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Evaluation of case-pack sizes in grocery retailing using a Markov Chain approach
Department of OperationsCatholic University Eichstätt-IngolstadtAuf der Schanz 49, 85049 Ingolstadt, Germany
Heinrich Kuhn
Thomas Wensing
Michael Sternbeck
SMMSO 2015, Volos, Greece
Optimizing case-pack sizes in grocery retailing using a Markov Chain approach
Department of OperationsCatholic University Eichstätt-IngolstadtAuf der Schanz 49, 85049 Ingolstadt, Germany
Heinrich Kuhn
Thomas Wensing
Michael Sternbeck
SMMSO 2015, Volos, Greece
Agenda
1. INTRODUCTION
2. MODEL DEVELOPMENT
3. OPTIMIZATION APPROACH
4. NUMERICAL STUDY
5. CONCLUSIONS AND FUTURE RESEARCH
Agenda
1. INTRODUCTION
2. MODEL DEVELOPMENT
3. OPTIMIZATION APPROACH
4. NUMERICAL STUDY
5. CONCLUSIONS AND FUTURE RESEARCH
Logistics costs distribution in retail chains
– 5 –
Warehousing
28%
(secondary) Transportation
24%
In-store Logistics
48%
1. INTRODUCTION
see Kuhn/Sternbeck (2013, OMR)
ConsumerDistribution center Transportation Store
Retail system
Case-Pack Size
– 6 –
Case-pack size of a productProducts delivered to stores are generally combined in case packs that are used as order and distribution unit
Case-pack sizes influence in-store processes
How to dimension the Case Pack (CP) from an in-store operations perspective?
Store
1. INTRODUCTION
DC
Instore Activities
– 8 –
SKU 1 SKU 2
Shelf on sales floor
Product delivery
Opening case pack
Filling the shelf
Storing overflow inventory
1. INTRODUCTION
SKU 1 SKU 2
Backroom store
Case pack
Single unit
Refilling when shelf space becomes available
Introductory Example
– 9 –
Status quoq = 12
Simulated scenarioq = 6
Number of units that have to be stored in the backroom was reduced substantially by 79.5% ̶ total instore logistics costs for this SKU were reduced by almost 48 %
1. INTRODUCTION
see Sternbeck (2014, JBE)
Shelf Size = 14 Shelf Size = 14
Two main research questions arise:
(A) How does case-pack size (q) influence
instore logistics costs ?
(B) What is the optimum
case-pack size (qopt) ?
Assuming an (r,s,nq) inventory policy
… and a given replenishment /delivery pattern
– 10 –
1. INTRODUCTION
1
1,5
2
2,5
3
3,5
4
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Costs
Case-pack size, q
qopt
Pattern Monday Tuesday Wednesday Thursday Friday Saturday1 X X X X X X2 X X X X X ---3 X --- X --- X X3 X --- X --- X ---4 X --- --- X --- ---5 --- --- X --- --- ---
0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14
Inventory
Time
(r,s,nq) policy
q 2q
r
s
Litrature
Relevance of Case-Pack Size in Retail Trade
van der Laan, Joost W., Optimal Case-pack Quantity of FMCG products, Retail Economics, September 2011, http://retaileconomics.com/case-pack-quantity/
Kuhn, H., Sternbeck M.G., Integrative retail logistics: An exploratory study, Operations Management Research, Vol. 6, No. 1, 2013, pp. 2-18.
(r,s,nq) Inventory Policy
Tempelmeier, H., Fischer, L., Approximation of the probability distribution of the customer waiting time under an (r, s, q) inventory policy in discrete time. IJPR 48 (21), 2010, 6275–6291.
Shang, K. H., Zhou, S. X., Optimal and heuristic echelon (r, nq, t) policies in serial inventory systems with fixed costs. Operations Research 58 (2), 2010, pp. 414–427.
Zheng, Y.-S., Chen, F., Inventory policies with quantized ordering. Naval Research Logistics Quarterly 39, 1992, pp. 285–305.
Optimum Case-Pack Size Models
Sternbeck, M.G., A store-oriented approach to determine order packaging quantities in grocery retailing, forthcoming in: Journal of Business Economics, 2015
Wen, N., Graves, St.C., Ren, Z.J., Ship-pack optimization in a two-echelon distribution system, European Journal of Operational Research, Vol. 220, No. 3, 2012, pp. 777-785
1. INTRODUCTION
– 11 –
Agenda
1. INTRODUCTION
2. MODEL DEVELOPMENT
3. OPTIMIZATION APPROACH
4. NUMERICAL STUDY
5. CONCLUSIONS AND FUTURE RESEARCH
Inventory Model – Control
– 13 –
Shelf Size 16Display Stock 8
q 5
l 3
r {Mo, We, Sa}
s (14,-,12,-,-,10)
2. MODEL DEVELOPMENT
demand
supply
• Stochastic demand
• Length of period = one day
• Lost sales
• Deterministic lead time (l)
• Replenishment policy:Generalization of the (r, s, nq) inventory policy:
- On certain days of the week (e.g., Mo, We, Sa)
- check the inventory position (IP) and place an order
- if IP is equal to or lower than the reorder level (s) for the particular weekday.
s141210
Inventory Model – Cost Drivers
– 14 –
Shelf Size: 16Display Stock: 8
q 5
l 3
r {Mo, We, Sa}
s (14,-,12,-,-,10)
What are the crucial instore logistics cost drivers ?
1) Physical Inventory
2) Display Stock Undershoot
3) Backroom Activity
4) Backroom Inventory
5) Case Pack Handling
2. MODEL DEVELOPMENT
demand
supply
(1)
(2)
(3)
(5)
(3,4)
Inventory Model – Cost Drivers
– 15 –
Shelf Size: 16Display Stock: 8
q 5
l 3
r {Mo, We, Sa}
s (14,-,12,-,-,10)
How do cost drivers develop with increasing case-pack size (q)
1) Physical Inventory ↑
2) Display Stock Undershoot ↓
3) Backroom Activity ↑ ↓
4) Backroom Inventory ↑
5) Case Pack Handling ↓
2. MODEL DEVELOPMENT
demand
supply
(1)
(2)
(3)
(5)
(3,4)
0
10
20
30
40
50
60
70
80
90
100
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
%
Case-pack size, q
Physical inventory
Backstore inventory
Under-shooting DS
Case-pack handling
Backstore activity
Inventory Model – Cost Function
– 16 –
1) Physical Inventory : I
2) Display Stock (DS) Undershoot (U )U := MAX(0, DS – I )
3) Backstore Activity (B1) B1 := 1 if B1>0 and A*>0
B1 := 0 otherwise
4) Backstore Inventory (B2)B2 := MAX(0, I – SP**)
5) Case Pack Handling (H) H := A/q
* New Arrivals (A)** Shelf Size (SP)
2. MODEL DEVELOPMENT
– 18 –
w_x, i,
n or 0
w_y, max(0,i–
d), n and 0, respectively
P(d=D) Case 1:
w_y is neither order issue nor
order arrival period
w_x, i, 0
w_y, max(0,i–
d), n*
P(d=D) Case 2: w_y is order issue period
n* = min(n|n*q ≥s-max(0, i – d))
w_x, i, n
w_y,i*,0
P(d=D) Case 3: w_y is order arrival period
i*= max(0, i – d+n*q)
Inventory Model – States and Transitions
Note: For a clearer presentation, we assume that only one order can be outstanding and order issue periods cannot be equal to arrival periods.
2. MODEL DEVELOPMENT
w : Mo, Tu, We, Th, Fr, Sa
sw = ( 14, - , 12, - , - , 10)Day
Open order of size n*q
Phys. inventory
Analytical Model - Approach
• Markov Chain– State defined by
• weekday (w)
• inventory level (I)• arriving or outstanding
case packs (n)
– Transitions follow from deterministic arrivals and stochastic demands
– 19 –
Solid arrows refer to transition probability 0.5, dotted arrows to certain transition.
q 2
l 1
r (-,x,-)
s (-,1,-)
D 0 or 1, equally distributed
2. MODEL DEVELOPMENT
Agenda
1. INTRODUCTION
2. MODEL DEVELOPMENT
3. OPTIMIZATION APPROACH
4. NUMERICAL STUDY
5. CONCLUSIONS AND FUTURE RESEARCH
0
10
20
30
40
50
60
70
80
90
100
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
%
Case-pack size, q
Physical inventory
Backstore inventory
Under-shooting DS
Case-pack handling
Backstore activity
Optimization Approach
– 22 –
3. Optimization Approach
In-store logistic costs ↑↓ when case pack size q
increase
1) Physical Inventory ↑
2) Display Stock Undershoot ↓
3) Backroom Activity ↑ ↓
4) Backroom Inventory ↑
5) Case Pack Handling ↓
Declining cost function (dec)
Total Cost Function
Increasing cost function (inc)
Optimization Procedure
Optimization Approach
– 23 –
0
0,1
0,2
0,3
0,4
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
final CUB
qmin qmax
C(q)
Cdec(q)
Cinc(q)
CP size, qqstartqopt
P3
Costs
P1
P4P2
First step-
Iteration left
Second step-
Iteration right
initial CUB
3. Optimization Approach
Agenda
1. INTRODUCTION
2. MODEL DEVELOPMENT
3. OPTIMIZATION APPROACH
4. NUMERICAL STUDY
5. CONCLUSIONS AND FUTURE RESEARCH
Numerical Study – Cases
4. Numerical Study
Stores (A, B, … , H)Products (1, 2, … , 6)
Changes of
current CP sizes
when
optimal CP sizes
are applied-7
23
-3
-19
1
-11
1
33
-2
-14
32
-10
29
-2
-6
6
-6 -6
-2
-13
9
-9 -8
-4
-9
4
-2
27
-7
29
-2
-14
-2
-16
-8
30
-1
-7
6
-3-6
15
-2
-8
9
-21
Product ID
65
21
4321
HGFEDCBA
Product ID Product category Sales category Sales (volume)Current case-
pack size
1 Pet food low sales 322 Personal care medium sales 123 Pet food high sales 324 Personal care very low sales 65 Personal care very high sales 166 Food high sales 12
Store Store size Store size (m2) Store salesA small medium salesB medium low salesC large medium salesD medium high salesE medium medium salesF medium very high salesG medium medium salesH medium high sales
Numerical Study – Product 1, Store H
– 26 –
4. Numerical Study
0
1
2
3
4
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Curr. unit
Case-pack size, q
Physical inventory (I)
Backroom inventory (B)
Undershooting DS (U)
Case-pack handling (H)
Backroom activity (A)
Total
Cost functions
Numerical Study – Product 1, Store H
– 27 –
q = 32
qopt = 11
4. Numerical Study
Shelf Size = 23
Shelf Size = 23
32
12
32
6
1612
1613
19
4
38
5
Product ID
Case
–pa
ck si
ze
21 43 65
Optimal case-pack size (one-size-for-all-stores) Current case-pack size
-23.3%
-3.2%
-20.7%
Total costs optimal case-pack
sizes tailored to each store
Total costs one case-pack
size for all stores
Total costs status quo
Numerical Study – One-Size-for-all-Stores
– 28 –
4. Numerical Study
Agenda
1. INTRODUCTION
2. MODEL DEVELOPMENT
3. OPTIMIZATION APPROACH
4. NUMERICAL STUDY
5. CONCLUSIONS AND FUTURE RESEARCH
Contribution, current status and further research
Optimizing case pack size Areas for further research
▪ The presented model fills a gap in literature and opens the opportunity to evaluate case-pack-sizes in respect to retail in-store logistics costs.
▪ Relevant in-store cost drivers are considered, i.e., physical inventory, display stock undershoot, backroom inventory, backroom activity, case pack handling.
▪ Numerical analyses of a real case study show that case-pack sizes should be reduced for some products as well as increased for other products.
5. CONCLUSIONS AND FUTURE RESEARCH
▪ Integration of unpacking and/or packing costs in the DC
▪ Integration of case pack size related picking costs in the DC
▪ Synchronization of case pack sizes across stores
▪ Agreements with fast moving consumer goods companies in respect of “optimal” case-pack sizes
– 30 –
…many thanks for your attention
– 32 –