introduction to linear programming - angelfire · introduction to linear programming....

Investigación de Operaciones

M. En C. Eduardo Bustos Farías 1

Overview of OR Modeling Approach

&Introduction to

Linear Programming



What is Operations Research?

• A field of study that uses computers, statistics, and mathematics to solve business problems.

• Also known as:– Management Science– Decision Science



OPERATIONS RESEARCH

• First applied to research on (military) problems

• Use of scientific knowledge through interdisciplinary team effort for the purpose of deciding the best utilization of limited resources



Origins of OR

• Management of WW II by England• A team of scientists• Worked on strategic and tactical problems• Land and air defenses



Decisions

• Most effective use of limited military resources– Deployment of radar defenses– Maximize bomber effectiveness



OR in the USA

• The US adopted OR methods in WW II based on Britain’s successes

• Following the war, industry adopted these OR methods and approaches

• The US assumed leadership in the development of OR as an academic discipline



The Simplex Method

• The first mathematical technique developed in OR was the simplex method of linear programming

• Developed in 1947 by George B. Dantzig

October 1999 issue of

ORMS Today Cover Photo



Progress in OR Applications

The development and application of the tools of OR are due in large part to the parallel development of digital computers to handle large scale computational problems



Example OR Application Areas

• Military planning• Industrial planning• Healthcare analysis• Financial institutional

and investment planning

• Governmental planning at all levels

• Transportation systems planning

• Logistics system design and operation

• Emergency system planning



Successful Applications of OR• Merril Lynch

– 5 million customers– 16,000 financial advisors– Developed a model to design product features

and pricing options to better reflect customer value

– Benefits: • $80 million increase in annual revenue• $22 billion increase in net assets



Successful Applications of OR• Jan de Wit Co.

– Brazil’s largest lily farmer– Annually plants 3.5 million bulbs and produces

420,000 pots & 220,000 bundles of lilies in 50 varieties.

– Developed model to determine what to plant, when to plant it, and how to sell it.

– Benefits:• 26% increase in revenue• 32% increase in contribution margin



Successful Applications of OR• Samsung Electronics

– Leading DRAM manufacturer– Semiconductor facilities cost $2-$3 billion– High equipment utilization is key– Developed comprehensive planning and

scheduling system to control WIP– Benefits:

• Cut cycle times in half• $1 billion increase in annual revenue



The Problem Solving Process

Identify Problem

Formulate & Implement

ModelAnalyze Model

Test Results

Implement Solution

unsatisfactoryresults



Phases of an OR Study• Define the problem, gather relevant data• Formulate a mathematical model• Develop a method for solving the model

(usually computer-based)• Test the model solution (validation), revise

as necessary• Prepare for ongoing applications of the

model in decision making• Implement



Define the Problem• Usually a very difficult process

– Vague, imprecise concepts of the problem– Data does not exist, or in an inappropriate form

• Determine– Objectives– Constraints– Interrelationships– Alternatives– Time constraints



Model Formulation

Modeling in ORA model in the sense used in OR is defined as an idealized representation of a real-life situation

Real-life system

Model



Model Formulation

Descriptive Model

The objective of a descriptive model is to provide the means for analyzing the behavior of an existing system for the purpose of improving its performance



Model Formulation

Prescriptive Model

The objective of a prescriptive model is to define the ideal structure of a future system, which includes functional relationships among its components



Model Formulation

General Model Classifications

• Iconic

• Analog

• Symbolic, or mathematical



Iconic ModelsModel Formulation

Iconic models represent the system by scaling it up or down, e.g., a toy airplane is an iconic model of a real one



Analog ModelsModel Formulation

Analog models require the substitution of one property for another for the ultimate purpose of achieving convenience in manipulating the model. After the problem is solved, the results are reinterpreted in terms of the original system.



Model Formulation

Symbolic, or Mathematical Models

Mathematical models employ a set of mathematical symbols to represent the decision variables of the system. These variables are related by appropriate mathematical functions to describe the behavior of the system. The solution of the problem is then obtained by applying well-developed mathematical techniques.



Additional OR Model TypesModel Formulation

• Simulation models– digital representations which “imitate” the

behavior of a system using a computer

• Heuristic models– some intuitive rules or guidelines are applied to

generate new strategies which yield improved solutions to the model



Categories of Mathematical ModelsModel Independent OR/MS

Category Form of f(.) Variables Techniques

Prescriptive known, known or under LP, Networks, IP,well-defined decision maker’s CPM, EOQ, NLP,

control GP, MOLP

Predictive unknown, known or under Regression Analysis, ill-defined decision maker’s Time Series Analysis,

control Discriminant Analysis

Descriptive known, unknown or Simulation, PERT,well-defined uncertain Queueing,

Inventory Models



StructureMath models have three basic sets of elements:

• Decision variables and parameters• Constraints or restrictions• Objective function(s)



Decision variables and parameters

– Decision variables are the unknowns which are to be determined from the solution of the model

– Parameters are the “known” decision variable coefficients and/or resource availability



Constraints or restrictions

– To account for the physical limitations of the system, the model must include constraints which limit the decision variables to their feasible (or permissible) values



Objective function– This defines the measure of effectiveness of the

system as a mathematical function of its decision variables

– The optimum solution to the model has been obtained if the corresponding values of the decision variables yield the best value of the objective function while satisfying all the constraints



Find the values of the decision variables

which will optimize:njx j ,...,2,1, =

n1,2,...,jfor 0 andm1,2,...,ifor ),...,,(

:subject to),...,,(

21

21

=≥=≤

=

j

ini

n

xbxxxg

xxxfz



The objective function and constraints in math model may take on many forms, depending upon the system being modeled

Functions may be linear or non-linear, the decision variablesmay be continuous or discrete, and parameters may be deterministic or stochastic.



Solution of the Model

MATHEMATICAL PROGRAMMINGMATHEMATICAL PROGRAMMING

Linear ProgrammingLinear Programming

Integer ProgrammingInteger Programming

Dynamic ProgrammingDynamic Programming

Stochastic ProgrammingStochastic Programming

Nonlinear programmingNonlinear programming




The Optimization Process• The optimal solution to a mathematical

model cannot generally be obtained in one step

• Rather it requires:– An initial solution– A set of computational rules (an algorithm)– Iteration of the algorithm to reach an optimal,

feasible solution (provided one exists)



Validation of the Model

Testing the Model Solution• Thoroughly check the model structure,

assumptions and process• Compare model solutions with historical data

from the existing system being modeled• Compare model solutions with forecasts for

planned new systems• Correct any errors in the formulation and re-solve

the model



Model Solution ReliabilityThe reliability of the solution obtained from the model depends on the validity of this model in representing the real system, i.e., how well the resulting solution actually applies to the assumed real system represented by the model (accuracy)

Real-life system

Model

Solution

Validation of the Model



Application of the Model

Preparing to Apply the Model

• Document the system for applying the model for use in decision making– Solution procedure– Operating procedures– Database and MIS interfaces– Interaction with other models for decision

support



Implementation of the Model

Implementation• Garner support of top management, operating

management, and the OR team• Explain the new system and its relationship to

operating realities• Indoctrinate user personnel• Monitor initial experiences and correct problems

discovered in usage• Maintain currency and accuracy of documentation



Introduction to Mathematical Programming

• We all face decision about how to use limited resources such as:

– Oil in the earth– Land for dumps– Time– Money– Workers



Mathematical Programming...

• MP is a field of management science that finds the optimal, or most efficient, way of using limited resources to achieve the objectives of an individual of a business.

• a.k.a. Optimization



Applications of Optimization

• Determining Product Mix• Manufacturing• Routing and Logistics• Financial Planning



Characteristics of Optimization Problems

• Decisions• Constraints• Objectives




MATHEMATICAL PROGRAMMINGMATHEMATICAL PROGRAMMING

Linear ProgrammingLinear Programming

Integer ProgrammingInteger Programming

Dynamic ProgrammingDynamic Programming

Stochastic ProgrammingStochastic Programming

Nonlinear programmingNonlinear programming



Linear Programming

• “Linear” means that all the mathematical functions in the model are required to be linear functions

• “Programming” is essentially a synonym for planning

• Thus, “linear programming” means planning using a mathematical model containing only linear functions



Concern of LP

LP deals with the problem of allocating limited resources among competing activities in an optimal manner



General Approach

Formulating and solving an LP model requires:– Optimizing (maximizing or minimizing) a

linear function of variables called the “objective function”

– Subject to a set of linear equalities and/or inequalities called “constraints”



Mathematical Expression of an LP Model

Max (or Min) Subject to:

and , j = 1, 2, ... , n

Z c x c x c x

a x a x a x ba x a x a x b

a x a x a x bx

n n

n n

n n

m m mn n m

j

= + + +

+ + + ≤ = ≥+ + + ≤ = ≥

+ + + ≤ = ≥≥ ∀

1 1 2 2

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

0

...

... ( , , )... ( , , )

... ( , , ) Con

stra

ints

Obj

eciv

eFu

nctio

n



Parameters and Variables

In the preceding formulation,– cj,bi, and aij, for (i=1,2,...,m; j=1,2,...,n) are

constants which are determined depending on the technology of the problem

– the xj’s are the decision variables– only one of the signs (<, =, >) holds for each

constraint



Canonical Form of an LP Model

Max (or Min)

S. T.

, i = 1,2,... ,m

, j = 1,2,..., n

Z c x

a x b

x

j jj

n

ij jj

n

i

j

=

≤=≥

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

≥

=

=

∑

∑

1

1

0



Matrix Form of LP ModelMax

Z c x

s t Ax bx

=

≤

≥

. .0

c c c cn=[ , ,..., ]1 2 x

xx

xn

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1

2

b

bb

bm

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1

2

A

a a aa a a

a a a

n

n

m m mn

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

11 12 1

21 22 2

1 2

0

00

0

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥



Observations

• cj is the increase or decrease in the overall measure of effectiveness (Z) that results from each unit increase or decrease in xj

• The number of relevant scarce resources is m, so that each of the first m linear inequalities corresponds to a constraint on the availability of one of these resources



Observations• bi is the amount of resource i available to

the n activities• aij is the amount of resource i consumed by

each unit of activity j• The left side of the constraint inequalities is

the total usage of the respective resource• The non-negativity constraints rule out the

possibility of negative activity levels



Formulation

The key to successful application of linear programming is the ability to recognizewhen a problem can be solved by linear programming, and to formulate the corresponding (and appropriate) model



5 Steps In Formulating LP Models:1. Understand the problem.2. Identify the decision variables.

X1=number of Aqua-Spas to produceX2=number of Hydro-Luxes to produce

3. State the objective function as a linear combination of the decision variables.

MAX: 350X1 + 300X2



5 Steps In Formulating LP Models(continued)

4. State the constraints as linear combinations of the decision variables.

1X1 + 1X2 <= 200 } pumps9X1 + 6X2 <= 1566 } labor12X1 + 16X2 <= 2880 } tubing

5. Identify any upper or lower bounds on the decision variables.

X1 >= 0X2 >= 0



Limitations of LP

ProportionalityThe objective function and the constraints are linear expressions of the decision variables. This means that the contribution of each activity is directly proportional to the level of the activity



Limitations of LP

AdditivityThe total measure of effectiveness (objective function) and each total resource usage (constraint) resulting from the joint performance of the activities must equal the respective sums of these quantities resulting from each activity being conducted individually



Limitations of LP

DivisibilityFractional levels of the decision variables must be permitted

DeterministicAll of the coefficients in the LP model

are assumed to be known constants( , , )a b cij i j



An Example LP ProblemBlue Ridge Hot Tubs produces two types of hot tubs: Aqua-Spas & Hydro-Luxes.

Aqua-Spa Hydro-LuxPumps 1 1Labor 9 hours 6 hoursTubing 12 feet 16 feetUnit Profit $350 $300

There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available.



LP Model for Blue Ridge Hot Tubs

MAX: 350X1 + 300X2

S.T.: 1X1 + 1X2 <= 2009X1 + 6X2 <= 156612X1 + 16X2 <= 2880X1 >= 0X2 >= 0



LP Formulation(Product Mix Problem)

Suppose we must decide on the number of units to be manufactured of two different products. The profits per unit of product 1 and product 2 are 2 and 5, respectively. Each unit of product 1 requires 3 machine hours and 9 units of raw material. Each unit of product 2 requires 4 machine hours and 7 units of raw material. The maximum available machine hours and raw material units are 200 and 300, respectively. A minimum of 20 units is required of product 1.



Product Mix Formulation

Max S.T. 200 300

Z x xx xx xx

x

= ++ ≤+ ≤

≥

≥

2 53 49 7

20

0

1 2

1 2

1 2

1

2

• x1 = number of units of product 1 to be produced

• x2 = number of units of product 2 to be produced



LP Formulation(Blending Problem)

The manager of an oil refinery must decide on the optimal mix of two possible blending processes, of which the inputs and outputs per production run of each process are as follows:

Input Output

Process Crude A Crude B Gasoline X Gasoline Y

1 5 3 5 8

2 4 5 4 4



LP Formulation(Blending Problem-continued)

The maximum amounts available of crudes A and B are 200 units and 150 units, respectively. Market requirements show that at least 100 units of gasoline X and 80 units of gasoline Y must be produced. The profits per production run from process 1 and process 2 are 3 and 4, respectively



Blending Problem Formulation

• x1 = number of production runs of process 1 to be made


Max s.t.

Z x xx xx xx xx xx x

= ++ ≤+ ≤+ ≥+ ≥≥ ≥

3 45 4 2003 5 1505 4 1008 4 80

0 0

1 2

1 2

1 2

1 2

1 2

1 2,



LP Formulation(Diet Problem)An individual wants to decide on the constituents of a diet which will satisfy his daily needs of proteins, fats, and carbohydrates at the minimum cost. Choices from 5 different types of foods can be made. The yields per unit of these foods are given in the following table:



Yields Per Unit

Food Type Proteins Fats Carbohydrates Cost/Unit

1 p1 f1 c1 d1

2 p2 f2 c2 d2

3 p3 f3 c3 d3

4 p4 f4 c4 d4

5 p5 f5 c5 d5

Min. DailyReq’mt.

P F C



Diet Problem FormulationMin s.t.

Z d x d x d x d x d xp x p x p x p x p x Pf x f x f x f x f x Fc x c x c x c x c x C

x x x x x

= + + + ++ + + + ≥+ + + + ≥+ + + + ≥

≥

1 1 2 2 3 3 4 4 5 5

1 1 2 2 3 3 4 4 5 5

1 1 2 2 3 3 4 4 5 5

1 1 2 2 3 3 4 4 5 5

1 2 3 4 5 0, , , ,

The decision variables represents the number of units used of the first, second, third, fourth and fifth type of food, respectively

x x x x x1 2 3 4 5, , , ,



The Graphical Method for Solving LP’s

If LP models have only two variables, they can be solved graphically:

• Plot all constraints by first plotting the equality relationship between resource usage and availability (the boundary), then determine the direction of the inequality if necessary– Plot the objective function and move it parallel to

itself, in the direction of optimization, until it last touches the feasible solution space.



Solving LP Problems: An Intuitive Approach

• Idea: Each Aqua-Spa (X1) generates the highest unit profit ($350), so let’s make as many of them as possible!

• How many would that be?– Let X2 = 0

• 1st constraint: 1X1 <= 200 • 2nd constraint: 9X1 <=1566 or X1 <=174• 3rd constraint: 12X1 <= 2880 or X1 <= 240

• If X2=0, the maximum value of X1 is 174 and the total profit is $350*174 + $300*0 = $60,900

• This solution is feasible, but is it optimal?• No!



Solving LP Problems:A Graphical Approach

• The constraints of an LP problem defines its feasible region.

• The best point in the feasible region is the optimal solution to the problem.

• For LP problems with 2 variables, it is easy to plot the feasible region and find the optimal solution.



X2

X1

250

200

150

100

50

00 50 100 150 200 250

(0, 200)

(200, 0)

boundary line of pump constraint

X1 + X2 = 200

Plotting the First Constraint



X2

X1

250

200

150

100

50

00 50 100 150 200 250

(0, 261)

(174, 0)

boundary line of labor constraint

9X1 + 6X2 = 1566

Plotting the Second Constraint



X2

X1

250

200

150

100

50

00 50 100 150 200 250

(0, 180)

(240, 0)

boundary line of tubing constraint 12X1 + 16X2 = 2880

Feasible Region

Plotting the Third Constraint



X2 Plotting A Level Curve of the Objective Function

X1

250

200

150

100

50

00 50 100 150 200 250

(0, 116.67)

(100, 0)

objective function 350X1 + 300X2 = 35000



A Second Level Curve of the Objective FunctionX2

X1

250

200

150

100

50

00 50 100 150 200 250

(0, 175)

(150, 0)





Using A Level Curve to Locate the Optimal SolutionX2

X1

250

200

150

100

50

00 50 100 150 200 250



optimal solution



Calculating the Optimal Solution• The optimal solution occurs where the “pumps” and

“labor” constraints intersect.• This occurs where:

X1 + X2 = 200 (1)and 9X1 + 6X2 = 1566 (2)

• From (1) we have, X2 = 200 -X1 (3)• Substituting (3) for X2 in (2) we have,

9X1 + 6 (200 -X1) = 1566which reduces to X1 = 122

• So the optimal solution is,X1=122, X2=200-X1=78

Total Profit = $350*122 + $300*78 = $66,100



Enumerating The Corner PointsX2

X1

250

200

150

100

50

00 50 100 150 200 250

(0, 180)

(174, 0)

(122, 78)

(80, 120)

(0, 0)

obj. value = $54,000



obj. value = $60,900obj. value = $0

Note: This technique will not work if the solution is unbounded.



Summary of Graphical Solution to LP Problems

1. Plot the boundary line of each constraint2. Identify the feasible region3. Locate the optimal solution by either:

a. Plotting level curvesb. Enumerating the extreme points (corner points)



Special Conditions in LP Models

• A number of special conditions may occur in LP problems:– Alternate Optimal Solutions – Redundant Constraints– Unbounded Solutions– Infeasibility



Example of Alternate Optimal SolutionsWhat if the price of Aqua-Spas generates a profit

of $450 instead of $350?X2

X1

250

200

150

100

50

00 50 100 150 200 250

450X1 + 300X2 = 78300objective function level curve

alternate optimal solutions



Example of a Redundant Constraint

What if 225 pumps are avaivale instead of 200?1X1 + 1X2 <= 225 Pumps

X2

X1

250

200

150

100

50

00 50 100 150 200 250

boundary line of tubing constraint

Feasible Region

boundary line of pump constraint

boundary line of labor constraint



Example of an Unbounded Solution

Consider the following problem:

MAX: Z= X1 + X2

S.T.:1X1 + 1X2 >= 400-1X1 + 2X2 <= 400X1 >= 0X2 >= 0



Example of an Unbounded SolutionX2

X1

1000

800

600

400

200

00 200 400 600 800 1000

X1 + X2 = 400

X1 + X2 = 600objective function

X1 + X2 = 800objective function

-X1 + 2X2 = 400



Example of Infeasibility

Consider the following problem:

MAX: Z= X1 + X2

S.T.:1X1 + 1X2 <= 1501X1 + 1X2 >= 200X1 >= 0X2 >= 0



Example of InfeasibilityX2

X1

250

200

150

100

50

00 50 100 150 200 250

X1 + X2 = 200

X1 + X2 = 150

feasible region for second constraint

feasible region for first constraint



Prototype ExampleThe WYNDOR GLASS CO. produces high-quality glass products including windows and galss doors. It has three plants. Aluminum frames and hardware are made in Plant 1, wood frames in Plant 2, and Plant 3 produces the glass and assembles the products.

Because of declining earnings, top management has decided to revamp the company’s product line. Unprofitable products are being discontinued, releasing production capacity to launch two new products having large sales potential.



Prototype Example

The two new products are:• Product 1: An 8-foot glass door with aluminum framing• Product 2: A 4 x 6 foot double-hung wood-framed window

An OR team has been assembled to recommend the product mix of these two new products which will maximize the profit from their sales. The team has assembled the data in the following table:



Prototype ExampleProduction Time/Batch (hrs.)

Product

Plant 1 2 Available ProductionTime/Week (hrs.)

1 1 0 4

2 0 2 12

3 3 2 18

Profit perbatch

$3,000 $5,000



ExampleMax

Z x xs t x

xx x

x x

= +≤≤

+ ≤≥

3 546

3 2 180

1 2

1

2

1 2

1 2

. .

,

2 4 6 8

2

4

68

x1

x2



Problems



EXAMPLE: An Investment Problem:Retirement Planning Services, Inc.

• A client wishes to invest $750,000 in the following bonds. Years to

Company Return Maturity RatingAcme Chemical 8.65% 11 1-ExcellentDynaStar 9.50% 10 3-GoodEagle Vision 10.00% 6 4-FairMicro Modeling 8.75% 10 1-ExcellentOptiPro 9.25% 7 3-Good

Sabre Systems 9.00% 13 2-Very Good



Investment Restrictions

• No more than 25% can be invested in any single company.

• At least 50% should be invested in long-term bonds (maturing in 10+ years).

• No more than 35% can be invested in DynaStar, Eagle Vision, and OptiPro.

• FORMULATE THE LP MODEL



EXAMPLE:A Blending Problem:The Agri-Pro Company

• Agri-Pro has received an order for 8,000 pounds of chicken feed to be mixed from the following feeds.

Nutrient Feed 1 Feed 2 Feed 3 Feed 4Corn 30% 5% 20% 10%Grain 10% 3% 15% 10%Minerals 20% 20% 20% 30%Cost per pound $0.25 $0.30 $0.32 $0.15

Percent of Nutrient in

• The order must contain at least 20% corn, 15% grain, and 15% minerals.

• FORMULATE THE LP MODEL



LP Formulation -Blending ProblemThe manager of an oil refinery must decide on the optimal mix of two possible blending processes, of which the inputs and outputs perproduction run of each process are as follows:

Input Output

Process Crude A Crude B Gasoline X Gasoline Y

1 5 3 5 8

2 4 5 4 4

The maximum amounts available of crudes A and B are 200 units and 150 units, respectively. Market requirements show that at least 100 units of gasoline X and 80 units of gasoline Y must be produced. The profits per production run from process 1 and process 2 are 3 and 4, respectively





Max s.t.

Z x xx xx xx xx xx x

= ++ ≤+ ≤+ ≥+ ≥≥ ≥

3 45 4 2003 5 1505 4 1008 4 80

0 0

1 2

1 2

1 2

1 2

1 2

1 2,

1

2

3

4

Solve the Blending Problem using the GRAPHICAL PROCEDURE

EXAMPLE:Blending Problem Formulation

introduction to linear programming - angelfire · introduction to linear programming....

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