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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India TRANSPORTATION MODEL 4 CHAPTER

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 3 Learning Objectives Structure special LP problems using the transportation and assignment models. Use the N.W. corner, Least Cost Method, VAM, stepping-stone and MODI method.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 4 Transportation Model - Characteristics Transportation problem deals with distribution of items from several sources to several destinations. Supply capacities and destination requirements are known and the cost of moving one unit from any source to any destination is also known. It aims at minimising the transportation cost. Only a single commodity can be moved.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 5 Transportation Problem Plant 1 50 tons Plant 2 80 tons Plant 3 70 tons Plant 4 140 tons Project B 180 tons Project C 90 tons Project A 70 tons 2 1 6 2 1 4 7 5 3 3 7 4

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 6 Setting up the Transportation Table Plants ProjectsAvailability ABC 127450 233180 354770 4162140 Demand7090180340 Cost of moving 1 ton from Plant 1 to Project A

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 7 Setting up the Transportation Table Check that demand and availability are equal or balanced. In this case they are equal. Develop an initial feasible solution. It must have cells occupied, i.e. cells representing routes along which the commodity is moved. m is the number of rows and n is the number of columns. assignments or allocations should be independent.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 8 Setting up the Transportation Table Independent allocations imply that it is not possible to start from an occupied cell and trace a path back to it by moving horizontally and vertically through other cells in such a manner that all cells at the corners of the path are occupied.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 9 North West Corner Rule Start in the upper left-hand cell and allocate units to shipping routes as follows: –Exhaust the availability of each row before moving down to the next row. –Exhaust the demand requirements of each column before moving to the next column to the right. –Check that all supply and demand requirements are met.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 10 Plant ProjectsAvailability ABC 150274 2203603180 3530440770 4161402 Demand7090180340 Initial Solution North West Corner Rule

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 11 Least Cost Method Allocate maximum number of units possible starting with the route with the least cost. In case of a tie, chose any one arbitrarily. Having done this, follow same logic till all required units have been moved.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 12 Plant ProjectsAvailability ABC 1220730450 233801 357047 4 16 2140 Demand7090180340 Initial Solution Least Cost Method

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 13 Vogel’s Approximation Method (VAM) For each row/column of table, find difference between two lowest costs. (Opportunity cost) Find greatest opportunity cost. Assign as many units as possible to lowest cost cell in row/column with greatest opportunity cost. Eliminate row or column which has been completely satisfied. Begin again, omitting eliminated rows/columns.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 14 Vogel’s Approximation Method 3401809070Dem- and 140210061404 7074 53 801 332 5047202301 CBA Avail- ability Projects Plant 2 2 1 1 111 2 1 1 122 5 5 1 12 5 1 33

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 15 Comparative Costs North West Corner Rule – Rs 1020 Least Cost Method – Rs 830 VAM – Rs 800. Check for further improvement by: –Stepping Stone Method –Modified Distribution (MODI)

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 16 Stepping Stone Method Select any unused cell to evaluate. Begin at this cell. Trace a closed path back to the original cell via cells that are currently being used (only horizontal or vertical moves allowed). Place + in unused cell; alternate - and + on each corner cell of the closed path. Calculate opportunity cost: add together the unit cost figures found in each square containing a -; subtract the unit cost figure in each square containing a +. Repeat above steps for each unused square. If opportunity cost of all unused cells is zero or negative, an optimal solution has been reached

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 17 Plant ProjectsAvailability ABC 15027Start450 -++1 2203603180 +- 3530440770 +- 4161402 Demand7090180340 Stepping Stone Method Plant 1 to Project C Route

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 18 Plant ProjectsAvailability ABC 150274 -5+1 2203603180 +5 3530440770 4161402 -2-7 Demand7090180340 Stepping Stone Method Opportunity cost of all unused cells

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 19 Select cell with highest positive opportunity cost. Begin at this cell. Trace a closed path back to the original cell via cells that are currently being used (only horizontal or vertical moves allowed). Place + in unused cell; alternate - and + on each corner cell of the closed path. Select the smallest quantity being shipped in the cells in the negative positions. Add this quantity to all cells with a positive sign and subtract it from all cells with a negative sign. If opportunity cost of all unused cells is zero or negative, an optimal solution has been reached, else recalculate opportunity cost of unused cells and repeat this step. Stepping Stone Method Opportunity cost of all unused cells

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 20 Plant ProjectsAvailability ABC 150274 -5+1 220360 20340180 -++5 3530 70440770 +- 4161402 -2-7 Demand7090180340 Stepping Stone Method – Developing an improved solution

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 21 Plant ProjectsAvailability ABC 150274 -5-4 2203 340 60180 -+ 357047 -5 42016140 120 2140 ++3-2- Demand7090180340 Stepping Stone Method – Developing an improved solution (2)

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 22 Plant ProjectsAvailability ABC 150274 -2 2320360180 -3 357047 -4-5 420161202140 -2 Demand7090180340 Stepping Stone Method – Developing an improved solution (3)

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 23 Stepping Stone Method – Final solution (3) As opportunity cost of all cells is negative an optimal solution is reached. –From Plant 1 send 50 tons to Project A –From Plant 2 send 20 tons to Project B and 60 tons to Project C –From Plant 3 send 70 tons to Project B –From Plant 4 send 20 tons to Project A and 120 tons to project C –Total cost Rs 760.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 24 MODI Compute the values of u i for each row and v j for each column: set u i + v j = C ij for all occupied or used cells. Set any one u i or v j value as zero Compute other u i and v j values. Compute the opportunity cost for each unused cell by the formula, Opportunity Cost = u i + v j - C ij Select the cell with the largest positive opportunity cost and proceed to solve the problem as you did using the stepping-stone method.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 25 Plant ProjectsAvailability ABC 150274 -5+1 2203603180 +5 3530440770 4161402 -2-7 Demand7090180340 uiui 0 2 1 -3 vjvj 522 MODI – Opportunity Cost of all unused cells

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 26 Plant ProjectsAvailability ABC 150274 -5+1 220360 20340180 -++5 3530 70440770 +- 4161402 -2-7 Demand7090180340 uiui 0 2 1 -3 vjvj 522 MODI – Improved solution (1)

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 27 Plant ProjectsAvailability ABC 150274 -5-4 2203 340 60180 -+-5 357047 42016140 1202140 ++3-2- Demand7090180340 uiui 0 2 1 2 vjvj 022 MODI – Improved solution (2)

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 28 Plant ProjectsAvailability ABC 150274 -2 2320360180 -3 357047 -4-5 420161202140 -2 Demand7090180340 uiui 0 -2 vjvj 352 MODI – Improved solution (3)

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 29 MODI – Final solution (3) As opportunity cost of all cells is negative an optimal solution is reached. –From Plant 1 send 50 tons to Project A –From Plant 2 send 20 tons to Project B and 60 tons to Project C –From Plant 3 send 70 tons to Project B –From Plant 4 send 20 tons to Project A and 120 tons to project C –Total cost Rs 760. Solution is the same as that obtained by stepping stone method.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 30 Transportation Model – Special Problems Unbalanced Problem –Demand Less than Supply –Demand Greater than Supply Degeneracy More Than One Optimal Solution

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 31 Unbalanced Problem When the demand and supply are not equal, the problem is unbalanced. A dummy source or a dummy destination is introduced to balance the problem. Since the dummy is only imaginary, the cost of moving from or to a dummy is zero.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 32 Unbalanced Problem – Demand greater than Supply Plants ProjectsAvailability ABC 127450 233180 354770 4162140 Demand90 180340 360

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 33 Introduce a dummy Plant and solve as a normal transportation problem Plants ProjectsAvailability ABC 127450 233180 354770 4162140 Dummy00020 Demand90 180360

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 34 Unbalanced Problem – Supply greater than Demand Plants ProjectsAvailability ABC 127450 2331100 354770 4162140 Demand7090180360 340

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 35 Introduce a dummy destination and proceed as in a normal case. Plants ProjectsAvailability ABCDummy 1274050 23310100 3547070 41620140 Demand709018020360

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 36 Degeneracy When the number of used cells is more or less than If the number of used cells is more, then either the formulation is incorrect or an improper assignment has been made.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 37 Degeneracy If the used cells are less than it is not possible to calculate all values. Such a situation may occur when demand and supply get exhausted simultaneously while making initial assignments. It may also occur when two cells become empty when moving quantities on the closed path. This is remedied by putting a zero or ε in a cell and treating it as used or occupied.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 38 Degeneracy – Initial solution Plant ProjectsAvailability ABC 150274 2 ε 38031 3510460770 4161202 Demand5090180320

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 39 Degeneracy – During operations OriginsDestinationsSupply ABCB 1310520 1 ε 51990 -++18 22103 4813043 +11+13+- 312075105-17 -8-17-19 Demand354618 1020527

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 40 Multiple Solutions If the opportunity cost of an unused cell is zero, multiple solutions are possible.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 41 Maximisation Problem For a maximisation problem, gains or profits are converted into opportunity loss by subtracting all values from the highest value. Since the problem is to maximise profits, it can be solved by minimising the opportunity loss.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 42 Maximisation Problem Plants Sofa SetsAvailability StandardDeluxeSuper Deluxe A69510451270800 B70010351275600 C68510501265700 Demand45010506002100 Contribution of 1 unit

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 43 Convert to opportunity loss matrix by subtracting all figures from 1265 i.e. the highest contribution. Plants Sofa SetsAvailability StandardDeluxeSuper Deluxe A69510451270800 B70010351275600 C68510501265700 Demand45010506002100 Plants Sofa SetsAvailability StandardDeluxeSuper Deluxe A5802305800 B5752400600 C59022510700 Demand45010506002100 Use this as the cost data and proceed as for a minimisation case. Substitute original values while calculating total profits.

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Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 44 Restriction on routes When certain routes cannot be used, either cross them out in the table indicating that they cannot be used or assign a very high cost to them (M).

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