Week 8: Transport Model
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1.Transport Model
The transportation model is a special class of linear programs that deals
with shipping a commodity from sources to destinations . The objective
is to determine the shipping schedule that minimizes the total shipping
cost while satisfying supply and demand limits. The application of the
transportation model can be extended to other areas of operation,
including inventory control, employment scheduling, and personnel
assignment.
2. Formulating Transportation Problems
transportation problem is represented by the network in figure (1), there
are m sources and n destinations, each represented by a node. the arcs
represent the routes linking the sources and the destinations. arc (i, j)
joining source i to destination j carries two pieces of information: the
transportation cost per unit, cij' and the amount shipped, xij' the amount
of supply at source i is ai and the amount of demand at destination j is bj .
the objective of the model is to determine the unknowns xij that will
minimize the total transportation cost while satisfying all the supply and
demand restrictions.
Figure 1: Representation of the transportation model with nodes and arcs
The relevant data can be formulated in a transportation tableau as:
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If total supply equals total demand then the problem is said to be a
balanced transportation problem.
Example 1:
MG Auto has three plants in Los Angeles, Detroit, and New Orleans, and
two major distribution centers in Denver and Miami. The capacities of the
three plants during the next quarter are 1000, 1500, and 1200 cars. The
quarterly demands at the two distribution centers are 2300 and 1400 cars.
The mileage chart between the plants and the distribution centers is given
in Table 1.The trucking company in charge of transporting the cars
charges 8 cents per mile per car. The transportation costs per car on the
different routes, rounded to the closest dollar, are given in Table 2.
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Table 1:Mileage Cgart
Denver
Miami
Los Angeles
1000
2690
Detroit
1250
1350
New Orleans
1275
850
Table 2:transportation cost per car
Denver(1)
Miami(2)
Los Angeles(1)
$80
$215
Detroit(2)
$100
$108
New Orleans (3)
$102
$68
The LP model of the problem is given as
Minimize z = 80x11 + 215x12 + 100x21 + 108x22 + 102x31 + 68x32
Subject to
X11+x12
=1000 (Los Angeles)
+x21+x22
=1500 (Detroit)
+x31+x32 =1200 (New Oreleans)
X11
+x21
+x31
=2300 (Denver)
X21+
x22+ x32 =1400 (Miami)
xij≥0 , i=1,2,3 ,j=1,2
These constraints are ail equations because the total supply from the three
sources (= 1000 + 1500 + 1200 = 3700cars) equals the total demand at
the two destinations (= 2300 + 1400 = 3700 cars).
The LP model can be solved by the simplex method. However, with the
special structure of the constraints we can solve the problem more
conveniently using the transportation tableau shown in Table (3)
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Table 3: transportation tableau of MG
Denver
Los Angeles
Miami
80
X11
100
Demand
215
1000
108
1500
X12
Detroit
New Orleans
Supply
X21
X22
102
X31
68
X32
2300
1200
1400
The optimal solution in Figure (2) calls for shipping 1000 cars from Los
Angeles to Denver, 1300 from Detroit to Denver, 200 from Detroit to
Miami, and 1200 from New Orleans to Miami. lhe associated minimum
transportation cost is computed as
1000 x $80 + 1300 x $100 + 200 x $108 + 1200 x $68 = $313,200.
Figure 2 : Optimal solution of MG Auto model
Example 2:
Powerco has three electric power plants that supply the needs of four
cities. Each power plant can supply the following numbers of kwh of
electricity: plant 1, 35 million; plant 2, 50 million; and plant 3, 40
million. The peak power demands in these cities as follows (in kwh): city
1, 45 million; city 2, 20 million; city 3, 30 million; city 4, 30 million. The
costs of sending 1 million kwh of electricity from plant to city is given in
the table below. To minimize the cost of meeting each city’s peak power
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demand, formulate a balanced transportation problem in a transportation
tableau and represent the problem as a LP model.
from
Plant 1
Plant 2
Plant 3
To
City1
$8
$9
$14
City2
$6
$12
$9
City3
$10
$13
$16
City4
$9
$7
$5
Solution:
Representation of the problem as a LP model
xij: number of (million) kwh produced at plant i and sent to city j.
min z = 8 x11 + 6 x12 + 10 x13 + 9 x14 + 9 x21 + 12 x22 + 13 x23 + 7 x24 + 14 x31
+ 9 x32 + 16 x33 + 5 x34
s.t.
x11 + x12 + x13 + x14 < 35 (supply constraints)
x21 + x22 + x23 + x24 < 50
x31 + x32 + x33 + x34 < 40
x11 + x21 + x31 > 45 (demand constraints)
x12 + x22 + x32 > 20
x13 + x23 + x33 > 30
x14 + x24 + x34 > 30
xij > 0 (i = 1, 2, 3; j = 1, 2, 3, 4)
formulation of the transportation problem in transport tableau as:
3. Balancing the Transportation Model
The transportation algorithm is based on the assumption that the model is
balanced, meaning that the total demand equals the total supply. If the
model is unbalanced, we can always add a dummy source or a dummy
destination to restore balance.
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3.1 Excess Supply
If total supply exceeds total demand, we can balance a transportation
problem by creating a dummy demand point that has a demand equal to
the amount of excess supply. Since shipments to the dummy demand
point are not real shipments, they are assigned a cost of zero. These
shipments indicate unused supply capacity.
3.2 Unmet Demand
If total supply is less than total demand, actually the problem has no
feasible solution. To solve the problem it is sometimes desirable to allow
the possibility of leaving some demand unmet. In such a situation, a
penalty is often associated with unmet demand. This means that a dummy
supply point should be introduced.
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