multi-retailer stochastic inventory ... - roberto rossi · multi-retailer stochastic inventory...

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Multi-retailer stochastic inventory control withservice level constraints and routing decisions

Dr. Maurizio Tomasella1

Dr. Roberto Rossi1

Prof. S. Armagan Tarim2

1The University of Edinburgh Business School, The University of Edinburgh, UK{maurizio.tomasella,roberto.rossi}@ed.ac.uk

2Department of Management, Hacettepe University, [email protected]

Rome, Italy, 3 July 2013

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Problem definition

I discrete finite timehorizon

I multiple retailersI single production facility

I infinite productioncapacity

I one (or more) vehicle(s)I infinite loading capacity

I single product itemI non-stationary

stochastic demand

I back orders if stock out

I negative orders notallowed

I zero delivery lead times

....

depot

vehicle(s)

retailer

retailer

retailer

I fixed transportation costb/w any nodes

I fixed ordering cost

I unit ordering cost

I unit holding cost

I service level constraint

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Problem definition (cont’d)

I ObjectiveI To minimise the expected total costs incurred - these include

I vehicle routing costsI replenishment costsI inventory costs

I Main IngredientsI a vehicle routing component, i.e. essentially

I a Travelling Salesman Problem (if 1 vehicle)I a Clover Leaf Problem (if more than 1vehicle)

I a Stochastic Lot Sizing component - at each retailer

I Inventory Control PoliciesI ref. J.H. Bookbinder and J.Y. Tan, Management Science, 34,

1096-1108, 1988I Static Uncertainty (SU)I Dynamic UncertaintyI Static-Dynamic Uncertainty (RS)

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Problem definition (cont’d)

I ObjectiveI To minimise the expected total costs incurred - these include

I vehicle routing costsI replenishment costsI inventory costs

I Main IngredientsI a vehicle routing component, i.e. essentially

I a Travelling Salesman Problem (if 1 vehicle)I a Clover Leaf Problem (if more than 1vehicle)

I a Stochastic Lot Sizing component - at each retailer

I Inventory Control PoliciesI ref. J.H. Bookbinder and J.Y. Tan, Management Science, 34,

1096-1108, 1988I Static Uncertainty (SU)I Dynamic UncertaintyI Static-Dynamic Uncertainty (RS)

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Literature

Deterministic

Authors Year Demand Routing Lot-sizing TechniqueStatic Dynamic TSP VRP IRP EOQ WW

Anily & Federgruen 1990,1994 X X X??? ??? X X X??? ??? X X X??? ??? X X X

Stochastic

Authors Year Demand Routing Lot-sizing TechniqueStationary Non stationary TSP VRP IRP SU SDU DU

Adelman 2004 X X X MILPYu et al. 2012 X X X MDP

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Notation

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Numerical study

I network notationI depot: node 1I retailers: nodes 2,. . . ,6

I planning horizonI 12 periods

I demandI normally distributedI σ/µ = 0.2

I initial stockI zero everywhere

I service levelI α = 0.95

I other parametersI v = 0I h = 1

....

depot

vehicle(s)

retailer

retailer

retailer

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Problem data - mean values of demand (µ)

PeriodRetailer 1 2 3 4 5 6 7 8 9 10 11 12

2 80 10 20 90 80 15 85 30 80 10 20 903 100 10 20 90 80 15 15 30 100 10 20 904 80 100 20 120 80 135 65 30 100 10 20 1905 80 10 20 90 230 15 65 30 100 10 20 906 80 10 20 90 80 15 85 30 100 10 20 190

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Problem data - fixed transportation costs

Intance 1

20

50

70

90

20

20

45

15

60

70

50

45

20

40

70

70

15

20

40

60

90

60

40

40

120

20

70

70

60

120

1

2

3

4

5

6

Instance 2

100

200

70

190

120

100

150

150

60

70

200

150

200

400

170

70

150

200

40

160

190

60

400

40

120

120

70

170

160

120

1

2

3

4

5

6

9

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RS: Instance 1

20

20

4015

40

70

1

2 3

4 5

6

20

20

15

50

40

40

201

2

3

4

5

6

2020 1

23 4 5

6

50

20

40

15

40

1

2

3

4

56

20 70

60

1

23

4

5

6

Re

tail

er

2R

eta

ile

r 3

Re

tail

er

4

Period 1 Period 4 Period 7 Period 9 Period 12

Replenishments


Replenishments

Re

tail

er

5R

eta

ile

r 6

Period 1

Period 4

Period 7

Period 9

Depot: node 1

Retailers: nodes 2..6

Period 12

1 2 3 4 5 6 7 8 9 10 11 12Period0

100

200

300

400Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

100

200

300

400

500

Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

100

200

300

400

500

600Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300Inventory level

E[TRC]: 15857

Solution time: 20s (R)

758s

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SU: Instance 1

Re

tail

er

2R

eta

ile

r 3

Re

tail

er

4


Replenishments

1 2 3 4 5 6 7 8 9 10 11 12Period0

100

200

300

400Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

100

200

300

400

500

Inventory level


Replenishments

Re

tail

er

5R

eta

ile

r 6

1 2 3 4 5 6 7 8 9 10 11 12Period0

100

200

300

400

500

600Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300Inventory level

20

20

4015

40

70

1

2 3

4 5

6

20

20

15

50

40

40

201

2

3

4

5

6

2020 1

23 4 5

6

Period 1

Period 4

Period 7

Period 9

Depot: node 1


2015

5040

40

1

2

3

4

56

20 70

60

1

23

4

5

6

Period 12

E[TRC]: 16836

Solution time: 10.78s

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RS: Instance 2

Re

tail

er

2R

eta

ile

r 3

Re

tail

er

4

Depot: node 1


1 2 3 4 5 6 7 8 9 10 11 12Period0

20

40

60

80

100

120

140

Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300Inventory level

Replenishments Replenishments

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300

350Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300

350Inventory level

Re

tail

er

6R

eta

ile

r 5

200

60

17070

40 70

1

2

3

4

5

6

Periods: 1, 4, 9, 12

100

6070

40

12

3

45

6

Period 2

12060

7040

70

1

23

4

5

6

Periods: 5, 7

7070 1

23

4

5 6

Period 6

E[TRC]: 6362

Solution time: 2293s (R)

850s

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SU: Instance 2

200

60

17070

40 70

1

2

3

4

5

6

70

150

170

40

60

120 1

2

3 4

5

6

Re

tail

er

2R

eta

ile

r 3

Re

tail

er

4

Depot: node 1

Retailers: nodes 2..6Replenishments Replenishments

Re

tail

er

6R

eta

ile

r 5

Periods: 1, 12

Periods: 4, 9

12060

7040

70

1

23

4

5

6

Periods: 5, 7

7070 1

23

4

5 6

Period 6

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300

350Inventory level

1 2 3 4 5 6 7 8 9 10 11 12Period0

50

100

150

200

250

300

350Inventory level

E[TRC]: 7885

Solution time: 62.59s

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Ongoing Work

I Experimental campaign being completed

I So far - RS up to 30% better than SU

I Delivery lead times being incorporated

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multi-retailer stochastic inventory ... - roberto rossi · multi-retailer stochastic inventory...

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