1. Modeling Service When Transport is restricted to Load-Driven
Pool Points in Retail Distribution
John Vande Vate
Spring 2007
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2. 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)
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3. 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
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4. 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
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5. Pool Points Asian Port Asian Factory Store US Port US DC Pool
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6. Pools Influence Trailer Fill Line Haul Delivery
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
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7. The Trade-offs Too Few Pools Too Many Pools High Delivery Costs
More moving inventory
Less waiting inventory
Too Many Pools
Low Delivery Costs
Less moving inventory
More waiting inventory
Spring 02 Vande Vate 7

8. OTD Dissected POS system records sale Transmitted to DC
Orders batched for efficient picking
Order picked
Trailer filled (Load driven)
Line Haul
Delivery
Constant
cost & time
Cost & Time depend on Pool Assignment

9. 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
Spring 02 Vande Vate 9

10. 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
Spring 02 Vande Vate 10

11. 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
Spring 02 Vande Vate 11

12. 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]*(Demand[prd,s]/DaysPerYear)*Assign[s, pool]
Reworded for clarity
Spring 02 Vande Vate 12

13. = Cubic Capacity of the Trailer
Make It Linear
Trailer Fill Time * sum{prd in PRODUCTS, s in STORES} CubicFt[prd]*(Demand[prd,s]/DaysPerYear)*Assign[s, pool]
= Cubic Capacity of the Trailer
Trailer Fill Time * Assign
What to do?
Reworded for clarity
Spring 02 Vande Vate 13

14. 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?
Spring 02 Vande Vate 14

15. 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]*(Demand[prd,s]/DaysPerYear)*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]

16. >= Cubic Capacity of the Trailer*Assign[store, pool]
Our Simple Model
var 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]*(Demand[prd,s]/DaysPerYear)*Assign[s, pool]
>= Cubic Capacity of the Trailer*Assign[store, pool]
Reworded for clarity
Max Time to Fill Trailer depends on the Store and the Pool:
Store Service Requirement, e.g., 3 days
Constant Order time, e.g., order processing, picking, etc.
Line haul time from DC to Pool
Delivery time from Pool to Store
Spring 02 Vande Vate 16

17. 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 Pool is open…
Ensure we get there in reasonable time (same time from there to all stores)
May have to handle a few special stores separately
Spring 02 Vande Vate 17

18. >= Cubic Capacity of the Trailer*Assign[store, pool]
A Simpler Model
var Assign{STORES, POOLS} binary;
Service Constraint for each pool:
Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES} CubicFt[prd]*(Demand[prd,s]/DayPerYear)*Assign[s, pool]
>= Cubic Capacity of the Trailer*Assign[store, pool]
Moved
var Open{POOLS} binary;
Reworded for clarity
Assign[store, pool]
Open[pool]
Max Time to Fill Trailer depends on the Store and the Pool:
Store Service Requirement, e.g., 3 days
Constant Order time, e.g., order processing, picking, etc.
Line haul time from DC to Pool
Delivery time from Pool to Store (depends on pool)
Spring 02 Vande Vate 18

19. >= Cubic Capacity of the Trailer*Assign[store, pool]
One DC Cube Only Model
var Assign{STORES, POOLS} binary;
Service Constraint for each pool:
Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES} CubicFt[prd]*(Demand[prd,s]/DaysPerYear)*Assign[s, pool]
>= Cubic Capacity of the Trailer*Assign[store, pool]
Moved
var Open{POOLS} binary;
Reworded for clarity
Max Time to Fill Trailer depends on the Pool:
Store Service Requirement, e.g., 3 days
Constant Order time, e.g., order processing, picking, etc.
Line haul time from DC to Pool
Delivery time from Pool to Store (depends on pool)
Spring 02 Vande Vate 19

20. >= Cubic Capacity of the Trailer*Open[pool]
More than One DC
Service Constraint for each pool:
Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES} CubicFt[prd]*(Demand[prd,s]/DaysPerYear)*Assign[s, pool]
>= Cubic Capacity of the Trailer*Open[pool]
What’s wrong with this?
Reworded for clarity
Spring 02 Vande Vate 20

21. 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.
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22. 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]
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23. What’s Wrong Now Service Constraint for each dc and pool:
Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}
CubicFt[prd]*(Ship[prd, dc, pool]/DaysPerYear)
>= Cubic Capacity of the Trailer*Open[pool]
Corrected
Reworded for clarity
Spring 02 Vande Vate 23

24. The Problem Does the DC serve the Pool?
Service Constraint for each dc and pool:
Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}
CubicFt[prd]*Ship[prd, dc, pool]/DaysPerYear
>= 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.
Corrected
Reworded for clarity
Spring 02 Vande Vate 24

25. 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]*Ship[prd, dc, pool] /DaysPerYear
>= 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]
Corrected
Reworded for clarity
Spring 02 Vande Vate 25

26. 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}
CubicFt[prd]*Ship[prd, dc, pool]/DaysPerYear
>= 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]
Corrected
Reworded for clarity
Spring 02 Vande Vate 26

27. 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}
CubicFt[prd]*Ship[prd, dc, pool]/DaysPerYear
>= Cubic Capacity of the Trailer*UseEdge[dc, pool]
Weight Service Constraint for each dc and pool:
Weight[prd]*Ship[prd, dc, pool]/DaysPerYear
>= Weight Limit of the Trailer*UseEdge[dc, pool]
Corrected
Reworded for clarity
Corrected
Reworded for clarity
Spring 02 Vande Vate 27

28. What’s Wrong Now? 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}
CubicFt[prd]*Ship[prd, dc, pool]/DaysPerYear
>= Cubic Capacity of the Trailer*UseEdge[dc, pool]
Weight Service Constraint for each dc and pool:
Weight[prd]*Ship[prd, dc, pool]/DaysPerYear
>= Weight Limit of the Trailer*UseEdge[dc, pool]
Corrected
Reworded for clarity
Corrected
Reworded for clarity
Spring 02 Vande Vate 28

29. Weight AND Cube Trailer departs when Don’t have to fill BOTH
Weight Limit is reached OR
Cubic Capacity is reached
Don’t have to fill BOTH
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30. 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)
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31. 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])
Spring 02 Vande Vate 31

32. A Full Model var UseEdge{DCS, POOLS} binary;
var CubeOut{DCS, POOLS} binary;
Cube Service Constraint for each dc and pool:
Max Time to Fill Trailer * sum{prd in PRODUCTS, s in STORES}
CubicFt[prd]*Ship[prd, dc, pool]/DaysPerYear
>= Cubic Capacity of the Trailer*(UseEdge[dc, pool]+ CubeOut[dc,pool]-1)
Weight Service Constraint for each dc and pool:
Weight[prd]*Ship[prd, dc, pool]/DaysPerYear
>= Weight Limit of the Trailer*(UseEdge[dc, pool]- CubeOut[dc,pool])

33. Added Detail The rest of the model
var Assign{STORES, POOLS} binary;
Meet Demand at each pool for each prd:
Sum{dc in DCS}Ship[prd, dc, pool] >=
Sum{s in STORES} Demand[prd,s]*Assign[s, 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]
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34. 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])

35. 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
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36. 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?
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37. 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)

38. 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
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39. The Whole Story var UseEdge{DCS, POOLS} binary;
var CubeOut{DCS, POOLS} binary;
var ServFail{POOLS} >= 0;
Cube 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]+ CubeOut[dc,pool]-1)
- Cubic Capacity of the Trailer*ServFail[pool];
Weight Service Constraint for each dc and 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]

40. Added Detail The rest of the model
var Assign{STORES, POOLS} binary;
Meet Demand at each pool for each prd:
Sum{dc in DCS}Ship[prd, dc, pool] >=
Sum{s in STORES} Demand[prd,s]*Assign[s, 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]
Spring 02 Vande Vate 40