Presentation on theme: "The Management Science Approach"— Presentation transcript:
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 variablesAn objective functionConstraintsFunctionalNonnegativity ConstraintsFirst create a model shellGather data -- consider time/cost issuesfor collecting, organizing, sorting datafor generating a solutionfor using the model
3 EXAMPLESuppose a motorhome compnay has inventory of motorhomes at their production facilities which they wish to transport to various retail dealerships -- at minimum costMotorhomes 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 dealershipConsider mileage, salaries, tolls, insurance, etc.
6 Definition of Decision Variables X1 = amount transported from P1 to R1X2 = amount transported from P1 to R2X3 = amount transported from P1 to R3X4 = amount transported from P2 to R1X5 = amount transported from P2 to R2X6 = amount transported from P2 to R3
7 Objective/Objective Function Objective -- Minimize Total Transportation CostIt costs $600 to drive a motorhome from P1 to R1How many will we send from P1 to R1?We don’t knowBut the symbol for the amount we ship from P1 to R1 is X1Thus the total cost of transporting motorhomes from P1 to R1 is $600X1Other costs are similarly figuredThus 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 R3Thus the total amount we transport from P1 is:X1 + X2 + X3What is the maximum we transport from P1?The supply which is 20Thus we have the following constraint for P1:X1 + X2 + X3 20Similarly for P2: X4 + X5 + X6 30
9 Retail Dealership Constraints Each dealership should receive exactly the number of orders it placedHow many motorhomes will R1 receiveIt will receive X1 from P1 and X4 from P2This should equal their order -- 12Thus, the constraint for S1 is:X1 + X4 = 12Similarly for dealerships R2 and R3:R2: X2 + X5 = 15R3: 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 0We 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 20X4 + X X 30X X = 12X X = 15X X = 22All X’s 0
12 Model Solution Choose an appropriate solution technique Generate model solutionTest/validate model resultsReturn to modeling if results are unacceptablePerform “what-if” analyses
13 Solution to the ModelOur 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.
15 Analysis From P1 send 12 to R1 and 8 to R3 1 motorhome remains in P2The total cost is $431 (in $100’s) or $43,100Any other transportation scheme would cost more.
16 Review SolutionWhen 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 timeThis solve/review process continues until the model produces “reasonable” results.If this does not happen the problem definition may have to be re-visitedAdditional experts can add input.The results are then ready for reporting and implementation.
17 Review Mathematical models consist of: Decision variablesObjective functionConstraintsModel shell should be built prior to collecting data.Models are solved and checked to see if results are reasonableRevision/additional input may be needed.Model results are reported to decision maker.