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 CanneryOutputWarehouseAllocation Bellingham75 TruckloadsSacramento80 Truckloads Eugene125 TruckloadsSalt Lake65 Truckloads Albert Lea100 TruckloadsRapid City70 Truckloads Total300 TruckloadsAlbuquerque85 Truckloads Total300 Truckloads
Case Study: P&T Company Shipping Cost/Truckload Warehouse CannerySacramentoSalt LakeRapid City AlbuquerqueSupply Bellingham$464$513$654$86775 Eugene$352$416$690$791125 Albert Lea$995$682$388$685100 Demand80657085
Mathematical Model for P&T Transportation Problem
Mathematical Model for P&T Transportation Problem Cont.
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 Source 1 Source 2 Source 3 Source m...... Destination 1 Destination 2 Destination 3 Destination n...... Supply S1 S2 S3 Sm Demand -D1 -D2 -D3 -Dn c 11 c 12 c 13 c 1n c 21 c 22 c 23 c 2n c 31 c 32 c 33 c 3n c m1 c m2 c m3 c mn
General Mathematical Model of Transportation Problems
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 x s,d stand for the amount allocated to supply row s and demand row d ◦ For x 1,1 select the minimum of the supply and demand for supply 1 and demand 1 ◦ If any supply is remaining then increment over to x s,d+1, otherwise increment down to x s+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
Algorithm for Transportation Simplex 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 (i,j) such that x ij 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 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 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
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 EmployeeWord Processing GraphicsPacketsRegistrationWage Ann35412740$14 Ian47453251$12 Joan39563643$13 Sean32512546$15
Assignment of Variables x 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