Operations management by stevenson 9th ch6s

Learning Objectivesto solution using linear programming description of a problem graphical method problems...  Used to obtain optimal solutions to problems that involve restrictions or

Linear Programming

Learning Objectives

to solution using linear programming

description of a problem

graphical method

problems

 Used to obtain optimal solutions to problems

that involve restrictions or limitations, such

as:

Linear Programming

 Linear programming (LP) techniques consist

of a sequence of steps that will lead to an

optimal solution to problems, in cases where

an optimum exists

Linear Programming

 Objective Function: mathematical statement

of profit or cost for a given solution

 Decision variables: amounts of either inputs

or outputs

 Feasible solution space: the set of all

feasible combinations of decision variables as

defined by the constraints

 Constraints: limitations that restrict the

available alternatives

Linear Programming Model

 Linearity: the impact of decision variables is

linear in constraints and objective function

 Divisibility: noninteger values of decision

variables are acceptable

 Certainty: values of parameters are known and

constant

 Nonnegativity: negative values of decision

variables are unacceptable

Linear Programming

Assumptions

1 Set up objective function and constraints

in mathematical format

2 Plot the constraints

3 Identify the feasible solution space

4 Plot the objective function

5 Determine the optimum solution

Graphical Linear Programming

Graphical method for finding optimal

solutions to two-variable problems

 Objective - profit

Inspection 2X1 + 1X2 <= 22 hours

X1, X2 >= 0

Linear Programming Example

Assembly Constraint 4X1 +10X2 = 100

0

2

4

6

8

10

12

Product X1

Linear Programming Example

Linear Programming Example

Add Inspection Constraint

2X1 + 1X2 = 22

0

5

10

15

20

25

Add Storage Constraint

3X1 + 3X2 = 39

0

5

10

15

20

25

Product X1

Assembly Storage

Inspection

Feasible solution space

Linear Programming Example

Add Profit Lines

0

5

10

15

20

25

Product X1

Linear Programming Example

 The intersection of inspection and storage

2X1 + 1X2 = 22

3X1 + 3X2 = 39

X1 = 9

X2 = 4

Z = $740

Solution

 Redundant constraint: a constraint that

does not form a unique boundary of the

feasible solution space

 Binding constraint: a constraint that forms

the optimal corner point of the feasible

solution space

Constraints

Solutions and Corner Points

coordinates of each corner point into the objective

function to determine which corner point is optimal

 Surplus: when the optimal values of

decision variables are substituted into a

greater than or equal to constraint and the

resulting value exceeds the right side value

 Slack: when the optimal values of decision

variables are substituted into a less than or

equal to constraint and the resulting value is

less than the right side value

Slack and Surplus

 Simplex: a linear-programming algorithm

that can solve problems having more than

two decision variables

Simplex Method

Figure 6S.15

MS Excel Worksheet for Microcomputer Problem

Figure 6S.17

MS Excel Worksheet Solution

 Range of optimality: the range of values for

which the solution quantities of the decision

variables remains the same

 Range of feasibility: the range of values for

the fight-hand side of a constraint over which

the shadow price remains the same

 Shadow prices: negative values indicating

how much a one-unit decrease in the original

Sensitivity Analysis