1. IE 311 Operations Research– I
Instructor: Dr. Mohamed Mostafa

2. Chapter 6: Distribution and Network Models

3. Introduction
A special class of linear programming problems is called Network Flow problems. Five different problems are considered:
Transportation Problems
Assignment Problems
Transshipment Problems
Shortest-Route Problems
Maximal Flow Problems

4. Transportation, Assignment, and Transshipment Problems
They are all network models, represented by a
set of nodes,
a set of arcs,
and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes.

5. In this chapter: Transportation & Assignment Problems
Network Representation
General LP Formulation
Solution with QM Software
Transshipment Problem

6. Transportation Problem
The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij.
The network representation for a transportation problem with two sources and three destinations is given on the next slide.

7. Transportation Problem
Network Representation 1 d1
c11 1
c12 s1
c13 2 d2
c21 2
c22 s2
c23 3 d3
Sources
Destinations

8. Transportation Problem
Linear Programming Formulation
Using the notation:
xij = number of units shipped from
origin i to destination j
cij = cost per unit of shipping from
si = supply or capacity in units at origin i
dj = demand in units at destination j

9. Transportation Problem
Linear Programming Formulation (continued) =
xij > 0 for all i and j

10. Solve as a maximization problem.
LP Formulation Special Cases
The objective is maximizing profit or revenue:
Minimum shipping guarantee from i to j:
xij > Lij
Maximum route capacity from i to j:
xij < Lij
Unacceptable route:
Remove the corresponding decision variable.
Solve as a maximization problem.

11. Acme Transportation Problem: Example #1
Acme Block Company has orders for 80 tons of
concrete blocks at three suburban locations as follows: Northwood tons,
Westwood tons, and
Eastwood tons.
Acme has two plants, each of which can produce 50 tons per week. Delivery cost per ton from each plant to each suburban location is shown on the next slide.
How should end of week shipments be made to fill
the above orders?

12. Delivery Cost Per Ton
Northwood Westwood Eastwood
Plant
Plant

13. Demand Unit Costs Supply 25 24 50 30 40 45 30 40 50 42 10 Origins
Network Representation
Demand
Unit Costs
Supply 1
North 25 24 1
Plant 1 50 30 40
2 West 45 30 2
Plant 2 40 50 42 3
East 10
Origins
Destinations

14. s.t. x11 + x12 + x13 < 50 (Plant 1 capacity)
Shipment Cost Minimization / LP Formulation
Decision variables: xij = Amount shipped for i = 1-2 and j = 1-3
Min Z = 24x x x x x x23
s.t. x11 + x12 + x13 < 50 (Plant 1 capacity)
x21 + x22 + x23 < 50 (Plant 2 capacity)
x11 + x21 = 25 (Northwood demand) x12 + x22 = 45 (Westwood demand)
x13 + x23 = 10 (Eastwood demand)
xij > 0 for i = 1-2 origins, j = 1-3 destinations

15. QM Input

16. QM Solution

18. Optimal Solution (Minimizing Cost)
From To Amount Cost
Plant 1 Northwood
Plant 1 Westwood ,350
Plant 2 Northwood
Plant 2 Eastwood
Total Cost = $2,490
Notes: Supply from Plant 2 is not fully utilized (excess shown in ‘Dummy’ column in QM). Also, nothing is shipped from Plant 1 to Eastwood or from Plant 2 to Westwood. The unit cost for that route must decrease more than the marginal cost (shown in QM) in order to be utilized.

19. QM Output (using “Transportation” Module)
Transportation Shipments Window
1) Which plant has excess supply? How much?
2) What the unit cost of shipments from Plant 1 to Eastwood would have to be in order for that route to be utilized (original cost = $40)
Marginal Costs Window

20. Transportation Problem: Example #2
The Navy has 9,000 pounds of material in Albany,
Georgia that it wishes to ship to three destinations:
San Diego, Norfolk, and Pensacola. They
require 4,000, 2,500, and 2,500 pounds,
respectively. There are three different carriers
and government regulations
require equal distribution of shipping
among the three carriers.

21. The shipping costs per pound for truck, railroad,
and airplane transit are shown below.
Formulate and solve a linear program to determine the shipping arrangements (mode, destination, and quantity) that will minimize the total shipping cost.
Destination
Mode San Diego Norfolk Pensacola
Truck $ $ $ 5
Railroad
Airplane

22. Decision Variables
We want to determine the pounds of material, xij , to be shipped by mode i to destination j. The following table summarizes the decision variables:
San Diego Norfolk Pensacola
Truck x x x13
Railroad x x x23
Airplane x x x33

23. Objective Function
Minimize the total shipping cost.
Min: (shipping cost per pound for each mode per destination pairing) x (number of pounds shipped by mode per destination pairing).
Min: 12x11 + 6x12 + 5x x x22 + 9x23
+ 30x x x33

24. Constraints
Equal use of transportation modes:
(1) x11 + x12 + x13 = 3000
(2) x21 + x22 + x23 = 3000
(3) x31 + x32 + x33 = 3000
Destination material requirements:
(4) x11 + x21 + x31 = 4000
(5) x12 + x22 + x32 = 2500
(6) x13 + x23 + x33 = 2500
Non-negativity of variables:
xij > 0, i = 1,2,3 and j = 1,2,3

25. QM Output

26. Solution Summary
San Diego will receive 1000 lbs. by truck and 3000 lbs. by airplane.
Norfolk will receive 2000 lbs. by truck and 500 lbs. by railroad.
Pensacola will receive 2500 lbs. by railroad.
The total shipping cost will be $142,000.

27. Practice Problem
Powerco has three electric power plants that supply the electric needs of four cities. The associated supply of each plant and demand of each city is given in the table. The cost of sending 1 million kwh of electricity from a plant to a city depends on the distance the electricity must travel. Determine how much energy should be supplied to the city by which plant?

29. Practice Problem

30. Costs to Haul from each site to each mill (round-trip costs) are given below:

31. Assignment Problem
An assignment problem seeks to minimize the total cost assignment of i workers to j jobs, given that the cost of worker i performing job j is cij.
It assumes all workers are assigned and each job is performed.
An assignment problem is a special case of a transportation problem in which all supplies and all demands are equal to 1; hence assignment problems may be solved as linear programs.
The network representation of an assignment problem with three workers and three jobs is shown on the next slide.

32. c11 c12 c13 Agents Tasks c21 c22 c23 c31 c32 c33
Network Representation
c11 1 1
c12
c13
Agents
Tasks
c21
c22 2 2
c23
c31
c32 3 3
c33

33. Linear Programming Formulation
Using the notation:
xij = if agent i is assigned to task j
0 otherwise
cij = cost of assigning agent i to task j

34. Linear Programming Formulation (continued)
xij > 0 for all i and j

35. LP Formulation Special Cases
Number of agents exceeds the number of tasks:
Number of tasks exceeds the number of agents:
Add enough dummy agents to equalize the number of
agents and the number of tasks. The objective function
coefficients for these new variable would be zero.
Extra agents simply remain unassigned.

36. LP Formulation Special Cases (continued)
The assignment alternatives are evaluated in terms of revenue or profit:
Solve as a maximization problem.
An assignment is unacceptable:
Remove the corresponding decision variable.
An agent is permitted to work t tasks:

37. Practice Problem
An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects.
Projects
Subcontractor A B C
Westside
Federated
Goliath
Universal
How should the contractors be assigned to minimize total mileage costs?

38. Network Representation50
West. A 36 16
Subcontractors
Projects 28
Fed. 30 B 18 35 32
Gol. C 20 25 25
Univ. 14

39. Linear Programming Formulation Min 50x11+36x12+16x13+28x21+30x22+18x23
s.t. x11+x12+x13 < 1
x21+x22+x23 < 1
x31+x32+x33 < 1
x41+x42+x43 < 1
x11+x21+x31+x41 = 1
x12+x22+x32+x42 = 1
x13+x23+x33+x43 = 1
xij = 0 or 1 for all i and j
Agents
Tasks
Note: If #agents = #tasks, first 4 constraints would have been “= 1” instead of “< 1”

40. QM Select the “Assignment” Module “Jobs” are the same as Projects
“Machines” are whatever/whoever is performing the Jobs
If assignments are being evaluated for
Cost  Select Minimize
Revenue/Profit  Select Maximize

42. Problem
Machineco has four machine and four jobs to be completed. Each machine must be assigned to complete one job. The time required to set up each machine for completing each job is shown below. Formulate the problem to minimize the total setup time needed to complete the four jobs.
Job1 Job2 Job3 Job4
Machine
Machine
Machine
Machine

43. Problem
Three professors must be assigned to teach six sections of Management Science. Each professor can teach up to two sections. Each has ranked the six time periods during which MS is taught. Higher the ranking, higher is the desirability of the professor to teach at that time. Determine an assignment of professors to sections.
9AM PM
Professor
Professor
Professor

44. Transshipment Problems

45. Transshipment Problem
Transshipment problems are transportation problems in which a shipment may move through intermediate nodes (transshipment nodes) before reaching a particular destination node.
The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide.

46. c36 c13 c37 s1 d1 c14 c15 c46 c47 Demand Supply c23 c24 c56 d2 s2 c25
Network Representation
c36 3
c13 1
c37 6 s1 d1
c14
c15
c46 4
c47
Demand
Supply
c23 2
c24
c56 7 d2 s2
c25 5
c57
Sources
Destinations
Intermediate Nodes

47. Linear Programming Formulation
Using the notation:
xij = number of units shipped from node i to node j
cij = cost per unit of shipping from node i to node j
si = supply at origin node i
dj = demand at destination node j

48. Linear Programming Formulation (continued)
xij > 0 for all i and j

49. Special Cases
Total supply not equal to total demand
Maximization objective function
Route capacities or route minimums
Unacceptable routes
The LP model modifications required here are
identical to those required for the special cases in
the transportation problem.

50. Transshipment Problem: Example
The Northside and Southside facilities of Zeron Industries supply three firms (Zrox, Hewes, Rockrite) with customized shelving for its offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc.
Currently weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for Rockrite. Both Arnold and Supershelf can supply at most 75 units to its customers.
Additional data is shown on the next slide.

51. Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are:
Zeron N Zeron S
Arnold
Supershelf
The costs to install the shelving at the various locations are:
Zrox Hewes Rockrite
Zeron N
Zeron S

52. 50 1 5 75 5 8 8 60 3 7 4 75 4 4 40 Network Representation Zrox Arnold
Zeron N 75
ARNOLD 5 8 8
Hewes 60
HEWES 3 7
Super
Shelf
Zeron S 4 75
WASH
BURN 4 4
Rock-
Rite 40

53. Decision Variables
xij = amount shipped from manufacturer i to supplier j
xjk = amount shipped from supplier j to customer k
where i = 1 (Arnold), 2 (Supershelf)
j = 3 (Zeron N), 4 (Zeron S)
k = 5 (Zrox), 6 (Hewes), 7 (Rockrite)
Objective Function
Minimize Overall Shipping Costs:
Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37
+ 3x45 + 4x46 + 4x47

54. Constraints
Amount Out of Arnold: x13 + x14 < 75
Amount Out of Supershelf: x23 + x24 < 75
Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0
Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0
Amount Into Zrox: x35 + x45 = 50
Amount Into Hewes: x36 + x46 = 60
Amount Into Rockrite: x37 + x47 = 40
Non-negativity of Variables: xij > 0, for all i and j.

55. QM Input as an LP

56. Zeron Shelving: Solution

57. Solution
Zrox
ZROX 50 50 75 1 5
Arnold
Zeron N 75
ARNOLD 5 25 8 8
Hewes 60 35
HEWES 3 7 4
Super
Shelf
Zeron S 40 75
WASH
BURN 4 75 4
Rock-
Rite 40

58. Example 2
Widgetco manufactures widgets at two factories, one in Memphis and one in Denver. The Memphis factory can produce upto 150 widgets per day, and Denver can produce upto 200. Widgets are shipped by air to consumers in LA and Boston. The customers in each city require 130 widgets per day. Because of the deregulation of air fares, the company believes that it may cheaper to first fly some widgets to NY or Chicago and then fly them to their final destinations. The costs of flying are provided on the next slide.
Formulate the problem to minimize the total cost of shipping the required widgets to its customers.

59. Memphis
Denver NY
Chicago LA
Boston $0 - $8
$13
$25
$28
$15
$12
$26 $6
$16
$17
$14