1. The Transportation and Assignment Problems
Chapter 8: Hillier and Lieberman
Dr. Hurley’s AGB 328 Course

2. Terms to Know
Sources, Destinations, Supply, Demand, The Requirements Assumption, The Feasible Solutions Property, The Cost Assumption, Dummy Destination, Dummy Source, Transportation Simplex Method, Northwest Corner Rule, Vogel’s Approximation Method, Russell’s Approximation Method, Recipient Cells, Donor Cells, Assignment Problems, Assignees, Tasks, Hungarian Algorithm

3. Case Study: P&T Company
P&T is a small family-owned business that processes and cans vegetables and then distributes them for eventual sale
One of its main products that it processes and ships is peas
These peas are processed in: Bellingham, WA; Eugene, OR; and Albert Lea, MN
The peas are shipped to: Sacramento, CA; Salt Lake City, UT; Rapid City, SD; and Albuquerque, NM

4. Case Study: P&T Company Shipping Data
Cannery
Output
Warehouse
Allocation
Bellingham
75 Truckloads
Sacramento
80 Truckloads
Eugene
125 Truckloads
Salt Lake
65 Truckloads
Albert Lea
100 Truckloads
Rapid City
70 Truckloads
Total
300 Truckloads
Albuquerque
85 Truckloads

5. Case Study: P&T Company Shipping Cost/Truckload
Warehouse
Cannery
Sacramento
Salt Lake
Rapid City
Albuquerque
Supply
Bellingham
$464
$513
$654
$867 75
Eugene
$352
$416
$690
$791
125
Albert Lea
$995
$682
$388
$685
100
Demand 80 65 70 85

6. Network Presentation of P&T Co. Problem
464 C1 W1 75
-80
513
867
654 W2
-65
352 C1
416
125
690 W3
791
-70
995
682 C1 W4
388
100
-85
685

7. Mathematical Model for P&T Transportation Problem

8. Mathematical Model for P&T Transportation Problem Cont.
Subject to:
𝑥 11 + 𝑥 𝑥 𝑥 =75
𝑥 21 + 𝑥 𝑥 𝑥 =125
𝑥 31 + 𝑥 𝑥 33 + 𝑥 34 =100
𝑥 𝑥 𝑥 =80
𝑥 𝑥 𝑥 =65
𝑥 𝑥 𝑥 =70
𝑥 𝑥 𝑥 =85
𝑥 𝑖𝑗 ≥0 (𝑖=1,2,3;𝑗=1,2,3,4)

9. Transportation Problems
Transportation problems are characterized by problems that are trying to distribute commodities from a any supply center, known as sources, to any group of receiving centers, known as destinations
Two major assumptions are needed in these types of problems:
The Requirements Assumption
The Cost Assumption

10. Transportation Assumptions
The Requirement Assumption
Each source has a fixed supply which must be distributed to destinations, while each destination has a fixed demand that must be received from the sources
The Cost Assumption
The cost of distributing commodities from the source to the destination is directly proportional to the number of units distributed

11. Feasible Solution Property
A transportation problem will have a feasible solution if and only if the sum of the supplies is equal to the sum of the demands.
Hence the constraints in the transportation problem must be fixed requirement constraints met with equality.

12. The General Model of a Transportation Problem
Any problem that attempts to minimize the total cost of distributing units of commodities while meeting the requirement assumption and the cost assumption and has information pertaining to sources, destinations, supplies, demands, and unit costs can be formulated into a transportation model

13. Visualizing the Transportation Model
When trying to model a transportation model, it is usually useful to draw a network diagram of the problem you are examining
A network diagram shows all the sources, destinations, and unit cost for each source to each destination in a simple visual format like the example on the next slide

14. Network Diagram Supply Demand Source 1 Destination 1 S1 -D1 Source 2
c1n
c13
Source 2
c21
Destination 2 S2
-D2
c22
c23
c2n
c31
Source 3
Destination 3 S3
c32
-D3
c33 .
c3n .
cm1
cm2
Source m
Destination n Sm
-Dn
cm3
cmn

15. General Mathematical Model of Transportation Problems
Minimize Z= 𝑖=1 𝑚 𝑗=1 𝑛 𝑐 𝑖𝑗 𝑥 𝑖𝑗
Subject to:
𝑗=1 𝑛 𝑥 𝑖𝑗 = 𝑠 𝑖 for I =1,2,…,m
𝑖=1 𝑚 𝑥 𝑖𝑗 = 𝑑 𝑗 𝑓𝑜𝑟 𝑗=1,2,…,𝑛
𝑥 𝑖𝑗 ≥0, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 𝑎𝑛𝑑 𝑗

16. Integer Solutions Property
If all the supplies and demands have integer values, then the transportation problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its decision variables
This implies that there is no need to add restrictions on the model to force integer solutions

17. Solving a Transportation Problem
When Excel solves a transportation problem, it uses the regular simplex method
Due to the characteristics of the transportation problem, a faster solution can be found using the transportation simplex method
Unfortunately, the transportation simplex model is not programmed in Solver

18. Modeling Variants of Transportation Problems
In many transportation models, you are not going to always see supply equals demand
With small problems, this is not an issue because the simplex method can solve the problem relatively efficiently
With large transportation problems it may be helpful to transform the model to fit the transportation simplex model

19. Issues That Arise with Transportation Models
Some of the issues that may arise are:
The sum of supply exceeds the sums of demand
The sum of the supplies is less than the sum of demands
A destination has both a minimum demand and maximum demand
Certain sources may not be able to distribute commodities to certain destinations
The objective is to maximize profits rather than minimize costs

20. Method for Handling Supply Not Equal to Demand
When supply does not equal demand, you can use the idea of a slack variable to handle the excess
A slack variable is a variable that can be incorporated into the model to allow inequality constraints to become equality constraints
If supply is greater than demand, then you need a slack variable known as a dummy destination
If demand is greater than supply, then you need a slack variable known as a dummy source

21. Handling Destinations that Cannot Be Delivered To
There are two ways to handle the issue when a source cannot supply a particular destination
The first way is to put a constraint that does not allow the value to be anything but zero
The second way of handling this issue is to put an extremely large number into the cost of shipping that will force the value to equal zero

22. Textbook Transportation Models Examined
P&T
A typical transportation problem
Could there be another formulation?
Northern Airplane
An example when you need to use the Big M Method and utilizing dummy destinations for excess supply to fit into the transportation model
Metro Water District
An example when you need to use the Big M Method and utilizing dummy sources for excess demand to fit into the transportation model

23. The Transportation Simplex Method
While the normal simplex method can solve transportation type problems, it does not necessarily do it in the most efficient fashion, especially for large problems.
The transportation simplex is meant to solve the problems much more quickly.

24. Finding an Initial Solution for the Transportation Simplex
Northwest Corner Rule
Let xs,d stand for the amount allocated to supply row s and demand row d
For x1,1 select the minimum of the supply and demand for supply 1 and demand 1
If any supply is remaining then increment over to xs,d+1, otherwise increment down to xs+1,d
For this next variable select the minimum of the leftover supply or leftover demand for the new row and column you are in
Continue until all supply and demand has been allocated

25. Finding an Initial Solution for the Transportation Simplex
Vogel’s Approximation Method
For each row and column that has not been deleted, calculate the difference between the smallest and second smallest in absolute value terms (ties mean that the difference is zero)
In the row or column that has the highest difference, find the lowest cost variable in it
Set this variable to the minimum of the leftover supply or demand
Delete the supply or demand row/column that was the minimum and go back to the top step

26. Finding an Initial Solution for the Transportation Simplex
Russell’s Approximation Method
For each remaining source row i, determine the largest unit cost cij and call it 𝑢 𝑖
For each remaining destination column j, determine the largest unit cost cij and call it 𝑣 𝑖
Calculate ∆ 𝑖𝑗 = 𝑐 𝑖𝑗 − 𝑢 𝑖 − 𝑣 𝑗 for all xij that have not previously been selected
Select the largest corresponding xij that has the largest negative ∆ij
Allocate to this variable as much as feasible based on the current supply and demand that are leftover

27. Algorithm for Transportation Simplex Method
Construct initial basic feasible solution
Optimality Test
Derive a set of ui and vj by setting the ui corresponding to the row that has the most amount of allocations to zero and solving the leftover set of equations for cij = ui + vj
If all cij – ui – vj ≥ 0 for every (i,j) such that xij is nonbasic, then stop. Otherwise do an iteration.

28. Algorithm for Transportation Simplex Method Cont.
An Iteration
Determine the entering basic variable by selecting the nonbasic variable having the largest negative value for cij – ui – vj
Determine the leaving basic variable by identifying the chain of swaps required to maintain feasibility
Select the basic variable having the smallest variable from the donor cells
Determine the new basic feasible solution by adding the value of the leaving basic variable to the allocation for each recipient cell.
Subtract this value from the allocation of each donor cell

29. Assignment Problems
Assignment problems are problems that require tasks to be handed out to assignees in the cheapest method possible
The assignment problem is a special case of the transportation problem

30. Characteristics of Assignment Problems
The number of assignees and the number of task are the same
Each assignee is to be assigned exactly one task
Each task is to be assigned by exactly one assignee
There is a cost associated with each combination of an assignee performing a task
The objective is to determine how all of the assignments should be made to minimize the total cost

31. General Mathematical Model of Assignment Problems
Minimize Z= 𝑖=1 𝑛 𝑗=1 𝑛 𝑐 𝑖𝑗 𝑥 𝑖𝑗
Subject to:
𝑗=1 𝑛 𝑥 𝑖𝑗 =1 for I =1,2,…,m
𝑖=1 𝑛 𝑥 𝑖𝑗 =1 𝑓𝑜𝑟 𝑗=1,2,…,𝑛
𝑥 𝑖𝑗 𝑖𝑠 𝑏𝑖𝑛𝑎𝑟𝑦, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 𝑎𝑛𝑑 𝑗

32. Modeling Variants of the Assignment Problem
Issues that arise:
Certain assignees are unable to perform certain tasks.
There are more task than there are assignees, implying some tasks will not be completed.
There are more assignees than there are tasks, implying some assignees will not be given a task.
Each assignee can be given multiple tasks simultaneously.
Each task can be performed jointly by more than one assignee.

33. Assignment Spreadsheet Models from Textbook
Job Shop Company
Better Products Company
We will examine these spreadsheets in class and derive mathematical models from the spreadsheets

34. Hungarian Algorithm for Solving Assignment Problems
Step 1: Find the minimum from each row and subtract from every number in the corresponding row making a new table
Step 2: Find the minimum from each column and subtract from every number in the corresponding column making a new table
Step 3: Test to see whether an optimal assignment can be made by examining the minimum number of lines needed to cover all the zeros
If the number of lines corresponds to the number of rows, you have the optimal and you should go to step 6
If the number of lines does not correspond to the number of rows, go to step 4

35. Hungarian Algorithm for Solving Assignment Problems Cont.
Step 4: Modify the table by using the following:
Subtract the smallest uncovered number from every uncovered number in the table
Add the smallest uncovered number to the numbers of intersected lines
All other numbers stay unchanged
Step 5: Repeat steps 3 and four until you have the optimal set

36. Hungarian Algorithm for Solving Assignment Problems Cont.
Step 6: Make the assignment to the optimal set one at a time focusing on the zero elements
Start with the rows and columns that have only one zero
Once an optimal assignment has been given to a variable, cross that row and column out
Continue until all the rows and columns with only one zero have been allocated
Next do the columns/rows with two non crossed out zeroes as above
Continue until all assignments have been made

37. In Class Activity (Not Graded)
Attempt to find an initial solution to the P&T problem using the a) Northwest Corner Rule, b) Vogel’s Approximation Method, and c) Russell’s Approximation Method
8.1-3b, set up the problem as a regular linear programming problem and solve using solver, then set the problem up as a transportation problem and solve using solver

38. In Class Activity (Not Graded)
Solve the following problem using the Hungarian method.

39. Case Study: Sellmore Company Cont.
The assignees for the task are:
Ann
Ian
Joan
Sean
A summary of each assignees productivity and costs are given on the next slide.

40. Case Study: Sellmore Company Cont.
Required Time Per Task
Employee
Word Processing
Graphics
Packets
Registration
Wage
Ann 35 41 27 40
$14
Ian 47 45 32 51
$12
Joan 39 56 36 43
$13
Sean 25 46
$15

41. Assignment of Variables
xij
i = 1 for Ann, 2 for Ian, 3 for Joan, 4 for Sean
j = 1 for Processing, 2 for Graphics, 3 for Packets, 4 for Registration

42. Mathematical Model for Sellmore Company

43. Mathematical Model for Sellmore Company Cont.