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Assignment and Transportation Problem Amit Kumar Bardhan Faculty of Management Studies University of Delhi
Step1.  Find a bfs. Find the minimum element in each row of the  mxm  cost matrix. Construct a new matrix by subtracting from each cost the minimum cost in its row. For this new matrix, find the minimum cost in each column. Construct a new matrix (reduced cost matrix) by subtracting from each cost the minimum cost in its column. Step2.  Draw the minimum number of lines (horizontal and/or vertical) that are needed to cover all zeros in the reduced cost matrix. If m lines are required, an optimal solution is available among the covered zeros in the matrix. If fewer than m lines are required, proceed to step 3. Step3.  Find the smallest nonzero element (call its value k) in the reduced cost matrix that is uncovered by the lines drawn in step 2. Now subtract k from each uncovered element of the reduced cost matrix and add k to each element that is covered by two lines. Return to step2. Assignment Problems: Hungarian method
Example 1: Assignment Problem Personnel Assignment: Time (hours) Job 1 Job 2 Job 3 Job 4 Person 1   14   5  8  7 Person 2   2   12    6  5 Person 3   7  8    3   9 Person 4   2   4   6  10
Example 2: Swimmer Selection The coach of a swim team needs to assign swimmers to a 200-meters medley relay team to send to the Olympics. Since most of his best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and the best time (in seconds) they have achieved in each of the strokes (for 50 meters) are: From: Hillier and Hillier
Swimmer Selection  [contd.] 31 28 29 26 29 Freestyle 33 30 38 28 33 Butterfly 41 34 42 33 43 Breaststroke 35 37 33 32 37 Backstroke Vir Ali Vipin Ajay Mark Stroke
The coach wishes to determine how to assign four swimmers to the four different strokes to minimize the sum of corresponding best times. Hest the coach in team selection. Swimmer Selection  [contd.]
Swimmer Selection  [Answer] Freestyle  ->  Mark Butterfly  ->  Ajay Breaststroke  ->  Ali   Backstroke  ->   Vipin
Example:  Flight Assignment ‘ YO Air Co.’ is a small commuter airline running 6 flights per day from New Delhi to surrounding tourist destinations.  Flight crews are all based in New Delhi, working flights to various locations and returning on the next flight home.  Taking into account complex work rules, and pay incentives, YO Air schedulers have constructed the 8 possible work patterns detailed in the following table. Each row of the table marks the flight that could be covered in a particular pattern and the daily cost per crew (in lakhs of Rupees)
Flight Assignment  [contd.] 1.54 √ √ √ 8 0.84 √ √ 7 1.12 √ √ 6 1.32 √ √ √ 5 1.60 √ √ √ 4 1.52 √ √ √ 3 0.96 √ √ 2 1.40 √ √ 1 6 5 4 3 2 1 Cost Flight Work Pattern
Flight Assignment  [contd.] The company wants to choose a minimum total cost collection of work patterns that covers all flights exactly once. How to find the best assignments?
Transportation Problem
Power Co: Shipping costs, Supply, and Demand for Powerco Example Transportation Tableau 30 30 20 45 Demand (Million kwh) 40 5 16 9 14 Plant 3 50 7 13 12 9 Plant 2 35 9 10 6 8 Plant 1 Supply (Million kwh) City 4 City 3 City 2 City 1 To From
Powerco Power Plant: Formulation Define Variables: Let Xij = number of (million kwh) produced at plant i and  sent to city j. Objective Function: Min z = 8*X11 + 6*X12 + 10*X13 + 9*X14   + 9*X21 + 12*X22 + 13*X23 + 7*X24   + 14*X31 + 9*X32 + 16*X33 + 5*X34 Supply Constraints: X11 + X12 + X13 + X14 < = 35  (Plant 1)   X21 + X22 + X23 + X24 < = 50  (Plant 2)   X31 + X32 + X33 + X34 < = 40  (Plant 3)
Powerco Power Plant: Formulation (Cont’d.) Demand Constraints: X11 + X21 +X31 > = 45 (City 1)   X12 + X22 +X32 > = 20 (City 2)   X13 + X23 +X33 > = 30 (City 3)   X14 + X24 +X34 > = 30 (City 4) Nonnegativity Constraints: Xij > = 0  (i=1,..,3; j=1,..,4) Balanced? Total Demand = 45 + 20 + 30 + 30 = 125 Total Supply  = 35 + 50 + 40 = 125 Yes, Balanced Transportation Problem! (If problem is  unbalanced, add dummy supply or demand as required.)
Example: Cost-less Corp Ltd. The Cost-less Corp Ltd. supplies its four retail outlets from its four plants. The shipping cost per shipment from each plant to each retail outlet is given below  200 300 100 200 4 100 200 400 300 3 300 100 900 200 2 200 400 600 500 1 Plant D C B A Retail outlet Unit shipping cost
Example: Cost-less Corp Ltd.  [contd.] Plant 1, 2, 3 and 4 make 10, 20, 20 and 10 shipments respectively. Retail outlets 1, 2, 3 and 4 need to receive 20, 10, 10 and 20 shipments per month respectively. The distribution manager wants to determine the best plan for how many shipments to send from each plant to respective retail outlets each month. The objective is to minimize the total shipping cost.
Solving transportation problem Can be formulated as Linear Program and solved through simplex method / solver Modified Simplex method applied directly on the table Step 1 : Find the Initial basic feasible solution (North-West Corner rule, Least Cost Method,  Vogel’s Approximation method) Step 2 : UV Method / Stepping stone method
Transshipment Problems A transportation problem allows only shipments that go directly from supply points to demand points. In many situations, shipments are allowed between supply points or between demand points. Sometimes there may also be points (called transshipment points) through which goods can be transshipped on their journey from a supply point to a demand point. Fortunately, the optimal solution to a transshipment problem can be found by solving a transportation problem.
The following steps describe how the optimal solution to a transshipment problem can be found by solving a transportation problem. Step 1 : If necessary, add a dummy demand point (with a supply of 0 and a demand equal to the problem’s excess supply) to balance the problem. Shipments to the dummy and from a point to itself will be zero. Let s= total available supply. Step 2 : Construct a transportation tableau as follows: A row in the tableau will be needed for each supply point and transshipment point, and a column will be needed for each demand point and transshipment point. Solving Transshipment Problems
Each supply point will have a supply equal to it’s original supply, and each demand point will have a demand to its original demand. Let s= total available supply. Then each transshipment point will have a supply equal to (point’s original supply)+s and a demand equal to (point’s original demand)+s. This ensures that any transshipment point that is a net supplier will have a net outflow equal to point’s original supply and a net demander will have a net inflow equal to point’s original demand. Although we don’t know how much will be shipped through each transshipment point, we can be sure that the total amount will not exceed s.
Transshipment Example Widgetco manufactures widgets at two factories, one in Memphis and one in Denver. The Memphis factory can produce as 150 widgets, and the Denver factory can produce as many as 200 widgets per day.  Widgets are shipped by air to customers in LA and Boston. The customers in each city require 130 widgets per day. Because of the deregulation of airfares, Widgetco believes that it may be cheaper first fly some widgets to NY or Chicago and then fly them to their final destinations. The cost of flying a widget are shown next. Widgetco wants to minimize the total cost of shipping the required widgets to customers.
Transportation Tableau Associated with the Transshipment Example NY  Chicago  LA  Boston  Dummy  Supply Memphis  $8  $13  $25  $28  $0  150 Denver  $15  $12  $26  $25  $0  200 NY  $0  $6  $16  $17  $0  350 Chicago  $6  $0  $14  $16  $0  350 Demand  350  350  130  130  90   Supply points: Memphis, Denver Demand Points: LA Boston Transshipment Points: NY, Chicago  The problem can be solved using the transportation simplex method
Example - 1 General Motors produces cars at L.A. and Detroit and has a warehouse in Atlanta. The company supplies cars to customers in Houston and Tampa. The cost of shipping a car between points is given in table. (‘-’ means the shipment is not allowed) L.A. can produce as many as 1100 cars and Detroit can produce as many as 2900 cars. Houston must receive 2400 cars and Tampa must receive 1500 cars.  Formulate a balanced transportation problem that can be used to minimize the shipping cost incurred in meeting demands at Houston and Tampa. Modify the answer to (a) If shipments between LA and Detroit are not allowed  Modify the answer to (a) if shipments between Houston and Tampa are allowed at a cost of $5.
Example - 1 - 0 121 109 89 Houston 0 - 82 117 210 Tampa 78 113 0 115 105 Atlanta 119 110 111 0 145 Detroit 225 90 100 140 0 L.A. Tampa Houston Atlanta Detroit L.A. From To ($)
Example – 1a (answer) 100 5,500 6,400 4,000 4,000 4,000 4,000 0 0 M 82 117 210 Tampa 4,000 0 M 0 121 109 89 Houston 4,000 0 78 113 0 115 105 Atlanta 6,900 0 119 110 111 0 145 Detroit 5,100 0 225 90 100 140 0 L.A Dummy Tampa Houston Atlanta Detroit L.A.
Example – 1b (answer) 100 5,500 6,400 4,000 4,000 4,000 4,000 0 0 M 82 117 210 Tampa 4,000 0 M 0 121 109 89 Houston 4,000 0 78 113 0 115 105 Atlanta 6,900 0 119 110 111 0 M Detroit 5,100 0 225 90 100 M 0 L.A Dummy Tampa Houston Atlanta Detroit L.A.
Example – 1c (answer) 100 5,500 6,400 4,000 4,000 4,000 4,000 0 0 5 82 117 210 Tampa 4,000 0 5 0 121 109 89 Houston 4,000 0 78 113 0 115 105 Atlanta 6,900 0 119 110 111 0 145 Detroit 5,100 0 225 90 100 140 0 L.A Dummy Tampa Houston Atlanta Detroit L.A.
Limitations of Transportation Problem One commodity ONLY: any one product supplied and demanded at multiple locations  Merchandise Electricity, water Invalid for multiple commodities: (UNLESS transporting any one of the multiple commodities is completely independent of transporting any other commodity and hence can be treated by itself alone) Example: transporting product 1 and product 2 from the supply points to the demand points where the total amount (of the two products) transported on a link is subject to a capacity constraint Example: where economy of scale can be achieved by transporting the two products on the same link at a larger total volume and at a lower unit cost of transportation
Limitations of Transportation Problem Difficult to generalize the technique to accommodate (these are generic difficulty for “mathematical programming,” including linear and non-linear programming Economy of scale the per-unit cost of transportation on a link decreasing with the volume (nonlinear and concave; there is a trick to convert a “non-linear program with a piecewise linear but convex objective function to a linear program; no such tricks exists for a piecewise linear but concave objective function) Fixed-cost: transportation usually involves fixed charges. For example, the cost of truck rental (or cost of trucking in general) consists of a fixed charge that is independent of the mileage and a mileage charge that is proportional to the total mileage driven. Such fixed charges render the objective function NON-LINEAR and CONCAVE and make the problem much more difficult to solve
Assignment 1 A company must meet the following demands for a product: January 30 units; February, 30 units; March 20 units. Demand may be backlogged at a cost of Rs. 5 per unit per month. All demand must be met by the end of March. Thus, if one unit of January demand is met during March, a backlogging cost of 5 (2) = 10 is incurred. Monthly production capacity and unit production cost during each month are given in the table. A holding cost of Rs. 20 per unit is assessed on the inventory at the end of each month.  Formulate a balanced transportation problem that could be used to determine, how to minimize the total cost (including backlogging, holding and production costs) of meeting demand. Use the transportation simplex to determine how to meet each month’s demand.
Question 1… Rs. 410 35 March Rs. 420 30 February Rs. 400 35 January Unit production cost Production capacity Month
Assignment - 2 Identify the limitations of transportation problem as discussed in the class. How can we overcome these limitations. Give one specific example. (within 350 words)

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Assign transportation

  • 1. Assignment and Transportation Problem Amit Kumar Bardhan Faculty of Management Studies University of Delhi
  • 2. Step1. Find a bfs. Find the minimum element in each row of the mxm cost matrix. Construct a new matrix by subtracting from each cost the minimum cost in its row. For this new matrix, find the minimum cost in each column. Construct a new matrix (reduced cost matrix) by subtracting from each cost the minimum cost in its column. Step2. Draw the minimum number of lines (horizontal and/or vertical) that are needed to cover all zeros in the reduced cost matrix. If m lines are required, an optimal solution is available among the covered zeros in the matrix. If fewer than m lines are required, proceed to step 3. Step3. Find the smallest nonzero element (call its value k) in the reduced cost matrix that is uncovered by the lines drawn in step 2. Now subtract k from each uncovered element of the reduced cost matrix and add k to each element that is covered by two lines. Return to step2. Assignment Problems: Hungarian method
  • 3. Example 1: Assignment Problem Personnel Assignment: Time (hours) Job 1 Job 2 Job 3 Job 4 Person 1 14 5 8 7 Person 2 2 12 6 5 Person 3 7 8 3 9 Person 4 2 4 6 10
  • 4. Example 2: Swimmer Selection The coach of a swim team needs to assign swimmers to a 200-meters medley relay team to send to the Olympics. Since most of his best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and the best time (in seconds) they have achieved in each of the strokes (for 50 meters) are: From: Hillier and Hillier
  • 5. Swimmer Selection [contd.] 31 28 29 26 29 Freestyle 33 30 38 28 33 Butterfly 41 34 42 33 43 Breaststroke 35 37 33 32 37 Backstroke Vir Ali Vipin Ajay Mark Stroke
  • 6. The coach wishes to determine how to assign four swimmers to the four different strokes to minimize the sum of corresponding best times. Hest the coach in team selection. Swimmer Selection [contd.]
  • 7. Swimmer Selection [Answer] Freestyle -> Mark Butterfly -> Ajay Breaststroke -> Ali Backstroke -> Vipin
  • 8. Example: Flight Assignment ‘ YO Air Co.’ is a small commuter airline running 6 flights per day from New Delhi to surrounding tourist destinations. Flight crews are all based in New Delhi, working flights to various locations and returning on the next flight home. Taking into account complex work rules, and pay incentives, YO Air schedulers have constructed the 8 possible work patterns detailed in the following table. Each row of the table marks the flight that could be covered in a particular pattern and the daily cost per crew (in lakhs of Rupees)
  • 9. Flight Assignment [contd.] 1.54 √ √ √ 8 0.84 √ √ 7 1.12 √ √ 6 1.32 √ √ √ 5 1.60 √ √ √ 4 1.52 √ √ √ 3 0.96 √ √ 2 1.40 √ √ 1 6 5 4 3 2 1 Cost Flight Work Pattern
  • 10. Flight Assignment [contd.] The company wants to choose a minimum total cost collection of work patterns that covers all flights exactly once. How to find the best assignments?
  • 12. Power Co: Shipping costs, Supply, and Demand for Powerco Example Transportation Tableau 30 30 20 45 Demand (Million kwh) 40 5 16 9 14 Plant 3 50 7 13 12 9 Plant 2 35 9 10 6 8 Plant 1 Supply (Million kwh) City 4 City 3 City 2 City 1 To From
  • 13. Powerco Power Plant: Formulation Define Variables: Let Xij = number of (million kwh) produced at plant i and sent to city j. Objective Function: Min z = 8*X11 + 6*X12 + 10*X13 + 9*X14 + 9*X21 + 12*X22 + 13*X23 + 7*X24 + 14*X31 + 9*X32 + 16*X33 + 5*X34 Supply Constraints: X11 + X12 + X13 + X14 < = 35 (Plant 1) X21 + X22 + X23 + X24 < = 50 (Plant 2) X31 + X32 + X33 + X34 < = 40 (Plant 3)
  • 14. Powerco Power Plant: Formulation (Cont’d.) Demand Constraints: X11 + X21 +X31 > = 45 (City 1) X12 + X22 +X32 > = 20 (City 2) X13 + X23 +X33 > = 30 (City 3) X14 + X24 +X34 > = 30 (City 4) Nonnegativity Constraints: Xij > = 0 (i=1,..,3; j=1,..,4) Balanced? Total Demand = 45 + 20 + 30 + 30 = 125 Total Supply = 35 + 50 + 40 = 125 Yes, Balanced Transportation Problem! (If problem is unbalanced, add dummy supply or demand as required.)
  • 15. Example: Cost-less Corp Ltd. The Cost-less Corp Ltd. supplies its four retail outlets from its four plants. The shipping cost per shipment from each plant to each retail outlet is given below 200 300 100 200 4 100 200 400 300 3 300 100 900 200 2 200 400 600 500 1 Plant D C B A Retail outlet Unit shipping cost
  • 16. Example: Cost-less Corp Ltd. [contd.] Plant 1, 2, 3 and 4 make 10, 20, 20 and 10 shipments respectively. Retail outlets 1, 2, 3 and 4 need to receive 20, 10, 10 and 20 shipments per month respectively. The distribution manager wants to determine the best plan for how many shipments to send from each plant to respective retail outlets each month. The objective is to minimize the total shipping cost.
  • 17. Solving transportation problem Can be formulated as Linear Program and solved through simplex method / solver Modified Simplex method applied directly on the table Step 1 : Find the Initial basic feasible solution (North-West Corner rule, Least Cost Method, Vogel’s Approximation method) Step 2 : UV Method / Stepping stone method
  • 18. Transshipment Problems A transportation problem allows only shipments that go directly from supply points to demand points. In many situations, shipments are allowed between supply points or between demand points. Sometimes there may also be points (called transshipment points) through which goods can be transshipped on their journey from a supply point to a demand point. Fortunately, the optimal solution to a transshipment problem can be found by solving a transportation problem.
  • 19. The following steps describe how the optimal solution to a transshipment problem can be found by solving a transportation problem. Step 1 : If necessary, add a dummy demand point (with a supply of 0 and a demand equal to the problem’s excess supply) to balance the problem. Shipments to the dummy and from a point to itself will be zero. Let s= total available supply. Step 2 : Construct a transportation tableau as follows: A row in the tableau will be needed for each supply point and transshipment point, and a column will be needed for each demand point and transshipment point. Solving Transshipment Problems
  • 20. Each supply point will have a supply equal to it’s original supply, and each demand point will have a demand to its original demand. Let s= total available supply. Then each transshipment point will have a supply equal to (point’s original supply)+s and a demand equal to (point’s original demand)+s. This ensures that any transshipment point that is a net supplier will have a net outflow equal to point’s original supply and a net demander will have a net inflow equal to point’s original demand. Although we don’t know how much will be shipped through each transshipment point, we can be sure that the total amount will not exceed s.
  • 21. Transshipment Example Widgetco manufactures widgets at two factories, one in Memphis and one in Denver. The Memphis factory can produce as 150 widgets, and the Denver factory can produce as many as 200 widgets per day. Widgets are shipped by air to customers in LA and Boston. The customers in each city require 130 widgets per day. Because of the deregulation of airfares, Widgetco believes that it may be cheaper first fly some widgets to NY or Chicago and then fly them to their final destinations. The cost of flying a widget are shown next. Widgetco wants to minimize the total cost of shipping the required widgets to customers.
  • 22. Transportation Tableau Associated with the Transshipment Example NY Chicago LA Boston Dummy Supply Memphis $8 $13 $25 $28 $0 150 Denver $15 $12 $26 $25 $0 200 NY $0 $6 $16 $17 $0 350 Chicago $6 $0 $14 $16 $0 350 Demand 350 350 130 130 90 Supply points: Memphis, Denver Demand Points: LA Boston Transshipment Points: NY, Chicago The problem can be solved using the transportation simplex method
  • 23. Example - 1 General Motors produces cars at L.A. and Detroit and has a warehouse in Atlanta. The company supplies cars to customers in Houston and Tampa. The cost of shipping a car between points is given in table. (‘-’ means the shipment is not allowed) L.A. can produce as many as 1100 cars and Detroit can produce as many as 2900 cars. Houston must receive 2400 cars and Tampa must receive 1500 cars. Formulate a balanced transportation problem that can be used to minimize the shipping cost incurred in meeting demands at Houston and Tampa. Modify the answer to (a) If shipments between LA and Detroit are not allowed Modify the answer to (a) if shipments between Houston and Tampa are allowed at a cost of $5.
  • 24. Example - 1 - 0 121 109 89 Houston 0 - 82 117 210 Tampa 78 113 0 115 105 Atlanta 119 110 111 0 145 Detroit 225 90 100 140 0 L.A. Tampa Houston Atlanta Detroit L.A. From To ($)
  • 25. Example – 1a (answer) 100 5,500 6,400 4,000 4,000 4,000 4,000 0 0 M 82 117 210 Tampa 4,000 0 M 0 121 109 89 Houston 4,000 0 78 113 0 115 105 Atlanta 6,900 0 119 110 111 0 145 Detroit 5,100 0 225 90 100 140 0 L.A Dummy Tampa Houston Atlanta Detroit L.A.
  • 26. Example – 1b (answer) 100 5,500 6,400 4,000 4,000 4,000 4,000 0 0 M 82 117 210 Tampa 4,000 0 M 0 121 109 89 Houston 4,000 0 78 113 0 115 105 Atlanta 6,900 0 119 110 111 0 M Detroit 5,100 0 225 90 100 M 0 L.A Dummy Tampa Houston Atlanta Detroit L.A.
  • 27. Example – 1c (answer) 100 5,500 6,400 4,000 4,000 4,000 4,000 0 0 5 82 117 210 Tampa 4,000 0 5 0 121 109 89 Houston 4,000 0 78 113 0 115 105 Atlanta 6,900 0 119 110 111 0 145 Detroit 5,100 0 225 90 100 140 0 L.A Dummy Tampa Houston Atlanta Detroit L.A.
  • 28. Limitations of Transportation Problem One commodity ONLY: any one product supplied and demanded at multiple locations Merchandise Electricity, water Invalid for multiple commodities: (UNLESS transporting any one of the multiple commodities is completely independent of transporting any other commodity and hence can be treated by itself alone) Example: transporting product 1 and product 2 from the supply points to the demand points where the total amount (of the two products) transported on a link is subject to a capacity constraint Example: where economy of scale can be achieved by transporting the two products on the same link at a larger total volume and at a lower unit cost of transportation
  • 29. Limitations of Transportation Problem Difficult to generalize the technique to accommodate (these are generic difficulty for “mathematical programming,” including linear and non-linear programming Economy of scale the per-unit cost of transportation on a link decreasing with the volume (nonlinear and concave; there is a trick to convert a “non-linear program with a piecewise linear but convex objective function to a linear program; no such tricks exists for a piecewise linear but concave objective function) Fixed-cost: transportation usually involves fixed charges. For example, the cost of truck rental (or cost of trucking in general) consists of a fixed charge that is independent of the mileage and a mileage charge that is proportional to the total mileage driven. Such fixed charges render the objective function NON-LINEAR and CONCAVE and make the problem much more difficult to solve
  • 30. Assignment 1 A company must meet the following demands for a product: January 30 units; February, 30 units; March 20 units. Demand may be backlogged at a cost of Rs. 5 per unit per month. All demand must be met by the end of March. Thus, if one unit of January demand is met during March, a backlogging cost of 5 (2) = 10 is incurred. Monthly production capacity and unit production cost during each month are given in the table. A holding cost of Rs. 20 per unit is assessed on the inventory at the end of each month. Formulate a balanced transportation problem that could be used to determine, how to minimize the total cost (including backlogging, holding and production costs) of meeting demand. Use the transportation simplex to determine how to meet each month’s demand.
  • 31. Question 1… Rs. 410 35 March Rs. 420 30 February Rs. 400 35 January Unit production cost Production capacity Month
  • 32. Assignment - 2 Identify the limitations of transportation problem as discussed in the class. How can we overcome these limitations. Give one specific example. (within 350 words)