The Transportation and Assignment Problems Chapter 8: Hillier and Lieberman 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 Simplex Method, Northwest Corner Rule, Vogel’s Approximation Method, Russell’s Approximation 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

Case Study: P&T Company Shipping Cost/Truckload Warehouse Cannery Sacramento Salt Lake Rapid City Albuquerque Supply Bellingham $464 $513 $654 $867 75 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 C1 W1 75 -80 513 867 654 W2 -65 352 C1 416 125 690 W3 791 -70 995 682 C1 W4 388 100 -85 685

Mathematical Model for P&T Transportation Problem

Mathematical Model for P&T Transportation Problem Cont. Subject to: 𝑥 11 + 𝑥 12 + 𝑥 13 + 𝑥 14 =75 𝑥 21 + 𝑥 22 + 𝑥 23 + 𝑥 24 =125 𝑥 31 + 𝑥 32 + 𝑥 33 + 𝑥 34 =100 𝑥 11 + 𝑥 21 + 𝑥 31 =80 𝑥 12 + 𝑥 22 + 𝑥 32 =65 𝑥 13 + 𝑥 23 + 𝑥 33 =70 𝑥 14 + 𝑥 24 + 𝑥 34 =85 𝑥 𝑖𝑗 ≥0 (𝑖=1,2,3;𝑗=1,2,3,4)

Transportation Problems Transportation problems are characterized by problems that are trying to distribute commodities from a 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 fixed supply which must be distributed to destinations, while each destination has a fixed 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 fixed 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 examining A network diagram shows all the sources, destinations, and unit cost for each source to each destination in a simple visual format like the example on the next slide

Network Diagram Supply Demand Source 1 Destination 1 S1 -D1 Source 2 c1n c13 Source 2 c21 Destination 2 S2 -D2 c22 c23 c2n c31 Source 3 Destination 3 S3 c32 -D3 c33 . c3n . cm1 cm2 Source m Destination n Sm -Dn cm3 cmn

General Mathematical Model of Transportation Problems Minimize Z= 𝑖=1 𝑚 𝑗=1 𝑛 𝑐 𝑖𝑗 𝑥 𝑖𝑗 Subject to: 𝑗=1 𝑛 𝑥 𝑖𝑗 = 𝑠 𝑖 for I =1,2,…,m 𝑖=1 𝑚 𝑥 𝑖𝑗 = 𝑑 𝑗 𝑓𝑜𝑟 𝑗=1,2,…,𝑛 𝑥 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 Excel solves a transportation problem, it uses the regular simplex method Due to the characteristics of the transportation problem, a faster solution can be found using the transportation simplex method Unfortunately, the transportation simplex 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 simplex method can solve the problem relatively efficiently With large transportation problems it may be helpful to transform the model to fit the transportation simplex model

Issues That Arise with Transportation Models Some of the issues that may arise are: The sum of supply exceeds the sums of demand The sum of the supplies is less than the sum of demands A destination has both a minimum demand and maximum demand Certain sources may not be able to distribute commodities to certain destinations The objective is to maximize 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 excess 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 extremely large number into the cost of shipping that will force the value to equal zero

Textbook Transportation Models Examined P&T A typical transportation problem Could there be another formulation? Northern Airplane An example when you need to use the Big M Method and utilizing dummy destinations for excess supply to fit into the transportation model Metro Water District An example when you need to use the Big M Method and utilizing dummy sources for excess demand to fit into the transportation model

The Transportation Simplex Method While the normal simplex method can solve transportation type problems, it does not necessarily do it in the most efficient fashion, especially for large problems. The transportation simplex is meant to solve the problems much more quickly.

Finding an Initial Solution for the Transportation Simplex Northwest Corner Rule Let xs,d stand for the amount allocated to supply row s and demand row d For x1,1 select the minimum of the supply and demand for supply 1 and demand 1 If any supply is remaining then increment over to xs,d+1, otherwise increment down to xs+1,d For this next 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 Simplex Vogel’s Approximation 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 Simplex Russell’s Approximation Method For each remaining source row i, determine the largest unit cost cij and call it 𝑢 𝑖 For each remaining destination column j, determine the largest unit cost cij and call it 𝑣 𝑖 Calculate ∆ 𝑖𝑗 = 𝑐 𝑖𝑗 − 𝑢 𝑖 − 𝑣 𝑗 for all xij that have not previously been selected Select the largest corresponding xij 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 Simplex Method Construct initial basic feasible solution Optimality Test Derive a set of ui and vj by setting the ui corresponding to the row that has the most amount of allocations to zero and solving the leftover set of equations for cij = ui + vj If all cij – ui – vj ≥ 0 for every (i,j) such that xij is nonbasic, then stop. Otherwise do an iteration.

Algorithm for Transportation Simplex Method Cont. An Iteration Determine the entering basic variable by selecting the nonbasic variable having the largest negative value for cij – ui – vj 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 exactly one task Each task is to be assigned by exactly 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 Minimize Z= 𝑖=1 𝑛 𝑗=1 𝑛 𝑐 𝑖𝑗 𝑥 𝑖𝑗 Subject to: 𝑗=1 𝑛 𝑥 𝑖𝑗 =1 for I =1,2,…,m 𝑖=1 𝑛 𝑥 𝑖𝑗 =1 𝑓𝑜𝑟 𝑗=1,2,…,𝑛 𝑥 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 Textbook Job Shop Company Better Products Company We will examine 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 examining 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 Next 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 Approximation Method, and c) Russell’s Approximation Method 8.1-3b, 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 next slide.

Case Study: Sellmore Company Cont. Required Time Per Task Employee Word Processing Graphics Packets Registration Wage Ann 35 41 27 40 $14 Ian 47 45 32 51 $12 Joan 39 56 36 43 $13 Sean 25 46 $15

Assignment of Variables xij 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

Mathematical Model for Sellmore Company Cont.