Linear Programming

1
Linear Programming

• Building Good Linear Models
• And
• Example 1
• Sensitivity Analyses, Unit Conversion, Summation
  Variables

2
Building Good ModelsA Check List

• Determine in general terms what the objective is
  (the objective function) and what factors are
  under the decision makers 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.)
• 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.

3
Building Good ModelsA Check List

• 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.
• Indicate which variables are
• 0, unrestricted, 0, integer, binary

4
Variables and Constraints With Percentages

• 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)

5
Rewriting the Percentage Constraints

• 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.

6
Using Summation Variables and Summation
Constraints

• Define the summation variable, X5, to be the
  total production.
• Immediately add the following summation
  constraint 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

7
Summation Variables and Summation 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 1s 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

8
Example 1Galaxy 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

9
Example 1Constraints

• 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

10
Example 1Profit/Resource Requirements

11
Decision Variables(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

12
Objective Function

• Max Total Net Weekly Profit
• Max Total Gross Weekly Profit
  Weekly Cost of Overtime

Gross Weekly Profit Product Profit
Per Dozen Doz. Per Week Gross Profit Space
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 Cost Overtime Hour
Scheduled Per
Week 180 X5 180X5
OBJECTIVE FUNCTION MAX 16X1 15X2 20X3 22X4
180X5
13
Plastic Constraint

• Total Amount of Plastic Used Per Week
• Plastic Available Per Week

2X1 1X2 3X3 4X4
Total Amount of Plastic Used Per Week Plastic
Available Per Week
3000
2X1 1X2 3X3 4X4 3000
14
Production Time Constraint

• Total Production Minutes Used Per Week

3X1 4X2 5X3 6X4
60(40) 2400
60X5
3X1 4X2 5X3 6X4 60 X5 2400
15
Overtime Availability

• The Number of Overtime Hours Scheduled/Week
• The Number of Overtime Hours Available/Week

X5
The Number of Overtime Hours Scheduled/Week The
Number of Overtime Hours Available/Week
32
X5 32
16
Zapper Contract Constraint

• The number of dozen Zappers produced/wk
• The number of dozen required by contract

X2
The number of dozen Zappers produced/wk The
number of dozen required by contract
200
X2 200
17
Mix Constraints Summation 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
18
Mix Constraints

• Space 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

.5X6
X1
.4X6
X2
.4X6
X3
.4X6
X4
X1 - .5X6 0 X2 - .4X6 0 X3 - .4X6
0 X4 - .4X6 0
19
Minimum Total Production
The total number of dozen units produced/wk The
minimum production limit

The total number of dozen units produced/wk The
minimum production limit
X6
1000
X6 1000
20
The 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 Xs 0
21
(No Transcript)
22
Solution/Analysis

23
Sensitivity Anslysis

24
Review

• 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.