Abstract
This paper addresses the order picking problem in a material handling system consisting of multiple carousels and one picker. Carousels are rotatable closedloop storage systems for small items, where items are stored in bins along the loop. An order at carousels consists of n different items stored there. The objective is to find an optimal picking sequence to minimizing the total order picking time. After proving the problem to be strongly NPhard and deriving two characteristics, we develop a dynamic programming algorithm (DPA) for a special case (twocarousel storage system) and an improved nearest items heuristics (INIH) for the general problem. Experimental results verify that the solutions are quickly and steadily achieved and show their better performance.
1. Introduction
Carousels storage system is commonly referred to as an automated computer controlled system which is widely used to store and pick small, light, and highly demanded items, as an effective warehousing facility. It has been used in links of the supply chain: by manufacturers, distributors, and retailers. Standard carousel system consists of a carousel which may hold many different products stored in bins that rotate along a closed loop and a picker (either human or robotic) who occupies a fixed position at the front of the carousel (see Figure 1). To retrieve a product, the carousel system automatically rotates (clockwise or counterclockwise) the bin with the requested product to the position of the picker. Accordingly, it is referred to as a producttopicker system. The operator may effectively use the rotation time of the carousel for activities such as sorting, packaging, or labeling of the retrieved goods. To expand the merits and further reduce the waiting time for a picker, multiple carousels are integrated so that while a carousel is for picking, others can convey items simultaneously. Some of its advantages include increased efficiency derived from bringing parts to picker instead of sending picker to parts, high speed storage and retrieval, low installation and operating cost, and efficient space utilization.
While companies are finding it important to optimize the operation of their carousel systems to gain a competitive edge in such environments, there are quite a few technical publications on the planning and control of carousels. Bartholdi III and Platzman [1] consider sequencing of picks in a single order. They assume that the time needed by a (robotic) picker to move between bins within the same carrier (or shelf) is negligible compared to the time to rotate the carousel to the next carrier (or shelf). This assumption reduces the problem to finding the shortest Hamiltonian path on a circle. They present a linear time algorithm that finds an optimum solution. Wen and Chang [2] also consider sequencing picks in a single order. They assume that the time to move between bins within the same carrier or shelf may not be neglected. They present three heuristics for this situation, based upon the algorithm in Bartholdi III and Platzman [1]. Several authors have considered the situation where the orderpicker consecutively picks multiple orders thereby completing all picks in an order before commencing with the next order; that is, all picks in an order are performed consecutively. Ghosh and Wells [3] and van den Berg [4] present efficient dynamic programming algorithms that find an optimal pick sequence for picking multiple orders when the sequence of the orders is fixed (the sequence of the picks in the orders is free). Bartholdi III and Platzman [1] consider the problem when the order sequence is free, yet picks within the same order must be performed consecutively. They impose the extra constraint that each order is picked along its shortest spanning interval and present a heuristic for the problem with the extra constraint. Van den Berg [4] presents a polynomial time algorithm that solves the problem with the extra constraint to optimality. The author also shows that the solution of the algorithm for the problem with the extra constraint is at most 1.5 revolutions of the carousel above a lower bound for the problem without the extra constraint.
In this paper, we focus on the deterministic single order picking problem in a multiple carousels system with a single order picker in which carousels are arrayed in the picking area, each with one picking station, and are independently controlled. We will present a formal problem description in Section 2. In Section 3, the problem under study is proved to be strongly NPhard. Some classical characteristics are put forward in Section 4 and an approximation algorithm is constructed for a special case (double carousel system) of the problem in Section 5. In Section 6, an improved nearest neighbor heuristics (INIH) is designed. We present our conclusions in the last section.
2. Problem Description
In the multiple carousels system depicted in Figure 2, there are multiple identical carousels, in which each one rotates individually either clockwise or counterclockwise. There is only one picker moving among the front of each carousel performing picking operations so as to collect all items from the picking points to the I/O (Input/Output) station. It is obvious to find that the picking time of the picker has to be considered explicitly, which is different from single carousel order picking problems. We are given an order with items randomly placed in the system. The objective is to find a picking scheduling to minimize the total picking time. We assume the following.(1)Each bin stores only one type of item without shortage during picking operation. There may be a few bins in each column, but the moving time among bin levels is negligible.(2)Rotation speed of the carousels is constant. It means that there is no acceleration or retarding time considered.(3)The capacity of the picker is large enough to contain the total number of items of any order.
The notation used throughout this paper is defined as follows: : number of the carousels; : the distance between any two adjacent carousels; : number of columns in each carousel (assuming that the distance between any two adjacent columns is 1); : number of items to be picked in carousel in one order, and ; : set of items in carousel ; : location of item in carousel ; : picking time of item ; : a feasible sequence of order picking, in which is denoted as the th item picked; : the last item picked in the th carousel before item is picked; : the travel time from item to item ; : the total picking time when all items required are picked; : moving speed of the picker along the picking points (it is easy to know that the moving time is between two adjacent carousels); : relative rotation speed of carousels, without loss of generality we normalize as 1.
By using a mapping process, we can obtain an exact twodimensional array for all storage units in carousels where indicates the carousel number and represents which column the unit locates in that carousel, ; . Therefore, item location is denoted as in the system.
As each carousel rotates individually either clockwise or counterclockwise, the travel time from item to item in the same carousel (denoted as ) is And if item and item do not exist in the same carousel, the travel time from carousel where item locates to another carousel where item locates (denoted as ) is Note that the array will change dynamically after an item is picked by rotating to the picking point. For example, if one item located at is picked, another item location at in the same carousel will become , where (if ) or (if ).
Assume there is a feasible sequence ; the operation time of picking item after (denoted as ) is And the total operation time until all items are picked (denoted as ) is , where item means the start picking point. Hence, the problem studied in the paper is to minimize by finding the best feasible sequence of the order required. It can be simply modeled as follows: When , it is a special case of the problem. We denote the problem under study by P and the special case by SP.
3. NPHardness
In this section we first prove that the special case SP is strongly NPhard by a reduction from the 3partition problem, which is known to be NPcomplete in the strong sense. Then, it is easy to know that the general problem P is NPhard because SP is a case of P.
3.1. 3Partition
Given positive integers and a set of integers with and for . Does there exist a partition such that , , and ?
Theorem 1. Problem SP is strongly NPhard.
Proof. For any given instance of 3partition, construct an instance of SP as follows:
Without loss of generality, assume that . Call items partition items and remaining items enforcing items. We now show that there exists a solution such that for this instance if and only if there exists a desired 3partition.
If Part. Suppose that there exists a desired solution for 3partition. Let be the corresponding partition item sets. Construct a picking sequence . It is easy to see that .
Only If Part. Let be a solution such that . Let and be the minimum total rotating time and total picking time of carousel 2, respectively. It is easy to see that , which means that carousel 2 has no idle time in . Let be the set of items picked on carousel 1 during time period while carousel 2 rotates from to in . To keep carousel 2 from waiting for the picker, we know that .
Let be the total picking time on carousel 1. Then it is clear that there are one enforcing item and at most 3 partition items in , and the total picking time of the partition items . On the other hand, if , for any , it means that at least one partition item is picked after picking the last item in carousel 2. Then , a contradiction.
Theorem 2. Problem P is strongly NPhard.
SP is a case of P, and SP is strongly NPhard. It is easy to know that problem P is strongly NPhard.
4. Classical Characteristics
Lemma 3. Considering the bidirectional single carousel, the optimal picking sequence switched direction at most once [1].
Supposing there are items to be picked in a certain carousel, three types of picking paths will exist as below (see Figure 3). It is obvious that the result of Lemma 3 can facilitate depriving the optimal sequence (linear time complexity method) for a single carousel system.(1)Travel in clockwise or counter clockwise direction until all items are picked (see Figures 3(a) and 3(b)).(2)Travel from location 0 to in a counterclockwise direction; then backward pass location from 0 to in a clockwise direction (see Figure 3(c)).(3)Travel from location 0 to in a clockwise direction; then backward pass location from 0 to in a counterclockwise direction (see Figure 3(d)).
(a)
(b)
(c)
(d)
Lemma 4. If there is an optimal sequence in multiple carousels system, the subsequence of one certain carousel is also optimal.
Proof. Prove it by reduction to absurdity. Let sequence be a feasible solution of the problem in which any subsequence on the specified carousel is optimal with the total operation time in the problem. Consider
And the relevant total operation time is
where the superscript on letter (the same below) indicates the carousel number.
The subsequence on the th carousel is
Assume that there exist an optimal solution in which the subsequence on the specified carousel is not optimal, such as the subsequence on the th carousel is . Consider
And the overall optimal solution is
with the total operation time
There must be .
Taking the position of and in the solution into account, three cases are discussed according to the relations between them and the adjacent positions.(1) The position of item is adjacent to item in the global optimal sequence. (i) They are not in the carousel where items and locate. It is easy to know that the positions exchanging of and will result in the carousel rotating once more at least according to Lemma 3. And we can make it as follows:
That is to say
where . Therefore, from (7) and (11), assumption contradiction.(ii) They are in the carousel where and locate. That is to say that and . The travel distance will be repeated, when the positions of items and in the optimal subsequence according to Lemma 3. Thus,
Combined with (7) and (11), we can make , assumption contradiction.(2) The position of item is not adjacent to in the global optimal sequence. in other words, and are not in the carousel where locates.(i) Items and locate in the same carousel. Hence,
The rest of formulas (7) and (11) are equivalent. (ii) Items and do not locate in the same carousel. So we have the following conclusions:
As a result, we can derive that from formulas (7), (11), (15), and (16), assumption contradiction
Through the above analysis of two cases, we can know that this conclusion of Lemma 4 is established.
In light of Lemma 4, we notice that the amount of the feasible solutions will decrease sharply to be given an order of sequencing items, where denotes amount of carousels in the system and represents the item quantity required in the th carousel . As a result, this conclusion can help us devise better or optimal solution by combining optimal subsequences of each of the carousels which is easy to find.
5. Heuristics
5.1. A Dynamic Programming Algorithm for a Special Case (Double Carousel System)
In the double carousel system, there are two identical carousels. Each carousel rotates individually either clockwise or counterclockwise. There is only a single picker that performs the picking operations. In this section, we develop a dynamic programming algorithm (DPA) for fixed picking sequences on two carousels. Assume that (see Figure 4) is not visited by the optimal path. Then there are two possible forms of the optimal path.(1)Travel from 0 to in a counterclockwise direction; then backward pass from 0 to in a clockwise direction (see Figure 4(a)).(2)Travel from 0 to in a clockwise direction; then backward pass from 0 to in a counterclockwise direction (see Figure 4(b)).
(a)
(b)
The optimal path on each carousel can be found by testing each of distinct arcs with the minimum distance, which is .
For fixed picking paths on each carousel, an optimal picking path can be obtained with the following dynamic programming method;(1)determine the optimal picking sequence for each carousel;(2)find the dominant carousel with longer picking time, and fix the picking schedule for the carousel;(3)schedule operations for another carousel as soon as possible with the constraint of the dominant schedule.
5.2. INIH for General Problem
Based on the previous analysis of two classical characteristics, we develop an improved nearest items heuristics (INIH) algorithm to solve the general problem in which all items required are stored randomly in carousels.
Let ordinal set be the optimal sequence of the th carousel, where , =, and the coordinate of element is (see Algorithm 1).

The sequence of all elements of the ordinal set is the order picking solution. It can improve the operation’s efficiency and search space of the general nearest items heuristics effectively so as to get better effect. It is easy to know that the complexity of INIH is linear of its scale according algorithm complexity theory [5]. And it is acceptable on computing time.
6. Performance Study
To explore the computational performance of DPA and INIH algorithms comparing other methods such as conventional search (CS) and nearest item heuristics (NIH), we construct two groups of tests on personal computer PIII 1.5 G Hz by C language. Consider a practical multiple carousels system with the operational characteristics specified as follows:
The two groups of items request: (10, 15, 20, 30, 50) and (100, 150, 200, 300, 500); items located on carousel: coordinate is from randomly and coordinate is from function ; the distance of any two adjacent bins on carousel = 0.5 m; m/s, m/s, and m; Number of bins per carousel is 100, carousels quantity for INIH, and for DPA. , , is from randomly, and .
Three indexes are tested: average order picking time (Ot), standard difference (SD), and algorithm operating time (Ts). The testing results are specified as Tables 1 and 2.
Based on the previous analysis of testing data, we can obtain that DPA and INIH are better than CS and NIH for multiple carousels system. And the quality of INIH and DPA’s solution is well distinctive gradually without no more computing time and standard difference when the problem becomes of larger scale and more complicated. To be summarized, the computing efficiency and effectiveness of INIH and DPA are better accepted and will support strongly in practice.
7. Conclusions
Order picking has long been identified as the most laborintensive and costly activity for almost every warehouse or distribution center. A bad order picking can lead to unsatisfactory service and high operational cost for the warehouse and consequently for the whole supply chain. In order to operate efficiently, the order picking process needs to be robustly designed and optimally controlled. In this paper we formulate the order picking optimization of multiple carousels system and give the mathematical model. According to the characteristics, an approximation algorithm (DPA) is presented for the special case of two carousels, and an improved nearest neighbor algorithm (INIH) is constructed for general problem. Finally, simulation tests are developed on the two heuristics. Experimental results verify that the optimum solutions are quickly and steadily achieved and show their better performance.
As a generalization of our problem, there are a number of issues with different layout and rotating modes which are of interest for further research, for example, the multiple orders sequencing problem in the same environment. It is interesting to investigate that.
Acknowledgments
This work is supported by Science Foundation of Ministry of Education of China (09YJC630088) and Fund of the third phase (2010B, 20111211QN07) project of “211 Engineering” program on National Key Disciplines Development in College of Economics and Management, South China Agricultural University.