6s-1 Linear Programming

CHAPTER6s

Linear Programming

6s-2 Linear Programming

Used to obtain optimal solutions to problems that involve restrictions or limitations, such as: Materials Budgets Labor Machine time

Linear ProgrammingLinear Programming

6s-3 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 ProgrammingLinear Programming

6s-4 Linear Programming

Objective: the goal of an LP model is maximization or minimization

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

Parameters: numerical values

Linear Programming ModelLinear Programming Model

6s-5 Linear Programming

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 AssumptionsLinear Programming Assumptions

6s-6 Linear Programming

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 ProgrammingGraphical Linear Programming

6s-7 Linear Programming

Objective - profitMaximize Z=60X1 + 50X2

Subject toAssembly 4X1 + 10X2 <= 100 hours

Inspection 2X1 + 1X2 <= 22 hours

Storage 3X1 + 3X2 <= 39 cubic feet

X1, X2 >= 0

Linear Programming ExampleLinear Programming Example

6s-8 Linear Programming

Assembly Constraint4X1 +10X2 = 100

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Linear Programming ExampleLinear Programming Example

6s-9 Linear Programming

Linear Programming ExampleLinear Programming Example

Add Inspection Constraint2X1 + 1X2 = 22

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6s-10 Linear Programming

Add Storage Constraint3X1 + 3X2 = 39

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AssemblyStorage

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Feasible solution space

Linear Programming ExampleLinear Programming Example

6s-11 Linear Programming

Add Profit Lines

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Linear Programming ExampleLinear Programming Example

6s-12 Linear Programming

The intersection of inspection and storage Solve two equations in two unknowns

2X1 + 1X2 = 223X1 + 3X2 = 39

X1 = 9X2 = 4Z = $740

SolutionSolution

6s-13 Linear Programming

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

ConstraintsConstraints

6s-14 Linear Programming

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 SurplusSlack and Surplus

6s-15 Linear Programming

Figure 6S.15

MS Excel Worksheet for MS Excel Worksheet for Microcomputer ProblemMicrocomputer Problem

6s-16 Linear Programming

Figure 6S.17

MS Excel Worksheet SolutionMS Excel Worksheet Solution