Transportation Problems

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```					Transportation
Problems
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.
Formulation
M        N
Min           c     ij   xij
i 1    j 1
N
s.t.   x
j 1
ij    Si         f or i  1,  , M

M

x
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
E4
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
example).
   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?
Inequalities
 Use <= for shipments from DCs.
 Use >= for shipments to customers.
 Do   we really need to?
   What do we do if supply is greater than
demand?
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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 views: 12 posted: 8/27/2012 language: English pages: 20