3 Case Study: P&T Company P&T is a small family-owned business that processes and cans vegetables and then distributes them for eventual saleOne of its main products that it processes and ships is peasThese peas are processed in: Bellingham, WA; Eugene, OR; and Albert Lea, MNThe peas are shipped to: Sacramento, CA; Salt Lake City, UT; Rapid City, SD; and Albuquerque, NM
4 Case Study: P&T Company Shipping Data CanneryOutputWarehouseAllocationBellingham75 TruckloadsSacramento80 TruckloadsEugene125 TruckloadsSalt Lake65 TruckloadsAlbert Lea100 TruckloadsRapid City70 TruckloadsTotal300 TruckloadsAlbuquerque85 Truckloads
5 Case Study: P&T Company Shipping Cost/Truckload WarehouseCannerySacramentoSalt LakeRapid CityAlbuquerqueSupplyBellingham$464$513$654$86775Eugene$352$416$690$791125Albert Lea$995$682$388$685100Demand80657085
6 Network Presentation of P&T Co. Problem 464C1W175-80513867654W2-65352C1416125690W3791-70995682C1W4388100-85685
7 Mathematical Model for P&T Transportation Problem
9 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 destinationsTwo major assumptions are needed in these types of problems:The Requirements AssumptionThe Cost Assumption
10 Transportation Assumptions The Requirement AssumptionEach source has a fixed supply which must be distributed to destinations, while each destination has a fixed demand that must be received from the sourcesThe Cost AssumptionThe cost of distributing commodities from the source to the destination is directly proportional to the number of units distributed
11 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.
12 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
13 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 examiningA 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
15 General Mathematical Model of Transportation Problems Minimize Z= 𝑖=1 𝑚 𝑗=1 𝑛 𝑐 𝑖𝑗 𝑥 𝑖𝑗Subject to:𝑗=1 𝑛 𝑥 𝑖𝑗 = 𝑠 𝑖 for I =1,2,…,m𝑖=1 𝑚 𝑥 𝑖𝑗 = 𝑑 𝑗 𝑓𝑜𝑟 𝑗=1,2,…,𝑛𝑥 𝑖𝑗 ≥0, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 𝑎𝑛𝑑 𝑗
16 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 variablesThis implies that there is no need to add restrictions on the model to force integer solutions
17 Solving a Transportation Problem When Excel solves a transportation problem, it uses the regular simplex methodDue to the characteristics of the transportation problem, a faster solution can be found using the transportation simplex methodUnfortunately, the transportation simplex model is not programmed in Solver
18 Modeling Variants of Transportation Problems In many transportation models, you are not going to always see supply equals demandWith small problems, this is not an issue because the simplex method can solve the problem relatively efficientlyWith large transportation problems it may be helpful to transform the model to fit the transportation simplex model
19 Issues That Arise with Transportation Models Some of the issues that may arise are:The sum of supply exceeds the sums of demandThe sum of the supplies is less than the sum of demandsA destination has both a minimum demand and maximum demandCertain sources may not be able to distribute commodities to certain destinationsThe objective is to maximize profits rather than minimize costs
20 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 excessA slack variable is a variable that can be incorporated into the model to allow inequality constraints to become equality constraintsIf supply is greater than demand, then you need a slack variable known as a dummy destinationIf demand is greater than supply, then you need a slack variable known as a dummy source
21 Handling Destinations that Cannot Be Delivered To There are two ways to handle the issue when a source cannot supply a particular destinationThe first way is to put a constraint that does not allow the value to be anything but zeroThe 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
22 Textbook Transportation Models Examined P&TA typical transportation problemCould there be another formulation?Northern AirplaneAn example when you need to use the Big M Method and utilizing dummy destinations for excess supply to fit into the transportation modelMetro Water DistrictAn example when you need to use the Big M Method and utilizing dummy sources for excess demand to fit into the transportation model
23 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.
24 Finding an Initial Solution for the Transportation Simplex Northwest Corner RuleLet xs,d stand for the amount allocated to supply row s and demand row dFor x1,1 select the minimum of the supply and demand for supply 1 and demand 1If any supply is remaining then increment over to xs,d+1, otherwise increment down to xs+1,dFor this next variable select the minimum of the leftover supply or leftover demand for the new row and column you are inContinue until all supply and demand has been allocated
25 Finding an Initial Solution for the Transportation Simplex Vogel’s Approximation MethodFor 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 itSet this variable to the minimum of the leftover supply or demandDelete the supply or demand row/column that was the minimum and go back to the top step
26 Finding an Initial Solution for the Transportation Simplex Russell’s Approximation MethodFor each remaining source row i, determine the largest unit cost cij and call it 𝑢 𝑖For each remaining destination column j, determine the largest unit cost cij and call it 𝑣 𝑖Calculate ∆ 𝑖𝑗 = 𝑐 𝑖𝑗 − 𝑢 𝑖 − 𝑣 𝑗 for all xij that have not previously been selectedSelect the largest corresponding xij that has the largest negative ∆ijAllocate to this variable as much as feasible based on the current supply and demand that are leftover
27 Algorithm for Transportation Simplex Method Construct initial basic feasible solutionOptimality TestDerive 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 + vjIf all cij – ui – vj ≥ 0 for every (i,j) such that xij is nonbasic, then stop. Otherwise do an iteration.
28 Algorithm for Transportation Simplex Method Cont. An IterationDetermine the entering basic variable by selecting the nonbasic variable having the largest negative value for cij – ui – vjDetermine the leaving basic variable by identifying the chain of swaps required to maintain feasibilitySelect the basic variable having the smallest variable from the donor cellsDetermine 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
29 Assignment ProblemsAssignment problems are problems that require tasks to be handed out to assignees in the cheapest method possibleThe assignment problem is a special case of the transportation problem
30 Characteristics of Assignment Problems The number of assignees and the number of task are the sameEach assignee is to be assigned exactly one taskEach task is to be assigned by exactly one assigneeThere is a cost associated with each combination of an assignee performing a taskThe objective is to determine how all of the assignments should be made to minimize the total cost
31 General Mathematical Model of Assignment Problems Minimize Z= 𝑖=1 𝑛 𝑗=1 𝑛 𝑐 𝑖𝑗 𝑥 𝑖𝑗Subject to:𝑗=1 𝑛 𝑥 𝑖𝑗 =1 for I =1,2,…,m𝑖=1 𝑛 𝑥 𝑖𝑗 =1 𝑓𝑜𝑟 𝑗=1,2,…,𝑛𝑥 𝑖𝑗 𝑖𝑠 𝑏𝑖𝑛𝑎𝑟𝑦, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 𝑎𝑛𝑑 𝑗
32 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.
33 Assignment Spreadsheet Models from Textbook Job Shop CompanyBetter Products CompanyWe will examine these spreadsheets in class and derive mathematical models from the spreadsheets
34 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 tableStep 2: Find the minimum from each column and subtract from every number in the corresponding column making a new tableStep 3: Test to see whether an optimal assignment can be made by examining the minimum number of lines needed to cover all the zerosIf the number of lines corresponds to the number of rows, you have the optimal and you should go to step 6If the number of lines does not correspond to the number of rows, go to step 4
35 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 tableAdd the smallest uncovered number to the numbers of intersected linesAll other numbers stay unchangedStep 5: Repeat steps 3 and four until you have the optimal set
36 Hungarian Algorithm for Solving Assignment Problems Cont. Step 6: Make the assignment to the optimal set one at a time focusing on the zero elementsStart with the rows and columns that have only one zeroOnce an optimal assignment has been given to a variable, cross that row and column outContinue until all the rows and columns with only one zero have been allocatedNext do the columns/rows with two non crossed out zeroes as aboveContinue until all assignments have been made
37 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 Method8.1-3b, 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
38 In Class Activity (Not Graded) Solve the following problem using the Hungarian method.
39 Case Study: Sellmore Company Cont. The assignees for the task are:AnnIanJoanSeanA summary of each assignees productivity and costs are given on the next slide.
40 Case Study: Sellmore Company Cont. Required Time Per TaskEmployeeWord ProcessingGraphicsPacketsRegistrationWageAnn35412740$14Ian47453251$12Joan39563643$13Sean2546$15
41 Assignment of Variables xiji = 1 for Ann, 2 for Ian, 3 for Joan, 4 for Seanj = 1 for Processing, 2 for Graphics, 3 for Packets, 4 for Registration