The Transportation Model

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The Transportation Model Formulations
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The Transportation Model The transportation model
is a special class of LPPs that deals with
transporting(shipping) a commodity from sources
(e.g. factories) to destinations (e.g.
warehouses). The objective is to determine the
shipping schedule that minimizes the total
shipping cost while satisfying supply and demand
limits. We assume that the shipping cost is
proportional to the number of units shipped on a
given route.
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We assume that there are m sources 1,2, , m and
n destinations 1, 2, , n. The cost of shipping
one unit from Source i to Destination j is cij.
We assume that the availability at source i is
ai (i1, 2, , m) and the demand at
the destination j is bj (j1, 2, , n). We make
an important assumption the problem is a
balanced one. That is
That is, total availability equals total demand.
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We can always meet this condition by introducing
a dummy source (if the total demand is more than
the total supply) or a dummy destination (if the
total supply is more than the total demand). Let
xij be the amount of commodity to be shipped from
the source i to the destination j.
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Thus the problem becomes the LPP
Minimize
subject to
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Thus there are m?n decision variables xij and mn
constraints. Since the sum of the first m
constraints equals the sum of the last n
constraints (because the problem is a balanced
one), one of the constraints is redundant and we
can show that the other mn-1 constraints are LI.
Thus any BFS will have only mn-1 nonzero
variables. Though we can solve the above LPP by
Simplex method, we solve it by a special
algorithm called the transportation algorithm.
We present the data in an m?n tableau as
explained below.
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Destination
1 2 . . n
Supply
1 2 . . m
Source
Demand
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Formulation of Transportation Models Example
5.1-2 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, 1300 and 1200 cars. The
quarterly demands at the two distribution centers
are 2300 and 1400 cars. The transportation cost
per car from Los Angeles to Denver and Miami are
80 and 215 respectively. The corresponding
figures from Detroit and New Orleans are 100, 108
and 102, 68 respectively.
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Formulate the transportation Model.
Since the total demand 3700 gt 3500 (Total
supply) we introduce a dummy supply with
availability 3700-3500200 units to make the
problem a balanced one. If a destination receives
u units from the dummy source, it means that that
destination gets u units less than what it
demanded. We usually put the cost per unit of
transporting from a dummy source as zero (unless
some restrictions are there). Thus we get the
transportation tableau
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Destination
Denver Miami Supply
Los Angeles
Source
Detroit
New Orleans
Dummy
Demand
We write inside the (i,j) cell the amount to be
shipped from source i to destination j. A blank
inside a cell indicates no amount was shipped.
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Problem 5 Problem Set 5.1A Page 169 In the
previous problem, penalty costs are levied at the
rate of 200 and 300 for each undelivered car at
Denver and Miami respectively. Additionally no
deliveries are made from the Los Angeles plant to
the Miami distribution center. Set up the
transportation model.
The above imply that the "cost" of transporting a
car from the dummy source to Denver and Miami are
respectively 200 and 300. The second condition
means we put a "high" transportation cost from
Los Angeles to Miami. We thus get the tableau
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Destination
Denver Miami Supply
Los Angeles
Source
Detroit
New Orleans
Dummy
Demand
Note M indicates a very "big" positive number.
In TORA it is denoted by "infinity".
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Problem 8 Problem Set 5.1A Page 170 Three
refineries with daily capacities of 6,5, and 8
million gallons, respectively, supply three
distribution areas with daily demands of 4,8, and
7 million gallons, respectively.Gasoline is
distributed to the three distribution areas
through a network of pipelines. The
transportation cost is 10 cents per 1000 gallons
per pipeline mile. The table below gives the
mileage between the refineries and the
distribution areas. Refinery 1 is not connected
to the distribution area 3.
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Distribution Area
1 2 3
1 120 180 -
Refinery
2 300 100 80
3 200 250 120
Construct the associated transportation model.
(Solution in the next slide)
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Destination
Distribution Area
1 2 3 Supply
Source
1
Refinery
2
3
Demand
The problem is a balanced one. M indicates a very
"big" positive number.
The total cost will be 1000
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Problem 10 Problem Set 5.1A Page 170 In the
previous problem, suppose that the daily demand
at area 3 drops to 4 million gallons. Surplus
production at refineries 1 and 2 is diverted to
other distribution areas by truck. The
transportation cost per 100 gallons is 1.50 from
refinery 1 and 2.20 from refinery 2. Refinery 3
can divert its surplus production to other
chemical processes within the plant. Formulate
the problem as a transportation model.
We introduce a dummy destination. Solution
follows.
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Destination
Distribution Area
1 2 3 Dummy Supply
Source
1
2
Refinery
3
Demand
M indicates a very "big" positive number.
The total cost will be 1000
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Problem 11 Problem Set 5.1 A Pages
170-171 Three orchards supply crates of oranges
to four retailers. The daily demand at the four
retailers is 150,150,400, and 100 crates,
respectively. Supply at the three orchards is
dictated by available regular labor and is
estimated at 150, 200, and 250 crates daily.
However, both orchards 1 and 2 have indicated
that they could supply more crates, if necessary
by using overtime labor. Orchard 3 does not offer
this option. The transportation costs (in
dollars) per crate from the orchards to the
retailers are given in Table below.
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Retailer
1 2 3 4
1 1 2 3 2
Orchard
2 2 4 1 2
3 1 3 5 3
Formulate the problem as a transportation model.
Since the orchards 1, 2 can supply more crates
with overtime labor, we increase their capacities
to 150200350 and 200200400 respectively (as
initially the total supply fell short by 200).
But then to balance the problem we add a dummy
destination. The tableau follows.
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Destination
Retailer
1 2 3 4 Dummy
Supply
Source
Orchard
1
2
3
Demand
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Problem 8.1-9 from Hillier and Lieberman
(Introduction to Operations Research, 7th
Edition) The Build-Em-Fast Company has agreed to
supply its best customer with three widgets
during each of the next 3 weeks, even though
producing them will require some overtime work.
The relevant production data are as follows
Week Max Production Max Production Prod
Cost / unit
Regular Time Overtime
Regular Time
1 2
2 300
2 3
2 500
3 1
2 400
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The cost / unit produced overtime for each week
is 100 more than for regular time. The cost of
storage is 50 / unit for each week it is stored.
There is already an inventory of 2 widgets on
hand currently, but the company does not want to
retain any widgets in inventory after the 3
weeks. Formulate the problem as a transportation
problem.
There are 6 sources namely widgets produced
regular time and overtime for the three weeks.
Also there are 3 destinations viz. demand for
the three weeks.
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We let xij as the number of units produced
regular time in week (i1)/2 for use in week j
(i1,3,5 j1,2,3). We let xij as the number of
units produced overtime in week i/2 for use in
week j (i2,4,6 j1,2,3). Thus
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To make these equalities we add a dummy
destination and let xi4 as the amount
transported from Source i to this dummy. Thus
the availabilities at the 6 sources are
2,2,3,2,1,2 respectively.
The demands at the three destinations (demand
for the three weeks) are
1,3,3 respectively (as the initial inventory is 2
widgets).
To make the problem balanced we add
demand 12 7 5 at the dummy destination.
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The cost per unit, cij are as follows
These are written in a transportation tableau.
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Destination
Demand for Week

1 2 3 Dummy Supply
1
Prod.week1 Reg time
Source
2
Prod.week1 Over time
3
Prod.week2 Reg time
Prod.week2 Over time
4
Prod.week3 Reg time
5
Prod.week3 Over time
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Demand