3 Characteristics of an Optimization Problem

General description

KPiller Illustration

Decisions that must be made; represented by decision variables

How many of each product to make next period (two decision variables)

Constraints or restrictions that limit the available decision alternatives

Limited department hours; orders that must be filled; minimum hours for labor (5 constraints)

A goal or objective that needs to be maximized or minimized

Maximize next period’s total profit z

Linear Programming (LP)

A mathematical programming problem is one that seeks to maximize or minimize an objective function subject to constraints.

If both the objective function and the constraints are linear, the problem is referred to as a linear programming problem.

KPiller Algebraic Formulation

Maximize 5000 X + 4000 Y (profit)

Subject to10 X + 15 Y <= 150 (dept A hours)

20 X + 10 Y <= 160 (dept B hours)

30 X + 10 Y >= 135 (testing hours)

X >= 1 (current order)

Y >= 3 (current order)

where X, Y are decision variables that represent the production quantity of E-Supremes and F-Supremes, respectively

Linear functions are functions in which each variable appears in a separate term raised to the first power and multiplied by a constant, which may be 0.

A feasible solution is an assignment of values to the decision variables that satisfies all the problem's constraints.

The objective does not affect the feasibility of the problem.

An optimal solution is a feasible solution that results in the largest possible objective function value, z, when maximizing or smallest z when minimizing.

Important LP Terminology

3 Steps of Linear Programming Model Formulation

Spreadsheet Based Algebraic

Model Solution Graphical Analysis Simplex Method (LINDO, CPLEX, etc.) Excel Solver Add-in

Sensitivity Analysis

Goals For Spreadsheet Design

Communication - A spreadsheet's primary business purpose is that of communicating information to managers.

Reliability - The output a spreadsheet generates should be correct and consistent.

Auditability - A manager should be able to retrace the steps followed to generate the different outputs from the model in order to understand the model and verify results.

Modifiability - A well-designed spreadsheet should be easy to change or enhance in order to meet dynamic user requirements.

Spreadsheet Design Guidelines

Organize the data, then build the model around the data. Do not embed numeric constants in formulas! Things which are logically related should be physically

related. Use formulas that can be copied: layout repetitive headings

in same order and in same row/column orientation. Apply appropriate use of absolute and relative cell references.

Column/row totals should be close to the columns/rows being totaled.

The English-reading eye scans left to right, top to bottom. Use color, shading, borders and protection to distinguish

changeable parameters from other model elements. Use text boxes and cell notes to document various

elements of the model.

Solver Modeling Requirements All components of the optimization problem

must be programmed on the same worksheet. Solver’s settings are saved with the worksheet.

Solver’s constraint dialog box will not let you enter formulas. All formulas and calculations must be done in the worksheet. The constraint dialog box just compares cells in the current worksheet to determine feasibility and optimality.

Hours Required per E-Supreme

Hours Required per F-Supreme

Total Hours Used

Total Hour Constraints

Department A 10 15 25 <= 150 hours

Department B 20 10 30 <= 160 hours

Finishing Dept 30 10 40 >=135 hours

E-Supreme F-Supreme

Current Orders 1 3

Quantity Made 1 1

Sales Price/Unit $ 20,000 $ 24,000

Variable Cost/Unit $ 15,000 $ 20,000

Profit per Unit $ 5,000 $ 4,000

Total Profit $ 5,000 $ 4,000

GRAND TOTAL PROFIT: $ 9,000

Programming the 3 Optimization Components in Solver

General description

Solver Terminology

Decision variables Changing Cells

Left Hand Sides (LHS) of Constraints

Cell Reference in Constraint Dialog box

Right Hand Sides (RHS) of Constraints

Constraint in Constraint Dialog box

Objective function Set Target Cell (click on Max or Min button)

Every linear program falls in one of three categories: It is infeasible

It has a unique optimal solution or alternate optimal solutions (different ways of achieving the same maximum or minimum objective value)

It has an objective function that can be increased without bound

Types of LP Solutions

“Ideal” Solver Result Message Solver found a solution. All

constraints and optimality conditions are satisfied Solver has identified an optimal solution for the problem you have formulated. Note that there may be alternative optimal solutions possible but Solver will just show you one possible solution.

Examples of Linear Programming Applications Production Planning:

several products multiperiod demand limited period resources want minimal production costs or maximum

profitability Transportation/Distribution Problems:

different routes limited supply at several sources demand requirements at various locations want minimal transportation costs

More Examples of LP Applications Investment Planning:

several investment alternatives risk and capital restrictions want maximum expected return

Labor Scheduling: full-time and part-time workforce multi-period staffing requirements workforce staffing restrictions want minimum total labor cost