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Transportation and Assignment Problems

Apr 07, 2019

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Transportation and Assignment Problems. The Transportation Model Solution of a Transportation Problem The Assignment Model Solution of the Assignment Model. Transportation and Assignment Problems Overview.

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Transportation and Assignment Problems • The Transportation Model • Solution of a Transportation Problem • The Assignment Model • Solution of the Assignment Model PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Transportation and Assignment ProblemsOverview - Part of a larger class of linear programming problems known as network flow models. - Possess special mathematical features that enabled development of very efficient, unique solution methods. - Methods are variations of traditional simplex procedure. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Transportation Model Characteristics • A product is transported from a number of sources to a number of destinations at the minimum possible cost. • Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product. • The linear programming model has constraints for supply at each source and demand at each destination. • All constraints are equalities in a balanced transportation model where supply equals demand. • Constraints contain inequalities in unbalanced models where supply does not equal demand. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Transportation Model ExampleProblem Definition and Data - Problem:How many tons of wheat to transport from each grain elevator to each mill on a monthly basis in order to minimize the total cost of transportation ? - Data: Grain ElevatorSupplyMill Demand 1. Kansas City 150 A. Chicago 200 2. Omaha 175 B. St.Louis 100 3. Des Moines 275 C. Cincinnati 300 Total 600 tons Total 600 tons PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Transportation Model ExampleModel Formulation minimize Z = $6x1A + 8x1B + 10x1C + 7x2A + 11x2B + 11x2C + 4x3A + 5x3B + 12x3C subject to x1A + x1B + x1C = 150 x2A + x2B + x2C = 175 x3A + x3B+ x3C = 275 x1A + x2A + x3A = 200 x1B + x2B + x3B = 100 x1C + x2C + x3C = 300 xij  0 where xij = tons of wheat from each grain elevator, i, i = 1, 2, 3, to each mill j, j = A,B,C Network of transportation routes for wheat shipments PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Solution of the Transportation ModelTableau Format • Transportation problems are solved manually within a tableau format. • Each cell in a transportation tableau is analogous to a decision variable that indicates the amount allocated from a source to a destination. • The supply and demand values along the outside rim of a tableau are called rim values. The Transportation Tableau PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Solution of the Transportation ModelSolution Methods • Transportation models do not start at the origin where all decision values are zero; they must instead be given an initial feasible solution. • Initial feasible solution determination methods include: • - northwest corner method • - minimum cell cost method • - Vogel’s Approximation Method • Methods for solving the transportation problem itself include: • - stepping-stone method and • - modified distribution method. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

- In the northwest corner method the largest possible allocation is made to the cell in the upper left-hand corner of the tableau , followed by allocations to adjacent feasible cells. - The initial solution is complete when all rim requirements are satisfied. - Transportation cost is computed by evaluating the objective function: Z = $6x1A + 8x1B + 10x1C + 7x2A + 11x2B + 11x2C + 4x3A + 5x3B + 12x3C = 6(150) + 8(0) + 10(0) + 7(50) + 11(100) + 11(25) + 4(0) + 5(0) + !2(275) = $5,925 The Northwest Corner Method The Initial NW Corner Solution PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Northwest Corner MethodSummary of Steps 1. Allocate as much as possible to the cell in the upper left-hand corner, subject to the supply and demand conditions. 2. Allocate as much as possible to the next adjacent feasible cell. 3. Repeat step 2 until all rim requirements are met. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Minimum Cell Cost Method(1 of 3) - In the minimum cell cost method as much as possible is allocated to the cell with the minimum cost followed by allocation to the feasible cell with minimum cost. The Initial Minimum Cell Cost Allocation The Second Minimum Cell Cost Allocation PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Minimum Cell Cost Method(2 of 3) - The complete initial minimum cell cost solution; total cost = $4,550. - The minimum cell cost method will provide a solution with a lower cost than the northwest corner solution because it considers cost in the allocation process. The Initial Solution PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Minimum Cell Cost MethodSummary of Steps(3 of 3) 1. Allocate as much as possible to the feasible cell with the minimum transportation cost, and adjust the rim requirements. 2. Repeat step 1 until all rim requirements have been met. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Vogel’s Approximation Method (VAM)(1 of 5) - Method is based on the concept of penalty cost or regret. - A penalty cost is the difference between the largest and the next largest cell cost in a row (or column). - In VAM the first step is to develop a penalty cost for each source and destination. - Penalty cost is calculated by subtracting the minimum cell cost from the next higher cell cost in each row and column. The VAM Penalty Costs PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Vogel’s Approximation Method (VAM)(2 of 5) - VAM allocates as much as possible to the minimum cost cell in the row or column with the largest penalty cost. The Initial VAM Allocation PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Vogel’s Approximation Method (VAM)(3 of 5) - After each VAM cell allocation, all row and column penalty costs are recomputed. The Second VAM Allocation PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Vogel’s Approximation Method (VAM)(4 of 5) - Recomputed penalty costs after the third allocation. The Third VAM Allocation PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Vogel’s Approximation Method (VAM)(5 of 5) - The initial VAM solution; total cost = $5,125 - VAM and minimum cell cost methods both provide better initial solutions than does the northwest corner method. The Initial VAM Solution PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Vogel’s Approximation Method (VAM)Summary of Steps 1. Determine the penalty cost for each row and column. 2. Select the row or column with the highest penalty cost. 3. Allocate as much as possible to the feasible cell with the lowest transportation cost in the row or column with the highest penalty cost. 4. Repeat steps 1, 2, and 3 until all rim requirements have been met. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(1 of 12) - Once an initial solution is derived, the problem must be solved using either the stepping-stone method or the modified distribution method (MODI). - The initial solution used as a starting point in this problem is the minimum cell cost method solution because it had the minimum total cost of the three methods used. The Minimum Cell Cost Solution PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(2 of 12) - The stepping-stone method determines if there is a cell with no allocation that would reduce cost if used. +1 The Allocation of One Ton to Cell 1A PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(3 of 12) - Must subtract one ton from another allocation along that row. The Subtraction of One Ton from Cell 1B PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(4 of 12) - A requirement of this solution method is that units can only be added to and subtracted from cells that already have allocations, thus one ton must be added to a cell as shown. The Addition of One Ton to Cell 3B and the Subtraction of One Ton from Cell 3A PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(5 of 12) - An empty cell that will reduce cost is a potential entering variable. - To evaluate the cost reduction potential of an empty cell, a closed path connecting used cells to the empty cells is identified. The Stepping-Stone Path for Cell 2A PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(6 of 12) - The remaining stepping-stone paths and resulting computations for cells 2B and 3C. The Stepping-Stone Path for Cell 2B The Stepping-Stone Path for Cell 3C PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(7 of 12) - After all empty cells are evaluated, the one with the greatest cost reduction potential is the entering variable. - A tie can be broken arbitrarily. The Stepping-Stone Path for Cell 1A PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(8 of 12) - When reallocating units to the entering variable (cell), the amount is the minimum amount subtracted on the stepping-stone path. - At each iteration one variable enters and one leaves (just as in the simplex method). The Second Iteration of the Stepping-Stone Method PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(9 of 12) - Check to see if the solution is optimal. The Stepping-Stone Path for Cell 2A The Stepping-Stone Path for Cell 1B PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(10 of 12) - Continuing check for optimality. The Stepping-Stone Path for Cell 2B The Stepping-Stone Path for Cell 3C PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(11 of 12) - The stepping-stone process is repeated until none of the empty cells will reduce costs (i.e., an optimal solution). - In example, evaluation of four paths indicates no cost reductions, therefore Table 19 solution is optimal. - Solution and total minimum cost : x1A = 25 tons, x2C = 175 tons, x3A = 175 tons, x1C = 125 tons, x3B = 100 tons Z = $6(25) + 8(0) + 10(125) + 7(0) + 11(0) + 11(175) + 4(175) + 5(100) + 12(0) = $4,525 PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution Method(12 of 12) - A multiple optimal solution occurs when an empty cell has a cost change of zero and all other empty cells are positive. - An alternate optimal solution is determined by allocating to the empty cell with a zero cost change. - Alternate optimal total minimum cost also equals $4,525. The Alternative Optimal Solution PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Stepping-Stone Solution MethodSummary of Steps 1. Determine the stepping-stone paths and cost changes for each empty cell in the tableau. 2. Allocate as much as possible to the empty cell with the greatest net decrease in cost. 3. Repeat steps 1 and 2 until all empty cells have positive cost changes that indicate an optimal solution. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

- MODI is a modified version of the stepping-stone method in which math equations replace the stepping-stone paths. - In the table, the extra left-hand column with the ui symbols and the extra top row with the vj symbols represent values that must be computed. - Computed for all cells with allocations : ui + vj = cij = unit transportation cost for cell ij. The Modified Distribution Method (MODI)(1 of 6) The Minimum Cell Cost Initial Solution PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Modified Distribution Method (MODI)(2 of 6) - Formulas for cells containing allocations: x1B: u1 + vB = 8 x1C: u1 + vC = 10 x2C: u2 + vC = 11 x3A: u3 + vA = 4 x3B: u3 + vB = 5 - Five equations with 6 unknowns, therefore let u1 = 0 and solve to obtain: vB = 8, vC = 10, u2 = 1, u3 = -3, vA= 7 The Initial Solution with All ui and vj Values PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Modified Distribution Method (MODI)(3 of 6) - Each MODI allocation replicates the stepping-stone allocation. - Use following to evaluate all empty cells: cij - ui - vj = kij where kij equals the cost increase or decrease that would occur by allocating to a cell. - For the empty cells in Table 26: x1A: k1A = c1A - u1 - vA = 6 - 0 - 7 = -1 x2A: k2A = c2A - u2 - vA = 7 - 1 - 7 = -1 x2B: k2B = c2B - u2 - vB = 11- 1 - 8 = +2 x3C: k3C = c3C - u3 -vC = 12 - (-3) - 10 = +5 PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Modified Distribution Method (MODI)(4 of 6) - After each allocation to an empty cell, the ui and vj values must be recomputed. The Second Iteration of the MODI Solution Method PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

- Recomputing ui and vj values: x1A: u1 + vA = 6, vA = 6 x1C: u1 + vC = 10, vC = 10 x2C: u2 + vC = 11, u2 = 1 x3A: u3 + vA = 4, u3 = -2 x3B: u3 + vB = 5, vB = 7 The Modified Distribution Method (MODI)(5 of 6) The New ui and vj Values for the Second Iteration PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Modified Distribution Method (MODI)(6 of 6) - Cost changes for the empty cells, cij - ui - vj = kij; x1B: k1B = c1B - u1 - vB = 8 - 0 - 7 = +1 x2A: k2A = c2A - u2 - vA = 7 - 1 - 6 = 0 x2B: k2B = c2B - u2 - vB = 11 - 1 -7 = +3 x3C: k2B = c2B - u3 - vC = 12 - (-2) - 10 = +4 - Since none of the values are negative, solution obtained is optimal. - Cell 2A with a zero cost change indicates a multiple optimal solution. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Modified Distribution Method (MODI)Summary of Steps 1. Develop an initial solution. 2. Compute the ui and vj values for each row and column. 3. Compute the cost change, kij, for each empty cell. 4. Allocate as much as possible to the empty cell that will result in the greatest net decrease in cost (most negative kij) 5. Repeat steps 2 through 4 until all kij values are positive or zero. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Unbalanced Transportation Model(1 of 2) - When demand exceeds supply a dummy row is added to the tableau. An Unbalanced Model (Demand . Supply) PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Unbalanced Transportation Model(2 of 2) - When supply exceeds demand, a dummy column is added to the tableau. - The dummy column (or dummy row) has no effect on the initial solution methods or the optimal solution methods. An Unbalanced Model (Supply . Demand) PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

- In a transportation tableau with m rows and n columns, there must be m + n - 1 cells with allocations; if not, it is degenerate. - The tableau in the figure does not meet the condition since 3 + 3 -1 = 5 cells and there are only 4 cells with allocations. Degeneracy(1 of 3) The Minimum Cell Cost Initial Solution PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Degeneracy(2 of 3) - In a degenerate tableau, all the stepping-stone paths or MODI equations cannot be developed. -To rectify a degenerate tableau, an empty cell must artificially be treated as an occupied cell. The Initial Solution PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Degeneracy(3 of 3) - The stepping-stone path s and cost changes for this tableau: 2A 2C 1C 1A x2A: 7 - 11 + 10 - 6 = 0 2B 2C 1C 1B x2B: 11 - 11 + 10 - 8 = + 2 3B 1B 1A 3A x3B: 5 - 8 + 6 - 4 = - 1 3C 1C 1A 3A x3C: 12 - 10 + 6 - 4 = + 4 The Second Stepping-Stone Iteration PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Prohibited Routes - A prohibited route is assigned a large cost such as M. - When the prohibited cell is evaluated, it will always contain the cost M, which will keep it from being selected as an entering variable. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Transportation Model ExampleQM for Windows Solution PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Transportation Model ExampleQM for Windows Solution (continued) PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Assignment ModelCharacteristics • Special form of linear programming model similar to the transportation model. • Supply at each source and demand at each destination limited to one unit. • In a balanced model supply equals demand. • In an unbalanced model supply does not equal demand. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Assignment ModelExample Problem Definition and Data Problem: Assign four teams of officials to four games in a way that will minimize total distance traveled by the officials. Supply is always one team of officials, demand is for only one team of officials at each game. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

The Assignment ModelExample Problem Model Formulation Minimize Z = 210xAR + 90xAA + 180xAD + 160xAC + 100xBR + 70xBA + 130xBD + 200xBC + 175xCR + 105xCA + 140xCD + 170xCC + 80xDR + 65xDA + 105xDD +120xDC subject to xAR + xAA + xAD+ xAC = 1 xBR + xBA + xBD + xBC = 1 xCR + xCA+ xCD + xCC = 1 xDR + xDA + xDD + xDC = 1 xAR + xBR + xCR + xDR = 1 xAA + xBA + xCA + xDA = 1 xAD+ xBD + xCD + xDD = 1 xAC + xBC + xCC + xDC = 1 xij 0 PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

Solution of the Assignment Model(1 of 7) - An assignment problem is a special form of the transportation problem where all supply and demand values equal one. - Example: assigning four teams of officials to four games in a way that will minimize distance traveled by the officials. The Travel Distances to Each Game for Each Team of Officials PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS

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Transportation and Assignment Problems After completing this chapter, you should be able to To learn more about the book this website supports, please visit its . and . is one of the many fine businesses of . You must be a registered user to view the in this website. If you already have a username and password, enter it below. If your textbook came with a card and this is your first visit to this site, you can to register. Username: Password: '); document.write(''); } // --> ( ) .'); } else{ document.write('This form changes settings for this website only.'); } //--> Send mail as: '); } else { document.write(' '); } } else { document.write(' '); } // --> '); } else { document.write(' '); } } else { document.write(' '); } document.write(' TA email: '); } else { document.write(' '); } } else { document.write(' '); } // --> Other email: '); } else { document.write(' '); } } else { document.write(' '); } // --> "Floating" navigation? 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Engineeringbro February 11, 2023 March 10, 2024 3 mins read Post author: Engineeringbro Post published: February 11, 2023 Post category: Blog Post comments: 0 Comments lets understand the Difference between transportation and assignment problems? Transportation problems and assignment problems are two types of linear programming problems that arise in different applications. The main difference between transportation and assignment problems is in the nature of the decision variables and the constraints. If you’re unable to see the whole table kindly convert the mobile view to the desktop view   Assignment Problem Minimization or maximization of the cost of transporting goods from one source to another Maximization of the total profit or minimization of the total cost in assigning tasks to individuals Nature of problem Involves transporting goods from sources to destinations Involves assigning tasks to individuals Number of sources and destinations Multiple sources and destinations An equal number of sources and destinations Availability and demand Each source and destination have a supply or demand value Each task has only one individual who can perform it Decision variables Amount of goods transported from each source to each destination Binary variables indicate whether an individual is assigned a task or not Constraints Capacity constraints on sources and demand constraints on destinations Each individual can only perform one task Solution method Transportation simplex method, northwest corner rule, Vogel’s approximation method Hungarian algorithm, brute force method Example Transporting goods from factories to warehouses Assigning tasks to employees or jobs to machines Additional Different between Transportation and Assignment Problems are as follows :  Decision Variables: In a transportation problem, the decision variables represent the flow of goods from sources to destinations. Each variable represents the quantity of goods transported from a source to a destination. In contrast, in an assignment problem, the decision variables represent the assignment of agents to tasks. Each variable represents whether an agent is assigned to a particular task or not. Constraints: In a transportation problem, the constraints ensure that the supply from each source matches the demand at each destination and that the total flow of goods does not exceed the capacity of each source and destination. In contrast, in an assignment problem, the constraints ensure that each task is assigned to exactly one agent and that each agent is assigned to at most one task. Objective function: The objective function in a transportation problem typically involves minimizing the total cost of transportation or maximizing the total profit of transportation. In an assignment problem, the objective function typically involves minimizing the total cost or maximizing the total benefit of assigning agents to tasks. In summary, The transportation problem is concerned with finding the optimal way to transport goods from sources to destinations, while the assignment problem is concerned with finding the optimal way to assign agents to tasks. Both problems are important in operations research and have numerous practical applications. Checkout  Home page  for more informative content and Follow us on  facebook  for more  Please Share This Share this content Opens in a new window You Might Also Like Domestic electrolux refrigeration system – Working, Components MPSO technique? Dial caliper: Definition, Least Count, Drawing, Parts, How to use Leave a reply cancel reply. Preferences Transportation and Assignment Problems - PowerPoint PPT Presentation Transportation and Assignment Problems Transportation and assignment problems for the model on the previous page note that: xij=1 if machine i is assigned to meet the demands of job j xij=0 if machine i is ... – powerpoint ppt presentation. 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. 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. 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. 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 Since we want to minimize the total cost of shipping from plants to cities Minimize Z 8X116X1210X139X14 9X2112X2213X237X24 14X319X3216X335X34 Since each supply point has a limited production capacity X11X12X13X14 lt 35 X21X22X23X24 lt 50 X31X32X33X34 lt 40 X11X21X31 gt 45 X12X22X32 gt 20 X13X23X33 gt 30 X14X24X34 gt 30 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) Min Z 8X116X1210X139X149X2112X2213X237X24 S.T. X11X12X13X14 lt 35 (Supply Constraints) X11X21X31 gt 45 (Demand Constraints) 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. Xij number of units shipped from supply point i to demand point j If Total supply equals to total demand, the problem is said to be a balanced transportation problem There are two basic methods Northwest Corner Method Vogels 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). 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. In this section we will explain how the simplex algorithm is used to solve 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. 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 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. Setup times (Also called the cost matrix) According to the setup table Machincos problem can be formulated as follows (for i,j1,2,3,4) 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. In general the LP formulation is given as PowerShow.com is a leading presentation sharing website. It has millions of presentations already uploaded and available with 1,000s more being uploaded by its users every day. Whatever your area of interest, here you’ll be able to find and view presentations you’ll love and possibly download. And, best of all, it is completely free and easy to use. You might even have a presentation you’d like to share with others. If so, just upload it to PowerShow.com. We’ll convert it to an HTML5 slideshow that includes all the media types you’ve already added: audio, video, music, pictures, animations and transition effects. Then you can share it with your target audience as well as PowerShow.com’s millions of monthly visitors. And, again, it’s all free. About the Developers PowerShow.com is brought to you by  CrystalGraphics , the award-winning developer and market-leading publisher of rich-media enhancement products for presentations. Our product offerings include millions of PowerPoint templates, diagrams, animated 3D characters and more. Transportation and Assignment Problems Cite this chapter. James K. Strayer 2   Part of the book series: Undergraduate Texts in Mathematics ((UTM)) 1301 Accesses Transportation and assignment problems are traditional examples of linear programming problems. Although these problems are solvable by using the techniques of Chapters 2–4 directly, the solution procedure is cumbersome; hence, we develop much more efficient algorithms for handling these problems. In the case of transportation problems, the algorithm is essentially a disguised form of the dual simplex algorithm of 4§2. Assignment problems, which are special cases of transportation problems, pose difficulties for the transportation algorithm and require the development of an algorithm which takes advantage of the simpler nature of these problems. This is a preview of subscription content, log in via an institution to check access. Access this chapter Available as PDF Read on any device Instant download Own it forever Compact, lightweight edition Dispatched in 3 to 5 business days Free shipping worldwide - see info Durable hardcover edition Tax calculation will be finalised at checkout Purchases are for personal use only Institutional subscriptions Unable to display preview.  Download preview PDF. Author information Authors and affiliations. Department of Mathematics, Lock Haven University, Lock Haven, PA, 17745, USA James K. Strayer You can also search for this author in PubMed   Google Scholar Rights and permissions Reprints and permissions Copyright information © 1989 Springer Science+Business Media New York About this chapter Strayer, J.K. (1989). Transportation and Assignment Problems. In: Linear Programming and Its Applications. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1009-2_7 Download citation DOI : https://doi.org/10.1007/978-1-4612-1009-2_7 Publisher Name : Springer, New York, NY Print ISBN : 978-1-4612-6982-3 Online ISBN : 978-1-4612-1009-2 eBook Packages : Springer Book Archive Share this chapter Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative Publish with us Policies and ethics Find a journal Track your research LIVE Course for free Ask a Question What is the difference between Assignment Problem and Transportation Problem? operations research Please log in or register to add a comment. It is used to optimize the transportation cost. It is about assigning finite source to finite destination (one source is alloted to one destination). Number of Source and demand may or may not be equal. Number of source and number of destination must be equal. If demand and supply are not equal, then transportation problem is known as Unbalanced Transportation Problem. If number of rows and number of columns are not equal, then the assignment problem is known as Unbalanced Assignment Problem. It requires to step to solve: Find Initial Solution using North West, Least Cost or Vogel Approximation Find Optimal Solution using MODI method. It requires only one step to solve. Hungarian Method is sufficient to find the optimal solutions. The assignment problem is a special case of the transportation problem. The differences are given below. 1. This is about reducing cost of transportation merchandise 1. This is about assigning finite sources to finite destinations where only one destination is allotted for one source with minimum cost 2. Number of sources and number of demand need not be equal 2. Number of sources and the number of destinations must be equal 3. If total demand and total supply are not equal then the problem is said to be unbalanced. 3. If the number of rows are not equal to the number of columns then problems are unbalanced. 4. It requires 2 stages to solve Getting initial basic feasible solution, by NWC, LCM, VAM and optimal solution by MODI method 4. It has only one stage. Hungarian method is sufficient for obtaining an optimal solution Find MCQs & Mock Test JEE Main 2025 Test Series NEET Test Series Class 12 Chapterwise MCQ Test Class 11 Chapterwise Practice Test Class 10 Chapterwise MCQ Test Class 9 Chapterwise MCQ Test Class 8 Chapterwise MCQ Test Class 7 Chapterwise MCQ Test Related questions Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. 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However, the Transportation algorithm is not very useful to solve this model because of degeneracy. Comparison Chart The problem may have a rectangular matrix or a square matrix. The assignment algorithm can not be used to solve the transportation model. The rows and columns may have any number of allocations depending on the rim conditions. The rows and columns must have one-to-one allocation. Because of this property, the matrix must be a square matrix. The basic feasible solution is obtained by the northwest corner method or LCM method or VAM The basic feasible solution is obtained by the Hungarian method or Flood’s technique or by Assignment algorithm. The optimality test is given by the stepping stone method or by the MODI method. The optimality test is given by drawing a minimum number of horizontal and vertical lines to cover all the zeros in the matrix. The rim requirement may have any positive numbers. The optimality test is given by drawing a minimum number of horizontal and vertical lines to cover all the zeros in the matrix. The transportation algorithm can be used to solve the assignment model. The assignment algorithm can not be used to solve the transportation model. Similarities Both are special types of linear programming problems. Both have an objective function, structural constraints, and non-negativity constraints. And the relationship between variables and constraints is linear. The coefficients of variables in the solution will be either 1 or zero in both cases. Both are basically minimization problems. For converting them into maximization problems same procedure is used. More Difference Difference between Lagrangian and Eulerian Approach Difference between Line Standards and End Standards Snapsolve any problem by taking a picture. Try it in the Numerade app? Industrial Engineering OR QUE 2.doc 142 IMAGES Transportation and Assignment Differences between Assignment Problem and Transportation Problem PPT PPT Write difference between a transportation problem and Assignment PPT VIDEO The Transportation Model & The Assignment Model Transportation & assignment problem 4- Transportation Planning 4- Transportation Planning Assignment Problem (special cases) What’s the assignment? COMMENTS Transportation model and assignment model Various steps are given Step 1 Select the North West (upper left-hand) corner cell of the transportation table and allocate as many units as possible equal to the minimum between available supply and demand requirement i.e., min (S1, D1). Step 2 Adjust the supply and demand numbers in the respective rows And columns allocation. Unit.5. transportation and assignment problems Unit.5. transportation and assignment problems - Download as a PDF or view online for free ... Column D1 is thus crossed out. Maximum difference is 1 in row S3 and in column D3. Select arbitrarily S3 and allot the least cost cell (S1, D2) 5 units. Cross out row S1 for it is already exhausted. Now, we have only one row S3 and two columns D2 and ... The Transportation and Assignment Problems The Assignment Problem. The Assignment Problem • Can use simplex method or transportation simplex method to solve • Recommendation: use specialized solution procedures for the assignment problem • Will be more efficient for large problems • Example: Pages 353-356 of the text. Transportation and Assignment Problems 2. Repeat step 1 until all rim requirements have been met. PN5033 - TRANSPORTATION AND ASSIGNMENT PROBLEMS. Vogel's Approximation Method (VAM) (1 of 5) - Method is based on the concept of penalty cost or regret. - A penalty cost is the difference between the largest and the next largest cell cost in a row (or column). PDF CHAPTER 15 TRANSPORTATION AND ASSIGNMENT PROBLEMS 7. Identify the relationship between assignment problems and transportation problems. 8. Formulate a spreadsheet model for an assignment problem from a description of the problem. 9. Do the same for some variants of assignment problems. 10. Give the name of an algorithm that can solve huge assignment problems that are well Transportation and Assignment Problems Describe the characteristics of assignment problems. Identify the relationship between assignment problems and transportation problems. Formulate a spreadsheet model for an assignment problem from a description of the problem. Do the same for some variants of assignment problems. Give the name of an algorithm that can solve huge assignment ... Difference between transportation and assignment problems? The transportation problem is concerned with finding the optimal way to transport goods from sources to destinations, while the assignment problem is concerned with finding the optimal way to assign agents to tasks. Both problems are important in operations research and have numerous practical applications. PPT Transportation and Assignment Problems For the model on the previous page note that: Xij=1 if machine i is assigned to meet the demands of job j Xij=0 if machine i is ... - A free PowerPoint PPT presentation (displayed as an HTML5 slide show) on PowerShow.com - id: 44b38a-MDY0Z ... difference between the two smallest shipping costs in the row ... Difference Between Transportation Problem and Assignment Problem The transportation problem is commonly approached through simplex methods, and the assignment problem is addressed using specific algorithms like the Hungarian method. In this article, we will learn the difference between transportation problems and assignment problems with the help of examples. The Transportation and Assignment Problems The Simplex Method for Transportation Problems. Illustrative Examples and a Note on Degeneracy. The Simplex Tableau Associated with a Transportation Tableau. The Assignment Problem: (Kuhn's) Hungarian Algorithm. Alternating Path Basis Algorithm for Assignment Problems. A Polynomial-Time Successive Shortest Path Approach for Assignment Problems PDF Transportation and Assignment Problems Assignment problems, which are special cases of transportation problems, pose difficulties for the transportation algorithm and require the development of an algorithm which takes advantage of the simpler nature of these problems. § 1. An Example; The Balanced Transportation Problem We begin with a typical example of a transportation problem. Difference between Transportation Problem vs Assignment ... hey everyone,this is sachin here. welcome to my youtube channel - sachin education hub. all commerce notes are provided here. online classes also available :... Difference Between Transportation and Assignment Problem Difference-Between-Transportation-And-Assignment-Problem - Free download as Word Doc (.doc), PDF File (.pdf), Text File (.txt) or read online for free. This document outlines the key concepts and methods in operations research. It covers topics such as linear programming formulation and graphical solution methods, simplex method, duality theory, transportation and assignment problems, integer ... What is the difference between Assignment Problem and Transportation Transportation Problem: Assignment Problem: 1. This is about reducing cost of transportation merchandise: 1. This is about assigning finite sources to finite destinations where only one destination is allotted for one source with minimum cost Transportation and transshipment problems The document discusses transportation and transshipment problems, describing transportation problems as involving the optimal distribution of goods from multiple sources to multiple destinations subject to supply and demand constraints. It presents the formulation of transportation problems as linear programming problems and provides examples ... Difference Between Assignment and Transportation Model Comparison Between Assignment and Transportation Model With Tabular Form. The Major Difference Between Assignment and Transportation model is that Assignment model may be regarded as a special case of the transportation model. However, the Transportation algorithm is not very useful to solve this model because of degeneracy. What the differences between The Transportation and Assignment Problem The assignment problem wh ich finds many allocations in all ocation and scheduling. For example: In a ss igning salesman to different regions vehicles and drives to differe nt routes. What is the difference between assignment problems and transportation Assignment problems involve assigning a set of tasks or jobs to a set of workers or machines, with the objective of minimizing the total cost or time required to complete all tasks. This is typically done by creating a matrix of costs or times for each worker-machine combination, and then finding the optimal assignment that minimizes the total ... OR QUE 2.doc Document OR QUE 2.doc, Subject Industrial Engineering, from Michigan State University, Length: 2 pages, Preview: c) Differences between Transportation Problem and Assignment Problem: Nature of Problem: Transportation Problem: In transportation problems, the assignment essay homework paper phd report resume speech thesis