1. Transportation, Transshipment and Assignment Models

2. Learning Objectives Structure special LP network flow models.
Set up and solve transportation models
Extend basic transportation model to include transshipment points.
Set up and solve facility location and other application problems as transportation models.
Set up and solve assignment models

3. Overview
Part of a larger class of linear programming problems are known as network flow models.
They possess special mathematical features that enabled the development of very efficient, unique solution methods.

4. Transportation Model
Transportation problem deals with the distribution of goods from several points of supply to a number of points of demand. They arise when a cost-effective pattern is needed to ship items from origins that have limited supply to destinations that have demand for the goods.
Resources to be optimally allocated usually involve a given capacity of goods at each source and a given requirement for the goods at each destination.
Most common objective of the transportation problem is to schedule shipments from sources to destinations so that total production and transportation costs are minimized

5. Transshipment Model
An extension of transportation problems is called transshipment problem in which a point can have shipments that both arrive as well as leave.
Example would be a warehouse where shipments arrive from factories and then leave for retail outlets
It may be possible for a firm to achieve cost savings (economies of scale) by consolidating shipments from several factories at a warehouse and then sending them together to retail outlets.

6. Assignment Model
Assignment problem refers to a class of LP problems that involve determining most efficient assignment of:
People to projects,
Salespeople to territories,
Contracts to bidders,
Jobs to machines, and so on
Objective is to minimize total cost or total time of performing tasks at hand, although a maximization objective is also possible.

7. Transportation Model

8. Transportation Model Problem definition
There are m sources. Source i has a supply capacity of Si.
There are n destinations.The demand at destination j is D j.
Objective:
To minimize the total shipping cost of supplying the
destinations with the required demand from the available supplies at the sources.

9. The Transportation Model
Characteristics
A product is to be transported from a number of sources to a number of destinations at the minimum possible cost.
Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product.
The linear programming model has constraints for supply at each source and demand at each destination.
All constraints are equalities in a balanced transportation model where supply equals demand.
Constraints contain inequalities in unbalanced models where supply is not equal to demand.

10. Transportation Model- Example 1
Executive Furniture Corporation

11. Transportation Model Example 1
Executive Furniture Corporation
Transportation Costs Per Desk

12. Transportation Model Example 1
Executive Furniture Corporation:LP Transportation Model Formulation
Objective: minimize total shipping costs =
5 XDA + 4 XDB + 3 XDC + 3 XEA + 2 XEB +
1 XEC + 9 XFA + 7 XFB + 5 XFC
Where: Xij = number of desks shipped from factory i to warehouse j
i = D (for Des Moines),
E (for Evansville), or
F (for Fort Lauderdale).
j = A (for Albuquerque),
B (for Boston), or
C (for Cleveland).

13. Transportation Model Example 1
Executive Furniture Corporation: Supply Constraints
Net flow at Des Moines = (Total flow in) - (Total flow out)
= (0) - (XDA + XDB + XDC)
Net flow at Des Moines =
-XDA - XDB - XDC = (Des Moines capacity) and
-XEA - XEB - XEC = (Evansville capacity)
-XFA - XFB - XFC = (Fort Lauderdale capacity)
Multiply each constraint by -1 and rewrite as:
XDA + XDB + XDC = 100 (Des Moines capacity)
XEA + XEB + XEC = 300 (Evansville capacity)
XFA + XFB + XFC = 300 (Fort Lauderdale capacity)

14. Transportation Model Example 1
Executive Furniture Corporation: Demand Constraints
Net flow at Albuquerque = (Total flow in) - (Total flow out)
= (XDA + XEA + XFA) - (0)
Net flow at Albuquerque =
XDA + XEA + XFA = 300 (Albuquerque demand) and
XDB + XEB + XFB = 200 (Boston demand)
XDC + XEC + XFC = 200 (Cleveland demand)

15. Transportation Model Example 1: The Optimum Solution
SHIP:
100 desks from Des Moines to Albuquerque,
200 desks from Evansville to Albuquerque,
100 desks from Evansville to Boston,
100 desks from Fort Lauderdale to Boston,
and 200 desks from Fort Lauderdale to Cleveland.
Total shipping cost is $3,000.

16. Transportation Model Example 2 Problem Definition and Data
Problem: How many tons of wheat to transport from each grain elevator to each mill on a monthly basis in order to minimize the total cost of transportation?
Data:
Grain Elevator Supply Mill Demand
1. Kansas City A. Chicago
2. Omaha B. St. Louis
3. Des Moines C. Cincinnati
Total tons Total tons

17. Transportation Model Example 2
Model Formulation

18. Transportation Model Example 2
Model Formulation
Minimize Z = $6x1A + 8x1B + 10x1C + 7x2A + 11x2B + 11x2C x3A + 5x3B + 12x3C
subject to:
x1A + x1B + x1C = 150
x2A + x2B + x2C = 175
x3A + x3B + x3C = 275
x1A + x2A + x3A = 200
x1B + x2B + x3B = 100
x1C + x2C + x3C = 300
xij  0
xij = tons of wheat from each grain elevator, i, i = 1, 2, 3, to each mill j, j = A,B,C

19. Transportation Model- Example 3
Carlton Pharmateuticals
Carlton Pharmaceuticals supplies drugs and other medical supplies.
It has three plants in: Cleveland, Detroit, Greensboro.
It has four distribution centers in: Boston, Richmond, Atlanta, St. Louis.
Management at Carlton would like to ship cases of a certain vaccine as economically as possible.

20. Data
Unit shipping cost, supply, and demand
Assumptions
Unit shipping cost is constant.
All the shipping occurs simultaneously.
The only transportation considered is between sources and destinations.
Total supply equals total demand.

21. Destinations Boston Sources Cleveland Richmond Detroit Atlanta
NETWORK
REPRESENTATION
Boston
Richmond
Atlanta
St.Louis
Destinations
Sources
Cleveland
Detroit
Greensboro
D1=1100 37 40 42 32 35 30 25 15 20 28
S1=1200
S2=1000
S3= 800
D2=400
D3=750
D4=750

22. The Mathematical Model
The structure of the model is:
Minimize <Total Shipping Cost> ST
[Amount shipped from a source] = [Supply at that source]
[Amount received at a destination] = [Demand at that destination]
Decision variables
Xij = amount shipped from source i to destination j.
where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis)

23. The supply constraints
Cleveland
S1=1200
X11
X12
X13
X14
Supply from Cleveland X11+X12+X13+X14 =
The supply constraints
Detroit
S2=1000
X21
X22
X23
X24
Supply from Detroit X21+X22+X23+X =
Boston
Greensboro
S3= 800
X31
X32
X33
X34
Supply from Greensboro X31+X32+X33+X34 =
D1=1100
Richmond
D2=400
Atlanta
D3=750
St.Louis
D4=750

24. The complete mathematical model=

25. Excel Optimal Solution

26. WINQSB Sensitivity Analysis
If this path is used, the total cost will increase by $5 per unit
shipped along it
Range of optimality

27. Range of feasibility
Shadow prices for plants - the cost savings realized for each extra case of vaccine
available at the plant
Shadow prices for warehouses - the cost incurred from having 1 extra case of vaccine
demanded at the warehouse

28. Interpreting sensitivity analysis results
Reduced costs
The amount of transportation cost reduction per unit that makes a given route economically attractive.
If the route is forced to be used under the current cost structure, for each item shipped along it, the total cost increases by an amount equal to the reduced cost.
Shadow prices
For the plants, shadow prices convey the cost savings realized for each extra case of vaccine available at plant.
For the warehouses, shadow prices convey the cost incurred from having an extra case demanded at the warehouse.

29. The company can use both regular time and overtime to produce skis.
Transportation Model- Example 4 Montpelier Ski Company: Using a Transportation model for production scheduling
Montpelier is planning its production of skis for the months of July, August, and September.
Production capacity and unit production cost will change from month to month.
The company can use both regular time and overtime to produce skis.
Production levels should meet both demand forecasts and end-of-quarter inventory requirement.
Management would like to schedule production to minimize its costs for the quarter.

30. Data:
Initial inventory = 200 pairs
Ending inventory required =1200 pairs
Production capacity for the next quarter is shown on the table
Holding cost rate is 3% per month per ski.
Production capacity, and forecasted demand for this quarter (in pairs of skis), and production cost per unit (by months)

31. Analysis of Unit costs Unit cost = [Unit production cost] +
Analysis of demand:
Net demand to satisfy in July = = 200 pairs
Net demand in August = 600
Net demand in September = = 2200 pairs
Analysis of Supplies:
Production capacities are thought of as supplies.
There are two sets of “supplies”:
Set 1- Regular time supply (production capacity)
Set 2 - Overtime supply
Initial inventory
Analysis of Unit costs
Unit cost = [Unit production cost] +
[Unit holding cost per month][the number of months stays in inventory]
Example: A unit produced in July in Regular time and sold in September costs 25+ (3%)(25)(2 months) = $26.50
Forecasted demand
In house inventory

32. Network representation
Production
Month/period
Network representation
Month
sold
July
R/T
July
R/T 25
25.75
26.50
1000
200
July
July
O/T
500 30
30.90
31.80 +M 26
26.78 +M 37 +M 29
Aug.
R/T
800 +M 32
32.96
Aug.
600
Demand
Production Capacity
Aug.
O/T
400
Sept.
2200
Sept.
R/T
400
Dummy
300
Sept.
O/T
200

33. Source: July production in R/T
Destination: July‘s demand.
Source: Aug. production in O/T
Destination: Sept.’s demand
Unit cost= $25 (production)
32+(.03)(32)=$32.96
Unit cost =Production+one month holding cost

34. Summary of the optimal solution
In July produce at capacity (1000 pairs in R/T, and 500 pairs in O/T). Store = 1300 at the end of July.
In August, produce 800 pairs in R/T, and 300 in O/T. Store additional = 500 pairs.
In September, produce 400 pairs (clearly in R/T). With 1000 pairs retail demand, there will be
( ) = 1200 pairs available for shipment
Inventory +
Production -
Demand

35. Unbalanced Transportation Problems
If supplies are not equal to demands, an unbalanced transportation model exists.
In an unbalanced transportation model, supply or demand constraints need to be modified.
There are two possible scenarios:
Total supply exceeds total requirement.
Total supply is less than total requirement.

36. Total Supply Exceeds Total Requirement
Total flow out of Des Moines ( XDA + XDB + XDC) should be
permitted to be smaller than total supply (100).
The constraint should be written as
-XDA - XDB - XDC >= (Des Moines capacity)
  -XEA - XEB - XEC >= (Evansville capacity)
-XFA - XFB - XFC >= (Fort Lauderdale capacity)
XDA + XDB + XDC <= 100
XEA + XEB + XEC <= 100
XFA + XFB + XFC <= 100

37. Total Supply Less Than Total Requirement
Total flow in to Albuquerque (that is, XDA + XEA + XFA) should
be permitted to be smaller than total demand (namely, 300).
This warehouse should be written as:
  XDA + XEA + XFA <= (Albuquerque demand)
XDB + XEB + XFB <= (Boston demand)
XDC + XEC + XFC <= (Cleveland demand)

38. Transportation Example 5: Formulation
Develop the linear programming model and solve using Excel:

39. Transportation Example 5: Formulation
Minimize Z = $8x1A + 5x1B + 6x1C + 15x2A + 10x2B + 12x2C x3A + 9x3B + 10x3C
subject to:
x1A + x1B + x1C = 120
x2A + x2B + x2C = 80
x3A + x3B + x3C = 80
x1A + x2A + x3A  150
x1B + x2B + x3B  70
x1C + x2C + x3C  100
xij  0

40. Transportation Model-Example 6
 Hardgrave Machine Company - New Factory Location
Produces computer components at its plants in Cincinnati, Kansas City, and Pittsburgh.
Plants not able to keep up with demand for orders at four warehouses in Detroit, Houston, New York, and Los Angeles.
Firm has decided to build a new plant to expand its productive capacity.
Two sites being considered:
Seattle, Washington and
Birmingham, Alabama.
Both cities attractive in terms: labor supply, municipal services, and ease of factory financing.

41. Transportation Model-Example 6
Hardgrave Machine Company: Demand Supply Data and Production Costs

42. Transportation Model-Example 6
Hardgrave Machine Company: Shipping Costs

43. Transshipment Model

44. Transshipment Model
In a Transshipment Problem flows can occur both out of and into the
same node in three ways:
 1. If total flow into a node is less than total flow out from node,
node represents a net creator of goods (a supply point).
- Flow balance equation will have a negative right hand
side (RHS) value.
2. If total flow into a node exceeds total flow out from node,
node represents a net consumer of goods, (a demand point).
- Flow balance equation will have a positive RHS value.
3. If total flow into a node is equal to total flow out from node,
node represents a pure transshipment point.
- Flow balance equation will have a zero RHS value.

45. The Transshipment Model
Characteristics
It is an extension of the transportation model.
Intermediate transshipment points are added between the sources and destinations.
Items may be transported from:
Sources through transshipment points to destinations
One source to another
One transshipment point to another
One destination to another
Directly from sources to to destinations
Some combination of these

46. Executive Furniture Corporation – Revisited
Assume it is possible for Executive Furniture to ship desks from Evansville factory to its three warehouses at very low unit shipping costs.
Consider shipping all desks produced at other two factories (Des Moines and Fort Lauderdale) to Evansville.
Consider using a new shipping company to move desks from Evansville to all its warehouses.

47. Executive Furniture Corporation - Revisited
Revised unit shipping costs are shown here.
Note Evansville factory shows up in both the “From” and “To” entries.

48. LP Model for theTransshipment Problem
Executive Furniture Corporation Revisited
LP Model for theTransshipment Problem
Two new additional decision variables for new shipping routes are to be added.
XDE= number of desks shipped from Des Moines to Evansville
XFE = number of desks shipped from Fort Lauderdale to Evansville
 Objective Function: minimize total shipping costs =
5XDA + 4XDB + 3XDC + 2XDE + 3XEA + 2XEB +
+1XEC + 9XFA + 7XFB + 5XFC + 3XFE

49. Executive Furniture Corporation Revisited
LP Model for theTransshipment Problem
Relevant flow balance equations written as:
 (0) - (XDA + XDB + XDC + XDE) = (Des Moines capacity)
(0) - (XFA + XFB + XFC + XFE) = (Fort Lauderdale capacity)
Supplies have been expressed as negative numbers in the RHS.
 Net flow at Evansville = (Total flow in) - (Total flow out)
= (XDE + XFE) - (XEA + XEB + XEC)
Net flow equals total number of desks produced (the supply) at Evansville.
Net flow at Evansville = (XDE + XFE) - (XEA + XEB + XEC) = -300
No change in demand constraints for warehouse requirements: 
XDA + XEA + XFA = (Albuquerque demand)
XDB + XEB + XFB = (Boston demand)
XDC + XEC + XFC = (Cleveland demand)

50. Transshipment Model Example 2 Problem Definition and Data
Extension of the transportation model in which intermediate transshipment points are added between sources and destinations. An example of a transshipment point is a distribution center or warehouse located between plants and stores
Data:

51. Transshipment Model Example 2 Problem Definition and Data

52. Transshipment Model Example 2
Model Formulation
Minimize Z = $16x x x x x x25 + 6x36 + 8x x38 + 7x x x48 + 4x56 + 5x57 + x58
subject to:
x13 + x14 + x15 = 300
x23+ x24 + x25 = 300
x36 + x46 + x56 = 200
x37+ x47 + x57 = 100
x38 + x48 + x58 = 300
x13 + x23 - x36 - x37 - x38 = 0
x14 + x24 - x46 - x47 - x48 = 0
x15 + x25 - x56 - x57 - x58 = 0
xij  0

53. Assignment Model

54. The Assignment Model Problem definition
m workers are to be assigned to m jobs
A unit cost (or profit) Cij is associated with worker i performing job j.
Minimize the total cost (or maximize the total profit) of assigning workers to jobs so that each worker is assigned a job, and each job is performed.

55. The Assignment Model Characteristics
It is a special form of linear programming models similar to the transportation model.
Supply at each source and demand at each destination is limited to one unit.
In a balanced model supply equals demand.
In an unbalanced model supply is not equal to demand.

56. The Assignment Model Assumptions
The number of workers is equal to the number of jobs.
Given a balanced problem, each worker is assigned exactly once, and each job is performed by exactly one worker.
For an unbalanced problem “dummy” workers (in case there are more jobs than workers), or “dummy” jobs (in case there are more workers than jobs) are added to balance the problem.

57. Assignment Model Example 1
Fix-It Shop Example
Received three new rush projects to repair:
(1) a radio,
(2) a toaster oven, and
(3) a broken coffee table.
Three workers (each has different talents and abilities).
Estimated costs to assign each worker to each of the three projects.

58. Assignment Model Example 1
Fix-It Shop
Rows denote people or objects to be assigned, and columns denote tasks or jobs assigned.
Numbers in table are costs associated with each particular assignment.

59. Assignment Model Example 1
Fix-It Shop: Assignment Alternatives and Costs
Owner's objective is to assign three projects to workers in a way that result is lowest total cost.

60. Assignment Model Example 1
Fix-It Shop
Owner's objective is to assign three projects to workers in a way that results in lowest total cost.

61. Assignment Model Example 1
Fix-It Shop
Formulate LP model -
Xij = “Flow” on arc from node denoting worker i to node denoting
project j.
Solution value will equal 1 if worker i is assigned to project j :
i = A (for Adams), B (for Brown), or C (for Cooper)
j = 1 (for project 1), 2 (for project 2), or 3 (for project 3)
Objective Function: minimize total assignment cost =
11XA1 + 14XA2 + 6XA3 + 8XB1 + 10XB2 + 11XB3 +
+ 9XC1 + 12XC2 + 7XC3

62. Assignment Model Example 1
Fix-It Shop
Constraints expressed using standard flow balance equations are as follows:
-XA1 - XA2 - XA3 = (Adams availability)
-XB1 - XB2 - XB3 = -1 (Brown availability)
-XC1 - XC2 - XC3 = -1 (Cooper availability)
XA1 + XB1 + XC1 = 1 (Project 1 requirement)
XA2 + XB2 + XC2 = 1 (Project 2 requirement)
XA3 + XB3 + XC3 = 1 (Project 3 requirement)

63. Assignment Model- Example 2
Ballston Electronics
Five different electrical devices produced on five production lines, are needed to be inspected.
The travel time of finished goods to inspection areas depends on both the production line and the inspection area.
Management wishes to designate a separate inspection area to inspect the products such that the total travel time is minimized.

64. Assignment Model- Example 2
Data: Travel time in minutes from assembly lines to inspection areas.

65. Assignment Model Example 2: Network Representation (3 of 3)
Assembly Line
Inspection Areas
S1=1
S2=1
S3=1
S4=1
S5=1
D1=1 1 A 2 B
D2=1 3 C
D3=1
D4=1 4 D
D5=1 5 E

66. Computer solutions
A complete enumeration is not efficient even for moderately large problems (with m=8, m! > 40,000 is the number of assignments to enumerate).
The Hungarian method provides an efficient solution procedure.
Special cases
A worker is unable to perform a particular job.
A worker can be assigned to more than one job.
A maximization assignment problem.

67. Assignment Model Example 3 Problem Definition and Data
Problem: Assign four teams of officials to four games in a way that will minimize total distance traveled by the officials. Supply is always one team of officials, demand is for only one team of officials at each game.
Data:

68. Assignment Model Example 3
Model Formulation
Minimize Z = 210xAR + 90xAA + 180xAD + 160xAC + 100xBR xBA + 130xBD + 200xBC + 175xCR + 105xCA xCD + 170xCC + 80xDR + 65xDA + 105xDD xDC
subject to:
xAR + xAA + xAD + xAC = 1 xij  0
xBR + xBA + xBD + xBC = 1
xCR + xCA + xCD + xCC = 1
xDR + xDA + xDD + xDC = 1
xAR + xBR + xCR + xDR = 1
xAA + xBA + xCA + xDA = 1
xAD + xBD + xCD + xDD = 1
xAC + xBC + xCC + xDC = 1

69. Summary
Three network flow models have been presented:
Transportation model deals with distribution of goods from several supplier to a number of demand points.
Transshipment model includes points that permit goods to flow both in and out of them.
Assignment model deals with determining the most efficient assignment of issues such as people to projects.