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

2. Building Good Models A Check List
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.)
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 Models A 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:
.7X X X X4 ≤ 0
-.3X1 + .7X X X4 ≤ 0
-.3X X2 + .7X X4 ≤ 0
-.3X X X3 + .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 + X X5 = 0
X X5 ≤ 0
X X5 ≤ 0
X X5 ≤ 0
X X5 ≤ 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 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”

8. Example 1 Galaxy 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 1 Constraints 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 1 Profit/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
Zappers
$26
$ 3 (1 lb)
$18
$ 5 4
DOZ
Space
Rays
$23
$ 2 (2 lb)
$ 5
=======
$16 3
Zappers
$25
$ 1 (1 lb)
$ 9
$15 4
DOZ
Big
Squirts
$29
$ 3 (3 lb)
$ 6
=======
$20 5
DOZ
Soakers
$36
$ 4 (4 lb)
$10
=======
$22 6
-$1/doz 1
50% Reduction

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 $ X X1
Zappers $ X X2
Big Squirts $ X X3
Soakers $ X X4
Weekly Cost of Overtime
Cost Per Overtime Hours Overtime Cost
Overtime Hour Scheduled Per Week
$ X X5
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 ≤
Total Regular Minutes Available +
Total Overtime
Minutes Scheduled
3X1 + 4X2 + 5X3 + 6X4
Total Production Minutes Used Per Week ≤
Total Regular Minutes Available +
Total Overtime
Minutes Scheduled
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 X1
.5X6 X2
.4X6 X3
.4X6 X4
.4X6
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 + 22X X5
s.t X X X X4 ≤ 3000 (Plastic)
3X X X X X5 ≤ 2400 (Time)
X5 ≤ (Overtime)
X2 ≥ (Contract)
X X X X X6 = (Sum)
X X6 = (Sp Ray Mix)
X X6 ≤ (Zapper Mix)
X X6 ≤ (Big Sq Mix)
X X6 ≤ (Soaker Mix)
X6 ≥ 1000 (Min Total)
All X’s ≥ 0

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

22. Solution/Analysis Produce Weekly All 32 overtime 565 dz. Space Rays
200 dz. Zappers
365 dz. Big Squirts
Soakers
Total = 1130 dozen
All 32 overtime
hours scheduled
Profit = $13,580
All time and overtime used.
Contract met exactly.
Exactly 50% Space Rays.
575 lbs. plastic unused.

23. Sensitivity Anslysis Solution will not change as long profit for
doz. Space Rays is
between $4 and $20
Profit per dozen Soakers
must increase by $2.50
(to $24.50) before it is
economically beneficial
to produce them.
Extra overtime production
hours will add $90
each to the profit.
This value is valid for total
overtime production hours
between and

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.