Linear ProgrammingLinear Programming

Building Good Linear ModelsAnd

Example 1Sensitivity Analyses, Unit Conversion,

Summation Variables

Building Good ModelsBuilding Good ModelsA Check ListA Check List

1. Determine in general terms what the objective is (the objective function) and what factors are under the decision maker’s control that can affect this objective (the decision variables).• Define decision variables using appropriate units

and time frame (cars per month, tons per production run, etc.)

2. List the restrictions (constraints) in short expressions (bulleted list).• Do not worry about listing all the variables or all the

constraints at the beginning. As the formulation progresses, if you find you need a new variable or another constraint add it at that time.

Building Good ModelsBuilding Good ModelsA Check ListA Check List

3. First formulate constraints in the form: (Some expression) has (some relation) to (another expression or a constant)

• Keep units on both sides of the relation the same

• If the RHS is an expression, do the algebra to rewrite the constraint as:

(Some expression involving only linear terms ) has (some relation) to (a constant)

• Use summation variables and constraints to simplify the input and make it more easily readable.• Summation variables are particularly useful when

there are many constraints involving percentages.

4. Indicate which variables are:• ≥ 0, unrestricted, ≤ 0, integer, binary

Variables and Constraints With Variables and Constraints With PercentagesPercentages

Suppose in the formulation of a particular problem involving the production of four different styles of televisions, the modeler wished to express that no model was to represent more than 30% of the total production.

• The total production is X1 + X2 + X3 + X4

• Valid expressions of the constraints:

X1 ≤ .3(X1 + X2 + X3 + X4)

X2 ≤ .3(X1 + X2 + X3 + X4)

X3 ≤ .3(X1 + X2 + X3 + X4)

X4 ≤ .3(X1 + X2 + X3 + X4)

Rewriting the Percentage Rewriting the Percentage ConstraintsConstraints

These constraints can be rewritten as: .7X1 - .3X2 - .3X3 - .3X4 ≤ 0

-.3X1 + .7X2 - .3X3 - .3X4 ≤ 0

-.3X1 - .3X2 + .7X3 - .3X4 ≤ 0

-.3X1 - .3X2 - .3X3 + .7X4 ≤ 0

• Correct but:– Input of many coefficients – could make mistakes – One of the factors affecting the speed of solving linear

programs is the number of non-zero entries in the formulation

– Looking at these constraints does not instantaneously convey (by inspection) that each TV is to represent no more than 30% of the total production.

Using Summation Variables and Using Summation Variables and Summation ConstraintsSummation Constraints

• Define the summation variable, Xsummation variable, X55, to be the total production.– Immediately add the following summation summation

constraintconstraint that says X5 is the total production

X5 = X1 + X2 + X3 + X4 or X1 + X2 + X3 + X4 – X5 = 0

• The constraints can now be written as:X1 + X2 + X3 + X4 - X5 = 0

X1 - .3X5 ≤ 0

X2 - .3X5 ≤ 0

X3 - .3X5 ≤ 0

X4 - .3X5 ≤ 0

Summation Variables and Summation Variables and Summation ConstraintsSummation Constraints

• In this form the problem– Is easier to input with less chance for input error– Involves many 0 coefficients, with many of the

remaining coefficients being 1’s – the computer likes this

– Is easily readable – you can tell the constraints are saying that no model should be more than 30% of the total production

• But this does add one more variable and one more constraint to the model.– This also affects solution speed– In the Solver dialogue box, make sure you include:

• The summation variable as part of the “Changing Cells”• The summation constraint as part of the “Add Constraints”

Example 1Example 1Galaxy Industries ExpansionGalaxy Industries Expansion

• Galaxy Industries is planning an expansion and a move to Juarez, Mexico where both material and labor costs are cheaper.– It will also produced two additional products –• Big Squirts and Soakers

– Costs/Selling Prices:• Plastic – now only $1/lb• Other miscellaneous variable costs reduced by 50%• Labor

– Sunk Cost for Regular Time– $180 more per hour for each overtime hour (labor, other)

• Selling Prices for Space Rays/Zappers – reduced by $1/dozen

Example 1Example 1ConstraintsConstraints

– Constraints:• Plastic Availability – 3000 lbs./week• Production time (Regular time) – 40 hours/week• Overtime Availability – Up to 32 hours/week• Must satisfy a Zapper contract – at least 200 dz./week• New Product Mix Constraints

– Space Rays = 50% of total production– (Zappers, Big Squirts, Soakers) each ≤ 40% of production

• Minimum total production – 1000 dz./week

Example 1Example 1Profit/Resource RequirementsProfit/Resource Requirements

Selling Price

Costs Plastic ($3/lb) Other Variable Costs Total Profit Per Dozen

Production Minutes

DOZ Space Rays

$24

$ 6 (2 lb)$10=======$ 8

3

DOZ

Zappers

$26

$ 3 (1 lb)$18=======$ 5

4

DOZ Big Squirts

$29

$ 3 (3 lb)$ 6=======$20

5

DOZ Space Rays

$23

$ 2 (2 lb)$ 5=======$16

3

DOZ

Zappers

$25

$ 1 (1 lb)$ 9=======$15

4

-$1/doz

1

50% Reduction

DOZ Soakers

$36

$ 4 (4 lb)$10=======$22

6

Decision VariablesDecision Variables(Initial)(Initial)

• X1 = # dozen Space Rays produced per week

• X2 = # dozen Zappers produced per week

• X3 = # dozen Big Squirts produced per week

• X4 = # dozen Soakers produced per week

• X5 = # overtime hours scheduled per week

Objective FunctionObjective Function• Max Total Net Weekly Profit =

• Max Total Gross Weekly Profit – Weekly Cost of

Overtime

Gross Weekly ProfitProduct Profit Per Dozen Doz. Per Week Gross ProfitSpace Rays $16 X1 16X1

Zappers $15 X2 15X2 Big Squirts $20 X3 20X3

Soakers $22 X4 22X4 Weekly Cost of Overtime

Cost Per Overtime Hours Overtime CostOvertime Hour Scheduled Per Week $180 X5 180X5

OBJECTIVE FUNCTIONMAX 16X1 + 15X2 + 20X3 + 22X4 – 180X5

Plastic ConstraintPlastic Constraint

Total Amount of Plastic Used Per Week

≤

Plastic Available Per Week

Total Amount of Plastic Used Per Week≤

Plastic Available Per Week

2X1 + 1X2 + 3X3 + 4X4

3000

2X1 + 1X2 + 3X3 + 4X4 ≤ 3000

Production Time ConstraintProduction Time ConstraintTotal Production Minutes Used Per Week

≤Total Regular

Minutes Available+

Total OvertimeMinutes Scheduled

Total Production Minutes Used Per Week≤

Total RegularMinutes Available

+Total Overtime

Minutes Scheduled

3X1 + 4X2 + 5X3 + 6X4

60(40) = 2400 60X5

3X1 + 4X2 + 5X3 + 6X4 – 60 X5 ≤ 2400

Overtime AvailabilityOvertime Availability

The Number of Overtime Hours Scheduled/Week

≤

The Number of Overtime Hours Available/Week

The Number of Overtime Hours Scheduled/Week

≤

The Number of Overtime Hours Available/Week

X5

32

X5 ≤ 32

Zapper Contract ConstraintZapper Contract Constraint

The number of dozen Zappers produced/wk

≥

The number of dozen required by contract

The number of dozen Zappers produced/wk≥

The number of dozen required by contract

X2

200

X2 ≥ 200

Mix Constraints –Mix Constraints –Summation Variable/ConstraintSummation Variable/Constraint

• The next set of constraints involve percentages of the total production.

• Define X6 = Total Weekly Production

• Total Weekly Production = X1 + X2 + X3 + X4

• Thus the summation constraint is:

X1 + X2 + X3 + X4 – X6 = 0

Mix ConstraintsMix ConstraintsSpace Rays = 50% of total production

Zappers ≤ 40% of total production

Big Squirts ≤ 40% of total production

Soakers ≤ 40% of total production

Space Rays = 50% of total production

Zappers ≤ 40% of total production

Big Squirts ≤ 40% of total production

Soakers ≤ 40% of total production

X1

X2

X3

X4

.5X6

.4X6

.4X6

.4X6

X1 - .5X6 = 0X2 - .4X6 ≤ 0X3 - .4X6 ≤ 0X4 - .4X6 ≤ 0

The total number of dozen units produced/wk

≥

The minimum production limit

Minimum Total ProductionMinimum Total Production

X6

1000

X6 ≥ 1000

The total number of dozen units produced/wk

≥

The minimum production limit

The Complete ModelThe Complete Model• Including the nonnegativity of the variables

the complete linear programming model is:

MAX 16X1 + 15X2 + 20X3 + 20X4 - 180X5

s.t. 2X1 + 1X2 + 3X3 + 4X4 ≤ 3000 (Plastic)

3X1 + 4X2 + 5X3 + 6X4 - 60X5 ≤ 2400 (Time)

X5 ≤ 32 (Overtime)

X2 ≥ 200 (Contract)

X1 + X2 + X3 + X4 - X6 = 0 (Sum)

X1 - .5X6 = 0 (Sp Ray Mix)

X2 - .4X6 ≤ 0 (Zapper Mix)

X3 - .4X6 ≤ 0 (Big Sq Mix)

X4 - .4X6 ≤ 0 (Soaker Mix)

X6 ≥ 1000 (Min Total)

All X’s ≥ 0

=SUMPRODUCT($C$3:$H$3,C5:H5)Drag down

Solution/AnalysisSolution/Analysis

Produce Weekly565 dz. Space Rays200 dz. Zappers365 dz. Big Squirts0 SoakersTotal = 1130 dozen

All 32 overtimehours scheduled All time and overtime used.

Contract met exactly.Exactly 50% Space Rays.575 lbs. plastic unused.Profit = $13,580

Sensitivity AnslysisSensitivity Anslysis

Solution will notchange as long profit for

doz. Space Rays is between $4 and $20

Profit per dozen Soakersmust increase by $2.50(to $24.50) before it is

economically beneficialto produce them.

Extra overtime productionhours will add $90each to the profit.

This value is valid for totalovertime production hoursbetween 16.67 and 40.67.

ReviewReview

• Tips on building mathematical models.

• Use of summation variables and constraints.

• Solving a linear program with various constraint types and a summation variable and constraint.

• Interpreting the output.