The Transportation Problem

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The Transportation Problem

• ARE 511

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Usually, we have a given capacity of goods at
each source and a given requirement for the goods
at each destination. The objective of such a
transportation problem is to schedule shipments
from sources to destinations so that total
transportation costs are minimized.
Although LP can be used to solve this type of
problem, more efficient special-purpose
algorithms have been developed for the
transportation application. As in the simplex
algorithm, they involve finding an initial
feasible solution and then making step-by-step
improvements until an optimal solution is
reached.
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Unlike the simplex method, the transportation
methods are fairly simple in terms of
computation. To begin the analysis of a
transportation problem, management must determine
time costs of shipping from each source to each
destination. Such data are usually presented in
tabular form, as seen in the Table for the ease
of the Executive Furniture Corp., a manufacturer
of office desks.
The next step is the setting up of a
transportation table its purpose is to summarize
all relevant data concisely and to keep track of
algorithm computations. We now construct a
transportation table and label its various
components.
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• DEVELOPING AN INITIAL SOLUTION THE NORTHWEST
  CORNER RULE
• Once the data have been arranged in tabular form,
  we must establish an initial feasible solution to
  the problem (just as we did in the first LP
  simplex table).
• One systematic procedure, known as the northwest
  corner rule, requires that we start in the upper
  left hand cell (or northwest corner) of the table
  and allocate units to shipping routes as follows
• Exhaust the supply ( factory supply) at each row
  before moving down to the next row.
• Exhaust the (warehouse) requirements of each
  column before moving to the next column, on the
  right.
• Check that all the supply and demands are met.

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• EXAMPLE 13.1
• We can use the northwest corner rule to find an
  initial feasible solution to the Executive
  Furniture Corp. problem shown in the Table.
• It takes five steps in this example to make the
  initial shopping assignments
• Assign 100 units from Des Moines to Albuquerque
  (exhausting Des Moines supply).
• Assign 200 units from Evansville to Albuquerque
  (exhausting Albuquerques demand).
• Assign 100 units from Evansville to Boston
  (exhausting Evansvilles supply).
• Assign 100 units from Ft. Lauderdale to Boston
  (exhausting Bostons demand).
• Assign 200 units from Ft. Lauderdale to Cleveland
  (exhausting Clevelands demand and Ft.
  Lauderdales supply).

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We can easily compute the cost of this shipping
assignment
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The solution given here is feasible since
demand-and-supply constraints are all satisfied.
It would be very lucky if this solution yielded
the minimal transportation cost for the problem,
however. It is more likely that one of the
iterative procedures designed to help reach an
optimal solution shall have to be employed.
First, try to use the method to solve Problem
13.1.
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The Stepping-Stone Method The stepping-stone
method is an iterative technique for moving from
an initial feasible solution to an optimal
solution. It is used to evaluate the cost
effectiveness of shipping goods via
transportation routes not currently in the
solution. Each unused cell, or square, in the
transportation table is tested by asking the
following question What would happen to total
shipping costs if one unit of product (for
example, one desk) were tentatively shipped on an
unused route ?
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• This testing of each unused square is conducted
  as follows
• Select an unused square to be evaluated.
• Beginning at this square, trace a closed path
  back to the original square via squares that are
  currently being used (only horizontal and
  vertical moves are allowed).
• Beginning with a plus () sign at the unused
  square, place alternate minus (-) signs and plus
  signs on each corner square of time closed path
  just traced.
• Calculate an improvement index by adding
  together the unit cost figures found in each
  square containing a plus sign and then
  subtracting the unit costs in each square
  containing a minus sign.
• Repeat steps l 4 until an improvement index has
  been calculated for all unused squares. If all
  indices computed are greater than or equal to
  zero, an optimal solution lies been reached. If
  not, it is possible to improve the current
  solution and decrease total shipping costs

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EXAMPLE 13.2 The stepping-stone method can be
applied to the Executive Furniture Corp. data in
Example 13.1 to evaluate unused shipping
routes. The four currently used routes are seen
to be Des Moines to Boston, Des Moines to
Cleveland, Evansville to Cleveland and Fort
Lauderdale to Albuquerque. Beginning with the
Des Moines to Boston route we first trace a
closed path using only currently occupied squares
(see the table), and then place alternate plus
signs and minus signs in the corners of this
path. To indicate more clearly the meaning of a
closed path, we see that only squares currently
used for shipping can be used in turning the
corners of the route being traced.
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Hence, the path Des Moines - Boston to Des Moines
- Albuquerque to Fort Lauderdale - Albuquerque to
Fort Lauderdale - Boston to Des Moines - Boston
would not be acceptable since the Fort Lauderdale
- Albuquerque square is currently empty. It can
be seen that only one closed route is possible
for each square that we wish to test . How to
decide which squares are given plus signs and
which minus signs? The answer Since we are
testing the cost effectiveness of the Des Moines
to Boston shipping route, we pretend we are
shipping one desk from Des Moines to Boston. This
is one more unit that we are sending between the
two cities, so we place a plus sign in the box.
But, if we ship one more unit than before from
Des Moines to Boston, we end up sending 101 desks
out of the Des Moines factory.
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That factorys capacity is only 100 units, hence
we must ship one desk less from Des Moines to
Albuquerque, - this change made to avoid
violating the limit constraint. To indicate that
the Des Moines to Albuquerque shipment has been
reduced, we place a minus sign in its box.
Continuing along the closed path, we notice that
we are no longer meeting the warehouse
requirements of 300 units. In fact, if the Des
Moines to Albuquerque shipment is reduced to 99
units the Evansville to Albuquerque load has to
be increased by one unit, to 201
desks. Therefore, we place a plus sign in that
box to indicate the increase. Finally, we note
that if the Evansville to Albuquerque route is
assigned 201 desks, the Evansville to Boston
route must be reduced by one unit, to 99 desks,
in order to maintain the Evansville factory
capacity constraint of 300 units.
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Thus, a minus sign is placed in the Evansville to
Boston box. We observe below that all four routes
on the closed path are hereby balanced in terms
of demand- supply limitations.
An improvement index for the Des Moines - Boston
route is now computed by adding unit costs in
squares with plus signs and subtracting costs in
squares with minus signs. Hence Des Moines
Boston index 4 - 5 8 - 4 3
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This means that for every desk shipped via the
Des Moines - Boston route, total transportation
costs will increase by 3 over their current
level. Let us now examine the Des Moines -
Cleveland unused route, one slightly more
difficult to trace with a closed path. Again, you
will notice that we turn each corner along the
path only at squares which represent existing
routes. The path can go through the Evansville
Cleveland box but cannot turn a corner or place
a or - sign there. Only an occupied square may
be used as a stepping stone
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Des Moines Cleveland index 3 - 5 8 - 4
7 - 5 4 Thus, opening this route will also
not lower our total shipping costs. The other two
routes may be evaluated in a similar
fashion. Evansville Cleveland index 3 - 4
7 - 5 1 (Closed path is EC EB FB
FC) Fort Lauderdale Albuquerque index 9 -
7 4 - 8 -2 (Closed path is FA FB EB
EA) Because the second index is negative, a
cost savings may be attained by making use of the
(currently unused) Fort Lauderdale to Albuquerque
route.
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We saw, in Example 13.2 that a better solution is
possible - this due to the presence of a negative
improvement index on one of the unused routes.
Each negative index represents the amount by
which total transportation costs could be
decreased if one unit or product were shipped by
that sourcedestination combination. The next
step, then, is to choose that route (unused
square) with the largest negative improvement
index. We can then ship the maximum allowable
number of units on that route and reduce the
total cost accordingly. The maximum quantity
that can be shipped on the new money-saving route
can be found by referring to the closed path of
plus signs and minus signs drawn for the route
and selecting the smallest number found in those
squares containing minus signs.
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To obtain a new solution, that number is added to
all squares on the closed path with plus signs
and subtracted from all squares on the path
assigned minus signs. One iteration of the
stepping-stone method is now complete. Again, we
must test to see if it is optimal or whether any
further improvements can be made. That is done
by evaluating each unused square as described in
the steps listed earlier.
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EXAMPLE 13.3 To improve Executive Furnitures
solution we make use of the improvement indices
calculated in Example 13.2. The largest (and
only) negative index is found on the Fort
Lauderdale to Albuquerque route. We repeat the
transportation table for the problem below
The maximum quantity that may be shipped on the
newly opened route (FA) is the smallest number
found in squares containing minus signs-in this
case, 100 units.
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Hence, we add 100 units to the 0 now being
shipped on route FA then proceed to subtract 100
from route FB, leaving zero in that square (but
still balancing the row total for F) then add
100 to route B yielding 200 and finally,
subtract 100 from route EA, leaving 100 units
shipped. Note that the new members still produce
the correct row and column totals as
required. The new solution is shown in the
following table
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Total shipping cost has been reduced by (100
units) X (2 saved/unit) 200,and is now
4,000, This cost figure can, of course, also be
derived by multiplying each unit shipping cost
times the number of units transported on its
route, namely, lOO(5) l00( 8) 100(9)
200(5) 4,000
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Demand Not Equal To Supply A situation occurring
quite frequently in real-world problems is the
case where total demand is not equal to total
supply. These unbalanced problems can be handled
easily by the solution procedures discussed above
if we first in introduce dummy sources or dummy
destinations. In the event that total supply is
greater than total demand, a dummy destination,
with demand exactly equal to the surplus, is
created. If total demand is greater than total
supply, we introduce a dummy source (factory)
with a supply equal to the excess of demand over
supply. In either ease, cost coefficients of zero
are assigned to each dummy location.
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EXAMPLE 13.4 Executive Furniture Increases the
rate of production of desks in its Des Moines
factory to 250. To reformulate this unbalanced
problem, we refer back to the data presented in
Example 13.1. The northwest corner rule is used
to find the initial feasible solution below.
? New Des Moines Capacity
total cost (250)(5) (50)(8) (200)(4)
50(3) 150(5) 150(0) 3,350
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Degeneracy In order for the stepping-stone method
to be applied to a transportation problem, a rule
pertaining to the number of shipping routes being
used must be observed. That rule may be stated as
follows The number of occupied squares in any
solution (initial or later) must be equal to the
number of rows in the table plus the number of
columns minus 1. When this rule is not met, the
solution is called degenerate. Usually,
degeneracy occurs when there are too few squares,
or shipping routes, being used. When this
happens, it becomes impossible to trace a closed
path for each unused square.
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You might observe that no problem discussed in
this unit thus far has been degenerate. The
original furniture problem, for example, had 5
assigned routes ( 3 rows or factories 3
columns or warehouses - 1). Example 13.4,
employing a dummy warehouse had 6 assigned routes
( 3 rows 4 columns - 1) and was also not
degenerate. To handle degenerate problems, we
artificially create an occupied cell that is, if
we place a zero (representing a fake shipment) in
one of the unused squares and then treat that
square as it were occupied. This square chosen,
it should be noted, must be in such a position as
to allow all stepping-stone paths to be chosen.
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EXAMPLE 13.5 Martin Shipping Co. has three
warehouses from which to supply its three major
retail customers in San Jose. Martins shipping
costs, warehouse supplies, and customer demands
are made in the following transportation table.
Initial shipping assignments are made in the
table by application of the northwest corner rule.
This initial solution is degenerate because it
violates the rule that the number of used squares
be equal to the number of rows the number of
columns.
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To correct the problem, we may place a 0 in the
unused square representing the shipping route
from warehouse 2 to customer 1. Now all
stepping-stone paths can be closed and
improvement indices computed.
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The Modi Method The MODI (modified distribution)
method allows us to compute indices for each
unused square without drawing all the closed
paths. Because of this, it can often provide
considerable time savings over the stepping-stone
method for solving transportation problems. In
applying the MODI method, we begin with an
initial solution obtained by using the northwest
corner rule. But now, we must compute a value
for each row (call the values R1, R2, R3 if there
are 3 rows) and for each column (K1, K2, K3) in
the transportation table.
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• In general, we let
• Ri value assigned to row i.
• Kj value assigned to column j.
• Cij cost in square ij (cost of shipping from
  source i to destination j).
• The MODI method then requires three steps
• To compute the values for each row and column,
  set Ri Kj Cij but only for those squares that
  are currently used or occupied.
• After all equations have been written, set R1
  0.
• Solve the system of equations for all R and K
  values.
• Compute the improvement index for each unused
  square by the formula Cij Ri Kj.
• Select the largest negative index and proceed to
  solve the problem as we did using the
  stepping-stone method.

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EXAMPLE 13.6 Given the initial solution to the
Executive furniture problem (from Example 13.1),
we can use the MODI method to calculate an
improvement index for each unused square. The
initial transportation table is repeated below.
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• We first set up an equation for each occupied
  square
• R1 K1 5
• R2 K1 8
• R2 K2 4
• R3 K2 7
• R3 K3 5
• Letting R1 0, we can easily solve, step by
  step, for K1, R2, K2, R3 and K3.
• 0 K1 5 ? K1 5
• R2 5 8 ? R2 3
• 3 K2 4 ? K2 1
• R3 1 7 ? R3 6
• 6 K3 5 ? K3 -1

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The improvement index for each unused cell is Cij
Ri Kj. Des Moines to Boston C12 R1 K2
4 0 1 3. Des Moines to Cleveland C13
R1 K3 3 0 ( - 1) 4. Evansville to
Cleveland C23 R2 K3 3 3 ( - 1)
1. Ft. Lauderdale to Albuquerque C31 R3 K1
9 6 5 - 2. Note that these indices are
exactly the same as the ones calculated in
Example 13.2. It is now necessary to trace only
one closed path, for Ft. Lauderdale to
Albuquerque, in order to proceed with the
solution procedures as used in the stepping-stone
method.
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• Homework problems
• Problem 13.1 Northwest Corner Method
• Problem 13.2 Stepping-stone Method
• Problem 13.3 Stepping-stone Method
• Problem 13.4 Dummy Sources Method
• Problem 13.5 Degeneracy
• Problem 13.1 MODI Method