multi-retailer stochastic inventory ... - roberto rossi · multi-retailer stochastic inventory...
TRANSCRIPT
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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
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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
Period 1 Period 4 Period 7 Period 9 Period 12
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
Period 1 Period 4 Period 7 Period 9 Period 12
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
Period 1 Period 4 Period 7 Period 9 Period 12
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
Retailers: nodes 2..6
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
Retailers: nodes 2..6
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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