翻譯原文-Optimal design of rack structure with modular cell in AS-RS.pdf

Int. J. Production Economics 98 (2005) 172178 Optimal design of rack structure with modular cell in AS/RS Young Hae Lee?, Moon Hwan Lee, Sun Hur Department of Industrial Engineering, Hanyang University, Ansan, Kyunggi-do, 425-791, Republic of Korea Available online 17 February 2005 Abstract In this paper, the model of AS/RS with the rack of modular cells is proposed fi rst. In general, under the concept of unit load, Automated Storage/Retrieval Systems (AS/RS) has the rack of equally sized cells. Many authors have studied the design of AS/RS with the rack of equally sized cells. However, they are inadequate and ineffi cient in meeting the various sizes of customers demands in todays business environment. Higher utilization and fl exibility of warehouse storage can be achieved by using AS/RS with the rack of modular cells. The best size of modular cell is determined as a decision variable and the effectiveness of the proposed model is also presented. The model developed in this research, is one type of AS/RS that is more fl exible to the size and has higher space utilization than those of existing rack structure, could be a very useful alternative for the storage of different unit load sizes. r 2005 Elsevier B.V. All rights reserved. Keywords: AS/RS; Modular cell; Equally sized cells; Flexibility and utilization 1. Introduction Automated Storage/Retrieval Systems (AS/RS) is widely used in numerous manufacturing fac- tories and distribution centers in the world. A typical AS/RS is composed of multiple parallel aisles of racks with storage cells (slots), a storage/ retrieval (S/R) machine for each aisle, and input/ output (I/O) station. The S/R machine moves simultaneously in horizontal and vertical direction in order to reduce the travel time, which is called as Tchebychev travel. The S/R machine can be operated under single and/or dual command. In a single command, only one operation of storage or retrieval of item is conducted. However, in a dual command both storage and retrieval of items are conducted during one cycle of S/R machine with an interleaving. There are various types of AS/RS with equally sized cells according to the size and volume of items to be handled, storage and retrieval meth- ods, and interaction of a S/R machine with the worker such as unit load AS/RS, mini-load AS/ RS, man-on-board AS/RS, automated item-retrie- val system, and deep-lane AS/RS (Groover, 1987). The design of AS/RS involves the determination of ARTICLE IN PRESS /locate/ijpe 0925-5273/$-see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2004.05.018 ?Corresponding author. Tel.: +82314005262; fax: +82316027730. E-mail address: yhleehanyang.ac.kr (Y.H. Lee). the number of S/R machines, their horizontal/ vertical velocities and travel times, the physical confi guration of the storage racks, etc. The design of warehouses has been studied basically with two approaches: analytical optimization methods and simulation. As for the analytical methods, Roberts and Reed (1972) presented an optimization model to determine the warehouse bay confi guration that minimizes the cost of handling and construction, ignoring the constraints on handling capacity of equipment and building sites. Karasawa et al. (1980)developedanon-linearmixedinteger programming (MIP) for a deterministic model of an AS/RS to minimize the total cost. Their cost model included storage racks, storage building, handling machines and land. The decision vari- ables involved were the number of cranes and the height and the length of rack. Optimization was performed as a function of suffi cient storage volume for all items and suffi cient the number of cranes to serve all storage and retrieval requests. Ashayeri et al. (1985) presented a mathematical model for the calculation of the optimal number of cranes and the optimal width and length of the warehouse subject to constraints on the constant crane velocities, the throughput and the length and width of building site. In the simulation methods, Bafna and Reed (1972) proposed a simulation program to evaluate the alternative design of high-rise automated warehouse systems. Rosenblatt and Roll (1984) applied a search procedure to a simulation model of an automated warehousing system to fi nd an optimum solution for minimizing the total cost of construction and operation. Perry et al. (1984) developed an optimum-seeking procedure and applied it interactively to simulated models of automated warehousing systems. Rosenblatt et al. (1993) proposed a recursive heuristic optimiza- tionsimulation model for obtaining optimal de- signs parameters for an AS/RS. This model fi nds the physical characteristics of the AS/RS, but the relationshipbetweendimensionofrackand capacity of S/R machine that could affect its performance was not considered in their model. Warehouse designs may be evaluated in several ways, and different measures of effectiveness have been considered by many authors. The most common ones are: throughput as measured by the number of pallets or orders handled per day (Bafna and Reed, 1972; Perry et al., 1984), average waiting time per customer/order or percentage of customers/orders waiting to be served beyond a prespecifi ed fi gure (Perry et al., 1983; Azadivar, 1989), are average travel time of a crane per single/ dual command (Graves et al., 1977; Han et al., 1987; Rosenblatt and Eynan, 1989; Lee, 1997). In most cases, measures of effectiveness are affected by the physical layout design and the method of operation of the AS/RS. As explained above, many authors have studied the optimal design of AS/RS with the rack of equally sized cells for using the concept of unit load. However, in terms of the fl exibility of storage capability, the existing rack confi guration using theconceptofunitloadis ineffi cientand inadequate for the storage of various types and various sizes of customers demands. Moreover, if the various sizes of products are to be stored in existing systems, the space utilization will be considerably decreased due to the increase of lost space in each cell. For the purpose of coping with the business environment that is changing rapidly, Lee et al. (1999) proposed the model of AS/RS with the rack of unequal sized cells. That is, in their model cells within the zone have the same size, but the sizes of cells in the different zones are different in height such that the rack can hold various types of load. Their model will be a good alternative for coping with those problems described above. However, if the quantity of storage demands for different sizes products fl uctuates in large, even the model proposed by Lee et al. (1999) will not basically be able to overcome infl exibility and low space utilization problems in existing rack structure of AS/RS. In order to resolve these drawbacks, in this paper the model of AS/RS with the rack of modular cells is proposed fi rst. Flexibility and space utilization of proposed model are compared to those of existing model through numerical examples. The remainder of this paper is organized as follows: we defi ne the rack structure with modular cells and describe assumptions used in this research in Section 2. Section 3 presents the ARTICLE IN PRESS Y.H. Lee et al. / Int. J. Production Economics 98 (2005) 172178173 optimization model for the determination of mod- ular cell size in height. In Section 4, the effectiveness of proposed model is analyzed with respect to the fl exibility, space utilization and cost. Finally, a summary of the research is presented in Section 5. 2. Assumptions and notations The structure of rack with modular cells to be modeled in this paper is depicted in Fig. 1. The rack structure with modular cells, considered in this research, is defi ned as follows: The rack can handle different sized loads in height, while it maximizes space utilization. The rack consists of modular cells, which allows the loads to occupy more than one cell according to their height. Modular cells have only 4 load-arms (brackets) and they have an open structure with same height (refer the Fig. 1). It is called opening cell because the top and bottom of the cell are not closed from bottom to top of the rack. Thus, the size of modular cell is determined by the size of products so as to minimize total lost spaces in the system. 2.1. Assumptions The following assumptions are made in this research. (1) The warehouse is divided into several aisles with racks on both sides. Thus, there are double racks between aisles and single rack along the walls. The rack has Nllevels and Nb bays of modular cells. (2) The number of S/R machines is equal to the number of aisles. No traversing is allowed between aisles, so they only can serve one aisle. (3) There are no technical problems for the construction of rack with modular cells pro- posed. However, applying this model to the storage of heavy product may be limited. (4) The products of different size and various types can be stored in the system, if the size of width and length of product is fi tted to that of modular cell. (5) The information for the size and expected storage volume of each product is known in advance. However, if the storage volume information of products is unknown, it is assumed that they have the same values, which are calculated by ni n=k: 2.2. Notations In this paper, the following notations are used. ntotal expected storage volume, n P ni nistorage volume of each product to be stored in the system, i 1;.;k hi; li; wiheight, length and width of load type i H; L; Wheight, length and width of an AS/RS, respectively Hm; Lm; Wmheight, length and width of modular cell, respectively Hf; Lf; Wfheight, length and width of cell in existing systems, respectively Na; Nl; Nbnumber of aisles, levels and bays of an AS/RS with modular cells N; N1; N2number of aisles, levels and bays in existing AS/RS, respectively Minumber of cell occupied by a load type i qiratio of nito n, qi ni=n Blength of load-arm (bracket) in the rack ARTICLE IN PRESS Hf Lm Hm bb (a)(b) Fig. 1. The structure of rack with modular cells. Y.H. Lee et al. / Int. J. Production Economics 98 (2005) 172178174 xd e defi ned as least integer greater than or equal to x (ceiling notation) xb c defi ned as greatest integer less that or equal to x (fl oor notation) c1; c2construction cost per modular cell and equally sized cell, respectively 3. Optimal design of rack with modular cell 3.1. Mathematical model On the basis of given parameters and variables, a mixed integer non-linear programming (non- linear MIP) model is developed for minimizing total lost spaces considering initial investment cost subject to several constraints. The objective func- tion is as follows. MinTLSm X k i1 nqiMiHm? hi 2NH ? NlHm 2NfL ? NbLm 2bg 4bNaNb,1 s.t. Min hipHmpMax hi,(2) X p i1 Minqip2NaNbNl and c1NaNbNlpc2NN1N2,3 X qi 1,(4) Wm;Lm max wi;max liif Nw aN l bXN l aN w b; max li;max wi;otherwise; ( (5) Mi hi Hm ? ;MiX1,(6) Mi;Na;Nb;Nl;Nw a;N l a;N w b;N l b40; Integer; where Na W 3Wm ? ;Nl H Hm ? ;Nb L Lm 2b ? , N W 3Wf ? ;N1 H Hf ? ;N2 L Lf ? Nw a W max 3wi jk ;Nla W max 3li jk ; Nw b L max wi jk ;Nlb L max li jk : The objective function minimizes lost spaces within each cell, in the rack height and rack length. Constraint (2) specifi es the physical bound of modular cell height. Constraint (3) is a limitation for total storage volume and cost required. Eq. (4) is a constraint on the storage volume of each product type i to be stored in the system. Eq. (5) specifi es the width and length of modular cells to maximizestoragevolumewithingivenarea physically. Constraint (6) indicates the number of cell required for the storage of a load size hi: Especially, the size of modular cell in height, Hm; is affected by construction cost per cell. The total lost space is in inverse proportion to the construc- tion cost over Hm: On the other hand, in case the different sized loads are stored in existing AS/RS, total lost space is calculated by Eq. (7). In this case, it is assumed that the height of equally sized cell, Hf; is greater than or equal to max hi: Min TLSf X k i1 nqiLfHf? hi 2NH ? NlHf 2NL ? NbLf.7 3.2. Algorithm The proposed mathematical model above is a non-linear mixed integer-programming problem (non-linear MIP). As we know, integer programs belong to a class of problems known as NP-hard. This means that there is no known algorithm for solving these problems such that the computa- tional effort at worst increases as a polynomial in ARTICLE IN PRESS Y.H. Lee et al. / Int. J. Production Economics 98 (2005) 172178175 the problem size. That is, we could formulate the optimization model such as the proposed model above and also the problem could be solved theoretically by using optimization tools such as GAMS, LINGO, C-PLEX, etc. But because it is a NP-hard problem, an heuristic algorithm was suggested in order to solve the problem more effectively. Even though we could obtain the optimal solution by investing so much time and resources, it does not have any special advantage and meaning in practice. Basically, this problem is to fi nd a best solution, which minimizes objective function among the feasible alternatives satisfying constraints. If the required storage volume of product is larger than the total storage volume of rack confi guration (PMinqi42NN1N2), or if it does not satisfy cost constraint, these alternatives of rack confi guration will be removed from the feasible alternative list. In order to solve effectively the model in this paper, heuristic algorithm is suggested as follows. Fig. 2 shows the procedure for deciding modular cell size. Algorithm for the decision of modular cell size Step 1: Decide the value of Wmand Lmby Eqs. (3)(6) and known product sizes. Go to step 2. Step 2: Determine the initial value (a) of Hmand incremental step size, b. Let Hm a and go to step 3. Step 3: Calculate the parameter of Miand rack dimension parameters, N, N1, N2, Na, Nl, Nb. Step4:Checkthefeasibilityoftherack confi guration created by Hm a. If it satisfi es feasibility constraints, add it to a feasible alter- native list and go to step 5. Otherwise, let Hm Hm+b and go to step 3. Repeat step 4, until the height of modular cells becomes less than or equal to max hi, Hmpmax hi. Step 5: Select a best alternative, TLSm?; in the feasible alternative list. The best solution can also be altered among feasible alternatives by decision- makers. 4. Numerical studies In this section, the effects of proposed model are investigated through numerical examples and their results are compared to that of previous model. It is also illustrated that how the objective function behaves for decision variables. In particular, the effect of the change of modular cell height and qi on TLSmis investigated. Space utilization as the measure of effectiveness for the proposed model is used and it is calculated by (TLSf?TLSm)/TLSf. The increase of space utilization will improve the storage capacity and fl exibility of the warehouse system. Thus, the increase of space utilization will come to the decrease of cost in terms of system construction and operation. The best solution of modular cell height is determined such that the total lost space over the Hmis minimized with satisfying storage volume required for the all type of loads and cost constraint. Tables 1 and 2 show the test data for three examples and the summary of solutions, respectively. ARTICLE IN PRESS Decide the value of Wm and Lm Let Hm = = min hi Calculate Mi , N, N1, N2 Na , Nl , Nb Mi nqi 2NN1N2 c1NN1N2 c2 Na Nl Nb qi = 1 Calculate objective function and add it to a feasible alternative list TLSM =(Hm , TLSM), (Hm , TLSM), Choose TLSM* = min (Hm , TLSM), (Hm , TLSM), Hm Max hi Stop Hm=Hm+ ? No No Yes Yes Initialize Fig. 2. The procedure for deciding modular cell size. Y.H. Lee et al. / Int. J. Production Economics 98 (2005) 172178176 Thenumberofalternativessatisfyingthe feasible condition is limited by Hm: It is also affected by the value of qifor each load type i: The performance of proposed model for space utiliza- tion is far better than that of existing model as shown in Table 2. It is increased as much as 25.4% in terms of space utilization when construction cost is considered. It means that the proposed model can store 119 unit loads more than the model of equally sized cells. This value, also, can be interpreted as the increase of shadow price or the reduction of risk for the storage demands of unexpected customers. As the rack size in height, Hm; is decided based on the required storage volume of each product type and size, the lost space is always less than that of present model which is considered in equally sized rack confi g- uration under the unit load concept. It means that the space utilization is improved in every case; there is only the difference of large or small. Fig. 3 shows the relationship between lost space and construction cost according to the change of Hm: As shown in Fig. 3, the construction cost can also be reduced considerably in comparison with the model of equally sized cells, Hf 1:2: As stated above, the optimal size of modular cells is affected by the storage volume of each product type i; rack confi guration and construction cost per cells. In addition to these advantages, the perfor- mance evaluation of proposed system could be easily estimated by existing model according to the storage policy. That is, the rack structure with modular cells can accommodate all types of storage policies that can be changed according to storagedemands.Consequently,allofthese advantages and effectiveness are entirely derived from the structural changes of rack of an AS/RS. ARTICLE IN PRESS Table 1 Test data for examples iEx. #1Ex. #2Ex. #3liwihi qinqiqinqiqinqi 1103150103150206300 2154725103150206300 3206300103150