**1. **Chapter 6: Distribution and Network Models

**2. **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

**3. **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.

**4. **In this chapter:

**5. **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.

**6. **Transportation Problem Network Representation

**8. **Transportation Problem Linear Programming Formulation (continued)

**9. ** 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.

**12. **Network Representation

**13. **Shipment Cost Minimization / LP Formulation
Decision variables: xij = Amount shipped for i = 1-2 and j = 1-3

**14. **QM Input

**15. **QM Solution

**26. **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?

**28. **Practice Problem

**30. **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.

**31. **Network Representation

**33. **Linear Programming Formulation (continued)

**39. **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

**43. **Transshipment Problems

**44. **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.

**45. **Network Representation

**49. ** 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. Transshipment Problem: Example

**51. **Network Representation

**52. **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

**53. **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.

**54. **QM Input as an LP

**57. **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.