The Transportation and Assignment Problems Chapter 9: Hillier and Lieberman Chapter 7: Decision Tools for Agribusiness Dr. Hurley s AGB 328 Course
Terms to Know Sources Destinations Supply Demand The Requirements Assumption The Feasible Solutions Property The Cost Assumption Dummy Destination Dummy Source Transportation Simple Method Northwest Corner Rule Vogel s Approimation Method Russell s Approimation Method Recipient Cells Donor Cells Assignment Problems Assignees Tasks Hungarian Algorithm
Case Study: P&T Company P&T is a small family-owned business that processes and cans vegetables and then distributes them for eventual sale One of its main products that it processes and ships is peas These peas are processed in: Bellingham WA; Eugene OR; and Albert Lea MN The peas are shipped to: Sacramento CA; Salt Lake City UT; Rapid City SD; and Albuquerque NM
Case Study: P&T Company Shipping Data Cannery Output Warehouse Allocation Bellingham 75 Truckloads Sacramento 80 Truckloads Eugene 125 Truckloads Salt Lake 65 Truckloads Albert Lea 100 Truckloads Rapid City 70 Truckloads Total 300 Truckloads Albuquerque 85 Truckloads Total 300 Truckloads
Case Study: P&T Company Shipping Cost/Truckload Warehouse Cannery Sacramento Salt Lake Rapid City Albuquerque Bellingham $464 $513 $654 $867 75 Supply Eugene $352 $416 $690 $791 125 Albert Lea $995 $682 $388 $685 100 Demand 80 65 70 85
Network Presentation of P&T Co. Problem 464 75 C1 513 W1-80 867 654 125 C1 791 352 416 690 W2-65 W3-70 995 682 100 C1 388 W4-85 685
Mathematical Model for P&T Transportation Problem 34 33 32 31 24 23 22 21 14 13 12 11 685 388 682 995 791 690 416 352 867 654 513 464 34 33 32 31 24 23 22 21 14 13 12 11 Minimize
Mathematical Model for P&T Transportation Problem Cont. Subject to: 11 + 12 + 13 + 14 = 75 21 + 22 + 23 + 24 = 125 31 + 32 + 33 + 34 = 100 11 + 21 + 31 = 80 12 + 22 + 32 = 65 13 + 23 + 33 = 70 14 + 24 + 34 = 85 ij 0 (i = 123; j = 1234)
Transportation Problems Transportation problems are characterized by problems that are trying to distribute commodities from any supply center known as sources to any group of receiving centers known as destinations Two major assumptions are needed in these types of problems: The Requirements Assumption The Cost Assumption
Transportation Assumptions The Requirement Assumption Each source has a fied supply which must be distributed to destinations while each destination has a fied demand that must be received from the sources The Cost Assumption The cost of distributing commodities from the source to the destination is directly proportional to the number of units distributed
Feasible Solution Property A transportation problem will have a feasible solution if and only if the sum of the supplies is equal to the sum of the demands. Hence the constraints in the transportation problem must be fied requirement constraints met with equality.
The General Model of a Transportation Problem Any problem that attempts to minimize the total cost of distributing units of commodities while meeting the requirement assumption and the cost assumption and has information pertaining to sources destinations supplies demands and unit costs can be formulated into a transportation model
Visualizing the Transportation Model When trying to model a transportation model it is usually useful to draw a network diagram of the problem you are eamining A network diagram shows all the sources destinations and unit cost for each source to each destination in a simple visual format like the eample on the net slide
Network Diagram Supply Demand S1 Source 1 c 11 c12 Destination 1 -D1 c 1n c 21 c 13 S2 Source 2 c 23 c 22 Destination 2 -D2 S3 Source 3... c 2n c31 c 32 c 33 c 3n c m1cm2 Destination 3... -D3 Sm Source m c m3 Destination n -Dn c mn
General Mathematical Model of Transportation Problems m n Minimize Z= i=1 j=1 c ij ij Subject to: n ij = s i for I =12 m m i=1 j=1 ij = d j for j = 12 n ij 0 for all i and j
Integer Solutions Property If all the supplies and demands have integer values then the transportation problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its decision variables This implies that there is no need to add restrictions on the model to force integer solutions
Solving a Transportation Problem When Ecel solves a transportation problem it uses the regular simple method Due to the characteristics of the transportation problem a faster solution can be found using the transportation simple method Unfortunately the transportation simple model is not programmed in Solver
Modeling Variants of Transportation Problems In many transportation models you are not going to always see supply equals demand With small problems this is not an issue because the simple method can solve the problem relatively efficiently With large transportation problems it may be helpful to transform the model to fit the transportation simple model
Issues That Arise with Transportation Models Some of the issues that may arise are: The sum of supply eceeds the sums of demand The sum of the supplies is less than the sum of demands A destination has both a minimum demand and maimum demand Certain sources may not be able to distribute commodities to certain destinations The objective is to maimize profits rather than minimize costs
Method for Handling Supply Not Equal to Demand When supply does not equal demand you can use the idea of a slack variable to handle the ecess A slack variable is a variable that can be incorporated into the model to allow inequality constraints to become equality constraints If supply is greater than demand then you need a slack variable known as a dummy destination If demand is greater than supply then you need a slack variable known as a dummy source
Handling Destinations that Cannot Be Delivered To There are two ways to handle the issue when a source cannot supply a particular destination The first way is to put a constraint that does not allow the value to be anything but zero The second way of handling this issue is to put an etremely large number into the cost of shipping that will force the value to equal zero
Tetbook Transportation Models Eamined P&T A typical transportation problem Could there be another formulation? Northern Airplane An eample when you need to use the Big M Method and utilizing dummy destinations for ecess supply to fit into the transportation model Metro Water District An eample when you need to use the Big M Method and utilizing dummy sources for ecess demand to fit into the transportation model
The Transportation Simple Method While the normal simple method can solve transportation type problems it does not necessarily do it in the most efficient fashion especially for large problems. The transportation simple is meant to solve the problems much more quickly.
Finding an Initial Solution for the Transportation Simple Northwest Corner Rule Let sd stand for the amount allocated to supply row s and demand row d For 11 select the minimum of the supply and demand for supply 1 and demand 1 If any supply is remaining then increment over to sd+1 otherwise increment down to s+1d For this net variable select the minimum of the leftover supply or leftover demand for the new row and column you are in Continue until all supply and demand has been allocated
Finding an Initial Solution for the Transportation Simple Vogel s Approimation Method For each row and column that has not been deleted calculate the difference between the smallest and second smallest in absolute value terms (ties mean that the difference is zero) In the row or column that has the highest difference find the lowest cost variable in it Set this variable to the minimum of the leftover supply or demand Delete the supply or demand row/column that was the minimum and go back to the top step
Finding an Initial Solution for the Transportation Simple Russell s Approimation Method For each remaining source row i determine the largest unit cost c ij and call it u i For each remaining destination column j determine the largest unit cost c ij and call it v i Calculate ij = c ij u i v j for all ij that have not previously been selected Select the largest corresponding ij that has the largest negative ij Allocate to this variable as much as feasible based on the current supply and demand that are leftover
Algorithm for Transportation Simple Method Construct initial basic feasible solution Optimality Test Derive a set of u i and v j by setting the u i corresponding to the row that has the most amount of allocations to zero and solving the leftover set of equations for c ij = u i + v j If all c ij u i v j 0 for every (ij) such that ij is nonbasic then stop. Otherwise do an iteration.
Algorithm for Transportation Simple Method Cont. An Iteration Determine the entering basic variable by selecting the nonbasic variable having the largest negative value for c ij u i v j Determine the leaving basic variable by identifying the chain of swaps required to maintain feasibility Select the basic variable having the smallest variable from the donor cells Determine the new basic feasible solution by adding the value of the leaving basic variable to the allocation for each recipient cell. Subtract this value from the allocation of each donor cell
Assignment Problems Assignment problems are problems that require tasks to be handed out to assignees in the cheapest method possible The assignment problem is a special case of the transportation problem
Characteristics of Assignment Problems The number of assignees and the number of task are the same Each assignee is to be assigned eactly one task Each task is to be assigned by eactly one assignee There is a cost associated with each combination of an assignee performing a task The objective is to determine how all of the assignments should be made to minimize the total cost
General Mathematical Model of Assignment Problems n n Minimize Z= i=1 j=1 c ij ij Subject to: n ij = 1 for I =12 m n i=1 j=1 ij = 1 for j = 12 n ij is binary for all i and j
Modeling Variants of the Assignment Problem Issues that arise: Certain assignees are unable to perform certain tasks. There are more task than there are assignees implying some tasks will not be completed. There are more assignees than there are tasks implying some assignees will not be given a task. Each assignee can be given multiple tasks simultaneously. Each task can be performed jointly by more than one assignee.
Assignment Spreadsheet Models from Tetbook Job Shop Company Better Products Company We will eamine these spreadsheets in class and derive mathematical models from the spreadsheets
Hungarian Algorithm for Solving Assignment Problems Step 1: Find the minimum from each row and subtract from every number in the corresponding row making a new table Step 2: Find the minimum from each column and subtract from every number in the corresponding column making a new table Step 3: Test to see whether an optimal assignment can be made by eamining the minimum number of lines needed to cover all the zeros If the number of lines corresponds to the number of rows you have the optimal and you should go to step 6 If the number of lines does not correspond to the number of rows go to step 4
Hungarian Algorithm for Solving Assignment Problems Cont. Step 4: Modify the table by using the following: Subtract the smallest uncovered number from every uncovered number in the table Add the smallest uncovered number to the numbers of intersected lines All other numbers stay unchanged Step 5: Repeat steps 3 and four until you have the optimal set
Hungarian Algorithm for Solving Assignment Problems Cont. Step 6: Make the assignment to the optimal set one at a time focusing on the zero elements Start with the rows and columns that have only one zero Once an optimal assignment has been given to a variable cross that row and column out Continue until all the rows and columns with only one zero have been allocated Net do the columns/rows with two non crossed out zeroes as above Continue until all assignments have been made
In Class Activity (Not Graded) Attempt to find an initial solution to the P&T problem using the a) Northwest Corner Rule b) Vogel s Approimation Method and c) Russell s Approimation Method 9.1-3b set up the problem as a regular linear programming problem and solve using solver then set the problem up as a transportation problem and solve using solver
In Class Activity (Not Graded) Solve the following problem using the Hungarian method.
Case Study: Sellmore Company Cont. The assignees for the task are: Ann Ian Joan Sean A summary of each assignees productivity and costs are given on the net slide.
Case Study: Sellmore Company Cont. Employee Word Processing Required Time Per Task Graphics Packets Registration Wage Ann 35 41 27 40 $14 Ian 47 45 32 51 $12 Joan 39 56 36 43 $13 Sean 32 51 25 46 $15
Assignment of Variables ij i = 1 for Ann 2 for Ian 3 for Joan 4 for Sean j = 1 for Processing 2 for Graphics 3 for Packets 4 for Registration
Mathematical Model for Sellmore Company 34 33 32 31 34 33 32 31 24 23 22 21 14 13 12 11 690 375 765 480 559 468 728 507 612 384 540 564 560 378 574 490 34 33 32 31 24 23 22 21 14 13 12 11 Minimize
Mathematical Model for Sellmore Company Cont. 1 1 1 1 0 1 0 1 0 1 0 1 1 1 1 1 : 44 34 24 14 43 33 23 13 42 32 22 12 41 31 21 11 34 33 32 31 34 33 32 31 24 23 22 21 14 13 12 11 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 to Subject