The Management Science Approach

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TheManagement 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

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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

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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 dont 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

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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

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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 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

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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 Xs ? 0

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The Complete Mathematical Model

• MIN 6X1 8X2 11X3 10X4 5X5 14X6 (in
  100s)
• S.T. X1 X2 X3
  ? 20
• X4 X5 X6 ? 30
• X1 X4
  12
• X2
  X5 15
• X3
  X6 22
• All Xs ?
  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
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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 100s) 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.