The Management Science Approach

1. TheManagement 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 ProductionFacilities Retail Dealerships TransportationCosts ($100’s) Demand Supply $6 R1 12 20 $10 P1 $8 R2 15 $5 $11 P2 30 $14 R3 22

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 600X1 + 800X2 + 1100X3 + 1000X4 + 500X5 + 1400X6

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 + 11X3 + 10X4 + 5X5 + 14X6 (in $100’s) S.T. X1 + X2 + X3 20 X4 + X5 + X6  30 X1 + X4 = 12 X2 + X5 = 15 X3 + X6 = 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 Total Transportation cost OUTPUT Total Shipped Total Received Amount Transported

15. Analysis • From P1 send 12 to R1 and 8 to R3 • From P2 send 15 to R2 and 14 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.