Presentation on theme: "Chapter 10, Part A Distribution and Network Models"— Presentation transcript:
1Chapter 10, Part A Distribution and Network Models Supply Chain ModelsTransportation ProblemTransshipment ProblemAssignment Problem
2Transportation, Transshipment, and Assignment 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.Transportation, transshipment, assignment, shortest-route, and maximal flow problems of this chapter as well as the PERT/CPM problems (in Chapter 13) are all examples of network problems.
3Transportation, Transshipment, and Assignment Problems Each of the problems of this chapter can be formulated as linear programs and solved by general purpose linear programming codes.For each of the 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.
4Supply Chain ModelsA supply chain describes the set of all interconnected resources involved in producing and distributing a product.In general, supply chains are designed to satisfy customer demand for a product at minimum cost.Those that control the supply chain must make decisions such as where to produce a product, how much should be produced, and where it should be sent.
5Transportation 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 network representation for a transportation problem with two sources and three destinations is given on the next slide.
6Transportation Problem Network Representation1d1c111c12s1c132d2c212c22s2c233d3SourcesDestinations
7Transportation Problem Linear Programming FormulationUsing the notation:xij = number of units shipped fromorigin i to destination jcij = cost per unit of shipping fromsi = supply or capacity in units at origin idj = demand in units at destination jcontinued
8Transportation Problem Linear Programming Formulation (continued)xij > 0 for all i and j
9Transportation Problem LP Formulation Special CasesTotal supply exceeds total demand:Total demand exceeds total supply:Add a dummy origin with supply equal to theshortage amount. Assign a zero shipping costper unit. The amount “shipped” from thedummy origin (in the solution) will not actuallybe shipped.Assign a zero shipping cost per unitMaximum route capacity from i to j:xij < LiRemove the corresponding decision variable.No modification of LP formulation is necessary.
10Transportation Problem LP Formulation Special Cases (continued)The objective is maximizing profit or revenue:Minimum shipping guarantee from i to j:xij > LijMaximum route capacity from i to j:xij < LijUnacceptable route:Remove the corresponding decision variable.Solve as a maximization problem.
11Transportation Problem: Example #1 Acme Block Company has orders for 80 tons ofconcrete blocks at three suburban locations as follows:Northwood tons, Westwood tons, andEastwood tons. Acme has two plants, each ofwhich can produce 50 tons per week. Delivery cost perton from each plant to each suburban location is shownon the next slide.How should end of week shipments be made to fillthe above orders?
12Transportation Problem: Example #1 Delivery Cost Per TonNorthwood Westwood EastwoodPlantPlant
14Transportation Problem: Example #2 The Navy has 9,000 pounds of material in Albany,Georgia that it wishes to ship to three installations:San Diego, Norfolk, and Pensacola. They require 4,000,2,500, and 2,500 pounds, respectively. Governmentregulations require equal distribution of shippingamong the three carriers.
15Transportation Problem: Example #2 The shipping costs per pound for truck, railroad,and airplane transit are shown on the next slide.Formulate and solve a linear program to determine theshipping arrangements (mode, destination, andquantity) that will minimize the total shipping cost.
16Transportation Problem: Example #2 DestinationMode San Diego Norfolk PensacolaTruck $ $ $ 5RailroadAirplane
17Transportation Problem: Example #2 Define the Decision VariablesWe want to determine the pounds of material, xij , to be shipped by mode i to destination j. The following table summarizes the decision variables:San Diego Norfolk PensacolaTruck x x x13Railroad x x x23Airplane x x x33
18Transportation Problem: Example #2 Define the Objective FunctionMinimize the total shipping cost.Min: (shipping cost per pound for each mode per destination pairing) x (number of pounds shipped by mode per destination pairing).Min: 12x11 + 6x12 + 5x x x22 + 9x23+ 30x x x33
19Transportation Problem: Example #2 Define the ConstraintsEqual use of transportation modes:(1) x11 + x12 + x13 = 3000(2) x21 + x22 + x23 = 3000(3) x31 + x32 + x33 = 3000Destination material requirements:(4) x11 + x21 + x31 = 4000(5) x12 + x22 + x32 = 2500(6) x13 + x23 + x33 = 2500Non-negativity of variables:xij > 0, i = 1, 2, 3 and j = 1, 2, 3
20Transportation Problem: Example #2 Computer OutputObjective Function Value =Variable Value Reduced Costxxxxxxxxx
21Transportation Problem: Example #2 Solution SummarySan Diego will receive 1000 lbs. by truckand 3000 lbs. by airplane.Norfolk will receive 2000 lbs. by truckand 500 lbs. by railroad.Pensacola will receive 2500 lbs. by railroad.The total shipping cost will be $142,000.
22Transshipment Problem Transshipment problems are transportation problems in which a shipment may move through intermediate nodes (transshipment nodes) before reaching a particular destination node.Transshipment problems can be converted to larger transportation problems and solved by a special transportation program.Transshipment problems can also be solved by general purpose linear programming codes.The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide.
23Transshipment Problem Network Representationc363c131c376s1d1c14c15c464c47DemandSupplyc232c24c567d2s2c255c57SourcesDestinationsIntermediate Nodes
24Transshipment Problem Linear Programming FormulationUsing the notation:xij = number of units shipped from node i to node jcij = cost per unit of shipping from node i to node jsi = supply at origin node idj = demand at destination node jcontinued
25Transshipment Problem Linear Programming Formulation (continued)xij > 0 for all i and jcontinued
26Transshipment Problem LP Formulation Special CasesTotal supply not equal to total demandMaximization objective functionRoute capacities or route minimumsUnacceptable routesThe LP model modifications required here areidentical to those required for the special cases inthe transportation problem.
27Transshipment Problem: Example The Northside and Southside facilities of Zeron Industries supply three firms (Zrox, Hewes, Rockrite) with customized shelving for its offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc.Currently weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for Rockrite. Both Arnold and Supershelf can supply at most 75 units to its customers.Additional data is shown on the next slide.
28Transshipment Problem: Example Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are:Zeron N Zeron SArnoldSupershelfThe costs to install the shelving at the various locations are:Zrox Hewes RockriteThomasWashburn
29Transshipment Problem: Example Network RepresentationZROXZrox5015ArnoldZeronN75ARNOLD588Hewes60HEWES37SuperShelfZeronS475WASHBURN44Rock-Rite40
30Transshipment Problem: Example Linear Programming FormulationDecision Variables Definedxij = amount shipped from manufacturer i to supplier jxjk = amount shipped from supplier j to customer kwhere i = 1 (Arnold), 2 (Supershelf)j = 3 (Zeron N), 4 (Zeron S)k = 5 (Zrox), 6 (Hewes), 7 (Rockrite)Objective Function DefinedMinimize Overall Shipping Costs:Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37+ 3x45 + 4x46 + 4x47
31Transshipment Problem: Example Constraints DefinedAmount Out of Arnold: x13 + x14 < 75Amount Out of Supershelf: x23 + x24 < 75Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0Amount Into Zrox: x35 + x45 = 50Amount Into Hewes: x36 + x46 = 60Amount Into Rockrite: x37 + x47 = 40Non-negativity of Variables: xij > 0, for all i and j.
32Transshipment Problem: Example Computer OutputObjective Function Value =Variable Value Reduced CostXXXXXXXXXX
33Transshipment Problem: Example SolutionZroxZROX50507515ArnoldZeronN75ARNOLD52588Hewes6035HEWES374SuperShelfZeronS4075WASHBURN4754Rock-Rite40
34Assignment ProblemAn assignment problem seeks to minimize the total cost assignment of m workers to m jobs, given that the cost of worker i performing job j is cij.It assumes all workers are assigned and each job is performed.An assignment problem is a special case of a transportation problem in which all supplies and all demands are equal to 1; hence assignment problems may be solved as linear programs.The network representation of an assignment problem with three workers and three jobs is shown on the next slide.
35Assignment Problem Network Representation c11 c12 c13 Agents Tasks c21
36Assignment Problem Linear Programming Formulation Using the notation: xij = if agent i is assigned to task j0 otherwisecij = cost of assigning agent i to task jcontinued
37Assignment Problem Linear Programming Formulation (continued) xij > 0 for all i and j
38Assignment Problem LP Formulation Special Cases Number of agents exceeds the number of tasks:Number of tasks exceeds the number of agents:Add enough dummy agents to equalize thenumber of agents and the number of tasks.The objective function coefficients for thesenew variable would be zero.Extra agents simply remain unassigned.
39Assignment Problem LP Formulation Special Cases (continued) The assignment alternatives are evaluated in terms of revenue or profit:Solve as a maximization problem.An assignment is unacceptable:Remove the corresponding decision variable.An agent is permitted to work t tasks:
40Assignment Problem: Example 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.ProjectsSubcontractor A B CWestsideFederatedGoliathUniversalHow should the contractors be assigned so that totalmileage is minimized?
41Assignment Problem: Example Network Representation50West.A3616SubcontractorsProjects28Fed.30B183532Gol.C202525Univ.14
42Assignment Problem: Example Linear Programming FormulationMin 50x11+36x12+16x13+28x21+30x22+18x23+35x31+32x32+20x33+25x41+25x42+14x43s.t. x11+x12+x13 < 1x21+x22+x23 < 1x31+x32+x33 < 1x41+x42+x43 < 1x11+x21+x31+x41 = 1x12+x22+x32+x42 = 1x13+x23+x33+x43 = 1xij = 0 or 1 for all i and jAgentsTasks
43Assignment Problem: Example The optimal assignment is:Subcontractor Project DistanceWestside CFederated AGoliath (unassigned)Universal BTotal Distance = 69 miles