The Management Science Approach Mathematical Modeling and Solution Procedures

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

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

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.

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

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

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

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

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

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

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

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

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.

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

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.

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.

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.