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

Computational Methods forManagement and Economics

Carla Gomes

Module 2

Introduction to LP

(Textbook – Hillier and Lieberman)

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) andE=mc² (Einstein’s famous equation of conservation

of energy into mass).

Mathematical Models

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.

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 x1 +

3 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.

Example of Optimization Problem

Minimize: Z = x12 + x2

2

Subject to: 2x1 – 2x2 = 6

x2 2

Z – objective;

x1,x2 - decision variables; two constraints;

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

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 1981-2001 2,000,000X faster!)

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.

Prototype Example – Wyndor Glass (Hillier and Liebermnan)

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:1. Should they go ahead with launching these two new products?2. If so, what should be the product mix?

Steps in setting up a LP

1. Determine and label the decision variables.

2. Determine the objective and use the decision variables to write an expression for the objective function.

3. Determine the constraints - feasible region.1. Determine the explicit constraints and write a functional

expression for each of them.

2. Determine the implicit constraints (e.g., nonnegativity constraints).

The LP Mathematical modelProblem 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.)

Objective Function

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

Algebraic Model for Wyndor Glass Co.

Let D = the number of doors to produceW = the number of windows to produce

Maximize P = 3 D + 5 Wsubject to

D ≤ 42W ≤ 123D + 2W ≤ 18

andD ≥ 0, W ≥ 0.

A graphical solution

1. Graph the line associated with each of the linear inequality constraints.

2. Determine on which side of each of these lines the feasible region must lie.

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:

Graphing the Product Mix

Pro

duct

ion

rate

(un

its

per

wee

k) f

or w

indo

ws

A product mix of

A product mix of

1

2

3

4

5

6

7

8

0

-1

-1-2 1 2 3 4 5 6 7 8

-2

Pro

duct

ion

rate

(un

its

per

wee

k) f

or w

indo

ws

Production rate (units per week) for doors

(4, 6)

(2, 3)

D = 4 and W = 6

D = 2 and W = 3

Origin

D

W

Graph Showing Non-Negativity Constraints:

D ≥ 0 and W ≥ 0P

rodu

ctio

n ra

te f

or w

indo

ws

8

6

4

2

2 4 6 80

Production rate for doors

Pro

duct

ion

rate

for

win

dow

s

D

W

Nonnegative Solutions Permitted by D ≤ 4

Prod

uctio

n ra

te f

or w

indo

ws

D

W

8

6

4

2

2 4 6 80Production rate for doors

Prod

uctio

n ra

te f

or w

indo

ws

D = 4

Nonnegative Solutions Permitted by 2W ≤ 12


8

6

4

2

2 4 6 80

2 W = 12

D

WProduction rate for windows

Boundary Line for Constraint 3D + 2W ≤ 18


8

6

4

2

2 4 6 80

10

(0, 9)

(2, 6)

(4, 3)

21_(1, 7 )

21_(3, 4 )

21_(5, 1 )

(6, 0)

3 D + 2 W = 18

D

WProduction rate for windows

Changing Right-Hand Side Creates Parallel Constraint Boundary Lines

12

10

8

6

4

2

0 2 4 6 8 10


D

W

3D + 2W = 24

3D + 2W = 18

3D + 2W = 12

Production rate for windows

Nonnegative Solutions Permitted by3D + 2W ≤ 18

8

6

4

0 2 4 6 8

10

2

Production rate for doorsD

W

3D + 2W = 18


Graph of Feasible Region

0 2 4 6 8

8

6

4

10

2

Feasible

region


W

2 W =12

D = 4

3 D + 2 W = 18


Graph of Feasible Region

0 2 4 6 8

8

6

4

10

2

Feasible

region


W

2 W =12

D = 4

3 D + 2 W = 18


Objective Function (P = 1,500)

0 2 4 6 8

8

6

4

2

Production rate

for windows


Feasible

regionP = 1500 = 300D + 500W

D

W

isoprofit line

Finding the Optimal Solution

0 2 4 6 8

8

6

4

2

Production rate

for windows


Feasible

region

(2, 6)

Optimal solution

10

W

D

P = 3600 = 300D + 500W

P = 3000 = 300D + 500W

P = 1500 = 300D + 500W

Our objective function is: maximize 3D+5W

The vector representing the gradient of the objective function is:

5

3

DW3

5

The line through the origin that contains this vector is:

isoprofit line

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.

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

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. )

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

Standard form of the LP model

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 problemsSmallest value minimization problems

• Multiple optimal solutions• No optimal solutions• Unbounded Z

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

f the problem has exactly one optimal solution it must be a CFP solutionIf the problem has multiple optimal solutions, at least two must be CPF

solutions

No Feasible Solutions – Why?

0 2 4 6 8

8

6

4

10

2

Feasible

region


W

2 W =12

D = 4

3 D + 2 W = 18


Maximize P = 3 D + 5 W

subject to

D ≤ 42W ≤ 123D + 2W ≤ 18

3 D + 5 W 50

and

D ≥ 0, W ≥ 0.

Previous Feasible Region

Multiple Optimal Solutions. Why?

0 2 4 6 8

8

6

4

10

2

Feasible

region


W

2 W =12

D = 4

3 D + 2 W = 18



subject to

D ≤ 42W ≤ 123D + 2W ≤ 18

and

D ≥ 0, W ≥ 0.

Every point on this line isAn optimal solution with P=18

Unbounded Objective Function. Why?

Prod

uctio

n ra

te f

or w

indo

ws

D

W

8

6

4

2

2 4 6 80Production rate for doors

Prod

uctio

n ra

te f

or w

indo

ws

D = 4


subject to

D ≤ 4

and

D ≥ 0, W ≥ 0.

(4,2) P=16

(4,4) P=20

(4,8) P=28

(4, ) P=

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.

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.

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

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 x1 for x1 = 0;

LP AssumptionsAdditivity

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.

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!!!

LP AssumptionsDivisibility

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!!!

LP AssumptionsCertaintyThe 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

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!

LP Assumptions - CommentsMathematical 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.

Additional Examples

• Giapetto’s Woodcarving (from Winston and Venkataramanan)

• Dorian Auto (from Winston and Venkataramanan)

• Radiation Therapy (from Hillier and Lieberman)

Giapetto’s Woodcarving, Inc.

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.

(from Winston)

3.2 – Graphical Solution to a 2-Variable LPBinding and Nonbinding constraints

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.

Once the optimal solution to an LP is found, it is useful to classify each constraint as being a binding or nonbinding constraint.

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.

3.2 – Graphical Solution to a 2-Variable LPConvex sets, Extreme Points, and LPA 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:

A E B

DC

A B

A B

(a) (b) (c) (d)


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).

A E B

DC

A B

A B

(a) (b) (c) (d)


A E B

DC

A B

A B

(a) (b) (c) (d)

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.


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.


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.

1.Any LP that has an optimal solution has an extreme point that is optimal.

Dorian Auto

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.

(from Winston)

- underlined text was modified

Documents

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