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The Management Science Approach Mathematical Modeling and Solution Procedures

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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

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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

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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.

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

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Definition of Decision Variables X 1 = amount transported from P1 to R1 X 2 = amount transported from P1 to R2 X 3 = amount transported from P1 to R3 X 4 = amount transported from P2 to R1 X 5 = amount transported from P2 to R2 X 6 = amount transported from P2 to R3

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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 X 1 –Thus the total cost of transporting motorhomes from P1 to R1 is $600X 1 –Other costs are similarly figured Thus the objective function is: MIN 600X X X X X X 6

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Production Facility Constraints 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 X 1 to R1, X 2 to R2, and X 3 to R3 Thus the total amount we transport from P1 is: X 1 + X 2 + X 3 –What is the maximum we transport from P1? The supply which is 20 Thus we have the following constraint for P1: X 1 + X 2 + X 3 20 X 4 + X 5 + X 6 30Similarly for P2: X 4 + X 5 + X 6 30

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Retail Dealership Constraints Each dealership should receive exactly the number of orders it placed –How many motorhomes will R1 receive It will receive X 1 from P1 and X 4 from P2 This should equal their order Thus, the constraint for S1 is: X 1 + X 4 = 12 Similarly for dealerships R2 and R3: R2: X 2 + X 5 = 15 R3: X 3 + X 6 = 22

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Nonnegativity Constraints We cannot transport a negative number of motorhomes from a production facility to a retail dealership. Thus: X 1 0, X 2 0, X 3 0, X 4 0, X 5 0, X 6 0 We write this simply as: All X’s 0

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The Complete Mathematical Model MIN 6X 1 + 8X X X 4 + 5X X 6 (in $100’s) S.T.X 1 + X 2 + X 3 20 X 4 + X 5 + X 6 30 X 1 + X 4 = 12 X 2 + X 5 = 15 X 3 + X 6 = 22 All X’s 0

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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

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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.

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

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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.

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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.

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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.

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