Transportation Problems

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Transportation Problems Powered By Docstoc
Dr. Ron Lembke
Transportation Problems
 Linear programming is good at solving
  problems with zillions of options, and
  finding the optimal solution.
 Could it work for transportation problems?
 Costs are linear, and shipment quantities
  are linear, so maybe so.
Defining Variables
 Define cij as the cost to ship one unit from
  i to j.
 Demand at location j is dj.
 Supply at DC i is Si
 Xij is the quantity shipped from DC i to
  customer j.
         M        N
  Min           c     ij   xij
         i 1    j 1
  s.t.   x
          j 1
                 ij    Si         f or i  1,  , M


         i 1
                 ij    d j f or j  1,  , N

         xij  0 f or all i, j
Transportation Method
You have 3 DCs, and need to deliver
 product to 4 customers.         D2

   A 10
   B 10
                                 F 12
   C 10

                                 G 11

Find cheapest way to satisfy all demand
Solving Transportation Problems

   Trial and Error            D    E    F   G
   Linear Programming
    – ooh, what’s that?!   A   10   9    8   7
   Tell me more!
                           B   10   11   4   5

                           C   8    7    4   8
Setting up LP
   Create a matrix of shipment costs (in grey in
   Create a matrix to hold the decision variables,
    shipment quantities (in yellow).
   Sum amount sent to each destination.
   Sum amount sent from each DC.
   Enter demands and supplies at each location.
   Compute total cost of shipments (in blue).
Using Solver
   If you don’t check “assume non-negative” we get
    the following results:

   Solver doesn’t converge to an optimal solution.
    Why not?
 Use <= for shipments from DCs.
 Use >= for shipments to customers.
     Do   we really need to?
   What do we do if supply is greater than
Product Shortages
 If total demand is greater than total supply,
  what happens?
 If demand in G is 15, we get this:
Product Shortages
 If demand at G is 15, there are no feasible
  solutions, much less a best one.
 We need to add a phantom source, Z, with
  huge capacity. Think of it as a supplier
  that ships empty boxes.
 Now supply can satisfy total demand.
Shortage Costs
   What cost should we use for supplier Z?
   It should be the last resort, so it should be higher
    than any real costs.
   The cost of a shipment from Z is really the cost
    of shorting the customer.
   If all customers are created equal, give them all
    the same shortage cost.
   If some are more important, give them higher
    shortage costs, and we’ll only short them as a
    last resort.
Shortage Solution
 Shortage is dealt with by shorting
  customer A, and B.
 Demand exceeds supply by 3 units. Our
  first choice is to short A, because they are
  the cheapest. We can only short them by
  2, their total demand.
 Next, short B by 1 unit.

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