# Layout Optimization of a Three Dimensional Order Picking Warehouse

## 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 Manuf...

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

ScienceDirect IFAC-PapersOnLine 48-3 (2015) 1155–1160

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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)

INCOM 2015 Venkitasubramony Rakesh et al. / IFAC-PapersOnLine 48-3 (2015) 1155–1160 May 11-13, 2015. Ottawa, Canada

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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