2 A network problem is one that can be represented by... 4.1 IntroductionA network problem is one that can be represented by...68910Nodes7Arcs10Function on Arcs
3 4.1 Introduction The importance of network models Many business problems lend themselves to a network formulation.Optimal solutions of network problems are guaranteed integer solutions, because of special mathematical structures. No special restrictions are needed to ensure integrality.Network problems can be efficiently solved by compact algorithms due to their special mathematical structure, even for large scale models.
4 Network Terminology Flow Directed/undirected arcs the amount sent from node i to node j, over an arc that connects them. The following notation is used:Xij = amount of flowUij = upper bound of the flowLij = lower bound of the flowDirected/undirected arcswhen flow is allowed in one direction the arc is directed (marked by an arrow). When flow is allowed in two directions, the arc is undirected (no arrows).Adjacent nodesa node (j) is adjacent to another node (i) if an arc joins node i to node j.
5 Network Terminology Path / Connected nodes Path :a collection of arcs formed by a series of adjacent nodes.The nodes are said to be connected if there is a path between them.Cycles / Trees / Spanning TreesCycle : a path starting at a certain node and returning to the same node without using any arc twice.Tree : a series of nodes that contain no cycles.Spanning tree : a tree that connects all the nodes in a network ( it consists of n -1 arcs).
6 4.2 The Transportation Problem Transportation problems arise when a cost-effective pattern is needed to ship items from origins that have limited supply to destinations that have demand for the goods.
7 The Transportation Problem Problem definitionThere are m sources. Source i has a supply capacity of Si.There are n destinations. The demand at destination j is Dj.Objective:Minimize the total shipping cost of supplying thedestinations with the required demand from the availablesupplies at the sources.
8 CARLTON PHARMACEUTICALS Carlton Pharmaceuticals supplies drugs and other medical supplies.It has three plants in: Cleveland, Detroit, Greensboro.It has four distribution centers in: Boston, Richmond, Atlanta, St. Louis.Management at Carlton would like to ship cases of a certain vaccine as economically as possible.
9 CARLTON PHARMACEUTICALS DataUnit shipping cost, supply, and demandAssumptionsUnit shipping costs are constant.All the shipping occurs simultaneously.The only transportation considered is between sources and destinations.Total supply equals total demand.ToFromBostonRichmondAtlantaSt. LouisSupplyCleveland$353040321200Detroit374042251000Greensboro40152028800Demand1100400750750
11 Destinations Boston Sources Cleveland Richmond Detroit Atlanta St.LouisDestinationsSourcesClevelandDetroitGreensboroD1=110037404232353025152028S1=1200S2=1000S3= 800D2=400D3=750D4=750
12 CARLTON PHARMACEUTICALS – Linear Programming Model The structure of the model is:Minimize Total Shipping CostST[Amount shipped from a source] £ [Supply at that source][Amount received at a destination] = [Demand at that destination]Decision variablesXij = the number of cases shipped from plant i to warehouse j.where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis)
13 The supply constraints ClevelandS1=1200X11X12X13X14Supply from Cleveland X11+X12+X13+X14 =The supply constraintsDetroitS2=1000X21X22X23X24Supply from Detroit X21+X22+X23+X24 =BostonGreensboroS3= 800X31X32X33X34Supply from Greensboro X31+X32+X33+X34 =D1=1100RichmondD2=400AtlantaD3=750St.LouisD4=750
14 CARLTON PHARMACEUTICAL – The complete mathematical model Total shipment out of a supply nodecannot exceed the supply at the node.=Total shipment received at a destinationnode, must equal the demand at that node.
15 CARLTON PHARMACEUTICALS Spreadsheet =SUMPRODUCT(B7:E9,B15:E17)=SUM(B7:E7)Drag to cells G8:G9=SUM(B7:E9)Drag to cells C11:E11
16 CARLTON PHARMACEUTICALS Spreadsheet MINIMIZE Total CostSHIPMENTSDemands are metSupplies are not exceeded
18 CARLTON PHARMACEUTICALS Sensitivity Report Reduced costsThe unit shipment cost between Cleveland and Atlanta must be reduced by at least $5, before it would become economically feasible to utilize itIf this route is used, the total cost will increase by $5 for each case shipped between the two cities.
19 CARLTON PHARMACEUTICALS Sensitivity Report Allowable Increase/DecreaseThis is the range of optimality.The unit shipment cost between Cleveland and Boston may increase up to $2 or decrease up to $5 with no change in the current optimal transportation plan.
20 CARLTON PHARMACEUTICALS Sensitivity Report Shadow pricesFor the plants, shadow prices convey the cost savings realized for each extra case of vaccine produced. For each additional unit available in Cleveland the total cost reduces by $2.
21 CARLTON PHARMACEUTICALS Sensitivity Report Shadow pricesFor the warehouses demand, shadow prices represent the cost savings for less cases being demanded. For each one unit decrease in demanded in Boston, the total cost decreases by $37.
22 Modifications to the transportation problem Cases may arise that require modifications to the basic model.Blocked routes - shipments along certain routes are prohibited.Remedies:Assign a large objective coefficient to the route (Cij = 1,000,000)
23 Modifications to the transportation problem Cases may arise that require modifications to the basic model.Blocked routes - shipments along certain routes are prohibited.Remedies:Assign a large objective coefficient to the route (Cij = 1,000,000)Add a constraint to Excel solver of the form Xij = 0Shipments on a Blocked Route = 0
24 Modifications to the transportation problem Cases may arise that require modifications to the basic model.Blocked routes - shipments along certain routes are prohibited.Remedies:Assign a large objective coefficient to the route (Cij = 1,000,000)Add a constraint to Excel solver of the form Xij = 0Do not include the cell representing the rout in the Changing cellsOnly Feasible Routes Included in Changing CellsCell C9 is NOT IncludedShipments from Greensboroto Cleveland are prohibited
25 Modifications to the transportation problem Cases may arise that require modifications to the basic model.Minimum shipment - the amount shipped along a certain route must not fall below a pre-specified level.Remedy: Add a constraint to Excel of the form Xij ³ BMaximum shipment - an upper limit is placed on the amount shipped along a certain route.Remedy: Add a constraint to Excel of the form Xij £ B
26 MONTPELIER SKI COMPANY Using a Transportation model for production scheduling Montpelier is planning its production of skis for the months of July, August, and September.Production capacity and unit production cost will change from month to month.The company can use both regular time and overtime to produce skis.Production levels should meet both demand forecasts and end-of-quarter inventory requirement.Management would like to schedule production to minimize its costs for the quarter.
27 MONTPELIER SKI COMPANY Data:Initial inventory = 200 pairsEnding inventory required =1200 pairsProduction capacity for the next quarter = 400 pairs in regular time.= 200 pairs in overtime.Holding cost rate is 3% per month per ski.Production capacity, and forecasted demand for this quarter (in pairs of skis), and production cost per unit (by months)
28 MONTPELIER SKI COMPANY Analysis of demand:Net demand in July = = 200 pairsNet demand in August = 600Net demand in September = = 2200 pairsAnalysis of Supplies:Production capacities are thought of as supplies.There are two sets of “supplies”:Set 1- Regular time supply (production capacity)Set 2 - Overtime supplyInitial inventoryIn house inventoryForecasted demand
29 MONTPELIER SKI COMPANY Analysis of Unit costsUnit cost = [Unit production cost] +[Unit holding cost per month][the number of months stays in inventory]Example: A unit produced in July in regular time and sold in September costs 25+ (3%)(25)(2 months) = $26.50
32 MONTPELIER SKI COMPANY Summary of the optimal solutionIn July produce at capacity (1000 pairs in R/T, and 500 pairs in O/T). Store = 1300 at the end of July.In August, produce 800 pairs in R/T, and 300 in O/T. Store additional = 500 pairs.In September, produce 400 pairs (clearly in R/T). With 1000 pairs retail demand, there will be( ) = 1200 pairs available for shipment to Ski Chalet.Inventory +Production -Demand
33 4.3 The Capacitated Transshipment Model Sometimes shipments to destination nodes are made through transshipment nodes.Transshipment nodes may beIndependent intermediate nodes with no supply or demandSupply or destination points themselves.Transportation on arcs may be bounded by given bounds
34 The Capacitated Transshipment Model The linear programming model of this problem consists of:Flow on arcs decision variablesCost minimization objective functionBalance constraints on each node as follows:Supply node – net flow out does not exceed the supplyIntermediate node – flow into the node is equal to the flow outDemand node – net flow into the node is equal to the demandBound constraints on each arc. Flow cannot exceed the capacity on the arc
35 DEPOT MAX A General Network Problem Depot Max has six stores located in the Washington D.C. area.
36 DEPOT MAXThe stores in Falls Church (FC) and Bethesda (BA) are running low on the model 5A Arcadia workstation.DATA:-125FC-136BA
37 DEPOT MAXThe stores in Alexandria (AA) and Chevy Chase (CC) have an access of 25 units.DATA:-12+1015FCAA-13+1526BACC
38 DEPOT MAXThe stores in Fairfax and Georgetown are transshipment nodes with no access supply or demand of their own.DATA:-12FX+10135FCAADepot Max wishes to transport the available workstations to FC and BA at minimum total cost.GN-13+15246BACC
39 DEPOT MAXThe possible routes and the shipping unit costs are shown.DATA:5102061512711-12FX+1013FCFCAAGN-13+1524BABACC
40 DEPOT MAXDataThere is a maximum limit for quantities shipped on various routes.There are different unit transportation costs for different routes.
41 DEPOT MAX – Types of constraints 5102061512711-12+10135Supply nodes: Net flow out of the node] = [Supply at the node]X12 + X13 + X15 - X21 = 10 (Node 1) X21 + X24 - X = 15 (Node 2)Intermediate transshipment nodes: [Total flow out of the node] = [Total flow into the node]X34+X35 = X (Node3) X46 = X24 + X34 (Node 4)7Demand nodes: [Net flow into the node] = [Demand for the node]X15 + X35 +X65 - X56 = 12 (Node 5) X46 +X56 - X65 = 13 (Node 6)246-13+15
42 All variables are non-negative DEPOT MAXThe Complete mathematical modelMin 5X X X X X X34 + 7X X X56 + 7X65S.T. X12 + X13 + X15 – X £ 10- X X21 + X £ 17– X X34 + X = 0– X24 – X X = 0– X – X35 + X56 - X65 = -12-X46 – X56 + X65 = -13X12 £ 3; X15 £ 6; X21 £ 7; X24 £ 10; X34 £ 8; X35 £ 8; X46 £ 17; X56 £ 7; X65 £ 5All variables are non-negative
44 4.4 The Assignment Problem Problem definitionm workers are to be assigned to m jobsA unit cost (or profit) Cij is associated with worker i performing job j.Minimize the total cost (or maximize the total profit) of assigning workers to job so that each worker is assigned a job, and each job is performed.
45 BALLSTON ELECTRONICSFive different electrical devices produced on five production lines, are needed to be inspected.The travel time of finished goods to inspection areas depends on both the production line and the inspection area.Management wishes to designate a separate inspection area to inspect the products such that the total travel time is minimized.
46 BALLSTON ELECTRONICSData: Travel time in minutes from assembly lines to inspection areas.
49 BALLSTON ELECTRONICS – The Linear Programming Model Min 10X11 + 4X12 + … X X55S.T. X11 + X12 + X13 + X14 + X15 = 1X21 + X … X25 = 1… … … …X51 + X52+ X53 + X54 + X55 = 1All the variables are non-negative
50 BALLSTON ELECTRONICS – Computer solutions A complete enumeration is not an efficient procedure even for moderately large problems (with m=8, m! > 40,000 is the number of assignments to enumerate).The Hungarian method provides an efficient solution procedure.
51 BALLSTON ELECTRONICS – Transportation spreadsheet =SUMPRODUCT(B7:F11,B17:F217)=SUM(B7:F7)Drag to cells H8:H12=SUM(B7:B11)Drag to cells C13:F13
52 BALLSTON ELECTRONICS – Transportation spreadsheet Each Area is ServedEach Line is Assigned
54 The Assignments Model - Modifications Unbalanced problem: The number of supply nodes and demand nodes is unequal.Prohibitive assignments: A supply node should not be assigned to serve a certain demand node.Multiple assignments: A certain supply node can be assigned for more than one demand node .A maximization assignment problem.
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