Operations management 12th stevenson ch19 linear programming

 Used to obtain optimal solutions to problems that involve restrictions or limitations, such as:  Materials  Budgets  Labor  Machine time  Linear programming LP techniques consist

Chapter 19: Learning Objectives

You should be able to:

Linear Programming (LP)

 A powerful quantitative tool used by operations and other manages to obtain

optimal solutions to problems that involve restrictions or limitations

Applications include:

Establishing locations for emergency equipment and personnel to minimize response time

Developing optimal production schedules

Developing financial plans

Determining optimal diet plans

 Used to obtain optimal solutions to problems that involve

restrictions or limitations, such as:

 Materials

 Budgets

 Labor

 Machine time

 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

Model Formulation

1. List and define the decision variables (D.V.)

 These typically represent quantities

2. State the objective function (O.F.)

 It includes every D.V in the model and its contribution to profit (or cost)

3. List the constraints

 Right hand side value

 Relationship symbol (≤, ≥, or =)

 Left Hand Side

The variables subject to the constraint, and their coefficients that indicate how much of

the RHS quantity one unit of the D.V represents

4. Non-negativity constraints

Example– LP Formulation

(Objective function)

(Constraints)

(Nonnegativity constraints)

0 ,

,

units 10

1 Product pounds 100 5 6 7

Material hours 250 8 4 2

Labor Subject to (profit) 4 8 5

Maximize

produce to

3 product of

Quantity

produce to

2 product of

Quantity

produce to

1 product of

Quantity Variables

Decision

3 2 1 1

3 2

1

3 2

1

3 2

1 3

2 1

≥

≤

≤ +

+

≤ +

+

+ +









=

=

=

x x x x

x x

x

x x

x

x x

x x

x x

Graphical LP

 Graphical LP

 Procedure

1. Set up the objective function and the constraints in mathematical format

2. Plot the constraints

3. Indentify the feasible solution space

The set of all feasible combinations of decision variables as defined by the constraints

4. Plot the objective function

5. Determine the optimal solution

Linear Programming Example

Assembly Time/Unit

Inspection Time/Unit

Storage Space/Unit Profit/Unit

Available 100 hours 22 hours 39 cubic feet

Linear Programming Example

 Find the quantity of each

model to produce in order to maximize the profit

LP Example – Decision Variables

Decision Variables

Example– Graphical LP: Step 1

0 ,

feet cubic 39 3 3

Storage hours 22 1 2

Inspection hours 100 10 4

Assembly Subject to 50 60

Maximize

produce to

2 type of

quantity

produce to

1 type of

quantity Variables

Decision

2 1

2 1

2 1

2 1

2 1

2 1

≥

≤ +

≤ +

≤ +

+







=

=

x x

x x

x x

x x

x x

x x

Example– Graphical LP: Step 2

 Plotting constraints:

 Begin by placing the nonnegativity constraints on a graph

Example– Graphical LP: Step 2

 Plotting constraints:

1. Replace the inequality sign with an equal sign

2. Determine where the line intersects each axis

3. Mark these intersection on the axes, and connect them with a straight line

4. Indicate by shading, whether the inequality is greater than or less than

5. Repeat steps 1 – 4 for each constraint

Example– Graphical LP: Step 2

Example– Graphical LP: Step 2

Instructor Slides 19-20

Example– Graphical LP: Step 2

Example– Graphical LP: Step 2

3 3

39 cubic feet

Example– Graphical LP: Step 2

Example– Graphical LP: Step 3

 Feasible Solution Space

Feasible Region

Example– Graphical LP: Step 4

 Plotting the objective function line

Every point on this line represents a combination of the decision variables that result in the

same profit (in this case, to the profit you selected)

Example– Graphical LP: Step 4

Example– Graphical LP: Step 4

 As we increase the value for the objective function:

Example– Graphical LP: Step 5

 Where is the optimal solution?

 The optimal solution occurs at the furthest point (for a maximization problem) from the origin the isoprofit

can be moved and still be touching the feasible solution space

 This optimum point will occur at the intersection of two constraints:

 Solve for the values of x1 and x2 where this occurs

Solutions and Corner Points

 The solution to any problem will occur at one of the feasible solution space

corner points

Corner points occur at the intersections of constraints

 The intersection of inspection and storage

 Solve two equations with two unknowns

Slack and Surplus

 Binding Constraint

 If a constraint forms the optimal corner point of the feasible solution space, it is binding

 It effectively limits the value of the objective function

 If the constraint could be relaxed, the objective function could be improved

 Surplus

 When the value of decision variables are substituted into a ≥ constraint the amount by which the resulting

value exceeds the right-hand side value

 Slack

 When the values of decision variables are substituted into a ≤ constraint, the amount by which the resulting

value is less than the right-hand side

Linear Programming Example

Labor (hrs) Stone (tons) Lumber (board feet) Profit/Unit Model A 4,000 2 2,000 $ 3,000 Model B 10,000 3 2000 $ 6,000 Available 400,000 150 200,000

The Simplex Method

 Simplex method

 A general purpose linear programming algorithm that can be used to solve

problems having more than two decision variables

Computer Solutions

 MS Excel can be used to solve LP problems using its Solver routine

 Enter the problem into a worksheet

 Where there is a zero in Figure 19.15, a formula was entered

Solver automatically places a value of zero after you input the formula

 You must designate the cells where you want the optimal values for the decision

variables

Computer Solutions

Computer Solutions

 In Excel 2010, click on Tools on the top of the worksheet, and in that menu, click on

Solver

 Begin by setting the Target Cell

Computer Solutions

 Add a constraint, by clicking add

 Repeat the process for each system constraint

Computer Solutions

 For the nonnegativity constraints, enter the range of cells designated for the optimal

values of the decision variables

 Click on Options

 Click Solve

Computer Solutions

 The Solver Results menu will appear

 You will have one of two results

A Solution

In the Solver Results menu Reports box

 Highlight both Answer and Sensitivity

Solver Results

 Solver will incorporate the optimal values of the decision variables and the objective function into

your original layout on your worksheets

Answer Report

Sensitivity Report

Sensitivity Analysis

 Sensitivity Analysis

 Assessing the impact of potential changes to the numerical values of an LP model

 Three types of changes

Objective function coefficients

Right-hand values of constraints

Constraint coefficients We will consider these

 A change in the value of an O.F coefficient can cause a change in the optimal solution

Basic and Non-Basic Variables

Unless the non-basic variable’s coefficient increases by more than its reduced cost, it

will continue to be non-basic

RHS Value Changes

 Shadow price

 Amount by which the value of the objective function would change with a

one-unit change in the RHS value of a constraint

Range of values for the RHS of a constraint over which the shadow price remains the

same

Binding vs Non-binding Constraints

 Non-binding constraints

effect on the optimal solution

 Binding constraint

values and to a change in the value of the objective function