Performance

1
Performance
Mean 65
2
Solutions to Exam 2

• We have a single commodity in warehouses around
  the country and wish to ship it to our customers
  at minimum cost. The carrier charges us a cost
  per unit shipped that depends only on the
  location of the warehouse and the location of the
  customer.
• Transportation Model

3
Weight Cube

• We have several commodities in warehouses around
  the country and wish to ship them to our
  customers at minimum cost. The commodities have
  different unit weights and densities. The carrier
  charges us a cost per truck load shipped that
  depends only on the location of the warehouse and
  the location of the customer. The truck has both
  a weight limit and a cubic capacity. If we ship a
  truck that is only partially full, the carrier
  will charge us for the larger of the fraction of
  the weight limit of the truck that we use or the
  fraction of the cubic capacity of the truck that
  we use.
• General Linear Programming Model

4
Single Sourcing

• We have a single commodity in warehouses around
  the country and wish to ship it to our customers
  at minimum cost. The carrier charges us a cost
  per unit shipped that depends only on the
  location of the warehouse and the location of the
  customer. Each customer insists on receiving all
  of his demand from a single warehouse. Different
  customers may be served from different
  warehouses, but no customer can be served by more
  than one warehouse.
• General Mixed Integer Linear Programming Model

5
Sets and Parameters

• / The Plants /
• set PLANTS
• / The DCS /
• set DCS
• / The Cross Docks /
• set CROSSDOCKS
• / The capacity at each plant /
• param CapacityPLANTS
• / The Demand at each Cross Dock /
• param DemandCROSSDOCKS

6
Variables and Objective

• / The set of shipments possible /
• set EDGES (PLANTS cross DCS) union (DCS cross
  CROSSDOCKS)
• / The unit cost on each edge /
• param CostEDGES
• / The variables are the quantities shipped on
  each edge /
• var ShipEDGES gt 0
• / The Objective Minimize Freight Costs /
• minimize FreightCost
• sum(f,t) in EDGES Costf,tShipf,t

7
Constraints

• / Constraints Observe plant capacities /
• s.t. PlantCapacities plant in PLANTS
• sum(plant, dc) in EDGES Shipplant, dc lt
  Capacityplant
• / Constraints Meet Demand at each CROSSDOCK /
• s.t. MeetDemanddock in CROSSDOCKS
• sum(dc,dock) in EDGES Shipdc,dock gt
  Demanddock
• / Constraints Conserve Flow at DC's /
• s.t. ConserveFlowdc in DCS
• sum (plan, dc) in EDGES Shipplant, dc
  
• sum (dc, dock) in EDGES Shipdc, dock

8
Single Sourcing etc

• The model described above does not impose the
  single sourcing constraints at the DCs or the
  single destination for the plants. Heres how to
  do that -- merge the following with the previous
  model
• / Whether or not we use each edge /
• var UseEdgeEDGES binary
• s.t. DefineUseEdgePlant(plant, dc) in EDGES
  plant in PLANTS
• Shipplant, dc lt CapacityplantUseEdgep
  lant,dc
• s.t. DefineUseEdgeDock(dc, dock) in EDGES dock
  in CROSSDOCKS
• Shipdc, dock lt DemanddockUseEdgedc,
  dock

9
Single Sourcing etc.

• / Use one edge from each plant /
• s.t. SingleDCforPlant plant in PLANTS
• sumdc in DCS UseEdgeplant,dc 1
• / Use one edge to each Cross Dock /
• s.t. SingleDCforCrossDockdock in CROSSDOCKS
• sumdc in DCS UseEdgedc, dock 1

10
Building the System

• Disney is planning to build tunnels to connect
  its merchandise warehouse in EuroDisney to all
  of the stores in the park. For safety reasons,
  the company will not allow tunnels to connect
  except at the warehouse or at stores. It wishes
  to minimize the cost of building the tunnels.
• Minimum Spanning Tree Model

11
Using the System

• After the tunnels (described in d.) are built,
  Disney is concerned with getting special orders
  to the stores from the warehouse as quickly as
  possible using a combination of the tunnels and
  surface streets. It has estimates of the time
  required to travel through each tunnel and down
  each street.
• Shortest Path Model

12
Formulation

• Pratt Whitney is preparing to make final
  arrangements for delivery of jet engines from its
  final assembly plants to aircraft manufacturers
  sites.
• Each jet engine is transported in a 747. A 747
  can carry only a single engine at a time.
• Each engine costs 100 million. Pratt Whitney
  estimates inventory carrying cost at about
  25/year.

13
P W Formulation

• The set of PW Plants
• set PLANTS
• The set of customer manufacturing sites
• set SITES
• The capacities of the plants in engines per year
• param CapacityPLANTS
• The demands at the customer manufacturing sites
  in engines per year
• param DemandSITES

14

• The freight cost per engine from each plant to
  each customer site
• param FreightCostPLANTS, SITES
• The travel time in days from each plant to each
  customer site
• param TravelDaysPLANTS, SITES
• The Engine Cost in /engine
• param EngineCost
• The inventory carrying cost as a fraction of the
  value of the item charged per year
• param HoldingCost

15
The Heart of the Matter

• Calculated parameter, the freight and inventory
  cost incurred per engine shipped from each plant
  to each site in /engine
• param Costplant in PLANTS, site in SITES
• FreightCostplant, site
• HoldingCostEngineCost
• TravelDaysplant, site/365

16
The Model

• Variables Number of engines shipped from each
  plant to each customer manufacturing site
• This is a transportation problem (with integer
  data) so we don't need to specify that these
  variables be integral.
• var ShipPLANTS, SITES gt 0

17
The Model

• Objective Minimize annual freight and pipeline
  inventory costs
• minimize TotalCost
• sumplant in PLANTS, site in SITES
• Costplant, siteShipplant, site
• s.t. ObserveCapacity plant in PLANTS
• sumsite in SITES Shipplant, site
• lt Capacityplant
• s.t. MeetDemandsite in SITES
• sumplant in PLANTS Shipplant, site
• gt Demandsite

18
Set Covering/Location

• Intel Corporation is facing increasing pressure
  to provide consignment inventory to computer
  manufacturers in emerging markets including
  Eastern Europe. The company has 10 major
  customers in the region and holds options to
  lease at 5 sites in the region. It would like to
  exercise the fewest options necessary to
  guarantee it has a warehouse within 200 miles of
  each customer.
• Formulate an optimization model to identify which
  lease options Intel should exercise in order to
  ensure it has the fewest possible warehouses and
  has a warehouse within 200 miles of each
  customer.

19

• The set of Intel Sites
• set SITES
• The set of Customer Sites
• set CUSTS
• The distance data
• param DistSITES, CUSTS
• The allowed distance (200 miles)
• param MaxDist

20

• Calculated parameter is customer within 200
  miles of the site?
• Param Coverssite in SITES, cust in CUSTS
• if Distsite,cuts lt MaxDist
• then 1 else 0
• Variables Open the site or not
• var OpenSITES binary

21
The Model

• Objective Minimize the number of sites opened
• miminize OpenSites
• sum site in SITES Opensite
• s.t. CoverEachCustomercust in CUSTS
• sumsite in SITES Coverssite,
  custOpensite
• gt 1

22

• set PRODUCERS
• set PRODUCTS
• set PORTS
• set DCS
• set CROSSDOCKS
• param SupplyPRODUCERS, PRODUCTS
• param DemandCROSSDOCKS, PRODUCTS

23

• set EDGES dimen 2
• param CostEDGES
• var FlowEDGES, PRODUCTSgt0
• minimize FreightCost
• sum(f,t) in EDGES, prd in PRODUCTS
  Costf,tFlowf,t,prd

24

• s.t. RespectSupplyproducer in PRODUCERS,
• prd in PRODUCTS
• sum(producer, t) in EDGES, prd in PRODUCTS
  
• Flowproducer, t, prd
• lt Supplyproducer, prd
• s.t. MeetDemandcrossdock in CROSSDOCKS,
• prd in PRODUCTS
• sum(f, crossdock) in EDGES, prd in PRODUCTS
  
• Flowf, crossdock, prd
• gt Demandcrossdock, prd

25
The Problem

• s.t. ConserveFlowfac in PORTS union DCS
• sum(f, fac) in EDGES,
• prd in PRODUCTS Flowf, fac, prd
• sum(fac, t) in EDGES,
• prd in PRODUCTS Flowfac, t, prd