Transportation

Problems
 Transportation Problem
• A distribution-type problem in which supplies of goods that are held
at various locations are to be distributed to other receiving locations.

• The solution of a transportation problem will indicate to a manager

the quantities and costs of various routes and the resulting minimum
cost.

• Used to compare location alternatives in deciding where to locate

factories and warehouses to achieve the minimum cost distribution
configuration.

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 Formulating the Model
• A transportation problem

• Typically involves a set of sending locations, which are referred to

as origins, and a set of receiving locations, which are referred to as
destinations.

• To develop a model of a transportation problem, it is necessary to

have the following information:
1. Supply quantity (capacity) of each origin.
2. Demand quantity of each destination.
3. Unit transportation cost for each origin-destination route.
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 Schematic of a Transportation Problem

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 Transshipment Problems
•A transportation problem in which some locations
are used as intermediate shipping points, thereby
serving both as origins and as destinations.

•Involve the distribution of goods from intermediate

nodes in addition to multiple sources and multiple
destinations.

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 Assignment Problems

•Involve the matching or pairing of two sets of

items such as jobs and machines, secretaries
and reports, lawyers and cases, and so forth.

•Have different cost or time requirements for

different pairings.

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 Schematic of a Transportation Problem

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 Transportation Table

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 Special Cases of Transportation Problems

• Maximization
•Transportation-type problems that concern profits or
revenues rather than costs with the objective to maximize
profits rather than to minimize costs.

• Unacceptable Routes
•Certain origin-destination combinations may be
unacceptable due to weather factors, equipment
breakdowns, labor problems, or skill requirements that
either prohibit, or make undesirable, certain combinations
(routes).
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 Special Cases of Transportation Problems

• Unequal Supply and Demand

• Situations in which supply and demand are not equal such that it is
necessary to modify the original problem so that supply and demand
are equalized.

• Quantities in dummy routes in the optimal solution are not shipped

and serve to indicate which supplier will hold the excess supply, and
how much, or which destination will not receive its total demand, and
how much it will be short.

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 Problem 1
Wheat is harvested in the Midwest and stored in grain elevators in three different cities – Kansas city, Omaha, and Des Moines.
These grain elevators supply three flour mills, located in Chicago, St. Louis, and Cincinnati. Grain is shipped to the mills in
railroad cars, each car capable of holding 1 ton of wheat. Each grain elevator is able to supply 150, 175, 275 tons (i.e. railroad cars)
of wheat to the mills on a monthly basis, respectively. Each mill demands 200, 100, and 300 tons of wheat per month,
respectively.

The cost of transporting 1 ton of wheat from each grain elevator (source) to each mill (destination) differs, according to the
distance and rail system. The costs are shown in the following Table. Determine how many tons of wheat to transport from each
elevator to each mill monthly basis to minimize the total cost of transportation.

Grain Elevator Mill

Chicago St. Louis Cincinnati
Kansas City $6 $8 $10
Omaha 7 11 11
Des Moines 4 5 12
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 Formulation
𝑥𝑖𝑗 = Tons of wheat transported from each grain elevator 𝑖 (𝑖 = 1, 2, 3) to each mill 𝑗 (𝑗 = 1, 2, 3).

Minimize,
Total cost of transportation, 𝑍 = 6𝑥11 + 8𝑥12 + 10𝑥13 + 7𝑥21 + 11𝑥22 + 11𝑥23 +4𝑥31 + 5𝑥32 + 12𝑥33

Subject to constraints,

𝑥11 + 𝑥12 + 𝑥13 = 150 𝑥11 + 𝑥21 + 𝑥31 = 200

𝑥21 + 𝑥22 + 𝑥23 = 175 𝑥12 + 𝑥22 + 𝑥32 = 100

𝑥31 + 𝑥32 + 𝑥33 = 275 𝑥13 + 𝑥23 + 𝑥33 = 300

𝑥𝑖𝑗 ≥ 0

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 Exercise 2

The Easy Time Grocery chain operates in major metropolitan areas on the East Coast. The
stores have a “no frills” approach, with low overhead and high volume. They generally buy
their stock in volume at low prices. However, in some cases they actually buy stock at stores
in other areas and ship it in. They can do this because of high prices in the cities they operate
in compared with costs in other locations. One example is baby food. Easy Time purchases
baby food at stores in Albany, Birmingham, Claremont, Dover, and Edison and then trucks it
to six stores in and around New York city. The stores in the outlying areas know what Easy
Time is up to, so they limit the number of cases of baby food Easy Time can purchase. The
following table shows the profit Easy Time makes per case of baby food, based on where the
chain purchases it and at which store it is sold, plus available baby food per week at purchase
locations and the shelf space available at each Easy Time store per week:

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 Purchase Easy Time Store (Profit/case ($))
Location 1 2 3 4 5 6 Supply
Albany 9 8 11 12 7 8 26
Bingamton 10 10 8 6 9 7 40
Claremont 8 6 6 5 7 4 20
Dover 4 6 9 5 8 10 40
Edison 12 10 8 9 6 7 45
Demand 25 15 30 18 27 35

Determine where Easy Time should purchase baby food and how the food should be
distributed to maximize profit.

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 A Network Diagram of a Transshipment Problem

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 Schematic of a Transportation Problem

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 Transportation Table

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 Transshipment Problem

Cost of Shipping One Unit from the Farms to Warehouses

Cost of Shipping One Unit from the Warehouses to Projects

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 Transhipment Model

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 Using the Transportation Problem to Solve Aggregate
Planning Problems

• Involves creating a long-term production plan for achieving a

demand-supply balance.

• Aggregate planners usually avoid in terms of thinking of individual

products.

• Planners are concerned about the quantity and timing of production

to meet the expected demand.

• Aggregate planners attempt to minimize the production cost over the

planning horizon.
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 Transportation Table for Aggregate Planning Purposes

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 Exercise – Aggregate Planning

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 Transportation Table for the Aggregate Planning Problem

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 Using the Transportation Problem to Solve Location
Planning Problems

• Comparing transportation costs for alternative locations for new facilities to minimize total
cost.

• Provides planners an opportunity to assess the impact of each warehouse location on the
total distribution costs for the system.

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 Assignment Problem
A manager has prepared a table that shows the cost of performing each of five
jobs by each of five employees (see Table). According to this table, job I will cost $15 if
done by Al. $20 if it is done by Bill, and so on. The manager has stated that his goal is
to develop a set of job assignments that will minimize the total cost of getting all four
jobs done. It is further required that the jobs be performed simultaneously, thus
requiring one job being assigned to each employee.

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