Modeling Service

15.057 Spring 02 Vande Vate 11

Modeling Service

When Transport is restricted to Load-DrivenPool Points in Retail Distribution

John Vande VateSpring 2007


Retail Inventory• Single “Product”, many SKUs

• Style• Color• Size

• Broad Offering attracts customers • Depth in SKU avoids missed sales• Stock enough in each SKU to cover

replenishment time (OTD)


Service Requirement

• Keep OTD short• Reduce depth without losing sales• Increase breadth to attract more customers

and expand market• OTD requirements differ by store

– Manhattan, NY– Manhattan, KS


What’s in OTD

• POS system records sale• Transmitted to DC • Orders batched for efficient picking• Order picked• Trailer filled (Load driven)• Line Haul to Pool Point• Delivery


Pool PointsAsian Port

US Port

Pool

Store

Asian Factory

US DC


Pools Influence

• Trailer Fill– The greater the volume to the pool the faster the

trailer fills• Line Haul

– Is determined by the distance from the DC to the Pool

• Delivery– Messier


The Trade-offs

• Too Few Pools– High Delivery Costs– More moving inventory– Less waiting inventory

• Too Many Pools– Low Delivery Costs– Less moving inventory– More waiting inventory

OTD Dissected

• POS system records sale• Transmitted to DC • Orders batched for efficient picking• Order picked• Trailer filled (Load driven)• Line Haul• Delivery

Constantcost & time

Cost & Time depend on Pool Assignment


Line Haul

• Time & Cost Depend on – The Pool Assignment– Which DC’s serve the Pool

• NY DC to Chicago Pool• LA DC to Chicago Pool


Trailer Fill

• Time Depends on– The Pool Assignments

• Which pool this store is assigned to • What other stores are assigned to this pool• Rate at which the Pool draws goods

– How the Pool is served• The Rate at which the Pool draws goods from

each DC


Drilling Down on Service

• Simple Model– Cube only (trailers never reach weight limit)– One DC only (don’t split volumes to Pool)

• Many DC’s– Cube only

• Weight & Cube and Many DC’s• Soft Constraints:

– Infeasible is not an acceptable answer


Toward a Simple Model

Trailer Fill Time (for store) * Rate Trailer Fills = Cubic Capacity of the Trailer

• Rate Trailer Fills– Translate annual demand at stores assigned to

the pool into cubic feet per day– Rate to pool is:

sum{prd in PRODUCTS, s in STORES}

CubicFt[prd]/DaysPerYear*Demand[prd,s]*Assign[s, pool]


Make It LinearTrailer Fill Time * sum{prd in PRODUCTS, s in STORES}

CubicFt[prd]/DaysPerYear*Demand[prd,s]*Assign[s, pool] = Cubic Capacity of the Trailer

• Trailer Fill Time * Assign • What to do?


Can’t Know Trailer Fill Time

Trailer Fill Time * Rate Trailer Fills = Cubic Capacity of the Trailer

Max Time to Fill Trailer * Rate Trailer Fills ? Cubic Capacity of the Trailer

Why Cubic Capacity of the Trailer?

What does this accomplish?

What’s Wrong?

Max Time to Fill Trailer to store * Rate Trailer Fills Cubic Capacity of the Trailer

Max Time to Fill Trailer*(sum{prd in PRODUCTS, s in STORES} CubicFt[prd]/DaysPerYear*Demand[prd,s]*Assign[s,

pool]) Cubic Capacity of the Trailer

• What if the Store is not assigned to the Pool?!Max Time to Fill Trailer * Rate Trailer Fills Cubic

Capacity of the Trailer*Assign[store,pool]


Our Simple Modelvar Assign{STORES, POOLS} binary;

Service Constraint for each store and pool:Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}

CubicFt[prd]/DaysPerYear*Demand[prd,s]*Assign[s, pool] >= Cubic Capacity of the Trailer*Assign[store, pool]

Max Time to Fill Trailer depends on the Store and the Pool:Store Service Requirement, e.g., 3 daysConstant Order time, e.g., order processing, picking, etc.Line haul time from DC to PoolDelivery time from Pool to Store


Getting Practical• There are scores of Pools and THOUSANDS of

Stores• That means hundreds of thousands of service

constraints!• Can we just impose them at the Pools?• If the Pools open…

– Ensure we get there in reasonable time (same time from there to all stores)

– May have to handle a few special stores separately


A Simpler Modelvar Assign{STORES, POOLS} binary;

Service Constraint for each pool:Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}


var Open{POOLS} binary;

Assign[store, pool]Open[pool]

Max Time to Fill Trailer depends on the Store and the Pool:Store Service Requirement, e.g., 3 daysConstant Order time, e.g., order processing, picking, etc.Line haul time from DC to PoolDelivery time from Pool to Store (depends on pool)


One DC Cube Only Modelvar Assign{STORES, POOLS} binary;

Service Constraint for each pool:Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}


var Open{POOLS} binary;

Max Time to Fill Trailer depends on the Pool:Store Service Requirement, e.g., 3 daysConstant Order time, e.g., order processing, picking, etc.Line haul time from DC to PoolDelivery time from Pool to Store (depends on pool)


More than One DCService Constraint for each pool:

Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}

CubicFt[prd]/DaysPerYear*Demand[prd,s]*Assign[s, pool] >= Cubic Capacity of the Trailer*Open[pool]

• What’s wrong with this?


The Problem

• If the pool is served by several DC’s, trailers fill more slowly

• The “rate” from each DC is less than the total rate of demand at the pool.


How to Get the Rates

• How fast does the trailer for pool fill at dc?• Not the rate of demand at the pool• The rate of shipments from the dc to the pool• Translate shipments Ship[*, dc, pool]


What’s Wrong NowService Constraint for each dc and pool:Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}

CubicFt[prd]/DaysPerYear*Ship[prd, dc, pool] >= Cubic Capacity of the Trailer*Open[pool]


The ProblemService Constraint for each dc and pool:Max Time to Fill Trailer * sum{prd in PRODUCTS, s in

STORES}

CubicFt[prd]/DaysPerYear*Ship[prd, dc, pool] >= Cubic Capacity of the Trailer*Open[pool]

• Does the DC serve the Pool?• These constraints insist the service is

good from EVERY dc to EVERY open pool.


Fixing the Problem

var UseEdge{DCS, POOLS} binary;Service Constraint for each dc and pool:Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}

CubicFt[prd]/DaysPerYear*Ship[prd, dc, pool] >= Cubic Capacity of the Trailer*UseEdge[dc, pool]

Define UseEdge for each prd, dc and pool: Ship[prd, dc, pool] <= sum{s in STORES (that can be assigned to the pool)} Demand[prd,s]* UseEdge[dc, pool]


Several DCs Cube Only

var UseEdge{DCS, POOLS} binary;Service Constraint for each dc and pool:Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}


Define UseEdge for each prd, dc and pool: Ship[prd, dc, pool] <= sum{s in STORES (that can be assigned to the pool)} Demand[prd,s]* UseEdge[dc, pool]


Weight & Cube

var UseEdge{DCS, POOLS} binary;

Cube Service Constraint for each dc and pool:Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}


Weight Service Constraint for each dc and pool:Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}

Weight[prd]/DaysPerYear*Ship[prd, dc, pool] >= Weight Limit of the Trailer*UseEdge[dc, pool]


What’s Wrong Now?

var UseEdge{DCS, POOLS} binary;




Weight[prd]/DaysPerYear*Ship[prd, dc, pool] >= Weight Limit of the Trailer*UseEdge[dc, pool]


Weight AND Cube

• Trailer departs when – Weight Limit is reached OR– Cubic Capacity is reached

• Don’t have to fill BOTH


How to Fix It?

• The Cubic Capacity Constraint Applies If– We use the edge from the dc to the pool:

UseEdge[dc, pool] = 1 AND– We Cube Out that trailer first: New Variable CubeOut[dc,pool] = 1

• How to capture this?• (UseEdge[dc,pool] + CubeOut[dc,pool] -1)


How to Fix It?

• The Weight Limit Constraint Applies If– We use the edge from the dc to the pool:

UseEdge[dc, pool] = 1 AND– We DO NOT Cube Out that trailer first: CubeOut[dc,pool] = 0

• How to capture this?• (UseEdge[dc,pool] - CubeOut[dc,pool])

A Full Modelvar UseEdge{DCS, POOLS} binary;var CubeOut{DCS, POOLS} binary;


CubicFt[prd]/DaysPerYear*Ship[prd, dc, pool] >= Cubic Capacity of the Trailer*(UseEdge[dc, pool]+ CubeOut[dc,pool]-1)


Weight[prd]/DaysPerYear*Ship[prd, dc, pool] >= Weight Limit of the Trailer*(UseEdge[dc, pool]- CubeOut[dc,pool])

Cube or Weight

• How do we know which constraint should apply?…..>= Cubic Capacity of the Trailer*(UseEdge[dc, pool]+ CubeOut[dc,pool]-1)

or ….>= Weight Limit of the Trailer*(UseEdge[dc, pool]- CubeOut[dc,pool])


What if it’s not feasible• Infeasible is not a very helpful answer• Want an answer that is “as close as possible”• Sequential Optimization:

– Minimize Service Failures– Minimize Cost subject to Best Achievable

Service•


Service Failures

• Can’t express them in terms of time• Express them in terms of Capacity• How much of the trailer is left unfilled at

the end of the available time?

Unfilled Capacity

• Max Time to Fill Trailer * Rate Trailer Fills Cubic Capacity of the Trailer*Binary Switch

• Scale this: Max Time to Fill Trailer * Rate Trailer Fills Cubic Capacity of Trailer

Binary Switch (0 or 1)

- Service Failure (fraction of trailer unfilled)


Sequential Optimization

• First Objective:– Minimize the sum of the Service Failures– Could weight Service Failures

• Second Objective:– Minimize Cost– s.t. Service Failures <= Best Possible

The Whole Storyvar UseEdge{DCS, POOLS} binary;var CubeOut{DCS, POOLS} binary;var ServFail{POOLS} >= 0;


CubicFt[prd]/DaysPerYear*Ship[prd, dc, pool] >= Cubic Capacity of the Trailer*(UseEdge[dc, pool]+ CubeOut[dc,pool]-1)

- Cubic Capacity of the Trailer*ServFail[pool];


Weight[prd]/DaysPerYear*Ship[prd, dc, pool] >= Weight Limit of the Trailer*(UseEdge[dc, pool]- CubeOut[dc,pool]) - Weight Limit of the Trailer*ServFail[pool]

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Modeling Service