The Transportation and
Assignment Problems
Chapter 9: Hillier and Lieberman
Chapter 7: Decision Tools for Agribusiness
Dr. Hurley’s AGB 328 Course
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
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
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
Total
300 Truckloads
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
Network Presentation of P&T Co.
Problem
464
75
C1
W1
-80
W2
-65
W3
-70
W4
-85
513
867
654
352
416
125 C1
690
791
995
682
388
100 C1
685
Mathematical Model for P&T
Transportation Problem
464 x11  513x12  654 x13  867 x14 
Minimize 352 x21  416 x22  690 x23  791x24 
x11, x12 , x13 , x14
x21, x22 , x23 , x24 995 x  682 x  388 x  685 x
31
32
33
34
x31, x32 , x33 , x34
Mathematical Model for P&T
Transportation Problem Cont.

Subject to:
𝑥11 + 𝑥12 + 𝑥13 + 𝑥14
= 75
𝑥21 +𝑥22 + 𝑥23 + 𝑥24
= 125
𝑥31 +𝑥32 + 𝑥33 + 𝑥34 = 100
𝑥11
+ 𝑥21
𝑥12
+𝑥22
𝑥13
𝑥𝑖𝑗 ≥ 0
+ 𝑥31
𝑥14
(𝑖 = 1,2,3; 𝑗 = 1,2,3,4)
= 80
+ 𝑥32
+𝑥23
+𝑥24
= 65
+ 𝑥33
= 70
+ 𝑥34 = 85
Transportation Problems
Transportation problems are characterized by
problems that are trying to distribute
commodities from 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
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
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.
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
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
Network Diagram
Demand
Supply
S1
c11
Source 1
c12
c13
c1n
S2
c21
c22
Source 2
c23
c2n
S3
Source 3
.
.
.
Sm
c31
Source m
c32
c33
c3n
cm1
cm2
cm3
cmn
Destination 1
Destination 2
Destination 3
-D1
-D2
-D3
.
.
.
Destination n
-Dn
General Mathematical Model of
Transportation Problems
𝑛
Minimize Z= 𝑚
𝑖=1 𝑗=1 𝑐𝑖𝑗 𝑥𝑖𝑗
Subject to:
𝑛
𝑗=1 𝑥𝑖𝑗 = 𝑠𝑖 for I =1,2,…,m
𝑚
𝑥𝑖𝑗 = 𝑑𝑗 𝑓𝑜𝑟 𝑗 = 1,2, … , 𝑛
𝑖=1
𝑥𝑖𝑗 ≥ 0, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 𝑎𝑛𝑑 𝑗
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
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
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

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
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
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
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
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.

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
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
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 𝑢𝑖
◦ For each remaining destination column j,
determine the largest unit cost cij and call it 𝑣𝑖
◦ Calculate ∆𝑖𝑗 = 𝑐𝑖𝑗 − 𝑢𝑖 − 𝑣𝑗 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
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.
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
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

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
General Mathematical Model of
Assignment Problems
Minimize Z= 𝑛𝑖=1 𝑛𝑗=1 𝑐𝑖𝑗 𝑥𝑖𝑗
Subject to:
𝑛
𝑗=1 𝑥𝑖𝑗 = 1 for I =1,2,…,m
𝑛
𝑥𝑖𝑗 = 1 𝑓𝑜𝑟 𝑗 = 1,2, … , 𝑛
𝑖=1
𝑥𝑖𝑗 𝑖𝑠 𝑏𝑖𝑛𝑎𝑟𝑦, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 𝑎𝑛𝑑 𝑗
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.
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
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
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
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
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

In Class Activity (Not Graded)

Solve the following problem using the
Hungarian method.
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.
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
32
51
25
46
$15
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
Mathematical Model for Sellmore
Company
490 x11  574 x12  378 x13  560 x14 
Minimize 564 x21  540 x22  384 x23  612 x24 
x11 , x12 , x13 , x14
x21 , x22 , x23 , x24 507 x31  728 x32  468 x33  559 x34 
x31 , x32 , x33 , x34
480 x31  765 x32  375 x33  690 x34
Mathematical Model for Sellmore
Company Cont.
Subject to :
x11  x12  x13  x14  1
1  x11, x12 , x13 , x14  0
x21  x22  x23  x24  1 1  x21 , x22 , x23 , x24  0
x31  x32  x33  x34  1 1  x31 , x32 , x33 , x34  0
x41  x42  x43  x44  1 1  x31 , x32 , x33 , x34  0
x11  x21  x31  x41  1
x12  x22  x32  x42  1
x13  x23  x33  x43  1
x14  x24  x34  x44  1