linear programming

Linear Programming

Dr Ravindra Singh

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Contents

• Introduction• History • Applications • Linear programming model• Example of Linear Programming Problems • Graphical Solution to Linear Programming

Problem• Sensitivity analysis

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Introduction

• Linear Programming is a mathematical modeling

technique used to determine a level of operational activity

in order to achieve an objective.

• Mathematical programming is used to find the best or

optimal solution to a problem that requires a decision or

set of decisions about how best to use a set of limited

resources to achieve a state goal of objectives.

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• Steps involved in mathematical programming

– Conversion of stated problem into a mathematical model that

abstracts all the essential elements of the problem.

– Exploration of different solutions of the problem.

– Find out the most suitable or optimum solution.

• Linear programming requires that all the mathematical

functions in the model be linear functions.

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LP Model Formulation

• Decision variables

– mathematical symbols representing levels of activity of an operation

• Objective function

– a linear relationship reflecting the objective of an operation

– most frequent objective of business firms is to maximize profit

– most frequent objective of individual operational units (such as a

production or packaging department) is to minimize cost

• Constraint

– a linear relationship representing a restriction on decision making

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History of linear programming

• It started in 1947 when G. B. Dantzig design the “simplex

method” for solving linear programming formulations of

U.S. Air Force planning problems.

• It soon became clear that a surprisingly wide range of

apparently unrelated problems in production management

could be stated in linear programming terms and solved by

the simplex method.

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Applications

The Importance of Linear Programming• Hospital management • Diet management• Manufacturing• Finance (investment)• Advertising• Agriculture

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The Galaxy Industries Production Problem

• Galaxy manufactures two drug combination of same drug:– X1 – X2

• Resources are limited to– 1000 pounds raw material.– 40 hours of production time per week.

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• Marketing requirement– Total production cannot exceed 700 dozens.

– Number of dozens of X1cannot exceed number

of dozens of X2 by more than 350.

• Technological input– X1 requires 2 pounds of raw material and

3 minutes of labor per dozen.– X2 requires 1 pound of raw material and

4 minutes of labor per dozen.

The Galaxy Industries Production Problem

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• The current production plan calls for: – Producing as much as possible of the more profitable

product, X1 ($8 profit per dozen).– Use resources left over to produce X2 ($5 profit

per dozen), while remaining within the marketing guidelines.

• The current production plan consists of:

X1 = 450 dozenX2 = 100 dozenProfit = $4100 per week

The Galaxy Industries Production Problem

8(450) + 5(100)

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Management is seeking a production schedule that will increase the company’s profit.

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• Decisions variables:

– X1 = Weekly production level of X1 (in dozens)

– X2 = Weekly production level of X2 (in dozens).

• Objective Function:

– Weekly profit, to be maximized

The Galaxy Linear Programming Model

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Max 8X1 + 5X2 (Weekly profit)

subject to

2X1 + 1X2 £ 1000 (Raw Material)

3X1 + 4X2 £ 2400 (Production Time)

X1 + X2 £ 700 (Total production)

X1 - X2 £ 350 (Mix)

Xj> = 0, j = 1,2 (Non negativity)

The Galaxy Linear Programming Model

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The Graphical Analysis of Linear Programming

The set of all points that satisfy all the constraints of the model is called

a

FEASIBLE REGION

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Using a graphical presentation

we can represent all the constraints,

the objective function, and the three

types of feasible points.

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The non-negativity constraints

X2

X1

Graphical Analysis – the Feasible Region

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1000

500

Feasible

X2

Infeasible

Production Time3X1+4X2 £ 2400

Total production constraint: X1+X2 £ 700 (redundant)

500

700

The Raw material constraint2X1+X2 £ 1000

X1

700

Graphical Analysis – the Feasible Region

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1000

500

Feasible

X2

Infeasible

Production Time3X1+4X2 £ 2400

Total production constraint: X1+X2 £ 700 (redundant)

500

700

Production mix constraint:X1-X2 £ 350

The Raw Material constraint2X1+X2 £ 1000

X1

700

Graphical Analysis – the Feasible Region

• There are three types of feasible pointsInterior points.Boundary points.Extreme points.

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The search for an optimal solution

Start at some arbitrary profit, say profit = $2,000...

Then increase the profit, if possible...

...and continue until it becomes infeasible

Profit =$4360500

700

1000

500

X2

X1

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Summary of the optimal solution

X1 = 320 dozen

X2 = 360 dozen

Profit = $4360

– This solution utilizes all the plastic and all the production

hours.

– Total production is only 680 (not 700).

– X1 production exceeds X2 production by only 40 dozens.

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– If a linear programming problem has an optimal solution, an extreme point is optimal.

Extreme points and optimal solutions

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• For multiple optimal solutions to exist, the objective function must be parallel to one of the constraints

Multiple optimal solutions

•Any weighted average of optimal solutions is also an optimal solution.

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Sensitivity Analysis of the Optimal Solution

• Is the optimal solution sensitive to changes in input parameters?

• Possible reasons for asking this question:– Parameter values used were only best estimates.– Dynamic environment may cause changes.– “What-if” analysis may provide economical and

operational information.

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• Range of Optimality– The optimal solution will remain unchanged as long as

• An objective function coefficient lies within its range of optimality • There are no changes in any other input parameters.

– The value of the objective function will change if the

coefficient multiplies a variable whose value is nonzero.

Sensitivity Analysis of Objective Function Coefficients.

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A linear programming model can

provide an insight and an intelligent

solution to the problem.

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REFERENCES

• www.math.ucla.edu/~tom/LP.pdf• www.sce.carleton.ca/faculty/chinneck/po/Chapter2.• www.markschulze.net/LinearProgramming.pdf• web.ntpu.edu.tw/~juang/ms/Ch02.• cmp.felk.cvut.cz/~hlavac/Public/.../Linear

%20Programming-1.ppt• www.slideshare.net/nagendraamatya/linear-

programming

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Thank you