DISTRIBUTION AND NETWORK MODELS (1/2) MANGT 521 (B): Quantitative Management Chapter 6 DISTRIBUTION AND NETWORK MODELS (1/2)

Chapter 6 Distribution and Network Models Transportation Problem Network Representation General LP Formulation Assignment Problem Transshipment Problem Shortest-Route Problem

Transportation, Assignment, and Transshipment Problems A network model is one which can be represented by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes. Examples of network problems: Transportation, assignment, transshipment, shortest-route, and maximal flow problems of this chapter as well as the minimal spanning tree and PERT/CPM problems (in Project Management courses).

Transportation, Assignment, and Transshipment Problems Each of the four problems of this chapter can be formulated as linear programs and solved by general purpose LP computer package. For each of the four problems, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be in terms of integer values for the decision variables. However, there are many computer packages that contain separate computer codes for these problems which take advantage of their network structure.

1. Transportation Problem The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij. The # of constraints in a transportation LP formulation = (# of origins) + (# of destinations) = m + n The network representation for a transportation problem with two sources and three destinations is given on the next slide.

1. Transportation Problem Network Representation 1 d1 c11 1 c12 s1 c13 2 d2 c21 2 c22 s2 c23 3 d3 Sources Destinations

1. Transportation Problem A General LP Model Using the notation: xij = number of units shipped from origin i to destination j cij = cost per unit of shipping from si = supply or capacity in units at origin i dj = demand in units at destination j continued

1. Transportation Problem A General LP Model (continued) To obtain a feasible solution in a transportation problem, “total supply ≥ total demand” xij > 0 for all i and j

Transportation Problem Example: Foster Generations Foster Generators operates plants in Cleveland, Ohio; Bedford, Indiana; and York, Pennsylvania. Production capacities over the next three-month planning period for one particular type of generator are as follows:

Transportation Problem Example: Foster Generations The firm distributes its generators through four regional distribution centers located in Boston, Chicago, St. Louis, and Lexington; the three-month forecast of demand for the distribution centers is as follows:

Transportation Problem Example: Foster Generations The cost for each unit shipped on each route is also given as follows: Management would like to determine how much of its production should be shipped from each plant to each distribution center.

Transportation Problem Example: Foster Generations Network Representation

Transportation Problem Example: Foster Generations LP Formulation The objective of the transportation problem is to minimize the total transportation cost: Therefore, the objective function is:

Transportation Problem Example: Foster Generations Consider supply constraints Total # of units shipped from Cleveland: Total # of units shipped from Bedford: Total # of units shipped from York:

Transportation Problem Example: Foster Generations Consider demand constraints Four demand constraints are needed to ensure that destination demands will be satisfied:

Transportation Problem Example: Foster Generations Combining the objective function and constraints into one model provides a 12-variable, 7-constraint LP formulation of the Foster Generators’ transportation problem:

Transportation Problem Example: Foster Generations Solution Summary

Transportation Problem Example: Foster Generations Network Representation of Optimal Solution

Transportation Problem Variations Variations of the basic transportation model may involve one or more of the following situations: Total supply not equal to total demand Route capacities or route minimums Unacceptable routes Can be easily accommodated with slight modifications

Transportation Problem Variations Total supply not equal to total demand “Total supply > total demand” No modification in the LP formulation is necessary. Excess supply will appear as slack (i.e. unused supply or amount not shipped from the origin).

Transportation Problem Variations Total supply not equal to total demand (cont’d) “Total supply < total demand” The LP Model of a transportation problem will NOT have a feasible solution. Add a dummy origin with supply equal to the shortage amount. Assign a zero (0) shipping cost per unit to the dummy origin. The amount “shipped” from the dummy origin (in the solution) will not actually be shipped. The destination(s) showing shipments being received from the dummy origin will be the destinations experiencing a shortfall, or unsatisfied demand.

Transportation Problem Variations Route capacities or route minimums Also can accommodate capacities or minimum quantities for one or more of the routes (“capacitated transportation problem”) Maximum route capacity from i to j: xij < Lij Fosters Generators example (Example 1): If the York-Boston route (from origin 3 to destination 1) had a capacity of 1,000 units because of limited space availability on its normal mode of transportation, the following route capacity constraint should be added to the existing LP model: x31 < 1,000

Transportation Problem Variations Route capacities or route minimums (cont’d) Minimum shipping guarantee from i to j: xij > Mij Fosters Generators example (Example 1): If the Bedford-Chicago route (from origin 2 to destination 2) had a previously committed order of at least 2,000 units, the following route minimum constraint should be added to the existing LP model: x22 > 2,000

Transportation Problem Variations Unacceptable routes Establishing a route from every origin to every destination may not be possible. Simply drop the corresponding arc from the network and remove the corresponding variable from the LP formulation. Fosters Generators example: If the Cleveland–St. Louis route (from origin 1 to destination 3) were unacceptable or unusable, the arc from Cleveland to St. Louis (x13) could be removed from the LP formulation.

2. Assignment Problem Typical assignment problems involve: Assigning jobs to machines, agents to tasks, sales personnel to sales, territories, contracts to bidders, etc. A special case of the transportation problem in which all supply and demand values equal to 1, and the amount shipped over each arc is either 0 or 1; hence assignment problems may be solved as linear programs. It assumes all workers are assigned and each job is performed.

2. Assignment Problem An assignment problem seeks to minimize the total cost, minimize time, or maximize profit assignment of m workers to n jobs, given that the cost of worker i performing job j is cij. The network representation of an assignment problem with three workers and three jobs is shown on the next slide.

2. Assignment Problem Network Representation c11 c12 c13 Agents Tasks

2. Assignment Problem A General LP Model Using the notation: xij = 1 if agent i is assigned to task j 0 otherwise cij = cost of assigning agent i to task j

2. Assignment Problem A General LP Model (continued) xij > 0 for all i and j If an agent is permitted to work for multiple (t) tasks at the same time:

Assignment Problem: Example #1 Fowle Marketing Research Fowle Marketing Research has just received requests for market research studies from three new clients. The company faces the task of assigning a project leader (agent) to each client (task). Currently, three individuals have no other commitments and are available for the project leader assignments. Fowle’s management realizes, however, that the time required to complete each study will depend on the experience and ability of the project leader assigned. The three projects have approximately the same priority, and management wants to assign project leaders to minimize the total number of days required to complete all three projects. If a project leader is to be assigned to one client only, what assignments should be made?

Assignment Problem: Example #1 Fowle Marketing Research Management must first consider all possible project leader–client assignments and then estimate the corresponding project completion times. Estimated completion times (in days)

Assignment Problem: Example #1 Fowle Marketing Research Network Representation

Assignment Problem: Example #1 Fowle Marketing Research LP Formulation Using the notation: xij = 1 if project leader i is assigned to client j 0 otherwise where i = 1, 2, 3, and j = 1, 2, 3 The completion times for three project leaders: Thus, the objective function is:

Assignment Problem: Example #1 Fowle Marketing Research LP Formulation (cont’d) Constraints reflect the conditions that each project leader can be assigned to at most one client and that each client must have one assigned project leader. Thus: “# of project leaders = # of clients”: All the constraints could be written as “=“ When “# of project leaders ≥ # of clients”: All the project leader constraints must be written as “≤“

Assignment Problem: Example #1 Fowle Marketing Research Combining the objective function and constraints into one model provides a 9-variable, 6-constraint LP formulation of the Fowle Marketing Research’s assignment problem:

Assignment Problem: Example #1 Fowle Marketing Research Computer Solution Output Value of the optimal solution Optimal solution The change in the optimal value of the solution per unit increase in the RHS of the constraint. Terry is assigned to client 2 (x12 = 1), Carle is assigned to client 3 (x23 = 1), and McClymonds is assigned to client 1 (x31 = 1). The total completion time required is 26 days.

Assignment Problem: Example #1 Fowle Marketing Research Solution Summary

Assignment Problem Variations Similar to transportation problem Total # of agents not equal to the total number of tasks Unacceptable assignments Can be easily accommodated with slight modifications

Assignment Problem Variations Total # of agents not equal to the total number of tasks “# of agents > # of tasks” No modification in the LP formulation is necessary. Extra agents will appear as slack (i.e. unassigned agents).

Assignment Problem: Example #2 An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects. Projects Subcontractor A B C Westside 50 36 16 Federated 28 30 18 Goliath 35 32 20 Universal 25 25 14 How should the contractors be assigned so that total mileage is minimized?

Assignment Problem: Example #2 Network Representation 50 West. A 36 16 Subcontractors Projects 28 Fed. 30 B 18 35 32 Gol. C 20 25 25 Univ. 14

Assignment Problem: Example #2 Linear Programming Formulation Min 50x11+36x12+16x13+28x21+30x22+18x23 +35x31+32x32+20x33+25x41+25x42+14x43 s.t. x11+x12+x13 < 1 x21+x22+x23 < 1 x31+x32+x33 < 1 x41+x42+x43 < 1 x11+x21+x31+x41 = 1 x12+x22+x32+x42 = 1 x13+x23+x33+x43 = 1 xij = 0 or 1 for all i and j Agents Tasks

Assignment Problem: Example #2 The optimal assignment is: Subcontractor Project Distance Westside C 16 Federated A 28 Goliath (unassigned) Universal B 25 Total Distance = 69 miles

Assignment Problem Variations Total # of agents not equal to the total number of tasks (cont’d) “# of agents < # of tasks” The LP Model of a assignment problem will NOT have a feasible solution. Add enough dummy agents to equalize the number of tasks. The objective function coefficients for the dummy agents would be zero (0). No assignments will actually be made to the clients receiving dummy project leaders.

Assignment Problem Variations Unacceptable assignments When an agent does not have the experience necessary for one or more of the tasks Simply remove the corresponding decision variable from the LP formulation.