Operations Research Modeling Toolset Queueing Theory Markov Chains PERT/ CPM Network Programming Dynamic Programming Simulation Markov Decision Processes Inventory Theory Linear Programming Stochastic Programming Forecasting Integer Programming Decision Analysis
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Queueing Theory
Markov Chains
PERT/ CPM
Network Programming
Dynamic Programming
Simulation
Markov Decision Processes
Inventory Theory
Linear Programming
Stochastic Programming
Forecasting
Integer Programming
Decision Analysis
Nonlinear Programming
Game Theory
Special case: Project Management with PERT/CPM
Special case: Transportation and Assignment Problems
and many more
The Transportation Problem
Sources
Destinations
1
Supply s1
1
Demand d1
Supply s2
2
2
Demand d2
…
…
xij
n
Demand dn
Supply sm
m
Costs cij
Bellingham, WA, Eugene, OR, and Albert Lea, MN
Sacramento, CA, Salt Lake City, UT, Rapid City, SD, and Albuquerque, NM
1
2
3
3
2
1
4
Shipping cost per truckload
Let xij denote…
Minimize
subject to
For transportation problems, when every si and dj have an integer value, every BFS is integer valued.
Initialization
(Find initial CPF solution)
Is the current CPF solution optimal?
Yes
Stop
No
Move to a better adjacent CPF solution
Z = 10770
= (Second smallest cij in row/col)  (Smallest cij in row/col)
90
90
100
90
100
90
100
50
90
100
50
90
140
100
50
90
140
100
50
90
140
60
100
50
90
140
60
100
50
90
140
60
210
140
100
50
90
Z = 10330
using cij – ui – vj = 0 for xij basic, or ui + vj = cij
(let ui = 0 for row i with the largest number of basic variables)
60
140
210
140
100
50
90
(let ui = 0 for row i with the largest number of basic variables)
60
140
210
140
100
50
90
60
140
140
210
50
100
90
Sources
Destinations
N
Demand = 100
A
140
Supply = 200
60
S
Demand = 140
210
Supply = 350
B
140
E
Demand = 300
(shortage of 90)
100
Supply = 150
C
50
W
Demand = 250
Cost Z = 10330
100
100
40
300
10
150
90
100
100

+
40
300
10
150
+

90
?
100
100
40
210
100
150
90
100
100
40
210
100
150
90
+

100
100
+

210
100
40
?
150
90
140
60
40
210
100
150
90
140
60
40
210
100
150
90
Z = 10330
Sources
Destinations
N
Demand = 100
60
A
Supply = 200
140
S
Demand = 140
40
Supply = 350
B
210
E
Demand = 300
(shortage of 90)
100
Supply = 150
C
150
W
Demand = 250
Cost Z = 10330
Let xij denote…
Minimize
subject to
How to identify the optimal solution: