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TRANSCRIPT
Chapter 5
Transportation Model
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Kingdom Of Saudi Arabia
Al-Imam Muhammad Ibn Saud Islamic University
College of Computer and Information Sciences
Information System Department
IS 332: Decision Support System
From “Operations Research: An Introduction” Book
Done by: T.Alkhansaa Abuhashim
The Transportation Problem
The transportation problem is a special case
of a LP problem
Because of its characteristics, the
transportation problem can be solved very
efficiently with a special algorithm called the
transportation algorithm
The problem is concerned with specifying how
to disposition a single product from several
sources to several destinations at minimum
cost
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Supply
Capacity
from source
m
Demand
required from
by destination n
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Car Distribution Problem
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The MG Auto Company has plants in LA, Detroit, and New Orleans.
Its major distribution centers are located 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 transportation
cost per car mile (by train) is approximately 8 cents. The mileage
chart between the plants and distribution centers is as follows:
Denver Miami
Los Angeles 1000 2690
Detroit 1250 1350
New Orleans 1275 850
Done by: T.Alkhansaa Abuhashim
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The mileage chart can be translated to cost per car at the rate of 8 cents/mile, which gives us the cost in $’s per car for shipping.
Find the best strategy to send cars from the plants to the distribution centers.
Denver Miami
Los Angeles $80 $215
Detroit $100 $108
New Orleans $102 &68
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Let Xij = units sent from location i (1, 2, 3) which refers to L,N,D to destination j (1, 2) which are V,M.
Then, the problem can be stated as:
Minimize z = 80X11 + 215X12 + 100X21+ 108X22+102X31+68X32
Subject to:X11 + X12 = 1000 (LA)
X21+ X22 = 1500 (Detroit)X31+ X32 = 1200 (New O.)
X11 + X21 + X31 = 2300 (Denver)X12 + X22 +X32 = 1400 (Miami)
X11 , X12 , X21 , X22 , X31 , X32 0
Done by: T.Alkhansaa Abuhashim
Transportation Tableau
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Markets (Destinations)
SupplyDenver Miami
P
l
an
t
Los
Angeles X11
80
X12
215 1000
Detroit
X21
100
X22
108 1500
New
Orleans X31
102
X32
68 1200
Demand 2300 1400
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The optimal solution in the previous figure -
obtained by TORA- 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.
The associated minimum transportation cost is
computed as 1000 × $80 + 1300 × $100 + 200 ×
$108 + 1200 × $68 = $313,200.
Done by: T.Alkhansaa Abuhashim
The General Transportation Problem
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In general, a transportation problem can be expressed as:
Subject to:
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Balancing the Transportation Model
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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.
Done by: T.Alkhansaa Abuhashim
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Characteristics of the Transportation
Problem
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The transportation problem could be solved using the regular simplex method. However, because of its special characteristics, a more efficient procedure is used. The procedure is called the transportation (simplex) method.
The transportation problem is solved in two phases:
1. Determination of an initial basic feasible solution
2. Finding an optimal solution through the sequential improvement of the initial feasible solution
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Nontraditional Transportation Models
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The application of the transportation model is not
limited to transporting commodities between
geographical sources and destinations.
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Production-Inventory Control
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Boralis manufactures backpacks for serious hikers. The demand for its product occurs during March to June of each year. Boralis estimates the demand for the four months to be 100, 200, 180, and 300 units, respectively. The company uses part-time labor to manufacture the backpacks and, accordingly, its production capacity varies monthly. It is estimated that Boralis can produce 50,180, 280, and 270 units in March through June. Because the production capacity and demand for the different months do not match, a current month's demand may be satisfied in one of three ways: Current month's production. Surplus production in an earlier month. Surplus production in a later month (backordering).
Done by: T.Alkhansaa Abuhashim
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In the first case, the production cost per backpack
is $40. The second case incurs an additional
holding cost of $.50 per backpack per month. In
the third case, an additional penalty cost of $2.00
per backpack is incurred for each month delay.
Boralis wishes to determine the optimal
production schedule for the four months.
Done by: T.Alkhansaa Abuhashim
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Production-inventory
1. Production period i (source i)
2. Demand period j (destination j)
3. Production capacity of period I (Supply amount at source i)
4. Demand for period j (Demand at destination j)
5. Unit cost (production + inventory + penalty) in period i for period j
(Unit transportation cost from source i to destination j)
Done by: T.Alkhansaa Abuhashim
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Optimal solution of the production-
inventory model
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The transportation cost for unit from period i to period j is computed as
Production cost in i, i = j
cij = Production cost in i + holding cost from i to j, i<j
Production cost in i + penalty cost from i to j, i>j
For example:
c11= $40
c24 = $40 + ($.50 + $.5) =$41
c41 = $40 + ($2 + $2 + $2) = $46
The dashed lines indicate back-ordering, the dotted lines in indicate production for a future period, and the solid lines show production in a period for itself. The total cost is $31,455.
Done by: T.Alkhansaa Abuhashim