Travel-time models for flow-rack automated storage and retrieval ...

Int J Adv Manuf Technol (2005) 25: 979–987 DOI 10.1007/s00170-003-1932-3

ORIGINAL ARTICLE

Zaki Sari · Can Saygin · Noureddine Ghouali

Travel-time models for flow-rack automated storage and retrieval systems

Received: 25 February 2003 / Accepted: 27 August 2003 / Published online: 20 February 2004  Springer-Verlag London Limited 2004 Abstract In this paper, closed-form travel-time expressions for flow-rack automated storage and retrieval systems are developed. The expressions, which are based on a continuous approach, are compared for accuracy, via simulation, with exact models which are based on a discrete approach. There is no significant difference between the results obtained from the continuous-approach-based closed-form expressions and the ones from the discrete-approach-based exact solutions. The closed-form expressions are easy to calculate due to their simplistic forms, even without a computer, while the exact solutions are extremely complex. On the basis of computation time, the proposed closed-form expressions are extremely practical when compared with the discrete-approach-based expressions, which require extensive computation time. The closed-form travel-time expressions developed in this study can be used to (1) establish performance standards for existing AS/RS, (2) evaluate throughput performance for flow-rack AS/RS alternative design configurations, and (3) compare different storage techniques for improved system performance. Due to their simplistic, yet accurate, definitions, the closed-form expressions, as well as the results of this study, are applicable to industry.

E(DC): dual-cycle expected travel time E(RC): expected retrieval time E(V ): expected travel time between two points on rack face L, H, D: length, height and depth of the flow-rack AS/RS rack l, h, d: length, height and depth of each flow-rack AS/RS storage segment M: number of segments in a bin (i.e., number of layers in rack) m: layer rank Nl : number of bins in each row Nh : number of bins in each column th : horizontal travel time from the pickup/drop-off point to the farthest column th : horizontal travel time between two consecutive bins tv : vertical travel time from the pickup/drop-off point to the farthest row tv : vertical travel time between two consecutive bins Vh , Vv : horizontal and vertical speeds of storage/retrieval machine ρ: load rate (x, y): bin position (x, y)d : dwell-point position

Keywords Automated storage and retrieval systems (AS/RS) · Flow-Rack AS/RS · Inventory management

1 Introduction

Notations b: shape factor T: normalization factor E(SC): single-cycle expected travel time Z. Sari · N. Ghouali Universit´e de Tlemcen, Laboratoire d’Automatique de Tlemcen, B.P. 119, 13000 Tlemcen, Algeria C. Saygin (u) University of Missouri – Rolla, Engineering Management Department, 1870 Miner Circle, Rolla, MO 65409, USA E-mail: [email protected]

Automated storage and retrieval systems (AS/RS) have been widely used not only as alternatives to traditional warehouses but also as part of advanced manufacturing systems [1]. Improved inventory management and control, increased storage capacity to meet long-range plans, quick response to locate/store/retrieve items, and reduced labor cost due to automation are among the major advantages provided by AS/RS [2, 3]. A typical AS/RS is composed of storage racks, storage/retrieval (S/R) machines and pickup/drop-off (P/D) stations. Several types of AS/RS are distinguished based on size and the volume of inventory items. These different types include unit-load, mini-load, manon-board, deep-lane, automated item-retrieval system and flowrack systems.

980

Today’s severe competition requires manufacturing companies to adopt various technologies, such as lean manufacturing and just-in-time, to reduce inventory levels while responding to customer needs in a timely manner. Therefore, quick access to inventory items to reduce manufacturing lead-time is of critical value from the standpoint of production control. In addition, it is important to estimate the storage and retrieval time of inventory items to generate realistic production schedules and meet deadlines. With these two motivations, this paper presents closed-form expressions for the travel time of S/R machines and analyses these models for a variety of storage rack sizes and configurations for flow-rack AS/RS. Two basic approaches, continuous rack face approximation and discrete rack face, are used and compared via simulation. This paper is organised as follows. A literature survey is presented in Sect. 2. Section 3 includes the preliminary work and background knowledge on closed-form expressions. Detailed models are presented in Sect. 4. The simulation study carried out to validate the accuracy of the proposed closed-form expressions is described in Sect. 5. Finally, conclusions are given in Sect. 6.

2 Literature survey Various simulation-based studies that analyse the throughput performance of AS/RS exist in the literature [4–7]. These studies compare various operating policies for a given system configuration. Various analytical approaches to developing cost models for AS/RS have been proposed. Bozer and White [8] present a design package that uses Zollinger’s cost model [9]. Karaswa et al. [10] propose a cost model for single command cycles. The problem, which is formulated as a non-linear programming model, is solved by using Lagrangian multipliers. There is extensive research in the area of dwell points for S/R machines. Bozer and White [11] suggest static dwell-point rules, although they provide no quantitative comparison of their performance. Egbelu [12] presents a model for the dynamic positioning of S/R machines with the objective of minimising the expected travel time. In their study, Hwang and Lim [13] show that the formulation proposed by Egbelu [12] could also be applied to facility location problems. In another study, Egbelu and Wu [14] compare the performance of several dwellpoint rules, adopted from [12] and [11], using simulation. Peters et al. [15] develop a closed-form solution for dwell-point location under a variety of AS/RS configurations. Chang and Egbelu present formulations for pre-positioning of S/R machines to minimise the maximum system response time [16] and minimise the expected system response time [17] for multiaisle AS/RS. Development of expected travel-time (i.e., average travel time) models for S/R machines is another area of research. A comparative study based on expected travel time of S/R machines for randomised and dedicated storage policies has been presented by Hausman et al. [18]. The rack configuration has

been assumed to be square in time (i.e., horizontal maximum travel time is equal to vertical maximum travel time) with single and dual command cycles. An extension of [18] has been proposed by Graves et al. [19]. They present analytical and empirical results for various combinations of alternative storage assignment rules and scheduling policies. Each alternative is compared on the basis of the expected travel time of the S/R machine. Based on a continuous rack approximation approach, Bozer and White [11] present expressions for the expected cycle times of an AS/RS performing single and dual command cycles. Hwang and Lee [20] present travel-time models which include constant acceleration and deceleration rates with a maximum-velocity restriction. Chang et al. [21] propose traveltime models that consider various travel speeds with known acceleration and deceleration rates. Chang and Wen [22] extend the work presented in [21] by investigating the rack configuration problem. All these studies are basically focused on unitload AS/RS.

3 Background knowledge In their study, Bozer and White [11] develop expected cycle-time expressions for a unit-load AS/RS, based on a continuous rack face approximation approach, by using a statistical model. The assumptions made in their study are as follows: • The rack is considered to be a continuous rectangular pick face where the pickup/drop-off (P/D) station is located at the lower left-hand corner of the rack. • The S/R machine operates on either a single or dual command basis. • The rack length and height, as well as the S/R machine speed in the horizontal and vertical directions, are known. • The S/R machine travels simultaneously in the horizontal and vertical directions (i.e., Tchebychev travel). In calculating the travel time, constant velocities are used for horizontal and vertical travel. • Randomized storage is used (i.e., any point within the pick face is equally likely to be selected for storage or retrieval). • P/D times associated with load handling are ignored. These times are generally independent of the rack shape and travel velocity of the S/R machine and could be added subsequently to the travel-time expressions. Bozer and White define th as the horizontal travel time from the P/D station to the farthest column and tv as the vertical travel time from the P/D station to the farthest row. In their model, they define the normalisation factor and shape factor as T = max(th , tv ) and b = min(th /T, tv /T ), respectively, which implies that 0 < b ≤ 1. They assume that th > tv , T = th and b = tv /T . Due to a randomised storage policy, the expected location for storage or retrieval is assumed to be randomly distributed between 0 and 1 for the horizontal direction (i.e., x-axis) and between 0 and b for the vertical direction (i.e., y-axis). Therefore, if two random locations are represented by (x1 , y1 ) and (x2 , y2 ), then the normalised travel

981

time between these two locations is given by max(|x1 –x2 |, |y1 –y2 |). Using these expressions, the authors present the following equations [11]: For single-cycle expected travel time: E(SC) = T

b2 +1 . 3

(1)

For dual-cycle expected travel time: E(DC) = T

4 b2 b3 + − 3 2 30

.

(2)

From the preceding analysis it can be deduced that the expected travel time from one corner to any location of the rack, and similarly from any location to one corner of the rack, can be given as follows [15]: E1 = T

b2 1 + 6 2

.

(3)

Similarly, the expected travel time from the midpoint to any location on the rack, and similarly from any location to the midpoint of the rack, can be formulated as follows [15]: E2 =

T 2

b2 1 + 6 2

4 Problem description A flow-rack AS/RS includes a rack which consists of sloping bins, where items loaded by a storage machine at one end of the rack on the store face travel along sloping wheels or rollers to the other end of the rack on the pick face, to be retrieved by a retrieval machine. Both the retrieval and storage machines can travel on the x–y plane to reach any bin on the rack. A drop-off station and a pickup station are located at the pick face and the store face of the rack, respectively. A restoring conveyor is used to link the storage and the retrieval machines. As shown in Fig. 1, a rack, which consists of bins, has a length L, height H and depth D. The rack has Nl bins in each row and Nh bins in each column. Each bin has M storage segments, numbered from 0 to M − 1. Each segment has a storage capacity of one item. The length, height and depth of each segment are l = L/Nl , h = H/Nh and d = D/M, respectively. Segments with the same rank form a layer. The storage and retrieval machines have the same horizontal speed, Vh , and the same vertical speed, Vv . Therefore, the travel times, th , tv , th and tv , can be calculated as follows: th = L/vh tv = H/vv

.

(4)

The abovementioned models, which have been developed by Bozer and White [11] for unit-load AS/RS, forms the basis of the models presented in this paper. Their formulation has been extended to apply to flow-rack AS/RS. Fig. 1. Configuration of a Typical Flow-rack AS/RS

t h = l/vh = th /Nl t v = h/vv = tv /Nh . These basic equations are used in the following sections for developing travel-time models using a continuous rack face approximation approach and a discrete rack face approach.

982

4.1 Continuous rack face approximation approach For a storage operation, a storage machine in a flow-rack AS/RS operates exactly the same as a S/R machine in a unit-load AS/RS; therefore, the expected storage time can be calculated by using Eq. 1. However, a retrieval operation for a particular item requires that the retrieval machine remove all items stored in front of the requested item until it reaches the first bin on the pick face of the rack. Thus, the expected retrieval time of a particular item stored in layer m can be calculated as follows: E(RC) m = E(V1 ) + E(V2 ) + m.E(V3 ) ,

(5)

where m is the layer rank of the requested item (0 ≥ m ≥ M − 1), E(V1 ) is the expected travel time between dwell point and retrieval point, E(V2 ) is the expected travel time between retrieval point and drop-off station, and E(V3 ) is the expected travel time from the retrieval point to the restoring conveyor and back to the retrieval point. The optimal dwell-point locations for the S/R machines are the pickup station and the midpoint of the rack face [15]. Expressions for E(V1 ), E(V2 ) and E(V3 ) can be derived easily based on the previous work presented in [11]: T b2 1 + E(V1 ) = , (6) 2 6 2 2 1 b + , (7) E(V2 ) = T 6 2 2 b +1 . (8) E(V3 ) = T 3 Equation 5 can be rewritten by using Eqs. 6–8: 2 2 b b 3 + 1 + m.T +1 . E(RC) m = T 4 3 3

For each case defined above, an expected retrieval time expression can be derived based on Eq. 9. For Case 1, since only the very first storage layer is utilised (i.e., m = 0), then the expected retrieval time becomes 2 b 1 3 +1 for ρ < . (10) E(RC) = T 4 3 M For the other three cases, the number of storage layers that will be highly populated can be calculated by dividing the number of stored items by the layer storage capacity (i.e., Nl .Nh ). In other words, it can be stated that the last highly populated layer for a given load rate ρ is m = ρM − 1 .

(11)

The expected retrieval time for any item stored in any layer for a system with m layers which are highly populated is 1 E(RC) i . m +1 m

E(RC) =

(12)

i=0

Replacing Eq. 12 with Eq. 9, the expected retrieval time can be rewritten as 2 2 b m b 3 + 1 + .T +1 . (13) E(RC) = T 4 3 2 3 Finally, replacing Eq. 13 with Eq. 11, the expected retrieval-time expression for a flow rack AS/RS becomes 2 1 1 b 1 +1 + ρ.M for ≤ ρ ≤ 1 . (14) E(RC) = T 3 4 2 M

4.2 Discrete rack face approach (9)

The expected travel time for S/R machines depends on the rack configuration, the storage capacity and the number of items stored in the rack. Using the notation given for a typical flow rack AS/RS (Fig. 1), the storage capacity can be calculated as N = Nl .Nh .M. The load rate of a system, ρ, is the ratio of the number of items stored in the rack to the storage capacity. Therefore, ρ varies between 0 and 1. The storage density at each layer (i.e., 0 ≤ m ≤ M − 1) in the rack depends on the load rate and the number of storage segments, M, in each bin. Assuming randomised storage, various scenarios can be presented: Case 1: If ρ ≤ 1/M, then stored items are highly populated in storage layer #0. Case 2: If 1/M < ρ ≤ 2/M, then stored items are highly populated in storage layers #0 and #1. Case 3: If m/M < ρ ≤ (m + 1)/M, then storage layers #0 through #m contains most of the stored items. Case 4: If (M − 1)/M < ρ ≤ 1, then storage capacity is almost fully utilised.

The expected retrieval-time equations presented in Sect. 4.1 provide “approximate” values since the formulation is based on a continuous rack face approximation approach. Experimental or analytical methods which give exact solutions are required to prove the validity of these equations. For this purpose, an exact retrieval time model based on a discrete rack face approach is presented in this section. The exact retrieval time can be obtained by adding the travel times associated with each storage segment and then dividing the result by the number of segments [11]. Similar to the discussion presented in Sect. 4.1, the expected travel time for S/R machines depends on the rack configuration rack, the storage capacity and the number of items stored in the rack. To retrieve an item from a storage layer, all items preceding it must first be retrieved by the retrieval machine, then sent to the storage machine via the restoring conveyor, and finally stored back in the rack. As shown in Fig. 1, each storage bin has M segments, and the drop-off station is located at (x, y) = (0, 1), and the restoring conveyor is located parallel to the depth of the rack at (x, y) = (Nl + 1, 1). The dwell point of the retrieval machine is

983

the midpoint of the rack face, which can be defined as Nl + 1 Nh + 1 (x, y)d = , . 2 2

(15)

Using Tchebychev travel, the travel time between bins (i, j) and (i , j ) is t (i, j), (i , j ) = max th i − i , tv j − j . (16) The travel time can be broken down into three components: Nl + 1 Nh + 1 E 1 = max th i − , tv j − 2 2 between dwell point and bin (i, j), (17) E 2 = max th |0 − i| , tv |1 − j| between bin (i, j) and drop-off station, and E 3 = max th |Nl + 1 − i| , tv |1 − j|

(18)

between bin (i, j) and restoring conveyor .

(19)

Then, using a discrete rack face approach, the retrieval timefor an item in storage layer #z of storage bin (i, j) will be E D (RC)ijz = E 1 + E 2 + 2.z.E 3 .

(20)

The expected retrieval time for any item in bin (i, j) with m ij of its storage segments occupied by other items is E D (RC)

ij

m ij 1 = (E 1 + E 2 + 2.k.E 3 ) . m ij

(21)

k=0

Considering the whole rack with a storage capacity of N = M.Nl .Nh , the expected retrieval time for any item will be E D (RC)

m ij Nh Nl 1 1 = (E 1 + E 2 + 2.k.E 3 ) . Nl Nh m ij i=1 j=1

(22)

k=0

Finally, the expected retrieval time based on a discrete rack face approach can be obtained by inserting Eqs. 17–19 into Eq. 22: h l 1 = Nl · Nh i=1 j=1 

 i − Nl +1 , t j − Nh +1 max t v h 2 2 m ij  1    ×  + max t |0 − i| , t |1 − j| v   m ij h k=0 +2.k. max th |Nl + 1 − i| , tv |1 − j|

N

N

the discrete approach, is extremely complicated when compared with Eqs. 10 and 14. The filling arrangement corresponds to the specific location of items stored in storage bins. For a load rate equal to 1 (i.e., ρ = 1), the rack is fully utilised. When the load rate ρ is less than 1, then the possible number of rack filling arrangements becomes extremely large. Since the retrieval time based on a single filling arrangement does not accurately represent the expected retrieval time (i.e., the accuracy of expected retrieval time depends on the number of filling arrangements), a large number of randomised filling arrangements and the average of all the corresponding retrieval times must be considered to obtain the expected retrieval time. Equations 10 and 14 require only the total number of stored items (i.e., only the load rate, ρ, is required), but not their specific locations in the rack. On the other hand, Eq. 23 requires the specific locations of stored items (i.e., m ij ), which leads to a very high computation time to obtain the expected retrieval time.

4.3 Pickup/drop-off and acceleration/deceleration delays In the previous sections, pickup/drop-off (P/D) times and acceleration/deceleration (A/D) effects on the retrieval time were not considered. In a unit-load AS/RS, these time components can simply be assumed to be constants and added to the final expressions [11, 15], or they can be introduced into the model as functions [20–22]. This section investigates the influence of P/D times and A/D effects on the expected retrieval-time models developed using a continuous approach and a discrete approach, presented in Sects. 4.1 and 4.2, respectively. P/D times and A/D effects can be considered constant for items of comparable size and weight. The retrieval of an item is associated with one pickup, one drop-off, two accelerations and two decelerations. Let tc be the sum of the time components of these four factors. The expected retrieval-time models (i.e., Eqs. 5 and 22) can be modified by adding tc : E(RC) mtc = E(V1 ) + E(V2 ) + tc + m E(V3 ) + tc Continuous approach .

(24)

E D (RC)

where

Nh Nl

m ij = [ρ.N] ∼ = ρ.N .

(23)

After inserting E(RC) i into Eq. 12 by Eq. 24, the expected retrieval time expression for a flow-rack AS/RS with P/D times and A/D effects becomes 1 1 ρ.M + 1 b2 +1 + ρ.M + tc 3 4 2 2 Continuous approach .

E(RC) tc = T

(25)

i=1 j=1

The closed-form expressions in Eqs. 10 and 14 are based on the continuous approach. Given T , b and M, these expressions can easily be calculated due to their simplistic forms, even without a computer. On the other hand, Eq. 23, which is based on

The averaged sum of all P/D- and A/D-related time components can be defined as follows: TcC =

ρ.M + 1 tc 2

Continuous approach .

(26)

984

Similarly, Eq. 22 can be rewritten by adding tc as follows:

Rearranging the terms, Eq. 29 becomes equal to Eq. 26:

E D (RC) tc =

TcD =

1 Nl Nh

Nh Nl i=1 j=1

m ij

1 [(E 1 + E 2 + tc ) + k (2E 3 + tc )] m ij k=0

Discrete approach .

(27)

Equation 27 can be rearranged as the summation of Eq. 22 and terms related to tc : E D (RC)

tc

=

E D (RC) +

Nh Nl 1 tc m ij + 1 Nl Nh 2 i=1 j=1

Discrete approach .

(28)

The averaged sum of all P/D- and A/D-related time components using a discrete approach can be written as follows: TcD =

Nh Nl 1 tc m ij + 1 Discrete approach . Nl Nh 2

(29)

i=1 j=1

ρ.M + 1 tc = TcC . 2

(30)

Therefore, it can be concluded that the influence of P/D times and A/D effects is the same for both continuous and discrete approaches.

5 Simulation study Since the discrete approach is typically used in the literature to validate the accuracy of approximate approaches, the closedform expressions, developed in Sect. 4.1, are compared with the discrete expression, developed in Sect. 4.2. To achieve this goal, a simulation study is carried out to validate the accuracy of the closed-form expressions by investigating the difference between the continuous and discrete approaches. The impact of the number of filling arrangements on the accuracy of the discrete approach is investigated. The simulation model includes a 36-bin flow-rack AS/RS with a configuration of 6 × 6 and a load rate, ρ, of 70%. The discrete expected retrieval time (DERT), its standard deviation and the computation

Table 1. Standard deviation, discrete expected retrieval time (DERT), and Computation time with respect to number of filling arrangement (NFA) NFA

Standard deviation

DERT

2 000 4 000 6 000 8 000 10 000 12 000 14 000 16 000 20 000 24 000 28 000

0.2737 0.2738 0.2753 0.2764 0.2769 0.2744 0.2755 0.2736 0.2739 0.2732 0.2748

13.2448 13.2455 13.2466 13.2511 13.2453 13.2507 13.2460 13.2490 13.2461 13.2469 13.2474

Computation time 4 min 37 s 9 min 09 s 13 min 25 s 18 min 29 s 26 min 09 s 27 min 33 s 32 min 22 s 40 min 45 s 45 min 10 s 54 min 14 s 1 h 3 min

NFA

Standard deviation

DERT

Computation time

32 000 36 000 40 000 50 000 60 000 70 000 80 000 90 000 100 000 110 000 120 000

0.2750 0.2734 0.2725 0.2752 0.2744 0.2742 0.2740 0.2744 0.2742 0.2747 0.2755

13.2492 13.2481 13.2467 13.2495 13.2482 13.2480 13.2462 13.2481 13.2475 13.2488 13.2477

1 h 13 min 1 h 21 min 1 h 33 min 1 h 54 min 2 h 14 min 2 h 38 min 3 h 05 min 3 h 25 min 3 h 48 min 4 h 04 min 4 h 28 min

Table 2. Percent error between continuous and discrete expected retrieval times versus the load rate ρ, for a 36-bin flow-rack AS/RS M-N b 1.00

4 – 144 0.571 0.444

0.250

ρ (%) 10 20 25 30 40 50 60 70 80 85 90 100

1.00

6 – 216 0.571 0.444

0.250

1.00

8 – 288 0.571 0.444

0.250

1.00

10 – 360 0.571 0.444

0.250

Percent error between continuous and discrete expected retrieval times

13.3 9.44 3.57 1.83 0.945 0.339 0.454 0.430 1.08 0.917

10.4 6.23 1.43 0.885 2.20 2.94 3.37 3.49 3.61 3.73

8.38 4.70 0.986 2.68 3.57 4.85 4.96 4.98 5.62 5.47

8.27 4.63 1.03 2.74 3.61 4.90 5.00 5.01 5.64 5.50

9.49 5.22 2.70 1.19 0.153 0.869 0.709 1.04 1.19 0.836 1.06

6.22 1.60 0.285 1.96 2.94 3.40 3.66 3.81 4.11 3.90 3.95

4.77 0.626 1.83 3.36 4.65 5.40 5.29 5.64 5.77 5.46 5.69

4.78 0.539 1.87 3.40 4.74 5.45 5.32 5.67 5.81 5.46 5.71

3.70 2.01 1.18 0.174 0.732 1.05 1.19 0.913 1.17 1.02 1.13

1.58 0.709 1.93 3.11 3.59 3.76 3.90 3.96 3.99 4.05 4.07

0.823 2.51 3.27 4.72 5.30 5.61 5.8 5.56 5.81 5.70 5.80

0.864 2.52 3.32 4.74 5.32 5.65 5.83 5.58 5.83 5.71 5.82

13.4 2.04 0.603 0.157 0.764 0.952 1.03 1.09 1.14 1.14 1.15 1.18

10.5 0.678 2.65 2.93 3.55 3.82 3.92 4.03 4.08 4.24 4.12 4.14

8.52 2.48 3.90 4.68 5.30 5.57 5.67 5.78 5.80 5.82 5.82 5.87

8.43 2.57 3.93 4.71 5.35 5.56 5.74 5.77 5.80 5.83 5.86 5.88

985 Table 3. Percent error between continuous and discrete expected retrieval times versus the load rate ρ, for a 144-bin flow-rack AS/RS M-N b 1.00

4 – 576 0.563 0.250

0.111

ρ (%) 10 20 25 30 40 50 60 70 80 85 90 100

1.00

6 – 864 0.563 0.250

0.111

1.00

8 – 1152 0.563 0.250

0.111

1.00

10 – 1440 0.563 0.250

13.8 2.58 1.20 0.504 0.0118 0.188 0.246 0.273 0.283 0.282 0.295 0.297

11.7 11.3 11.8 0.562 0.162 0.677 0.884 1.24 0.738 1.59 1.96 1.42 2.17 2.56 1.99 2.37 2.72 2.17 2.42 2.83 2.22 2.46 2.86 2.29 2.47 2.88 2.30 2.48 2.89 2.32 2.50 2.90 2.32 2.51 2.92 2.33

0.111


13.7 11.6 9.35 7.25 4.85 2.80 2.36 0.302 1.01 1.03 0.476 1.59 0.0089 2.07 0.132 2.23 0.0845 2.18 0.231 2.34

9.43 11.2 11.8 5.72 6.90 7.41 3.62 2.44 2.95 1.26 0.0473 0.463 0.457 1.38 0.856 0.146 1.95 1.43 0.155 2.43 1.91 0.191 2.60 2.06 0.191 2.55 2.01 0.308 2.72 2.17 0.266

7.37 3.67 1.54 0.787 1.59 1.97 2.28 2.33 2.32 2.45 2.43

6.97 3.32 1.21 1.16 1.96 2.34 2.68 2.70 2.74 2.84 2.83

7.50 3.80 1.72 0.644 1.44 1.83 2.13 2.16 2.16 2.27 2.26

5.04 2.60 1.28 0.274 0.0322 0.156 0.195 0.322 0.262 0.292 0.285

2.96 0.556 0.789 1.77 2.15 2.28 2.34 2.47 2.43 2.48 2.48

2.64 0.173 1.12 2.18 2.52 2.69 2.75 2.90 2.84 2.89 2.89

3.14 0.709 0.632 1.61 1.99 2.10 2.17 2.29 2.26 2.32 2.30


0.36

4 – 900 0.111

0.040

1.00

0.36

6 – 1350 0.111

0.040

1.00

0.36

8 – 1800 0.111

0.040

ρ (%)


10 20 25 13.8 11.7 30 9.51 7.52 40 4.77 2.78 50 2.43 0.458 60 1.19 0.795 70 0.498 1.51 80 1.15 1.89 85 0.0011 2.02 90 0.0715 2.11 100 0.148 2.20

9.61 5.65 3.60 1.44 0.556 0.142 0.358 0.116 0.187 0.154 0.171

12.2 12.7 7.93 8.51 3.23 3.79 0.887 1.45 0.354 0.225 1.06 0.468 1.45 0.859 1.57 0.972 1.65 1.05 1.74 1.13

7.62 3.68 1.64 0.564 1.48 1.90 2.11 2.20 2.28 2.26 2.29

8.03 4.10 2.04 0.114 1.05 1.44 1.63 1.73 1.80 1.78 1.81

8.60 4.66 2.66 0.472 0.437 0.810 1.02 1.11 1.18 1.16 1.18

4.95 2.67 1.48 0.428 0.0487 0.0902 0.146 0.166 0.176 0.177 0.183

2.96 0.679 0.536 1.62 2.02 2.20 2.27 2.29 2.30 2.33 2.34

3.38 3.95 1.09 1.68 0.0759 0.482 1.16 0.583 1.55 0.943 1.70 1.11 1.78 1.16 1.80 1.17 1.82 1.18 1.83 1.19 1.84 1.20

1.00

10 – 2250 0.36 0.111

13.9 2.68 1.19 0.577 0.0430 0.106 0.169 0.181 0.179 0.207 0.186 0.190

11.8 12.2 12.8 0.687 1.12 1.69 0.807 0.370 0.206 1.45 0.984 0.397 2.01 1.56 0.934 2.19 1.72 1.09 2.26 1.82 1.14 2.31 1.83 1.19 2.34 1.84 1.20 2.37 1.88 1.23 2.36 1.86 1.21 2.37 1.87 1.21

0.040


4 – 1600 0.640 0.250

0.0400 1.00

6 – 2400 0.640 0.250

0.040

1.00

8 – 3200 0.640 0.250

0.0400 1.00

ρ (%)


10 20 25 30 40 50 60 70 80 85 90 100

9.66 5.83 3.65 1.49 0.601 0.203 0.0351 0.0540 0.0760 0.0830 0.0961

13.9 12.7 12.3 13.1 9.56 8.46 8.03 8.80 4.82 3.76 3.34 4.07 2.48 1.42 0.998 1.74 1.23 0.163 0.244 0.497 0.550 0.524 0.958 0.203 0.187 0.903 1.33 0.568 0.0666 1.02 1.46 0.694 0.0128 1.11 1.54 0.770 0.0832 1.19 1.63 0.850

8.57 4.77 2.58 0.415 0.473 0.893 1.09 1.17 1.20 1.22 1.24

8.13 4.36 2.17 0.498 0.898 1.33 1.54 1.62 1.66 1.68 1.70

8.90 5.09 2.90 0.764 0.151 0.560 0.737 0.832 0.856 0.867 0.883

time have been calculated for a variety of different numbers of filling arrangements (NFA), ranging from 2000 to 120 000. The results of this preliminary simulation study are displayed in Table 1. DERT values obtained for NFA between 2000 and

4.99 2.72 1.52 0.479 0.123 0.0163 0.0758 0.0906 0.0939 0.0967 0.103

3.91 1.65 0.433 0.613 0.997 1.14 1.22 1.23 1.24 1.25 1.26

3.51 1.23 0.0297 1.07 1.44 1.60 1.67 1.70 1.71 1.72 1.73

4.23 1.98 0.774 0.293 0.653 0.801 0.860 0.881 0.884 0.893 0.900

13.9 2.72 1.31 0.628 0.110 0.0272 0.0742 0.0939 0.100 0.106 0.107 0.107

10 – 4000 0.640 0.250

0.040

12.8 12.4 13.1 1.65 1.22 1.97 0.232 0.165 0.581 0.449 0.876 0.112 1.00 1.43 0.639 1.16 1.61 0.810 1.22 1.68 0.865 1.25 1.72 0.901 1.26 1.73 0.897 1.26 1.73 0.894 1.27 1.74 0.905 1.28 1.75 0.911

12 000 are accurate to three significant digits, while DERT values become accurate to four significant digits for NFA ranging from 14 000 to 120 000. Therefore, the larger the number of filling arrangements, the higher the accuracy.

986 Table 6. Percent error between continuous and discrete expected retrieval times versus the load rate ρ, for a 630-bin flow-rack AS/RS M-N b 0.700

4 – 2520 0.357 0.129

0.0143 0.700

ρ (%) 10 20 25 30 40 50 60 70 80 85 90 100

6 – 3780 0.357 0.129

0.014

0.700

8 – 5040 0.357 0.129

0.0143 0.700

10 – 6300 0.357 0.129

0.0143


13.1 12.6 8.84 8.34 4.11 3.63 1.77 1.29 0.526 0.0459 0.156 0.641 0.539 1.03 0.650 1.15 0.728 1.23 0.810 1.32

12.8 13.5 8.55 9.21 3.84 4.49 1.50 2.14 0.249 0.903 0.430 0.232 0.822 0.157 0.933 0.267 1.02 0.343 1.10 0.417

8.93 5.13 2.95 0.778 0.121 0.517 0.703 0.788 0.811 0.825 0.843

8.44 4.63 2.47 0.286 0.611 1.03 1.22 1.31 1.33 1.35 1.37

8.67 4.86 2.67 0.496 0.404 0.807 0.983 1.08 1.11 1.13 1.14

9.32 5.49 3.32 1.16 0.269 0.138 0.311 0.390 0.416 0.424 0.433

To validate the computation method used for the experiments displayed in Table 1, the flow-rack AS/RS has been simulated for a 100% load rate for 20 000 NFA. Since at a 100% load rate all filling arrangements are identical, then all DERT values must be equal and the standard deviation must be equal to zero for an accurate computation method. The standard deviation for the above simulation has been on the order of 10−12 . Therefore, the variation in the DERT values is not due to the computation method. To keep the computation time within an acceptable range, an NFA value of 20 000 and four-digit accuracy for DERT values have been used in the detailed simulation study, which includes a variety of rack sizes and configurations. Racks sizes include 36, 144, 225, 400 and 630 bins, each containing 4, 6, 8 and 10 layers, respectively. The shape factor, b, varies from 1 to 0.014 and the load rate, ρ, from 1/M to 1. The results, including the percent error between continuous and discrete approaches for different rack sizes, configurations and load rates, are displayed in Tables 2 through Table 6.

6 Conclusions In this paper, closed-form travel-time expressions for a flowrack AS/RS are presented. The expressions are developed using a continuous approach and are compared with a discrete approach for accuracy via simulation. The following conclusions can be reached based on the results of the simulation study: • Regardless of the rack size and configuration, the error at ρmin = 1/M varies between 11% and 14%, which is justified by the fact that the continuous approach is defined only for ρ ≥ 1/M and ρmin = 1/M, which is at the limit of this range. • For b = 1, (i.e., square-in-time system), the error, which is less than 2% for a 36-bin rack size with a load rate larger than 2ρmin = 2/M, is relatively small. The error decreases as the rack size increase. For a 400-bin rack size, the error is less than 0.1%. • For b < 1 and ρ > 2ρmin , the error varies depending on the rack size. It is large for small rack sizes, e.g., 5.9% for a 36-

4.30 2.01 0.809 0.245 0.621 0.760 0.817 0.835 0.848 0.853 0.860

3.82 1.51 0.324 0.740 1.13 1.28 1.34 1.36 1.38 1.38 1.40

3.99 1.71 0.527 0.541 0.905 1.05 1.12 1.14 1.16 1.15 1.16

4.64 2.36 1.18 0.128 0.217 0.350 0.405 0.421 0.435 0.437 0.441

13.2 2.01 0.596 0.0849 0.615 0.774 0.828 0.847 0.865 0.864 0.866 0.870

12.7 12.9 13.6 1.52 1.72 2.39 0.105 0.320 0.986 0.579 0.351 0.306 1.12 0.902 0.207 1.30 1.07 0.364 1.36 1.12 0.424 1.38 1.14 0.431 1.40 1.17 0.435 1.40 1.17 0.444 1.41 1.17 0.442 1.41 1.18 0.446

bin rack, and decreases as the rack size increases, e.g., 2.4% for a 630-bin rack. • The error does not seem to be very sensitive to the changes in the number of layers, M. • The error is shape dependent. It is at a minimum for b = 1 and increases as b decreases until the maximum error occurs at 0.25 < b < 0.36. For b < 0.25, the error decreases. This study also sheds light on AS/RS design. Such storage and retrieval systems are typically designed to operate at 85% of their total storage capacity [23, 24]. The closed-form expressions developed in this study yield very accurate results (i.e., minimal error) for a load rate range of 0.7 and 0.85, which can be defined as the zone of actual operation. In this zone, for b = 1 the error is less than 1.2% for a 36-bin rack configuration and less than 0.1% for a 400-bin rack configuration. On the other hand, for b < 1, the error decreases as the rack size increases, e.g., 2.9% for 144 bins, 2.4% for 225 bins, 1.8% for 400 bins, and 1.4% for 630 bins. However, for very small systems with b < 1, the error is significant, e.g., 5.8% for 36 bins. Therefore, the results, as well as the closed-form expressions, are highly applicable to industry due to their simplistic, yet accurate, formats. Overall, the results obtained by the continuous approach are very close to those obtained by the discrete approach. The discrete approach required over 3000 h to obtain the results tabulated in Tables 2 through 6, while the continuous approach took less than 10 min. Therefore, the closed-form expressions developed using the continuous approach are extremely practical due to the difference in computation time. However, the continuous approach is less accurate for predicting the expected retrieval time for small systems and for low load rates. The closed-form travel-time expressions developed in this study can be used to (1) establish performance standards for existing systems, (2) evaluate throughput performance for alternative flow-rack AS/RS design configurations, and (3) compare different storage techniques for improved system performance. Acknowledgement This study has been partially funded by the U.S. Department of State, award number S-ECAAS-02-GR-281 (PS).

987

References 1. Lee HF (1997) Performance analysis for automated storage and retrieval systems. IIE Trans 29(1):15–28 2. Allen SL (1992) A selection guide to AS/R systems. Ind Eng 24(1):28–31 3. Material Handling Institute (1997) Considerations for planning and installing an automated storage/retrieval system. AS/RS Document-100 7M 4. Barrett BG (1977) An empirical comparison of high-rise warehouse policies for operator-controlled stacker cranes. Logistics Research and Analysis, Eastman Kodak Company, Rochester, NY 5. Sand GM (1976) Stacker crane product handling systems. Eastman Kodak Company, Rochester, NY 6. Schwarz LB, Graves SC, Hausman WH (1978) Scheduling policies for automatic warehousing systems: simulation results. AIIE Trans 10(1):260–270 7. Koenig J (1980) Design and model the total system. Ind Eng 12(10):22–27 8. Bozer YA, White JA (1980) Optimum designs of automated storage/retrieval systems. In: Proceedings of the TIMS/ORSA joint national meeting, Washington, DC, May 1980 9. Zollinger HA (1975) Planning, evaluating and estimating storage systems. In: Proceedings of the seminar on advanced material handling, Purdue University, West Lafayette, IN, 29–30 April 1975 10. Karaswa Y, Nakayama H, Dohi S (1980) Trade-off analysis for optimal design automated warehouses. Int J Sys Sci 11(3):567–576 11. Bozer YA, White JA (1984) Travel-time models for automated storage/retrieval systems. IIE Trans 16(2):329–338 12. Egbelu PJ (1991) Framework for dynamic positioning of storage/retrieval machines in an automated storage/retrieval system, Int J Prod Res 29(1):17–37

13. Hwang H, Lim JM (1993) Deriving an optimal dwell point of the storage/retrieval machine in an automated storage/retrieval system. Int J Prod Res 31(8):2591–2602 14. Egbelu PJ, Wu CT (1993) A comparison of dwell point rules in automated storage/retrieval systems. Int J Prod Res 31(8):2515–2530 15. Peters BA, Smith JS, Hale TS (1996) Closed form models for determining the optimal dwell point location in automated storage and retrieval systems. Int J Prod Res 34(4):1757–1771 16. Chang SH, Egbelu PJ (1997) Relative pre-positioning of storage/retrieval machines in automated storage/retrieval systems to minimize maximum system response time. IIE Trans 29(2):303–312 17. Chang SH, Egbelu PJ (1997) Relative pre-positioning of storage/retrieval machines in automated storage/retrieval systems to minimize expected system response time. IIE Trans 29(2):313–322 18. Hausman WH, Schwarz LB, Graves SC (1976) Optimal storage assignment in automatic warehousing systems. Manage Sci 22(4):629–638 19. Graves SC, Hausman WH, Schwarz LB (1977) Storage retrieval interleaving in automatic warehousing systems. Manage Sci 23(7):935–945 20. Hwang H, Lee SB (1990) Travel-time models considering the operating characteristics of the storage and retrieval machine. Int J Prod Res 28(10):1779–1789 21. Chang DT, Wen UP, Lin JT (1995) The impact of acceleration/deceleration on travel-time models for automated storage retrieval systems. IIE Trans 27(1):108–111 22. Chang DT, Wen UP (1997) The impact on rack configuration on the speed profile of the storage and retrieval machine. IIE Trans 29(3):525– 531 23. White JA, Kinney HD (1982) Storage and warehousing. In: Salvendy G (ed) Handbook of industrial engineering. Wiley, New York 24. Groover MP (1987) Automation production systems and computer integrated manufacturing. Prentice-Hall, Englewood Cliffs, NJ, pp 404–416