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.

Transportation Problems
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
Defining Variables

x ij shipment from source i to customer j

c ij cost to ship one unit from i to j.

d j demand at destination j.

S i Supply at source i
Formulation
M
Min
N
 c
i 1
ij
x ij
j 1
N
s .t .
Shipments out
can‘t exceed
supply
Demand must
be satisfied

x ij  S i
for i  1,  , M
j 1
M

x ij  d
j
for j  1,  , N
i 1
x ij  0 for all i , j
Transportation Costs c ij

Need to know costs
from each i, to each j
D
E
F
G
A
10
9
8
7
B
10
11
4
5
C
8
7
4
8
Creating Transp LP in Excel
Demand and Supply Constraints
Overview – Setting up the LP





Create a matrix of shipment costs (in grey in
example).
Create a matrix to hold the decision variables,
shipment quantities (in yellow).
Compute total cost of shipments using
SUMPRODUCT (in red).
Sum amount sent to each destination, enter
demand at each destination
Sum amount sent from each DC, enter supplies
available at each source
Adding Multiple Constraints at Once
LP Formulated
Solver Options

If you don’t check “assume non-negative” we get
the following results:
Sending -15 units from A to D, Solver
thinks we would have costs of 10*(-15).
So we would earn $150 by sending -15
units?
Bottom Line: Pretty much always click
“Assume Non-negative.”
Optimal Solution
Inequality Notation
Use <= for shipments from Sources.
 Use >= for shipments to customers.

 Do

we really need to?
What if supply is greater than demand?
 No
problem. Only send as much as you have
to. Some units just don’t get sent.
 Problem we just solved had this.
Product Shortages
If total demand exceeds total supply?
 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 - Demand Exceeds Supply
To modify the LP formulation:
-add row for Z costs (Row 6)
-add row for Z shipments (Row 12)
-change SumProduct (Total Costs)
to include cost of shortages
-In Solver dialog box, change variables
-In Solver, change constraints to add Z
Shortage Solution
Shortage Solution
Shortage is dealt with by shorting
customers D and E.
 Demand exceeds supply by 3 units. Our
first choice is to short D, because they are
the cheapest. We can only short them by
2, their total demand.
 Next, short E by 1 unit.

So What?
Where do we use this?
 If we have finite supplies (always), and
estimates of demand (usually) and costs
of each shipment (usually)
 If you have the data, this finds the perfect
answer
 Former student working for 3PL had to
figure out how to satisfy expected needs
for this new software. How to get copies to
distributors?

Summary of Transportation
Problems
Wrote LP to describe Transportation
Problem
 Formulated LP in Excel, solved with Solver
 Dealing with excess supply is no problem
 Dealing with shortage of supply, we have
to add imaginary source of “empty” boxes
