Weight and Cube, Frequency

Extend Network Flows to Multicommodity

More than one product Different products share conveyance

capacityDelivery Schedules or Frequency

Separate Network Flows

Move two products through the network

E.g., CPU’s and MonitorsHow does this affect our

Transshipment Model?

With 2 Products

3 plants2 distribution centers2 customersMinimize shipping costs

Direct from plant to customer

Via DC

A Transshipment Model

The Sets The Plants set PLANTS; The Distribution Centers set DCS; The Customers set CUSTS; The Products set PRODS;

Transshipment Model

The Set of Edges We assume all Plant-DC, Plant-Customer,

DC-Customer edges are possible. Convenient to define a set of Edgesset EDGES := (PLANTS cross DCS) union (PLANTS cross CUSTS) union (DCS cross CUSTS);

A Transshipment Model

The Parameters The Supply at each plant of each product

param Supply{PLANTS, PRODS};

The Demand at each Customer for each productparam Demand{CUSTS, PRODS};

The Cost on each edge - cost of the truckparam Cost{EDGES};

A Transshipment Model

The Variables The volume shipped on each edge of each

product var Ship{EDGES, PRODS} >= 0;

The ConstraintsCombine ideas of Shortest paths (flow

conservation) with Transportation (meet supply and demand) for each Product

A Transshipment ModelFor each Plants.t. RespectSupply {plant in PLANTS,

prod in PRODS}: sum{(plant, t) in EDGES} Ship[plant,t, prod]<= Supply[plant, prod];

A Transshipment ModelFor each Customers.t. MeetDemand {cust in CUSTS, prod in PRODS}: sum{(f, cust) in EDGES} Ship[f, cust, prod]>= Demand[cust, prod];

A Transshipment ModelFor each DC: Conserve flows.t. ConserveFlow {dc in DCS, prod in PRODS}: sum{(f, dc) in EDGES} Ship[f,dc,prod] = sum{(dc, t) in EDGES} Ship[dc,t, prod];Flow into the DC = Flow out of the DC for each

product

Modeling Trucks

The cost per conveyance param Cost {EDGES};The number of conveyances

var Conveyances{EDGES} integer >= 0;Freight Cost in the Objective

minimize FreightCost: sum {(f,t) in EDGES} Cost[f,t]*Conveyances[f,t];

How many Conveyances?

Different commodities sharing a vehicle

Easy to figure vehicle capacity for single commodity

How to figure vehicle capacity for mixed loads?

Typical Approach

10 items

20 items

40% 60%

4 items12 items

Formulation

Blue Vol/10 + Red Vol/20 1To calculate the number of vehicles

required to carry the given volumesConveyances =

Blue Vol/10 + Red Vol/20

Several Capacities

Weight LimitSpace or CubeEnsure loads meet each limitVehicles Blue Vol/10 + Red

Vol/20

Vehicles Blue Vol/8 + Red Vol/25

Number that reaches the weight limit

Number that fills the cube

Counting Conveyances

Weight of each productparam Weight{PRODS};Cube of each productparam Cube{PRODS};Weight Limitparam WeightLimit;Cubic capacityparam CubicCap;

Counting Conveyances

Conveyances required by weights.t. ConveyByWeight{(f,t) in EDGES}: Conveyances[f,t] >= sum{prod in PRODS} Ship[f,t,prod]*Weight[prod]/WeightLimit;Conveyances required by cubes.t. ConveyByCube{(f,t) in EDGES}: Conveyances[f,t] >= sum{prod in PRODS} Ship[f,t,prod]*Cube[prod]/CubicCap;

FrequencySometimes conveyances depart on a

schedule even if they are not full Time definite delivery

LTL carriersParcel carriers

Lean Manufacturing Auto industry

Scheduled ServiceOcean carriersAir carriers“Pool operators”

Impacts

Number of conveyances required Equipment Utilization

Time product waits for carriage Service Inventory...

Number of Conveyances

One way to handleGiven number of departures in the

period modeledparam Frequency{EDGES};var Conveyances{(f,t) in EDGES} integer >= Frequency[f,t];

Writing these as bounds is better than adding separate constraints.

Inventory

So far, just FreightInventory ComponentWe focus on moving and waiting

inventoryCapture moving inventory as we did

in Total Cost

Moving Inventory

Holding Cost * Product Value * Daily Volume * Days of Transit = Pipeline inventory cost on edge

param HoldingCost;param Value{PRODS};param TransitDays{EDGES};Assuming Ship[f,t,prod] is daily

volume shipped from f to t of prod...

Pipeline Inventory

Sum{(f,t) in EDGES, prod in PRODS} HoldingCost*Value[prod]*Ship[f,t,prod]* TransitDays[f,t];

This is the annual cost to finance the value of products that are always on the road.

Waiting Inventory

Just looks at waiting for transportIgnores waiting for production and

consumption. Assumes smooth or uniform operationsIf carrier departs once every 5 days,

some parts arrive just after last departure some parts arrive just before next departure on average parts wait about 2.5 days add this to the transit time

Adding Waiting Time

If Frequency is given in departures per day (e.g., Frequency of 0.2 = 1 every 5 days)

1/Frequency = Days between departures

0.5/Frequency = Average days product waits for transport

Pipeline and Waiting Inventory

Sum{(f,t) in EDGES, prod in PRODS} HoldingCost*Value[prod]*Ship[f,t,prod]* (TransitDays[f,t]+0.5/Frequency[f,t]);

This assumes the system operates like a subway and pays inventory on riders from the time they put their token in at the first station till they exit the last station.

Add this to the objective to minimize freight and inventory costs