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Layout Optimization of a Three Dimensional Order Picking Warehouse
Proceedigs of the 15th IFAC Symposium on May 11-13, 2015. Canada Proceedigs of theOttawa, 15th IFAC Symposium on Information Control Problems in Manufacturing Information Control Problems in Manufacturing Available online at www.sciencedirect.com May 11-13, 2015. Ottawa, Canada May 11-13, 2015. Ottawa, Canada
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Layout Optimization of a Three Dimensional Order Picking Warehouse Layout Optimization of a Three Dimensional Order Picking Warehouse Layout Optimization of a Three Dimensional Order Picking Warehouse Venkitasubramony Rakesh and Gajendra. K Adil* Venkitasubramony Rakesh and Gajendra. K Adil* Venkitasubramony Rakesh and Gajendra. K Adil* Shailesh J Mehta School of Management, Indian Institute of Technology Bombay, Mumbai, India 400076 Shailesh J Mehta School of Management, Indian Institute of Technology Bombay, Shailesh J Mehta School of Management, Indian Institute of Technology Bombay, *Tel: (+91-22) 25767738; e-mail: [email protected] Mumbai, India 400076 Mumbai, India 400076 *Tel: (+91-22) 25767738; e-mail: [email protected] *Tel: (+91-22) 25767738; e-mail: [email protected] Abstract: The warehouse layout decision is important as it affects several aspects of warehouse performance as material handling cost, isspace cost andasstorage capacity. Weaspects developofanwarehouse algorithm Abstract: Thesuch warehouse layout decision important it affects several Abstract: Thesuch warehouse decision isspace important itdepth affects ofanwarehouse that determines lane depth,layout number of cost, storage levels, andseveral longitudinal width of a three performance as material handling cost lateral andasstorage capacity. Weaspects develop algorithm performance suchlane as material handling cost, space cost lateral and Wecosts. develop an algorithm dimensional order picking space andstorage material handling The algorithm is that determines depth,warehouse number ofminimizing storage levels, depthcapacity. and longitudinal width of a three that determines lane depth,warehouse numberofofvariation storage of levels, depth and longitudinal widthalgorithm of a three illustrated with an example. Effect aspace set lateral ofand parameters on the optimal decision is dimensional order picking minimizing material handling costs. layout The dimensional order picking warehouse minimizing material on handling costs. layout The algorithm is studied. illustrated with an example. Effect of variation of aspace set ofand parameters the optimal decision is illustrated with an example. Effect of variation of a set of parameters on the optimal layout decision is studied. warehouse design, order picking, warehouse layout Keywords: © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. studied. Keywords: warehouse design, order picking, warehouse layout Keywords: warehouse design, order picking, warehouse layout costs in designing a three dimensional warehouse and 1. INTRODUCTION illustrates how to compare alternatives using and the costs in designing a three design dimensional warehouse 1. INTRODUCTION costs inYoon designing a three dimensional warehouse and model. and Sharp (1996) propose a stage by stage Warehouse is an important component in the supply chain illustrates how to compare design alternatives using the 1. INTRODUCTION illustrates howand to Sharp compare design the procedure for analysis and(1996) design ofalternatives ordera picking of products. Bartholdi Hackman (2011) model. Yoon propose stageusing bysystem stage Warehouse is an importantand component in the supplydefine chain model. Yoon and Sharp (1996) propose a stage by stage Warehouse is an important component in the supply chain using orderforcharacteristics and system parameters. Caron warehouses “points in the chain (2011) where product procedure analysis and design of order picking system of products.as Bartholdi and supply Hackman define procedure forcharacteristics analysis and optimum design of number order picking system of products. and supply define et al. order (2000) relates the of aisles to pauses, however briefly, and isHackman touched”. Despite huge using and system parameters. Caron warehouses as Bartholdi “points in the chain (2011) where product using characteristics andCOI system parameters. Caron warehouses as “points the chain where product number of picks, shapethe of the (cube per order index) technology, labour andinspace costs associated with their et al. order (2000) relates optimum number of aisles to pauses, however briefly, and supply is touched”. Despite huge et al. etc. (2000) relates optimum number of analytical aisles to pauses, however and is touched”. Despite huge curve and VisCOI (2006) develop operation, warehouses still unavoidable due to reasons number of Roodbergen picks, shapethe of the (cube per order index) technology, labourbriefly, andare space costs associated with their number of picks, shape of the COI (cube per order index) technology, labour and space costs associated with their expressions for travel length for two different routing that include fluctuating demand and supply, cost curve etc. Roodbergen and Vis (2006) develop analytical operation, warehouses are still unavoidable due to reasons curve etc. and Roodbergen and Vis for (2006) develop operation, warehouses are still unavoidable due toservices. reasons heuristics usetravel the expressions to two optimize theanalytical layout of economics in bulk transportation and value expressions for length different routing that include fluctuating demand and added supply, cost expressions for travel length for two different routing that include fluctuating demand and supply, cost a warehouse. et expressions al. (2008) propose a particle swarm heuristics and Onut use the to optimize the layout of economics in bulk transportation and value added services. The warehouse layout decisionsand such as added height, width, heuristics andalgorithm use the to optimize thelevel layout of economics in bulk transportation value services. optimization designing a multi unita warehouse. Onut et expressions al.for (2008) propose a particle swarm lane aisle width, aisle such location location of The depth, warehouse layout cross decisions as and height, width, aoptimization warehouse. Onut et al. (2008) propose a particle swarm load warehouse employing based astorage. A detailed algorithm for class designing multi level unitThe depth, warehouse layout decisions such as and height, width, I/O point aisle are width, important asaisle they effect both capital lane cross location location of optimization algorithm for class designing multi level unitreview of warehouse design literature available Gu et load warehouse employing basedisastorage. A in detailed lane depth, aisle width, cross aisle location and location of investment well asboth operational I/O point (construction are importantof facility) as they aseffect capital load warehouse employing class based storage. A detailed al. (2010). review of warehouse design literature is available in Gu et I/O point are important as they effect both capital costs (space(construction costs, handling etc).as well as operational investment of costs facility) review of warehouse design literature is available in Gu et al. (2010). investment of costs facility) It can be seen from the literature review that there is no costs (space(construction costs, handling etc).as well as operational al. (2010). This a mathematical costs paper (spacepresents costs, handling costs etc).model and algorithm study optimizes length, width, of levels It can which be seen from thetheliterature reviewnumber that there is no to decide of the most important decisions in designing This paperfour presents a mathematical model and algorithm It be seen from literature reviewnumber that there is no andcanlane depth of the a the three dimensional order study which optimizes length, width, ofpicking levels This paper presents a mathematical model and algorithm ato warehouse system, namely, lane depth, number decide fourracking of the most important decisions in designing study which optimizes the length, width, number of levels warehouse. and lane depth of a three dimensional order picking decide of the most important decisions inwidth. designing of storagefour levels, lateral depth and longitudinal We ato warehouse racking system, namely, lane depth, number and lane depth of a three dimensional order picking warehouse. apresent warehouse racking system, namely, lane depth, number literature 1.1. Section 2 of storagerelevant levels, lateral depthinandsection longitudinal width. We warehouse. 2. PROBLEM STATEMENT of storagerelevant levels, lateral depth andsection longitudinal width. We elaborates the problem and in states the key 2. PROBLEM STATEMENT present literature 1.1.assumptions. Section 2 We consider 2. a PROBLEM rectangular STATEMENT warehouse with a racking present relevant literature in section 1.1. Section 2 Section 3 develops a mathematical for assumptions. the problem. elaborates the problem and states model the key system as shown in Fig 1. The picking aisle a fixed We consider a rectangular warehouse with has a racking elaborates theanproblem and states the themodel key In Section 4, algorithm to solve is presented. Section 3 develops a mathematical model for assumptions. the problem. We consider a rectangular warehouse with has a racking width to accommodate picker/vehicle movement. Each system as shown in Fig 1. The picking aisle a fixed Section 35 develops a different mathematical model for demonstrates the problem. Section discusses scenarios and In Section 4, an algorithm to solve the model is presented. system astwo shown in Fig 1. The picking aisle has adepth. fixed shelf has picking faces, each with a certain lane width to accommodate picker/vehicle movement. Each In Section 4,setanofalgorithm to on solve model is presented. effects of5 adiscusses parameters the the decisions. Conclusions Section different scenarios and demonstrates width to two accommodate picker/vehicle movement. Each A certain clearance iseachallowed for lane easedepth. of shelf has picking faces, with a certain Section 5 discusses different scenarios and demonstrates are presented 6 on the decisions. Conclusions effects of a setinofSection parameters shelf has two picking faces, each with a certain lane depth. storage/extraction. A certain clearance is allowed for ease of effects of a set of parameters on the decisions. Conclusions are presented in Section 6 A certain clearance is allowed for ease of 1.1 Literaturein Review storage/extraction. are presented Section 6 The following are key layout decisions: storage/extraction. 1.1 Literature Review Warehouse layout optimization has received considerable The following are key layout decisions: 1.1 Literature Review Lane depth oflayout each shelf The •following are key decisions: attention forhas a received few decades. Berry Warehouseamong layoutresearchers optimization considerable • Lane depth of each shelf Warehouse layout optimization has received considerable (1968) compares two warehouse block attention among researchers for a arrangements few decades. - Berry • Number of vertical Lane depth of each storage shelf levels in each shelf attention among researchers a arrangements fewrequirements decades. - Berry stacking and pallet onfor volume and (1968) compares tworacks warehouse block • Number of vertical storage levels in each shelf Warehouse lateral depth (1968) compares tworacks warehouse arrangements - block • Number of vertical storage levels in each shelf handling costs. Bassan et al.on(1980) develop an analytical stacking and pallet volume requirements and stacking and pallet racks on volume requirements and • Warehouse lateral depth model to decide the dimensions of a rectangular unit load handling costs. Bassan et al. (1980) develop an analytical • Warehouse longitudinal lateral depth width handling et al. (1980) an analytical warehouse and Bassan athezoned warehouse. Park and Webster model to costs. decide dimensions of a develop rectangular unit load • Warehouse longitudinal width model decide dimensions of a rectangular load (1989) todevelop that captures different kinds of • Warehouse longitudinal width warehouse and aathemodel zoned warehouse. Park and unit Webster warehouse and aa model zoned that warehouse. and kinds Webster (1989) develop captures Park different of (1989) develop a model that captures different kinds of
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3. MATHEMATICAL MODEL 3.1 Notations The following are the notations used in the model. a) Order Characteristics parameters
ot
Number of orders per inventory cycle
lc p
Number of order lines for entire case of product p per inventory cycle
uc p
Average
number
of
units
of
cases
of
product p demanded per line
li p
Number of order lines for individual items of product p per inventory cycle
ui p
Average
number
of
units
of
items
of
product p demanded per line
Np
Average number of picks per order
vc p
Volume of a case of product p (cu. ft)
vi p
Volume of an item of product p (cu. ft)
F
Maximum planned inventory volume (cu ft)
b) Warehouse attributes parameters
Fig.1 Top and side view of a three dimensional warehouse racking system The major assumptions in building the model are as follows
h
Height of each of storage level(ft)
α Pd
Clearance allowance, i.e. fraction of total volume to be left empty for ease of storage/retrieval Depth of a pallet (ft)
Aw
Aisle width mandated by design requirements (ft)
Vhor
Horizontal velocity of picker (ft/hr)
Vver
Vertical velocity of picker (ft/hr)
e
Extraction time factor, i.e. average time required to extract a product per feet of lane depth (hr/ft)
Cl
Cost of labour per hour (Rs/hr)
2.1 Assumptions a)
The warehouse employs a random storage policy.
Ca
b) The inventory levels are known as decisions of order quantity and reorder point are made a priori. c)
Manual order picking is employed in the warehouse using ‘S’ shaped/ traversal routing.
c) Decision Variables (ref to Fig 1)
d) The loading/unloading dock (I/O point) is located at the centre of the front wall of the warehouse and the shelves that stock items are perpendicular to the front wall, as shown in Fig 1. e)
The extraction time varies linearly with the lane depth.
f)
The extraction cost varies linearly with the number of units of a product picked.
Area cost in the form of rent or otherwise (Rs/sq ft).
N
Number of storage levels in the racking system
M
Number of picking aisles in the warehouse Lane depth (number of pallets)
Ld
y x L
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Lateral depth of the racking system (ft) Longitudinal width of the racking system (ft) Total aisle length measured along facings (ft) (L= M ×y)
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Ar
Aspect ratio (i.e. ratio of lateral depth, y , to longitudinal width, x )
x=
(8)
The average number of picks per order, N p , can be
(1)
calculated as
(2)
Np =
Volume of product that can be stocked in the racks is
V = L × ( 2 Ld × Pd ) × N × h (1 − α )
(7)
y = A × Ar
Using the above notation, the maximum planned inventory volume F (in cubic feet) can be calculated as
F = ( lc p × uc p × vc p ) + ( li p × ui p × vi p )
A Ar
1157
∑ ( lc
p
+ li p )
p
ot
(9)
The number of aisles, M, can be calculated as
3.3 Cost Components
M=
The decisions mentioned in the previous section have bearing on different kinds of costs for which we derive the expressions next.
M −1 N p N p −1 HT = 2 x + yM 1 − + 0.5 y N + 1 M p
The footprint area, A can be expressed as (3)
(11)
And the associated area cost can be calculated as
AC = Ca × L × ( ( 2 × Ld × Pd ) + Aw )
(10)
The expected horizontal distance traversed per order can be expressed according to Hall (1993) as
3.3.1 Area Cost (AC)
A = x × y = L × ( ( 2 × Ld × Pd ) + Aw )
L y
Thus the total horizontal travel cost is
HTC =
(4)
HT × ot × Cl VHor
(12)
3.3.2 Vertical Travel Cost (VTC) As random storage policy is assumed, the two way expected distance traversed by the picker in the vertical direction is N × h . This travel should be undertaken for each line item encountered. Thus the total vertical travel cost can be calculated as
N ×h VTC = Cl × × ∑ ( lc p + li p ) Vver p
3.3 Model The mathematical model can be formulated as follows. Minimize Total Cost,
C ( x, y, N, L d ) = AC + EC + VTC + HTC
Subject to
(5)
L × ( 2 Ld × Pd ) × N × h (1 − α ) = F
3.3.3 Extraction Cost (EC)
1 ≤ Ld ≤ Ld max 1 ≤ N ≤ N max y Ar min ≤ ≤ Ar max x
The extraction cost can be calculated as a product of extraction time factor, lane depth, total number of extractions and the cost of labour per unit time
EC = Cl × (e × Ld × Pd ) × ∑ ( lc p × uc p + li p × ui p ) p
and equations (3), (7), (8) &(10) x, y, L, M >0 N , Ld > 0 and integer
(6) 3.3. 4 Horizontal Travel Cost (HTC) We define aspect ratio Ar =
(13)
(14) (15) (16) (17) (18) (19)
The objective of layout design is to minimize the sum of all the costs given by (13). Constraint (14) makes sure that there is adequate space available to store all products. In addition, bounds are placed on lane depth, number of levels and aspect ratio represented by constraints (15), (16) and (17) respectively. Equations (3), (7) & (8) relate total area to lateral depth and longitudinal width. Equation (10) relates number of aisles to total storage length and lateral
y . The longitudinal width x
( x ) and lateral depth ( y ) of the warehouse can be expressed as a function of Aspect Ratio ( Ar ) and Area ( A ) as follows
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depth. Constraints (18) and (19) ensure non-negativity and integrality of variables.
Product Groups Number of lines per shift for cases Average case pick units Number of lines per shift that require item pick Average item pick units Average Case Volume Average Item volume Total number of customer orders per shift Height of a layer Clearance percentage Pallet depth (ft) Aisle Width (ft) Horizontal Velocity of picker (ft/hr) Vertical Velocity of picker (ft/hr) Cost of labor per unit time (Rs / hr) Extraction Constant (Time/unit lane depth) Rent per unit area (Rs/sqft)
4. SOLUTION APPROACH The solution to the above problem involves a step by step approach as shown in Exhibit 1. Step 1: Obtain values for order characteristics and warehouse attributes parameters Step 2: Calculate the product flow through the warehouse as shown in (1) Step 3: Vary Ld from 1 to Ld max Step 3.1: Vary
N from 1 to N max
Step 3.1.1: Calculate L using (14) Step 3.1.2 : Vary Ar from Ar min to
Ar max
•
Calculate
•
and (8), (9) and (10) respectively Calculate the total cost using (13)
Step-3.1.3
x , y , N p and M as per (7)
Get
the
minimum
* r
cost
* r
C ( A , N , Ld ) at the optimal A *
*
*
of A
*
N , and lane depth Ld
*
*
calculate A = 202991.25 sq ft,
7 400
5 300
4 27 2.7 400
5 8 0.8
10 125 1.3
3 33.30% 2 7 300 100 600 0.007 3
x = 1424.75 ft, y =
142.475 ft and N p = 4 , M = 129.5. The total Vertical travel cost is Rs 28,800, Extraction cost is Rs 82,320, Horizontal travel cost is Rs 1,876,522 and Area cost is Rs 608,973. Thus the total cost of operation turns out to be Rs 2,596,616.
4.1 Illustrative Example Consider a sample case with values of parameters shown in Table 1. Further, the aspect ratio is considered to be within the range of 0.1 and 2 since longer warehouses are better when number of picks is larger as per Hall (1993). In addition, upper bounds are specified for lane depth ( Ld max =5) and number of levels ( N = 20), to account for
In Step 3.1.2, the aspect ratio is varied from 0.1 to 2 and the cost is calculated at each value. Fig 2. shows the costs obtained for different Aspect ratios. We find that the minimum cost is Rs 2,384,327 is obtained at an Aspect Ratio of 0.3 for Ld = 1 , N = 1 .
physical limitations of the warehouse racking system. The procedure for obtaining ideal layout can be worked out as follows. Obtaining all parameters would constitute Step 1.
In Step 3.2, we obtain the minimum cost as shown above for different values of N . The results obtained are shown in Fig. 3. It is seen that the minimum cost is obtained at N = 11 with a cost of Rs 945,555. From the data obtained in previous step, the Aspect ratio for achieving this cost is found to be 0.3 for Ld = 1
As per Step 2, Total product volume, F is (500x5) + (200x7) + (100x5) + (100x4) + (400x5) + (300x10) = 147630
As per Step 4, the exercise is repeated for different values of Lane depth, ranging from 2 pallets to 5 pallets. The results are shown in Table 2. It is observed that for smaller lane depths, taller shelves perform better and for larger lane depths, shorter shelves perform better. The reason is, at a larger lane depth, more compaction is achieved per level and hence the optimal number of levels would decrease.
Ld = 1 , N = 1 (Step 3.1). L can
be calculated in Step 3.1.1 as
L=
5 100
For Ar = 0.1 , Using (3), (7), (8), (9) and (10), we can
*
Exhibit 1. Solution algorithm
We illustrate Step 3 for
3 100
N* .
Step 4 Get minimum cost C ( Ar , N , Ld ) at values * r ,
2 200
Table 1: Values of order characteristics and warehouse parameters for the example case
*
Step 3.2 Get minimum cost C ( Ar , N , Ld ) at optimal values of Ar and
1 500
F =18453.75 ft 2 Ld × N × h (1 − α )
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there may be multiple options for lane depth with almost similar cost. 5. EFFECT OF CHANGING PARAMETERS The layout decision is affected by changes in parameters. In this section, three such effects are illustrated. 5.1 Effect of area cost and extraction time constant on optimal lane depth The lane depth decision as shown in the illustration in section 4.1 is arrived at considering an extraction time constant of 0.007 hr/ft and area cost of Rs 3/sq ft. Consider a scenario where the extraction constant is higher, say 0.02 hr/ft and the area cost is lower, say Rs 1/sq ft. The effect is shown in Table 3. Here, smaller lane depths clearly outperform the larger ones; the reason being the effect of higher extraction costs would offset the area cost savings involved in larger lane depths. This is different from the illustration in Table 2, where a low lane depth option involves high area cost and a high lane depth option suffers from a high extraction cost, leading to lane depth of 2 as an optimal choice.
Fig 2. Variation of total cost with Aspect ratio at N = 1
C
N*
Ar*
1
945555.7
11
0.3
2
911597.1
9
0.3
3
943936.5
9
0.4
4
995847.4
9
0.4
5
1056426
8
0.4
C
N*
Ar*
1
1059276
10
0.3
2
1186601
9
0.3
3
1375332
8
0.3
4
1581275
8
0.4
5
1795914
8
0.4
Table 3. Variation of total cost across different levels and different lane depths when extraction constant is 0.02 hr/ft and area cost is Rs 1/sq ft
Fig 3. Variation of minimum costs across different N
Ld
Ld
5.2 Effect of number of stops per order on optimal aspect ratio In 4.1, the number of orders considered was 400. Consider a scenario where the same volume of product flow is spread over 100 orders. This would make the average number of picks per order. N p , four times higher, i.e. 16. The variation of total cost with Aspect ratio, for N =1 is shown in Fig 4. The minimum cost occurs at an Aspect ratio of 0.2 as opposed to 0.3 in the previous case. This is consistent with the observation of Hall (1993) that longer warehouses perform better when number of picks per order are large. In a warehouse with smaller aspect ratio, the number of aisles is larger, making it possible to skip certain aisles in the traversal routing, which will more than offset the increase in distance travelled in the longitudinal direction. Our model explicitly bounds the aspect ratio on the lower side, not allowing unreasonable shapes.
Table 2. Variation of total cost across different levels and different lane depths. It is seen that the optimal cost, C* of Rs 911,597occurs at a lane depth, Ld * = 2 pallets, number of storage levels,
N * = 9, longitudinal width, x* = 248.01 ft and lateral * depth, y = 74.40 ft. It is also observed that as the number of storage levels increases, the optimal lane depth decreases. Even at a particular number of storage levels,
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Fig 4. Variation of total cost with aspect ratio with and
N =1 Fig 5. Variation on total cost across different number of levels, when extraction time constant is high and rent per unit area is low
ot = 100
5.3 Effect of labour cost and area cost on optimal number of levels
REFERENCES
In Fig 3, it was seen that the cost falls rapidly as we increase the number of levels from 1 to around 11. Beyond 11 levels the cost increases. At lower levels, the effect of increased area cost is dominant. At higher levels the pick cost becomes dominant. Now consider a situation where the area cost is very low, at Rs 0.5/sq ft and labour cost is very high, at Rs 3000/sq ft. Fig. 5 shows the variation of cost across different levels at lane depth of 1. The optimal number of levels is 10, which is lower than in the illustration case. Also, as the number of levels is increased from the optimal the pick cost increase is more severe as the labour cost has increased. But if aisle width is increased to 15 (original value is 7), we find that the optimal number of levels shifts to 11, as an increased aisle width would increase the area cost, and thus an increased number of levels would compensate for that, thereby brining the cost back to optimal.
Bassan, Y., Roll, Y., & Rosenblatt, M. J. (1980). Internal layout design of a warehouse. AIIE Transactions, 12(4), 317-322. Bartholdi, J. J., & Hackman, S. T. (2008). Warehouse & Distribution Science: Release 0.89. Supply Chain and Logistics Institute. Berry, J. R. (1968). Elements of warehouse layout. The International Journal of Production Research, 7(2), 105121. Caron, F., Marchet, G., & Perego, A. (2000). Optimal layout in low-level picker-to-part systems. International Journal of Production Research, 38(1), 101-117. Gu, J., Goetschalckx, M., & McGinnis, L. F. (2010). Research on warehouse design and performance evaluation: A comprehensive review. European Journal of Operational Research, 203(3), 539-549.
6. CONCLUSION
Hall, R. W. (1993). Distance approximations for routing manual pickers in a warehouse. IIE transactions, 25(4), 7687.
An algorithm is presented that optimizes lane depth, number of levels, length and width of a warehouse. The effect of variation of a set of parameters on the cost is also studied. The generation of cost curves by varying different parameters would help the designer decide from a range of alternatives for design variables and not just the optimal value. It also helps in knowing the quantum of change in cost due to change in different parameters, which is generally difficult to predict due to interaction of multiple effects and trade-offs. Needless to say, it would make sense for the warehouse manager to understand how the cost changes with changes in multiple parameters. The study can be extended to optimize layout considering other storage and routing policies.
Önüt, S., Tuzkaya, U. R., & Doğaç, B. (2008). A particle swarm optimization algorithm for the multiple-level warehouse layout design problem. Computers & Industrial Engineering, 54(4), 783-799. Park, Y. H., & Webster, D. B. (1989). Modelling of threedimensional warehouse systems. the international journal of production research, 27(6), 985-1003. Roodbergen, K. J., & Vis, I. F. (2006). A model for warehouse layout. IIE transactions, 38(10), 799-811. Yoon, C. S., & Sharp, G. P. (1996). A structured procedure for analysis and design of order pick systems. IIE transactions, 28(5), 379-389.
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Layout Optimization of a Three Dimensional Order Picking Warehouse Layout Optimization of a Three Dimensional Order Picking Warehouse Layout Optimization of a Three Dimensional Order Picking Warehouse Venkitasubramony Rakesh and Gajendra. K Adil* Venkitasubramony Rakesh and Gajendra. K Adil* Venkitasubramony Rakesh and Gajendra. K Adil* Shailesh J Mehta School of Management, Indian Institute of Technology Bombay, Mumbai, India 400076 Shailesh J Mehta School of Management, Indian Institute of Technology Bombay, Shailesh J Mehta School of Management, Indian Institute of Technology Bombay, *Tel: (+91-22) 25767738; e-mail: [email protected] Mumbai, India 400076 Mumbai, India 400076 *Tel: (+91-22) 25767738; e-mail: [email protected] *Tel: (+91-22) 25767738; e-mail: [email protected] Abstract: The warehouse layout decision is important as it affects several aspects of warehouse performance as material handling cost, isspace cost andasstorage capacity. Weaspects developofanwarehouse algorithm Abstract: Thesuch warehouse layout decision important it affects several Abstract: Thesuch warehouse decision isspace important itdepth affects ofanwarehouse that determines lane depth,layout number of cost, storage levels, andseveral longitudinal width of a three performance as material handling cost lateral andasstorage capacity. Weaspects develop algorithm performance suchlane as material handling cost, space cost lateral and Wecosts. develop an algorithm dimensional order picking space andstorage material handling The algorithm is that determines depth,warehouse number ofminimizing storage levels, depthcapacity. and longitudinal width of a three that determines lane depth,warehouse numberofofvariation storage of levels, depth and longitudinal widthalgorithm of a three illustrated with an example. Effect aspace set lateral ofand parameters on the optimal decision is dimensional order picking minimizing material handling costs. layout The dimensional order picking warehouse minimizing material on handling costs. layout The algorithm is studied. illustrated with an example. Effect of variation of aspace set ofand parameters the optimal decision is illustrated with an example. Effect of variation of a set of parameters on the optimal layout decision is studied. warehouse design, order picking, warehouse layout Keywords: © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. studied. Keywords: warehouse design, order picking, warehouse layout Keywords: warehouse design, order picking, warehouse layout costs in designing a three dimensional warehouse and 1. INTRODUCTION illustrates how to compare alternatives using and the costs in designing a three design dimensional warehouse 1. INTRODUCTION costs inYoon designing a three dimensional warehouse and model. and Sharp (1996) propose a stage by stage Warehouse is an important component in the supply chain illustrates how to compare design alternatives using the 1. INTRODUCTION illustrates howand to Sharp compare design the procedure for analysis and(1996) design ofalternatives ordera picking of products. Bartholdi Hackman (2011) model. Yoon propose stageusing bysystem stage Warehouse is an importantand component in the supplydefine chain model. Yoon and Sharp (1996) propose a stage by stage Warehouse is an important component in the supply chain using orderforcharacteristics and system parameters. Caron warehouses “points in the chain (2011) where product procedure analysis and design of order picking system of products.as Bartholdi and supply Hackman define procedure forcharacteristics analysis and optimum design of number order picking system of products. and supply define et al. order (2000) relates the of aisles to pauses, however briefly, and isHackman touched”. Despite huge using and system parameters. Caron warehouses as Bartholdi “points in the chain (2011) where product using characteristics andCOI system parameters. Caron warehouses as “points the chain where product number of picks, shapethe of the (cube per order index) technology, labour andinspace costs associated with their et al. order (2000) relates optimum number of aisles to pauses, however briefly, and supply is touched”. Despite huge et al. etc. (2000) relates optimum number of analytical aisles to pauses, however and is touched”. Despite huge curve and VisCOI (2006) develop operation, warehouses still unavoidable due to reasons number of Roodbergen picks, shapethe of the (cube per order index) technology, labourbriefly, andare space costs associated with their number of picks, shape of the COI (cube per order index) technology, labour and space costs associated with their expressions for travel length for two different routing that include fluctuating demand and supply, cost curve etc. Roodbergen and Vis (2006) develop analytical operation, warehouses are still unavoidable due to reasons curve etc. and Roodbergen and Vis for (2006) develop operation, warehouses are still unavoidable due toservices. reasons heuristics usetravel the expressions to two optimize theanalytical layout of economics in bulk transportation and value expressions for length different routing that include fluctuating demand and added supply, cost expressions for travel length for two different routing that include fluctuating demand and supply, cost a warehouse. et expressions al. (2008) propose a particle swarm heuristics and Onut use the to optimize the layout of economics in bulk transportation and value added services. The warehouse layout decisionsand such as added height, width, heuristics andalgorithm use the to optimize thelevel layout of economics in bulk transportation value services. optimization designing a multi unita warehouse. Onut et expressions al.for (2008) propose a particle swarm lane aisle width, aisle such location location of The depth, warehouse layout cross decisions as and height, width, aoptimization warehouse. Onut et al. (2008) propose a particle swarm load warehouse employing based astorage. A detailed algorithm for class designing multi level unitThe depth, warehouse layout decisions such as and height, width, I/O point aisle are width, important asaisle they effect both capital lane cross location location of optimization algorithm for class designing multi level unitreview of warehouse design literature available Gu et load warehouse employing basedisastorage. A in detailed lane depth, aisle width, cross aisle location and location of investment well asboth operational I/O point (construction are importantof facility) as they aseffect capital load warehouse employing class based storage. A detailed al. (2010). review of warehouse design literature is available in Gu et I/O point are important as they effect both capital costs (space(construction costs, handling etc).as well as operational investment of costs facility) review of warehouse design literature is available in Gu et al. (2010). investment of costs facility) It can be seen from the literature review that there is no costs (space(construction costs, handling etc).as well as operational al. (2010). This a mathematical costs paper (spacepresents costs, handling costs etc).model and algorithm study optimizes length, width, of levels It can which be seen from thetheliterature reviewnumber that there is no to decide of the most important decisions in designing This paperfour presents a mathematical model and algorithm It be seen from literature reviewnumber that there is no andcanlane depth of the a the three dimensional order study which optimizes length, width, ofpicking levels This paper presents a mathematical model and algorithm ato warehouse system, namely, lane depth, number decide fourracking of the most important decisions in designing study which optimizes the length, width, number of levels warehouse. and lane depth of a three dimensional order picking decide of the most important decisions inwidth. designing of storagefour levels, lateral depth and longitudinal We ato warehouse racking system, namely, lane depth, number and lane depth of a three dimensional order picking warehouse. apresent warehouse racking system, namely, lane depth, number literature 1.1. Section 2 of storagerelevant levels, lateral depthinandsection longitudinal width. We warehouse. 2. PROBLEM STATEMENT of storagerelevant levels, lateral depth andsection longitudinal width. We elaborates the problem and in states the key 2. PROBLEM STATEMENT present literature 1.1.assumptions. Section 2 We consider 2. a PROBLEM rectangular STATEMENT warehouse with a racking present relevant literature in section 1.1. Section 2 Section 3 develops a mathematical for assumptions. the problem. elaborates the problem and states model the key system as shown in Fig 1. The picking aisle a fixed We consider a rectangular warehouse with has a racking elaborates theanproblem and states the themodel key In Section 4, algorithm to solve is presented. Section 3 develops a mathematical model for assumptions. the problem. We consider a rectangular warehouse with has a racking width to accommodate picker/vehicle movement. Each system as shown in Fig 1. The picking aisle a fixed Section 35 develops a different mathematical model for demonstrates the problem. Section discusses scenarios and In Section 4, an algorithm to solve the model is presented. system astwo shown in Fig 1. The picking aisle has adepth. fixed shelf has picking faces, each with a certain lane width to accommodate picker/vehicle movement. Each In Section 4,setanofalgorithm to on solve model is presented. effects of5 adiscusses parameters the the decisions. Conclusions Section different scenarios and demonstrates width to two accommodate picker/vehicle movement. Each A certain clearance iseachallowed for lane easedepth. of shelf has picking faces, with a certain Section 5 discusses different scenarios and demonstrates are presented 6 on the decisions. Conclusions effects of a setinofSection parameters shelf has two picking faces, each with a certain lane depth. storage/extraction. A certain clearance is allowed for ease of effects of a set of parameters on the decisions. Conclusions are presented in Section 6 A certain clearance is allowed for ease of 1.1 Literaturein Review storage/extraction. are presented Section 6 The following are key layout decisions: storage/extraction. 1.1 Literature Review Warehouse layout optimization has received considerable The following are key layout decisions: 1.1 Literature Review Lane depth oflayout each shelf The •following are key decisions: attention forhas a received few decades. Berry Warehouseamong layoutresearchers optimization considerable • Lane depth of each shelf Warehouse layout optimization has received considerable (1968) compares two warehouse block attention among researchers for a arrangements few decades. - Berry • Number of vertical Lane depth of each storage shelf levels in each shelf attention among researchers a arrangements fewrequirements decades. - Berry stacking and pallet onfor volume and (1968) compares tworacks warehouse block • Number of vertical storage levels in each shelf Warehouse lateral depth (1968) compares tworacks warehouse arrangements - block • Number of vertical storage levels in each shelf handling costs. Bassan et al.on(1980) develop an analytical stacking and pallet volume requirements and stacking and pallet racks on volume requirements and • Warehouse lateral depth model to decide the dimensions of a rectangular unit load handling costs. Bassan et al. (1980) develop an analytical • Warehouse longitudinal lateral depth width handling et al. (1980) an analytical warehouse and Bassan athezoned warehouse. Park and Webster model to costs. decide dimensions of a develop rectangular unit load • Warehouse longitudinal width model decide dimensions of a rectangular load (1989) todevelop that captures different kinds of • Warehouse longitudinal width warehouse and aathemodel zoned warehouse. Park and unit Webster warehouse and aa model zoned that warehouse. and kinds Webster (1989) develop captures Park different of (1989) develop a model that captures different kinds of
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3. MATHEMATICAL MODEL 3.1 Notations The following are the notations used in the model. a) Order Characteristics parameters
ot
Number of orders per inventory cycle
lc p
Number of order lines for entire case of product p per inventory cycle
uc p
Average
number
of
units
of
cases
of
product p demanded per line
li p
Number of order lines for individual items of product p per inventory cycle
ui p
Average
number
of
units
of
items
of
product p demanded per line
Np
Average number of picks per order
vc p
Volume of a case of product p (cu. ft)
vi p
Volume of an item of product p (cu. ft)
F
Maximum planned inventory volume (cu ft)
b) Warehouse attributes parameters
Fig.1 Top and side view of a three dimensional warehouse racking system The major assumptions in building the model are as follows
h
Height of each of storage level(ft)
α Pd
Clearance allowance, i.e. fraction of total volume to be left empty for ease of storage/retrieval Depth of a pallet (ft)
Aw
Aisle width mandated by design requirements (ft)
Vhor
Horizontal velocity of picker (ft/hr)
Vver
Vertical velocity of picker (ft/hr)
e
Extraction time factor, i.e. average time required to extract a product per feet of lane depth (hr/ft)
Cl
Cost of labour per hour (Rs/hr)
2.1 Assumptions a)
The warehouse employs a random storage policy.
Ca
b) The inventory levels are known as decisions of order quantity and reorder point are made a priori. c)
Manual order picking is employed in the warehouse using ‘S’ shaped/ traversal routing.
c) Decision Variables (ref to Fig 1)
d) The loading/unloading dock (I/O point) is located at the centre of the front wall of the warehouse and the shelves that stock items are perpendicular to the front wall, as shown in Fig 1. e)
The extraction time varies linearly with the lane depth.
f)
The extraction cost varies linearly with the number of units of a product picked.
Area cost in the form of rent or otherwise (Rs/sq ft).
N
Number of storage levels in the racking system
M
Number of picking aisles in the warehouse Lane depth (number of pallets)
Ld
y x L
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Lateral depth of the racking system (ft) Longitudinal width of the racking system (ft) Total aisle length measured along facings (ft) (L= M ×y)
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Ar
Aspect ratio (i.e. ratio of lateral depth, y , to longitudinal width, x )
x=
(8)
The average number of picks per order, N p , can be
(1)
calculated as
(2)
Np =
Volume of product that can be stocked in the racks is
V = L × ( 2 Ld × Pd ) × N × h (1 − α )
(7)
y = A × Ar
Using the above notation, the maximum planned inventory volume F (in cubic feet) can be calculated as
F = ( lc p × uc p × vc p ) + ( li p × ui p × vi p )
A Ar
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∑ ( lc
p
+ li p )
p
ot
(9)
The number of aisles, M, can be calculated as
3.3 Cost Components
M=
The decisions mentioned in the previous section have bearing on different kinds of costs for which we derive the expressions next.
M −1 N p N p −1 HT = 2 x + yM 1 − + 0.5 y N + 1 M p
The footprint area, A can be expressed as (3)
(11)
And the associated area cost can be calculated as
AC = Ca × L × ( ( 2 × Ld × Pd ) + Aw )
(10)
The expected horizontal distance traversed per order can be expressed according to Hall (1993) as
3.3.1 Area Cost (AC)
A = x × y = L × ( ( 2 × Ld × Pd ) + Aw )
L y
Thus the total horizontal travel cost is
HTC =
(4)
HT × ot × Cl VHor
(12)
3.3.2 Vertical Travel Cost (VTC) As random storage policy is assumed, the two way expected distance traversed by the picker in the vertical direction is N × h . This travel should be undertaken for each line item encountered. Thus the total vertical travel cost can be calculated as
N ×h VTC = Cl × × ∑ ( lc p + li p ) Vver p
3.3 Model The mathematical model can be formulated as follows. Minimize Total Cost,
C ( x, y, N, L d ) = AC + EC + VTC + HTC
Subject to
(5)
L × ( 2 Ld × Pd ) × N × h (1 − α ) = F
3.3.3 Extraction Cost (EC)
1 ≤ Ld ≤ Ld max 1 ≤ N ≤ N max y Ar min ≤ ≤ Ar max x
The extraction cost can be calculated as a product of extraction time factor, lane depth, total number of extractions and the cost of labour per unit time
EC = Cl × (e × Ld × Pd ) × ∑ ( lc p × uc p + li p × ui p ) p
and equations (3), (7), (8) &(10) x, y, L, M >0 N , Ld > 0 and integer
(6) 3.3. 4 Horizontal Travel Cost (HTC) We define aspect ratio Ar =
(13)
(14) (15) (16) (17) (18) (19)
The objective of layout design is to minimize the sum of all the costs given by (13). Constraint (14) makes sure that there is adequate space available to store all products. In addition, bounds are placed on lane depth, number of levels and aspect ratio represented by constraints (15), (16) and (17) respectively. Equations (3), (7) & (8) relate total area to lateral depth and longitudinal width. Equation (10) relates number of aisles to total storage length and lateral
y . The longitudinal width x
( x ) and lateral depth ( y ) of the warehouse can be expressed as a function of Aspect Ratio ( Ar ) and Area ( A ) as follows
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depth. Constraints (18) and (19) ensure non-negativity and integrality of variables.
Product Groups Number of lines per shift for cases Average case pick units Number of lines per shift that require item pick Average item pick units Average Case Volume Average Item volume Total number of customer orders per shift Height of a layer Clearance percentage Pallet depth (ft) Aisle Width (ft) Horizontal Velocity of picker (ft/hr) Vertical Velocity of picker (ft/hr) Cost of labor per unit time (Rs / hr) Extraction Constant (Time/unit lane depth) Rent per unit area (Rs/sqft)
4. SOLUTION APPROACH The solution to the above problem involves a step by step approach as shown in Exhibit 1. Step 1: Obtain values for order characteristics and warehouse attributes parameters Step 2: Calculate the product flow through the warehouse as shown in (1) Step 3: Vary Ld from 1 to Ld max Step 3.1: Vary
N from 1 to N max
Step 3.1.1: Calculate L using (14) Step 3.1.2 : Vary Ar from Ar min to
Ar max
•
Calculate
•
and (8), (9) and (10) respectively Calculate the total cost using (13)
Step-3.1.3
x , y , N p and M as per (7)
Get
the
minimum
* r
cost
* r
C ( A , N , Ld ) at the optimal A *
*
*
of A
*
N , and lane depth Ld
*
*
calculate A = 202991.25 sq ft,
7 400
5 300
4 27 2.7 400
5 8 0.8
10 125 1.3
3 33.30% 2 7 300 100 600 0.007 3
x = 1424.75 ft, y =
142.475 ft and N p = 4 , M = 129.5. The total Vertical travel cost is Rs 28,800, Extraction cost is Rs 82,320, Horizontal travel cost is Rs 1,876,522 and Area cost is Rs 608,973. Thus the total cost of operation turns out to be Rs 2,596,616.
4.1 Illustrative Example Consider a sample case with values of parameters shown in Table 1. Further, the aspect ratio is considered to be within the range of 0.1 and 2 since longer warehouses are better when number of picks is larger as per Hall (1993). In addition, upper bounds are specified for lane depth ( Ld max =5) and number of levels ( N = 20), to account for
In Step 3.1.2, the aspect ratio is varied from 0.1 to 2 and the cost is calculated at each value. Fig 2. shows the costs obtained for different Aspect ratios. We find that the minimum cost is Rs 2,384,327 is obtained at an Aspect Ratio of 0.3 for Ld = 1 , N = 1 .
physical limitations of the warehouse racking system. The procedure for obtaining ideal layout can be worked out as follows. Obtaining all parameters would constitute Step 1.
In Step 3.2, we obtain the minimum cost as shown above for different values of N . The results obtained are shown in Fig. 3. It is seen that the minimum cost is obtained at N = 11 with a cost of Rs 945,555. From the data obtained in previous step, the Aspect ratio for achieving this cost is found to be 0.3 for Ld = 1
As per Step 2, Total product volume, F is (500x5) + (200x7) + (100x5) + (100x4) + (400x5) + (300x10) = 147630
As per Step 4, the exercise is repeated for different values of Lane depth, ranging from 2 pallets to 5 pallets. The results are shown in Table 2. It is observed that for smaller lane depths, taller shelves perform better and for larger lane depths, shorter shelves perform better. The reason is, at a larger lane depth, more compaction is achieved per level and hence the optimal number of levels would decrease.
Ld = 1 , N = 1 (Step 3.1). L can
be calculated in Step 3.1.1 as
L=
5 100
For Ar = 0.1 , Using (3), (7), (8), (9) and (10), we can
*
Exhibit 1. Solution algorithm
We illustrate Step 3 for
3 100
N* .
Step 4 Get minimum cost C ( Ar , N , Ld ) at values * r ,
2 200
Table 1: Values of order characteristics and warehouse parameters for the example case
*
Step 3.2 Get minimum cost C ( Ar , N , Ld ) at optimal values of Ar and
1 500
F =18453.75 ft 2 Ld × N × h (1 − α )
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there may be multiple options for lane depth with almost similar cost. 5. EFFECT OF CHANGING PARAMETERS The layout decision is affected by changes in parameters. In this section, three such effects are illustrated. 5.1 Effect of area cost and extraction time constant on optimal lane depth The lane depth decision as shown in the illustration in section 4.1 is arrived at considering an extraction time constant of 0.007 hr/ft and area cost of Rs 3/sq ft. Consider a scenario where the extraction constant is higher, say 0.02 hr/ft and the area cost is lower, say Rs 1/sq ft. The effect is shown in Table 3. Here, smaller lane depths clearly outperform the larger ones; the reason being the effect of higher extraction costs would offset the area cost savings involved in larger lane depths. This is different from the illustration in Table 2, where a low lane depth option involves high area cost and a high lane depth option suffers from a high extraction cost, leading to lane depth of 2 as an optimal choice.
Fig 2. Variation of total cost with Aspect ratio at N = 1
C
N*
Ar*
1
945555.7
11
0.3
2
911597.1
9
0.3
3
943936.5
9
0.4
4
995847.4
9
0.4
5
1056426
8
0.4
C
N*
Ar*
1
1059276
10
0.3
2
1186601
9
0.3
3
1375332
8
0.3
4
1581275
8
0.4
5
1795914
8
0.4
Table 3. Variation of total cost across different levels and different lane depths when extraction constant is 0.02 hr/ft and area cost is Rs 1/sq ft
Fig 3. Variation of minimum costs across different N
Ld
Ld
5.2 Effect of number of stops per order on optimal aspect ratio In 4.1, the number of orders considered was 400. Consider a scenario where the same volume of product flow is spread over 100 orders. This would make the average number of picks per order. N p , four times higher, i.e. 16. The variation of total cost with Aspect ratio, for N =1 is shown in Fig 4. The minimum cost occurs at an Aspect ratio of 0.2 as opposed to 0.3 in the previous case. This is consistent with the observation of Hall (1993) that longer warehouses perform better when number of picks per order are large. In a warehouse with smaller aspect ratio, the number of aisles is larger, making it possible to skip certain aisles in the traversal routing, which will more than offset the increase in distance travelled in the longitudinal direction. Our model explicitly bounds the aspect ratio on the lower side, not allowing unreasonable shapes.
Table 2. Variation of total cost across different levels and different lane depths. It is seen that the optimal cost, C* of Rs 911,597occurs at a lane depth, Ld * = 2 pallets, number of storage levels,
N * = 9, longitudinal width, x* = 248.01 ft and lateral * depth, y = 74.40 ft. It is also observed that as the number of storage levels increases, the optimal lane depth decreases. Even at a particular number of storage levels,
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Fig 4. Variation of total cost with aspect ratio with and
N =1 Fig 5. Variation on total cost across different number of levels, when extraction time constant is high and rent per unit area is low
ot = 100
5.3 Effect of labour cost and area cost on optimal number of levels
REFERENCES
In Fig 3, it was seen that the cost falls rapidly as we increase the number of levels from 1 to around 11. Beyond 11 levels the cost increases. At lower levels, the effect of increased area cost is dominant. At higher levels the pick cost becomes dominant. Now consider a situation where the area cost is very low, at Rs 0.5/sq ft and labour cost is very high, at Rs 3000/sq ft. Fig. 5 shows the variation of cost across different levels at lane depth of 1. The optimal number of levels is 10, which is lower than in the illustration case. Also, as the number of levels is increased from the optimal the pick cost increase is more severe as the labour cost has increased. But if aisle width is increased to 15 (original value is 7), we find that the optimal number of levels shifts to 11, as an increased aisle width would increase the area cost, and thus an increased number of levels would compensate for that, thereby brining the cost back to optimal.
Bassan, Y., Roll, Y., & Rosenblatt, M. J. (1980). Internal layout design of a warehouse. AIIE Transactions, 12(4), 317-322. Bartholdi, J. J., & Hackman, S. T. (2008). Warehouse & Distribution Science: Release 0.89. Supply Chain and Logistics Institute. Berry, J. R. (1968). Elements of warehouse layout. The International Journal of Production Research, 7(2), 105121. Caron, F., Marchet, G., & Perego, A. (2000). Optimal layout in low-level picker-to-part systems. International Journal of Production Research, 38(1), 101-117. Gu, J., Goetschalckx, M., & McGinnis, L. F. (2010). Research on warehouse design and performance evaluation: A comprehensive review. European Journal of Operational Research, 203(3), 539-549.
6. CONCLUSION
Hall, R. W. (1993). Distance approximations for routing manual pickers in a warehouse. IIE transactions, 25(4), 7687.
An algorithm is presented that optimizes lane depth, number of levels, length and width of a warehouse. The effect of variation of a set of parameters on the cost is also studied. The generation of cost curves by varying different parameters would help the designer decide from a range of alternatives for design variables and not just the optimal value. It also helps in knowing the quantum of change in cost due to change in different parameters, which is generally difficult to predict due to interaction of multiple effects and trade-offs. Needless to say, it would make sense for the warehouse manager to understand how the cost changes with changes in multiple parameters. The study can be extended to optimize layout considering other storage and routing policies.
Önüt, S., Tuzkaya, U. R., & Doğaç, B. (2008). A particle swarm optimization algorithm for the multiple-level warehouse layout design problem. Computers & Industrial Engineering, 54(4), 783-799. Park, Y. H., & Webster, D. B. (1989). Modelling of threedimensional warehouse systems. the international journal of production research, 27(6), 985-1003. Roodbergen, K. J., & Vis, I. F. (2006). A model for warehouse layout. IIE transactions, 38(10), 799-811. Yoon, C. S., & Sharp, G. P. (1996). A structured procedure for analysis and design of order pick systems. IIE transactions, 28(5), 379-389.
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