Transportation, Assignment and Transshipment Problems

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Chapter 4

• Transportation, Assignment and Transshipment
  Problems

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Description

• A transportation problem basically deals with the
  problem, which aims to find the best way to
  fulfill the demand of n demand points using the
  capacities of m supply points. While trying to
  find the best way, generally a variable cost of
  shipping the product from one supply point to a
  demand point or a similar constraint should be
  taken into consideration.

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Formulating Transportation Problems
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Formulating Transportation Problems

• we have to determine how much electricity is sent
  from each plant to each city
• Decision Variable
• Xij Amount of electricity produced at plant i
  and sent to city

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Objective function

• minimize the total cost of shipping from plants
  to cities
• Minimize Z 8X11 6X12 10X139X14
• 9X21 12X2213X237X24
• 14X31 9X3216X335X34

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• Supply Constraints each supply point has a
  limited production capacity
• X11X12X13X14 lt 35
• X21X22X23X24 lt 50
• X31X32X33X34 lt 40
• Demand Constraints each supply point has a
  limited production capacity
• X11X21X31 gt 45
• X12X22X32 gt 20
• X13X23X33 gt 30
• X14X24X34 gt 30

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• Sign Constraints negative amount of electricity
  can not be shipped all Xijs must be non
  negative
• Xij gt 0 (i 1,2,3 j 1,2,3,4)

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LP Formulation of Powercos Problem

• Min Z 8X116X1210X139X149X2112X22
• 13X237X2414X319X3216X335X34
• S.T.
• X11X12X13X14 lt 35 (Supply
  Constraints)
• X21X22X23X24 lt 50
• X31X32X33X34 lt 40
• X11X21X31 gt 45 (Demand Constraints)
• X12X22X32 gt 20
• X13X23X33 gt 30
• X14X24X34 gt 30
• Xij gt 0 (i 1,2,3 j 1,2,3,4)

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General Description of a Transportation Problem

• A set of m supply points from which a good is
  shipped. Supply point i can supply at most si
  units.
• A set of n demand points to which the good is
  shipped. Demand point j must receive at least di
  units of the shipped good.
• Each unit produced at supply point i and shipped
  to demand point j incurs a variable cost of cij.

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Xij number of units shipped from supply point i
to demand point j
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Balanced Transportation Problem
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Balancing a TP if total supply exceeds total
demand

• If total supply gt total demand,
• adding dummy demand point.
• Since shipments to the dummy demand point are not
  real, they are assigned a cost of zero.

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Balancing a transportation problem if total
supply is less than total demand

• If total supply lt total demand (no feasible
  solution)
• one or more of the demand will be left unmet.
• a penalty cost is often associated with unmet
  demand

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Methods to find the bfs for a balanced TP

• There are three basic methods
• Northwest Corner Method
• Minimum Cost Method
• Vogels Method

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Northwest Corner Method
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Minimum Cost Method

• begin the minimum cost method, first we find the
  decision variable with the smallest shipping cost
  (Xij).
• Then assign Xij its largest possible value, which
  is the minimum of si and dj
• Cross out row i and column j and reduce the
  supply or demand of the noncrossed-out row or
  column by the value of Xij.
• Choose the cell with the minimum cost of shipping
  from the cells that do not lie in a crossed-out
  row or column and we will repeat the procedure.

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Vogels Method

• computing each row and column a penalty.
• penalty difference between the two smallest
  shipping costs in the row or column.
• Identify the row or column with the largest
  penalty.
• Find the first basic variable which has the
  smallest shipping cost in that row or column.
• Then assign the highest possible value to that
  variable, and cross-out the row or column as in
  the previous methods.
• Compute new penalties and use the same procedure