1. Chapter 7 Transportation, Assignment, and Transshipment Problems
Math

2. Special Types of LPs
Have used linear programming to formulate and solve two types of problem:
1. Determining optimal production processes (choosing a mix of products which maximize profits given limitations in resources)
2. Meeting minimum specifications (diet example: selecting foods which meet nutritional requirements at minimum cost. Others - allocating health care resources)
Amazing feature of this model is its versatility: many
different, seemingly unrelated, problems can be modeled as LPs

3. Transportation Problem
A given (homogeneous) commodity is to be shipped from a set of different supply centers to another set of different destinations E.g. you make Honda Accords at 5 factories around the US and distribute the cars to 100 dealers this month E.g. tanks from 10 military bases in the US are to be shipped to Iraq, Afghanistan, and Bosnia

4. Transportation Problem
Typical data format:
m supply centers
n destinations
si = amount of commodity at supply center i
dj = amount of commodity required by destination j
cij = cost to ship one unit from supply center i to destination j 1 2
... n
supply
c11
c12
c1n s1
c21
c22
c2n s2 m
cm1
cm2
cmn sm
demand d1 d2 dn

5. Example
Peas are canned in Washington, Oregon, and Minnesota and shipped to warehouses in California, Utah, South Dakota, and New Mexico warehouse cannery What are the decision variables? Objective function? Constraints? 1 2 3 4
supply
465
513
654
867 75
352
416
690
791
1 25
995
682
388
685
100
demand 80 65 70 85

6. AN LP!
xij = amount shipped from source i to destination j
minimize  cij xij (total shipping costs)
subject to xij = si (use up all the supply)
 xij = dj (meet all the demand) note: this assumes si =  dj
 Special form:
 x11 + x x1n = s1
x21 + x x2n = s2
...
xm1 + xm xmn = sm
x x xm1 = d1
x x xm2 = d2

7. Cannery
MINIMIZE 465 x11+513x12+654x13+867x14+352x21+416x22+690x23+791x x31+682x32+388x33+685x34 SUBJECT TO x11+x12+x13+x14 = 75 x21+x22+x23+x24 = 125 x31+x32+x33+x34 = 100 x11 +x21 +x31 = 80 x12 +x22 +x32 = 65 x13 +x23 +x33 = 70 x14 +x24 +x34 = 85.

8. Observations
Solutions must be integers Special format can be exploited in solving the lp si >  dj can be handled by using a dummy destination Suppose warehouse 4 only wants 65 units warehouse cannery The special form of the problem leads to efficiencies in solving Won't cover 1 2 3 4
dummy
supply
465
513
654
867 75
352
416
690
791
1 25
995
682
388
685
100
demand 80 65 70 20

9. Assignment Problem
How do you assign n people to n jobs? Assume there is a cost, cij, associated with person i doing job j Job Person How is this a transportation problem? 1 2
... n
c11
c12
c1n
c21
c22
c2n
cn1
cn2
cnn

10. Assignment Problem
Each person can do one job, si = 1 Each job is done by one person , dj = 1 cij = cost or benefit when person i does job j Job Person E.g. 5 students in a class divide a 5 problem exam 1 2
... n
supply
c11
c12
c1n
c21
c22
c2n
cn1
cn2
cnn
demand

11. Transshipment Problems
Shipments are allowed to go through intermediate destinations Can be solved as a transportation problem Won't cover