Transportation and Distribution Planning

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Transportation and Distribution Planning

• Matthew J. Liberatore
• John F. Connelly Chair in Management
• Professor, Decision and Information Techologies

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TRANSPORTATION PROBLEM

• Mathematical programming has been successfully
  applied to important supply chain problems.
• These problems address the movement of products
  across links of the supply chain (supplier,
  manufacturers, and customers).
• We now focus on supply chain applications in
  transportation and distribution planning.

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TRANSPORTATION PROBLEM

• A manufacturer ships TV sets from three
  warehouses to four retail stores each week.
  Warehouse capacities (in hundreds) and demand (in
  hundreds) at the retail stores are as follows
• Capacity Demand
• Warehouse 1 200 Store 1 100
• Warehouse 2 150 Store 2 200
• Warehouse 3 300 Store 3 125
• 650 Store 4 225
• 650

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TRANSPORTATION PROBLEM

• The shipping cost per hundred TV sets for each
  route is given below
• To
• From Store 1 Store 2 Store 3 Store 4
• warehouse 1 10 5 12 3
• warehouse 2 4 9 15 6
• warehouse 3 15 8 6 11

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TRANSPORTATION PROBLEM

• What are the decision variables?
• XIJnumber of TV sets (in cases) shipped from
  warehouse I to store J
• I is the index for warehouses (1,2,3)
• J is the index for stores (1,2,3,4)

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TRANSPORTATION PROBLEM

• What is the objective?
• Minimize the total cost of transportation which
  is obtained by multiplying the shipping cost by
  the amount of TV sets shipped over a given route
  and then summing over all routes
• OBJECTIVE FUNCTION 
• MIN 10X115X1212X13 3X14
• 4 X219X2215X23 6X24
• 15X318X32 6X3311X34

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TRANSPORTATION PROBLEM

• How are the supply constraints expressed?
• For each warehouse the amount of TV sets shipped
  to all stores must equal the capacity at the
  warehouse
• X11X12X13X14200 SUPPLY CONSTRAINT FOR
  WAREHOUSE 1
• X21X22X23X24150 SUPPLY CONSTRAINT FOR
  WAREHOUSE 2
• X31X32X33X34300 SUPPLY CONSTRAINT FOR
  WAREHOUSE 3

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TRANSPORTATION PROBLEM

• How are the demand constraints expressed?
• For each store the amount of TV sets shipped
  from all warehouses must equal the demand of the
  store
• X11X21X31100 DEMAND CONSTRAINT FOR STORE 1
• X12X22X32200 DEMAND CONSTRAINT FOR STORE 2
• X13X23X33125 DEMAND CONSTRAINT FOR STORE 3
• X14X24X34225 DEMAND CONSTRAINT FOR STORE 4

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TRANSPORTATION PROBLEM

• Since partial shipment cannot be made, the
  decision variables must be integer valued
• However, if all supplies and demands are
  integer-valued, the values of our decision
  variables will be integer valued

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TRANSPORTATION PROBLEM

• After solution in Solver
• The total shipment cost is 3500, and the optimal
  shipments are warehouse 1 ships 25 cases to
  store 2 and 175 to store 4 warehouse 2 ships 100
  to store 1 and 50 to store 4, and warehouse 3
  ships 175 to store 2 and 125 to store 3.
• The reduced cost of X11 is 9, so the cost of
  shipping from warehouse 1 to store 1 would have
  to be reduced by 9 before this route would be
  used

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UNBALANCED PROBLEMS

• Suppose warehouse 2 actually has 175 TV sets.
  How should the original problem be modified?
• Since total supply across all warehouses is now
  greater than total demand, all supply constraints
  are now lt
• Referring to the original problem, suppose store
  3 needs 150 TV sets. How should the original
  problem be modified?
• The demand constraints are now lt

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RESTRICTED ROUTE

• Referring to the original problem, suppose there
  is a strike by the shipping company such that the
  route from warehouse 3 to store 2 cannot be used.
• How can the original problem be modified to
  account for this change?
• Add the constraint
• X320

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WAREHOUSE LOCATION

• Suppose that the warehouses are currently not
  open, but are potential locations.
• The fixed cost to construct warehouses and their
  capacity values are given as
• WAREHOUSES FIXED COST CAPACITY
• Warehouse 1 125,000 300
• Warehouse 2 185,000 525
• Warehouse 3 100,000 325

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WAREHOUSE LOCATION

• How do we model the fact that the warehouses may
  or may not be open?
• Define a set of binary decision variables YI, I
  1,2,3, where warehouse I is open if YI 1 and
  warehouse I is closed if YI 0

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WAREHOUSE LOCATION

• How must the objective function change?
• Additional terms are added to the objective
  function which multiply the fixed costs of
  operating the warehouse by YI and summing over
  all warehouses I
• 125000Y1185000Y2100000Y3
• Why cant we use the current capacity
  constraints?
• Product cannot be shipped from a warehouse if it
  is not open. Since the capacity is available
  only if the warehouse is open, we multiply
  warehouse 1s capacity by Y1.

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WAREHOUSE LOCATION

• Also, we must make the YI variables binary
  integer
• Total fixed and shipping costs are 289,100
  warehouses 2 and 3 are open warehouse 2 ships
  100 to store 1, 225 to store 4 and warehouse 3
  ships 200 to store 2 and 125 to store 4