine the non-negativity constraints

16. Algebraic Model for Wyndor Glass Co.
Let D = the number of doors to produce
W = the number of windows to produce
Maximize P = 3 D + 5 W
subject to
D ≤ 4 2W ≤ 12 3D + 2W ≤ 18
and
D ≥ 0, W ≥ 0.

17. A graphical solution
Since this problem is two dimensional it is possible to provide a graphical solution. To graphically find the feasible region, we will follow these steps:
Graph the line associated with each of the linear inequality constraints.
Determine on which side of each of these lines the feasible region must lie.

18. Graphing the Product Mix
Figure 2.5 Graph showing the points (D, W) = (2, 3) and (D, W) = (4, 6) for the Wyndor Glass Co. product-mix problem.

19. Graph Showing Non-Negativity Constraints: D ≥ 0 and W ≥ 0
Figure 2.6 Graph showing that the constraints D ≥ 0 and W ≥ 0 rule out solutions for the Wyndor Glass Co. product-mix problem that are to the left of the vertical axis or under the horizontal axis.

20. Nonnegative Solutions Permitted by D ≤ 4
Figure 2.7 Graph showing that the nonnegative solutions permitted by the constraint D ≤ 4 lie between the vertical axis and the line where D = 4.

21. Nonnegative Solutions Permitted by 2W ≤ 12
Figure 2.8 Graph showing that the nonnegative solutions permitted by the constraint 2W ≤ 12 must lie between the horizontal axis and the constraint boundary line whose equation is 2W = 12.

22. Boundary Line for Constraint 3D + 2W ≤ 18
Figure 2.9 Graph showing that the boundary line for the constraint 3D + 2W ≤ 18 intercepts the horizontal axis at D = 6 and intercepts the vertical axis at W = 9.

23. Changing Right-Hand Side Creates Parallel Constraint Boundary Lines
Figure Graph showing that changing only the right-hand side of a constraint (such as 3D + 2W ≤ 18) creates parallel constraint boundary lines.

24. Nonnegative Solutions Permitted by 3D + 2W ≤ 18
Figure Graph showing that nonnegative solutions permitted by the constraint 3D + 2W ≤ 18 lie within the triangle formed by the two axes and this constraint’s boundary line, 3D + 2W = 18.

25. Graph of Feasible Region
Figure Graph showing how the feasible region is formed by the constraint boundary lines, where the arrows indicate which side of each line is permitted by the corresponding constraint.

26. Graph of Feasible Region
Figure Graph showing how the feasible region is formed by the constraint boundary lines, where the arrows indicate which side of each line is permitted by the corresponding constraint.

27. Objective Function (P = 1,500)
Figure Graph showing the line containing all the points (D, W) that give a value P = 1,500 for the objective function.
isoprofit line

28. Finding the Optimal Solution
Our objective function is:
maximize 3D+5W
The vector representing the
gradient of the objective
function is:
The line through the origin that contains this vector is:
Figure Graph showing three objective function lines for the Wyndor Glass Co. product-mix problem, where the top one passes through the optimal solution.
isoprofit line

29. Summary of the Graphical Method
Draw the constraint boundary line for each constraint. Use the origin (or any point not on the line) to determine which side of the line is permitted by the constraint.
Find the feasible region by determining where all constraints are satisfied simultaneously.
Determine the slope of one objective function line (perpendicular to its gradient vector). All other objective function lines will have the same slope.
Move a straight edge with this slope through the feasible region in the direction of improving values of the objective function (direction of the gradient). Stop at the last instant that the straight edge still passes through a point in the feasible region. This line given by the straight edge is the optimal objective function line.
A feasible point on the optimal objective function line is an optimal solution.

30. Terminology and Notation
Resources – m (plants)
Activities – n (2 products)
Wyndor Glass problem optimal product mix --- allocation of resources to activities i.e., choose the levels of the activities that achieve best overall measure of performance

31. Terminology and notation (cont.)
Z – value of the overall measure of performance; value of the objective function,
xj – level of activity j (for j = 1, 2, …, n)  decision variables
cj – increase in Z for each unit increase in the level of activity j; coefficient of objective function associated with activity j
bi – amount of resource i that is available (for i=1,2,…, m). Right-hand-side of constraint associated with resource i.
aij – amount of resource i consumed by each unit of activity j.  Technological coefficient.
(The values of cj, bi, and aij are the input constants for the model  the parameters of the model. )

32. Standard form of the LP model
xi >= 0 , (i =1,2,…,n)
Other forms:
Minimize Z (instead of maximizing Z)
   Some functional constraints have signs >= (rather than <=)
   Some functional constraints are equalities
   Some variables have unrestricted sign, i.e., they are not subject to the non-negativity constraints

33. Terminology of solutions in LP model
Solution – not necessarily the final answer to the problem!!!  
Feasible solution – solution that satisfies all the constraints
Infeasible solution – solution for which at least one of the constraints is violated
Feasible region – set of all points that satisfies all the constraints (possible to have a problem without any feasible solutions)
Binding constraint – the left-hand side and the right-hand side of the constraint are equal, I.e., constraint is satisfied in equality. Otherwise the constraint is nonbinding.
Optimal solution – feasible solution that has the best value of the objective function.
Largest value  maximization problems
Smallest value  minimization problems
Multiple optimal solutions
No optimal solutions
Unbounded Z

34. Corner Point Solutions
Corner-point feasible solution – special solution that plays a key role when the simplex method searches for an optimal solution. Relationship between optimal solutions and CPF solutions:
Any LP with feasible solutions and bounded feasible region 
(1)  the problem must possess CPF solutions and at least one optimal solution
(2)  the best CPF solution must be an optimal solution
If the problem has exactly one optimal solution it must be a CFP solution
If the problem has multiple optimal solutions, at least two must be CPF solutions

35. No Feasible Solutions – Why?
Maximize P = 3 D + 5 W
subject to
D ≤ 4 2W ≤ 12 3D + 2W ≤ 18
3 D + 5 W 50
and
D ≥ 0, W ≥ 0.
Previous
Feasible
Region

36. Multiple Optimal Solutions. Why?
Maximize P = 3 D + 2 W
subject to
D ≤ 4 2W ≤ 12 3D + 2W ≤ 18
and
D ≥ 0, W ≥ 0.
Every point on this line is
An optimal solution
with P=18

37. Unbounded Objective Function. Why?
(4, ) P=
Maximize P = 3 D + 2 W
subject to
D ≤ 4
and
D ≥ 0, W ≥ 0.
(4,8) P=28
Figure 2.7 Graph showing that the nonnegative solutions permitted by the constraint D ≤ 4 lie between the vertical axis and the line where D = 4.
(4,4) P=20
(4,2) P=16

38. LP Assumptions Proportionality –
LP Assumptions
Proportionality –
The contribution of each activity to the value of the objective function Z is proportional to the level of the activity xj as represented by the cjxj term;
The contribution of each activity to the left-hand side of each functional constraint is proportional to the level of the activity xj as represented by
the term aij.
This assumption implies that all the x terms of the linear equations cannot
have exponents greater than 1.
Note: if there is a term that is a product of different variables,
even though the proportionality assumption is satisfied,
the additivity assumption is violated.

39. Examples of violation of proportionality assumption
Case 1 - violation occurs
as a result of e.g., startup costs,
associated with product 1.
E.g., costs of setting up
the production, or distribution
of product 1.

40. Examples of violation of proportionality assumption
Case 2 - slope of the objective function for product 1 keeps increasing as x1 is increased –
there is an increasing marginal return. Economies of scale that sometimes
can be achieved at higher levels of production (e.g., more efficient machinery,
discounts for large purchases,etc).
Case 3 – opposite of case 2; the slope of the objective function for product 1 keeps decreasing as x1 is increased – there is a decreasing marginal return.
E.g., in order to sell higher volumes of product 1 require a major marketing campaign.
Case 2
Case 3

41. What to do when proportionality is violated?
Non linear programming;
To deal with case 1 (fixed charge problem) –
mixed integer programming.
3 x1 – 1 for x1 > 0;
3 x for x1 = 0;
Figure Graph showing the line containing all the points (D, W) that give a value P = 1,500 for the objective function.

42. LP Assumptions Additivity
The contribution of all variables to the objective function and to the left-hand side of the functional constraints has to be additive, i.e., it has to be the sum of the individual contributions of the respective activities – therefore crossproducts of variables are ruled out.

43. realm of non-linear programming!!!
Case 3 – the two products are complementary in some way that increases the
profit! E.g., a major marketing campaign required for product 2
will help product 1.
Case 4 – the two products are competitive in some way that decreases the profit! E.g., two products use the same equipment and that requires setting up costs
for going from one product to the other.
 What to do when additivity assumption violated:
realm of non-linear programming!!!

44. LP Assumptions Divisibility
Decision variables in an LP model are allowed to have any values, including noninteger values, that satisfy the functional and nonnegativity constraints. i.e., activities can be run at fractional levels.
What to do when divisibility assumption violated:
realm of integer programming!!!

45. LP Assumptions Certainty
The parameters of the model, (coefficients of the objective function and of the functional constraints, and the righ-hand sides of the functional constraints) are assumed to be known constants.
T Rarely the case – sometimes we use approximations  important to perform sensitivity analysis to identify sensitive parameters (the parameters that cannot be changed without changing the value of the objective function).
What to do when certainty assumption violated:
treat parameters as random variables

46. LP Assumptions - Comments
Quite often the assumptions are not 100% applicable!
Except for the divisibility assumption, minor disparities are to be expected. This is especially true for the certainty assumption, so sensitivity analysis normally is a must to compensate for the violation of this assumption!

47. LP Assumptions - Comments
Mathematical model --- abstractions of the real world problem.
Approximations and simplifying assumptions generally are required in order for the model to be tractable.
Adding too much detail and precision can make the model too complex for useful analysis of the problem.
Main issues  a correlation between the prediction of the model and what would actually happen in the real problem.

48. Additional Examples
Giapetto’s Woodcarving (from Winston and Venkataramanan)
Dorian Auto (from Winston and Venkataramanan)
Radiation Therapy (from Hillier and Lieberman)

49. Giapetto’s Woodcarving, Inc.
(from Winston)
Giapetto’s Woodcarving, Inc. manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto’s variable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapetto’s variable labor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor. A train requires 1 hour of finishing and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw material but only 100 finishing hours and 80 carpentry hours. Demand for trains is unlimited, but at most 40 soldiers are bought each week. Giapetto wants to maximize weekly profit (revenues – costs). Formulate a mathematical model of Giapetto’s situation that can be used to maximize Giapetto’s weekly profit.

50. 3.2 – Graphical Solution to a 2-Variable LP
Binding and Nonbinding constraints
Once the optimal solution to an LP is found, it is useful to classify each constraint as being a binding or nonbinding constraint.
A constraint is binding if the left-hand and right-hand side of the constraint are equal when the optimal values of the decision variables are substituted into the constraint. In the Giapetto LP, the finishing and carpentry constraints are binding.

51. 3.2 – Graphical Solution to a 2-Variable LP
A constraint is nonbinding if the left-hand side and the right-hand side of the constraint are unequal when the optimal values of the decision variables are substituted into the constraint. In the Giapetto LP, the demand constraint for wooden soldiers is nonbinding since at the optimal solution (x1 = 20), x1 < 40.

52. 3.2 – Graphical Solution to a 2-Variable LP
Convex sets, Extreme Points, and LP
A set of points S is a convex set if the line segment jointing any two pairs of points in S is wholly contained in S.
For any convex set S, a point p in S is an extreme point if each line segment that lines completely in S and contains the point P has P as an endpoint of the line segment.
Consider the figures (a) – (d) below:

53. 3.2 – Graphical Solution to a 2-Variable LP
For example, in figures (a) and (b) below, each line segment joining points in S contains only points in S. Thus is convex for (a) and (b). In both figures (c) and (d), there are points in the line segment AB that are not in S. S in not convex for (c) and (d).

54. 3.2 – Graphical Solution to a 2-Variable LP
In figure (a), each point on the circumference of the circle is an extreme point of the circle. In figure (b), A, B, C, and D are extreme points of S. Point E is not an extreme point since E is not an end point of the line segment AB.

55. 3.2 – Graphical Solution to a 2-Variable LP
Extreme points are sometimes called corner points, because if the set S is a polygon, the extreme points will be the vertices, or corners, of the polygon.
The feasible region for the Giapetto LP will be a convex set.

56. 3.2 – Graphical Solution to a 2-Variable LP
It can be shown that:
The feasible region for any LP will be a convex set.
The feasible region for any LP has only a finite number of extreme points.
Any LP that has an optimal solution has an extreme point that is optimal.

57. Dorian Auto - underlined text was modified (from Winston)
Dorian Auto manufactures luxury cars and trucks. The company believes that its most likely customers are high-income men and women. To reach these groups, Dorian Auto has embarked on an ambitious TV advertising campaign and has decided to purchase 1-minute commercial spots on two types of programs: comedy shows and football games. Each comedy commercial is seen by 7 million high-income women and 2 million high-income men. Each football commercial is seen by 2 million high-income women and 12 million high-income men. A 1-minute comedy ad costs $50,000, and a 1-minute football ad costs $100,000. Dorian would like the commercials to be seen by at least 28 million high-income women and 24 million high-income men. Use LP to determine how Dorian Auto can meet its advertising requirements at minimum cost.
- underlined text was modified