Linear Programming Contents Introduction History Applications Linear programming

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Linear Programming

Linear Programming

Contents • • • Introduction History Applications Linear programming model Example of Linear Programming

Contents • • • Introduction History Applications Linear programming model Example of Linear Programming Problems Graphical Solution to Linear Programming Problem • Sensitivity analysis 2

Introduction • Linear Programming is a mathematical modeling technique used to determine a level

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. 3

 • Steps involved in mathematical programming – Conversion of stated problem into a

• 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. 4

LP Model Formulation • Decision variables – mathematical symbols representing levels of activity of

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 5

History of linear programming • It started in 1947 when G. B. Dantzig design

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. 6

Applications The Importance of Linear Programming • • • Hospital management Diet management Manufacturing

Applications The Importance of Linear Programming • • • Hospital management Diet management Manufacturing Finance (investment) Advertising Agriculture 7

The Galaxy Industries Production Problem • Galaxy manufactures two drug combination of same drug:

The Galaxy Industries Production Problem • Galaxy manufactures two drug combination of same drug: – X 1 – X 2 • Resources are limited to – 1000 pounds raw material. – 40 hours of production time per week. 8

The Galaxy Industries Production Problem • Marketing requirement – Total production cannot exceed 700

The Galaxy Industries Production Problem • Marketing requirement – Total production cannot exceed 700 dozens. – Number of dozens of X 1 cannot exceed number of dozens of X 2 by more than 350. • Technological input – X 1 requires 2 pounds of raw material and 3 minutes of labor per dozen. – X 2 requires 1 pound of raw material and 4 minutes of labor per dozen. 9

The Galaxy Industries Production Problem • The current production plan calls for: – Producing

The Galaxy Industries Production Problem • The current production plan calls for: – Producing as much as possible of the more profitable product, X 1 ($8 profit per dozen). – Use resources left over to produce X 2 ($5 profit per dozen), while remaining within the marketing guidelines. • The current production plan consists of: X 1 = 450 dozen X 2 = 100 dozen Profit = $4100 per week 8(450) + 5(100) 10

Management is seeking a production schedule that will increase the company’s profit. 11

Management is seeking a production schedule that will increase the company’s profit. 11

The Galaxy Linear Programming Model • Decisions variables: – X 1 = Weekly production

The Galaxy Linear Programming Model • Decisions variables: – X 1 = Weekly production level of X 1 (in dozens) – X 2 = Weekly production level of X 2 (in dozens). • Objective Function: – Weekly profit, to be maximized 12

The Galaxy Linear Programming Model Max 8 X 1 + 5 X 2 (Weekly

The Galaxy Linear Programming Model Max 8 X 1 + 5 X 2 (Weekly profit) subject to 2 X 1 + 1 X 2 1000 (Raw Material) 3 X 1 + 4 X 2 2400 (Production Time) X 1 + X 2 700 (Total production) X 1 X 2 350 (Mix) Xj> = 0, j = 1, 2 (Non negativity) 13

The Graphical Analysis of Linear Programming The set of all points that satisfy all

The Graphical Analysis of Linear Programming The set of all points that satisfy all the constraints of the model is called a FEASIBLE REGION 14

Using a graphical presentation we can represent all the constraints, the objective function, and

Using a graphical presentation we can represent all the constraints, the objective function, and the three types of feasible points. 15

Graphical Analysis – the Feasible Region X 2 The non-negativity constraints X 1 16

Graphical Analysis – the Feasible Region X 2 The non-negativity constraints X 1 16

Graphical Analysis – the Feasible Region X 2 The Raw material constraint 2 X

Graphical Analysis – the Feasible Region X 2 The Raw material constraint 2 X 1+X 2 1000 700 Total production constraint: X 1+X 2 700 (redundant) 500 Infeasible Production Time 3 X 1+4 X 2 2400 Feasible 500 700 X 1 17

Graphical Analysis – the Feasible Region X 2 The Raw Material constraint 2 X

Graphical Analysis – the Feasible Region X 2 The Raw Material constraint 2 X 1+X 2 1000 700 Total production constraint: X 1+X 2 700 (redundant) 500 Production Time 3 X 1+4 X 2 2400 Infeasible Production mix constraint: X 1 -X 2 350 Feasible 500 700 X 1 Interior points. B. oundary points. Extreme points. • There are three types of feasible points 18

The search for an optimal solution X 2 1000 Start at some arbitrary profit,

The search for an optimal solution X 2 1000 Start at some arbitrary profit, say profit = $2, 000. . . Then increase the profit, if possible. . . and continue until it becomes infeasible 700 500 Profit =$4360 X 1 500 19

Summary of the optimal solution X 1 = X 2 = Profit = $4360

Summary of the optimal solution X 1 = X 2 = Profit = $4360 320 360 dozen – This solution utilizes all the plastic and all the production hours. – Total production is only 680 (not 700). – X 1 production exceeds X 2 production by only 40 dozens. 20

Extreme points and optimal solutions – If a linear programming problem has an optimal

Extreme points and optimal solutions – If a linear programming problem has an optimal solution, an extreme point is optimal. 21

Multiple optimal solutions • For multiple optimal solutions to exist, the objective function must

Multiple optimal solutions • For multiple optimal solutions to exist, the objective function must be parallel to one of the constraints • Any weighted average of optimal solutions is also an optimal solution. 22

Sensitivity Analysis of the Optimal Solution • Is the optimal solution sensitive to changes

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. 23

Sensitivity Analysis of Objective Function Coefficients. • Range of Optimality – The optimal solution

Sensitivity Analysis of Objective Function Coefficients. • 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. 24

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REFERENCES • • • www. math. ucla. edu/~tom/LP. pdf www. sce. carleton. ca/faculty/chinneck/po/Chapter 2.

REFERENCES • • • www. math. ucla. edu/~tom/LP. pdf www. sce. carleton. ca/faculty/chinneck/po/Chapter 2. www. markschulze. net/Linear. Programming. pdf web. ntpu. edu. tw/~juang/ms/Ch 02. cmp. felk. cvut. cz/~hlavac/Public/. . . /Linear%20 Progra mming-1. ppt • www. slideshare. net/nagendraamatya/linearprogramming 26

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