Transportation zMode Selection zRoute Selection yShortest Path yMinimum Spanning Tree yTransportation yAssignment yTSP yRoute Sequencing yTanker Scheduling

Mode Selection zShip zRail zTruck zPlane... Faster Cheaper More Variable

Mode that minimizes Total Cost zTransportation Cost zInventory Costs ySource yPipeline yDestination - including safety stock

Costs zTransportation Cost yCost per unit * Units zCost per unit y$/CWT (based on origin, destination, freight, weight) y$/Time (leased, dedicated transportation)

Inventory zAt the plant y1/2 “cycle quantity” zAt the warehouse y1/2 “cycle quantity” ySafety stock depends on lead time variability zIn the pipeline yAnnual Volume * Days in Transit Days per year

Example (page 187)

Multi-Modal Systems zShip from Japan to Long Beach zRail from Long Beach to Terminals zTruck from Terminals to Dealerships zWhere to change mode? zHow many channels to operate?

Route Selection zGetting From A to B zUnderlying Network yRoads yAirports yTelecommunication links zCosts of using each link zFind the cheapest (shortest) path

Example (page 192) A E D C B J H G F I

Shortest Path Model

Applicability zSingle Origin zSingle Destination zNo requirement to visit intermediate nodes zNo “negative cycles”

Tree of Shortest Paths zFind shortest paths from Origin to each node zSend n-1 units from origin zGet 1 unit to each destination

Shortest Path Problem

Minimum Spanning Tree zFind the cheapest total cost of edges required to tie all the nodes together A E D C B J H G F I

Greedy Algorithm zConsider links from cheapest to most expensive zAdd a link if it does not create a cycle with already chosen links zReject the link if it creates a cycle.

What’s the difference zShortest Path Problem yRider’s version yConsider the number of riders who will use it zSpanning Tree Problem yBuilder’s version yConsider only the cost of construction

Transportation Problem zSources with limited supply zDestinations with requirements zCost proportional to volume zMultiple sourcing allowed zExample: Regal Company (Tools)

PROTRAC Engine Distribution * * * * * * * Belgium Germany Netherlands The Hague Amsterdam Antwer p Nancy Liege Tilburg Leipzig Miles

Transportation Costs Unit transportation costs from harbors to plants Minimize the transportation costs involved in moving the engines from the harbors to the plants

A Transportation Model

Crossdocking z3 plants z2 distribution centers z2 customers zMinimize shipping costs yDirect from plant to customer yVia DC

A Network Model

Good News zLots of applications zSimple Model zOptimal Solutions Quickly zIntegral Data, Integral Answers

Bad News zWhat’s Missing? ySingle Homogenous Product yLinear Costs yNo conversions or losses y...

Homogenous Product

Linear Costs zNo Fixed Charges zNo Volume Discounts zNo Economies of Scale

Integer Models zCan model nearly everything zSometimes very difficult to solve zFoundation in Linear Models zExpand from there

The Rules zExactly like Linear Models except zSome decision variables restricted to yBinary - 0 or 1, Yes or No, True or False yIntegers

Steco Warehouse Location

A Linear Model zIgnore leasing for now -- all warehouses are open zObjective: Minimize Total Cost zDecision Variables: yNumber of trucks from each warehouse to each customer each month zConstraints yEnough trucks to each customer yNot too many trucks from each warehouse zRecognize this?

Making Discrete Decisions zNew Decision Variables yDo we lease warehouse or not -- binary zNew Constraints yEffective Capacity depends on whether or not warehouse is open. yWarehouse A effective capacity is x0 if we do not lease the warehouse x200 if we do. zThis is Linear!

An Integer Model

Special Case zAssignment Problem zSupply at each source 1 zRequirement at each destination 1 zMatch up suppliers with destinations zHow’s this different from single sourcing zAssigning workers to tasks

“Application” zTruckload shipping zLoads waiting at customers zTrucks sitting at locations zWhich truck should handle which load? zNo concern for what to do after that. zWhat are “sources” zWhat are “destinations” zWhat are costs?

Traveling Salesman Problem zVehicle at depot zCustomers to be served (visited) zVehicle must visit all and return to depot zMinimize travel cost

Example zFull Service Business zDriver at Service Center zAssigned vending machines to visit zWhat order should he visit to minimize the time to complete the work and get back to the depot?

Extensions zIf the “customers” involve transportation yCustomers = truck load shipments zIf more than one “salesman” involved yConstruct routes for the 7 drivers at the North Metro Service center zIf the vehicle has capacity yLTL deliveries zIf we intersperse pickups and deliveries zIf there are time windows on service

Basic TSP zData issues yEstimate distance by location yCalculate point to point distances yCalculate point to point costs

Heuristics zCluster first Route Second yBuild delivery zones with approximately equal work. yRoute a vehicle in each zone zClustering Approaches yAssign most distant blocks first ySweep ySpace-filling curve

Space-Filling Curve zEach point (X,Y) on the map zExpress X = string of 0’s and 1’s yX = 16.5 = *2 4 +0*2 3 +0*2 2 +0*2 1 +0*2 0 +1* *2 -2 zExpress Y = string of 0’s and 1’s yY = 9.75 = *2 4 +1*2 3 +0*2 2 +0*2 1 +1*2 0 +1* *2 -2 zSpace Filling Number - interleave bits y  (X,Y) =

Properties zEvery pair (X,Y) has a unique point  (X,Y) zEvery point on the line corresponds to a single point (X,Y) zIf  (X,Y) and  (X’, Y’) are close together, (X,Y) and (X’,Y’) tend to be close together.

Clustering zCompute  (X,Y) for each customer zSort the customers by their  values zTo build N routes yGive first 1/Nth of customers to first route yGive second 1/Nth of customers to second route y...

Routing zEach route visits the customers in order of their  values. z  defines a route on the plane that visits every point. We visit the customers in the same order as that route

Route First zBuild a single large route zAssign each vehicle a segment of the route

Routing Heuristics zSpace-Filling Curve zClarke-Wright Savings zNearest Neighbor zNearest Insertion zFarthest Insertion z2-interchange

2-Interchange

Route Sequencing

Tanker Scheduling

Consolidation zCross docking zMulti-stop shipments zLarger Orders zDelay shipments zConsolidate Production