Transportation and Assignment Problems

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Transportation and Assignment Problems

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Applications of Network Optimization
Applications
Physical analogof nodes
Physical analogof arcs
Flow
Communicationsystems
phone exchanges, computers, transmission facilit
ies, satellites
Cables, fiber optic links, microwave relay
links
Voice messages, Data, Video transmissions
Hydraulic systems
Pumping stationsReservoirs, Lakes
Pipelines
Water, Gas, Oil,Hydraulic fluids
Integrated computer circuits
Gates, registers,processors
Wires
Electrical current
Mechanical systems
Joints
Rods, Beams, Springs
Heat, Energy
Transportationsystems
Intersections, Airports,Rail yards
Highways,Airline routes Railbeds
Passengers, freight, vehicles, operators
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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

• Example 1 Powerco has three electric power
  plants that supply the electric needs of four
  cities.
• The associated supply of each plant and demand of
  each city is given in the table 1.
• The cost of sending 1 million kwh of electricity
  from a plant to a city depends on the distance
  the electricity must travel.

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Transportation tableau

• A transportation problem is specified by the
  supply, the demand, and the shipping costs. So
  the relevant data can be summarized in a
  transportation tableau. The transportation
  tableau implicitly expresses the supply and
  demand constraints and the shipping cost between
  each demand and supply point.

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Table 1. Shipping costs, Supply, and Demand for
Powerco Example
From To To To To To
From City 1 City 2 City 3 City 4 Supply (Million kwh)
Plant 1 8 6 10 9 35
Plant 2 9 12 13 7 50
Plant 3 14 9 16 5 40
Demand (Million kwh) 45 20 30 30
Transportation Tableau
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Solution

• Decision Variable
• Since we have to determine how much electricity
  is sent from each plant to each city
• Xij Amount of electricity produced at plant i
  and sent to city j
• X14 Amount of electricity produced at plant 1
  and sent to city 4

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2. Objective function

• Since we want to minimize the total cost of
  shipping from plants to cities
• Minimize Z 8X116X1210X139X14
• 9X2112X2213X237X24
• 14X319X3216X335X34

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3. Supply Constraints

• Since each supply point has a limited production
  capacity
• X11X12X13X14 lt 35
• X21X22X23X24 lt 50
• X31X32X33X34 lt 40

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4. Demand Constraints

• Since each supply point has a limited production
  capacity
• X11X21X31 gt 45
• X12X22X32 gt 20
• X13X23X33 gt 30
• X14X24X34 gt 30

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5. Sign Constraints

• Since a 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 8X116X1210X139X149X2112X2213X237X24
  
• 14X319X3216X335X34
• 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

1. A set of m supply points from which a good is
   shipped. Supply point i can supply at most si
   units.
2. 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.
3. 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

• If Total supply equals to total demand, the
  problem is said to be a balanced transportation
  problem

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Methods to find the bfs for a balanced TP

• There are two basic methods
• Northwest Corner Method
• Vogels Method

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1. Northwest Corner Method

• To find the bfs by the NWC method
• Begin in the upper left (northwest) corner of the
  transportation tableau and set x11 as large as
  possible (here the limitations for setting x11 to
  a larger number, will be the demand of demand
  point 1 and the supply of supply point 1. Your
  x11 value can not be greater than minimum of this
  2 values).

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According to the explanations in the previous
slide we can set x113 (meaning demand of demand
point 1 is satisfied by supply point 1).
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After we check the east and south cells, we saw
that we can go east (meaning supply point 1 still
has capacity to fulfill some demand).
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After applying the same procedure, we saw that we
can go south this time (meaning demand point 2
needs more supply by supply point 2).
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Finally, we will have the following bfs, which
is x113, x122, x223, x232, x241, x342
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3. Vogels Method

• Begin with computing each row and column a
  penalty. The penalty will be equal to the
  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.

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An example for Vogels MethodStep 1 Compute the
penalties.
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Step 2 Identify the largest penalty and assign
the highest possible value to the variable.
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Step 3 Identify the largest penalty and assign
the highest possible value to the variable.
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Step 4 Identify the largest penalty and assign
the highest possible value to the variable.
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Step 5 Finally the bfs is found as X110, X125,
X135, and X2115
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The Transportation Simplex Method

• In this section we will explain how the simplex
  algorithm is used to solve a transportation
  problem.

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How to Pivot a Transportation Problem

• Based on the transportation tableau, the
  following steps should be performed.
• Step 1. Determine (by a criterion to be developed
  shortly, for example northwest corner method) the
  variable that should enter the basis.
• Step 2. Find the loop (it can be shown that there
  is only one loop) involving the entering variable
  and some of the basic variables.
• Step 3. Counting the cells in the loop, label
  them as even cells or odd cells.

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• Step 4. Find the odd cells whose variable assumes
  the smallest value. Call this value ?. The
  variable corresponding to this odd cell will
  leave the basis. To perform the pivot, decrease
  the value of each odd cell by ? and increase the
  value of each even cell by ?. The variables that
  are not in the loop remain unchanged. The pivot
  is now complete. If ?0, the entering variable
  will equal 0, and an odd variable that has a
  current value of 0 will leave the basis. In this
  case a degenerate bfs existed before and will
  result after the pivot. If more than one odd cell
  in the loop equals ?, you may arbitrarily choose
  one of these odd cells to leave the basis again
  a degenerate bfs will result

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Assignment Problems

• Example Machineco has four jobs to be completed.
  Each machine must be assigned to complete one
  job. The time required to setup each machine for
  completing each job is shown in the table below.
  Machinco wants to minimize the total setup time
  needed to complete the four jobs.

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• Setup times
• (Also called the cost matrix)

Time (Hours) Time (Hours) Time (Hours) Time (Hours)
Job1 Job2 Job3 Job4
Machine 1 14 5 8 7
Machine 2 2 12 6 5
Machine 3 7 8 3 9
Machine 4 2 4 6 10
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The Model

• According to the setup table Machincos problem
  can be formulated as follows (for i,j1,2,3,4)

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• For the model on the previous page note that
• Xij1 if machine i is assigned to meet the
  demands of job j
• Xij0 if machine i is not assigned to meet the
  demands of job j
• In general an assignment problem is balanced
  transportation problem in which all supplies and
  demands are equal to 1.

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The Assignment Problem

• In general the LP formulation is given as
• Minimize

Each supply is 1
Each demand is 1