1. The Management Science Approach
Mathematical Modeling and Solution Procedures

2. Building a Mathematical Model
A mathematical model can consist of:
A set of decision variables
An objective function
Constraints
Functional
Nonnegativity Constraints
First create a model shell
Gather data -- consider time/cost issues
for collecting, organizing, sorting data
for generating a solution
for using the model

3. EXAMPLE
Suppose a motorhome compnay has inventory of motorhomes at their production facilities which they wish to transport to various retail dealerships -- at minimum cost
Motorhomes will be driven one at a time from a production facility to a dealership

4. Issues How many production facilities are there?
What is the supply at each facility?
How many dealerships desire the motorhomes?
How many did each dealership order?
Transportation costs between each production facility and each dealership
Consider mileage, salaries, tolls, insurance, etc.

5. Pictorial Model P2 P1 R3 R1 R2 $6 Demand 12 15 22 Supply 20 30 $5 $10
Production Facilities P1 R3 R1 R2
Retail
Dealerships $6
Transportation Costs ($100’s)
Demand 12 15 22
Supply 20 30 $5
$10
$14
$11 $8

6. Definition of Decision Variables
X1 = amount transported from P1 to R1
X2 = amount transported from P1 to R2
X3 = amount transported from P1 to R3
X4 = amount transported from P2 to R1
X5 = amount transported from P2 to R2
X6 = amount transported from P2 to R3

7. Objective/Objective Function
Objective -- Minimize Total Transportation Cost
It costs $600 to drive a motorhome from P1 to R1
How many will we send from P1 to R1?
We don’t know
But the symbol for the amount we ship from P1 to R1 is X1
Thus the total cost of transporting motorhomes from P1 to R1 is $600X1
Other costs are similarly figured
Thus the objective function is:
MIN 600X X X X X X6

8. Production Facility Constraints
From each production, we cannot transport more motorhomes from a facility than are available.
How many will we transport from P1?
We will transport X1 to R1, X2 to R2, and X3 to R3
Thus the total amount we transport from P1 is:
X1 + X2 + X3
What is the maximum we transport from P1?
The supply which is 20
Thus we have the following constraint for P1:
X1 + X2 + X3  20
Similarly for P2: X4 + X5 + X6  30

9. Retail Dealership Constraints
Each dealership should receive exactly the number of orders it placed
How many motorhomes will R1 receive
It will receive X1 from P1 and X4 from P2
This should equal their order -- 12
Thus, the constraint for S1 is:
X1 + X4 = 12
Similarly for dealerships R2 and R3:
R2: X2 + X5 = 15
R3: X3 + X6 = 22

10. Nonnegativity Constraints
We cannot transport a negative number of motorhomes from a production facility to a retail dealership.
Thus:
X1  0, X2  0, X3  0, X4  0, X5  0, X6  0
We write this simply as: All X’s  0

11. The Complete Mathematical Model
MIN 6X1 + 8X2 + 11X X4 + 5X5 + 14X6 (in $100’s)
S.T. X1 + X X  20
X4 + X X  30
X X = 12
X X = 15
X X = 22
All X’s  0

12. Model Solution Choose an appropriate solution technique
Generate model solution
Test/validate model results
Return to modeling if results are unacceptable
Perform “what-if” analyses

13. Solution to the Model
Our example fits the requirements for what is called a transportation model.
We can use an approach called linear programming or us a template that is specifically designed to solved these specially structured problems.

14. INPUT
Supplies
Demands
Unit Costs
OUTPUT
Total Shipped
Total Received
Amount Transported
Total
Transportation
cost

15. Analysis From P1 send 12 to R1 and 8 to R3
1 motorhome remains in P2
The total cost is $431 (in $100’s) or $43,100
Any other transportation scheme would cost more.

16. Review Solution
When the model is solved it should be reviewed to check for any obvious inconsistencies.
If the model is not performing as expected it can be changed at this time
This solve/review process continues until the model produces “reasonable” results.
If this does not happen the problem definition may have to be re-visited
Additional experts can add input.
The results are then ready for reporting and implementation.

17. Review Mathematical models consist of:
Decision variables
Objective function
Constraints
Model shell should be built prior to collecting data.
Models are solved and checked to see if results are reasonable
Revision/additional input may be needed.
Model results are reported to decision maker.