Layout model and Its Optimization For Man-to-part Order Picking System *

Abstract: It established the general layout model for picking area of man-to-part order picking system, and it has carried on the specific analysis to the average expected walk distance under the various storage strategies and path strategies through using some of the conclusions of previous studies. The layout model has universality, and the different average expected walk distances can generate different layout modules. According to the composition characteristics of the model, it uses a numerical example to show the method of solving for model and obtains the relevant conclusions of the layout of order picking system.

Key words: Man-to-part order picking system; layout; model; optimization

I. Introduction

Minimizing the operating time of order picking is the goal of all order picking systems, and this goal can be realized through many kinds of ways. Under the given warehouse layout conditions, it can shorten the order picking time and improve the order picking efficiency through adopting the optimized picking path, the orders’ partial picking and appropriate storage strategy. However, whether the layout of the order picking system itself is good or not also is a crucial factor which can affect the order picking efficiency, the difference is that the former belongs to operational level and the latter belongs to the design level. The system layouts which have different qualities may be almost same in construction costs, but their impacts on future operations will be very different. Sometimes the ratio between the actual removal expense and the effective removal expense of the facility’s layout which does not meet the design principles of the logistics system even reaches 332% [1]. Therefore, in the stage of initial planning and layout, it should make comprehensive consideration and planning to hope to play the maximum performance of the order picking system. Based on this point, this paper studies the structure and layout problems of low level man-to-part order picking system under the condition of some given operation strategies to help the designers to bring the order picking efficiency into the consideration in the design process and improve the flexibility of the order picking system.

II. The order picking system and operational assumptions

It considers the low level man-to-part level walk picking system. The pickers walk or drive to the picking area and remove the needed products from the storage position. The pickers and the car can walk from the two opposite directions in the laneway; the goods are stored on the shelves on both sides of the laneway. The relevant assumptions are as follows:

(1) The picking area is composed of a series of determined rectangular shelves, each shelf stores at least more than one type of products.

(2) The maximum cargo space’ height is less than 2 meters [2], the pickers can remove the products in any positions without using any tools.

(3) Ignore the additional time on the vertical direction which is caused by the shelf’s height.

(4) The narrow laneway picking system, namely the pickers can remove the items on both two sides of the shelf at the same time, and it do not need the additional time of shifting from one side of the shelf to another.

(5) A picking order is equal to an order (the order does not be picked) or a batch of orders (batch picking).

(6) There is no demand correlation among the picked products, namely the probability of a product which appears in the order is not affected by the other products in the order.

III. The layout model for picking area

The working time of picking is composed of the following three parts[2-3]: ① the walking time from the beginning to the first item to be picked, the shifting time between each item to be picked and the time from the last item to be picked to the end can be regarded as the sum of the walking time in the storage channel and the lateral channel; ② the processing time in the storage position point: such as the time of finding goods, removing the goods and documents processing; ③ the management time of the path’s start and end: the time of managing and starting the task, for example, the time of obtaining and depositing the picking equipments (the car, roll cage etc.), and the time of obtaining the picking orders. It supposes that the time of management and processing has nothing to do with the storage position strategy and the path strategy, and the walking time is the monotonic increasing function of the walking distance, the goal of minimum picking time is equal to the minimum picking distance[4-7]. Here takes the minimum average walking distance as the goal to establish the layout model of the picking area.

IV. The analysis of the average expected walking distance

4.1 The average walking distance under the random storage strategy

The random distribution strategy except to its many advantages, such as the high space utilization and the strong flexibility, usually also is the benchmarking of improving the other storage strategy, besides, the random storage is used more in practice, when the market changes too quick and it can obtain the demand frequency of products statistics, it usually only uses the random storage strategy. Here firstly analyzes the walking distance under the random storage strategy, and then carries on analysis to the walking distance under the other strategies on the basis of this.

It supposes Dc to express the distance which walks along x direction in front and rear channels in Figure 1, and Da expresses the distance which walks along y direction of storage shelf, the area of the storage area A=xy, r=y/x is called the warehouse’s shape factor. Because the starting and the ending are at the same location, no matter which path strategy is used, the walking distance of the x direction is only related to the number of distribution laneways of the products to be picked, its expression can unified be expressed as[124]:

The walking distance in the picking laneway along the y direction has the relationship with the path strategy, and it is expressed by the mark E[D■■], here x expresses the appropriate path strategy. Thus the pickers’ average expected walking distance in picking area can be expressed by E[D■■] which is related to the path strategy and E[D■■] which has no relationship with the path strategy two parts, that is: D■■(n,y,d)=E[D■■]+E[D■■].

It can respectively determine the expressions of E[D■■] according to the different path strategies in bibliography [2-4, 8], such as the traversing strategy, the midpoint strategy and the maximum gap strategy and so on.

4.2 The average walking distance under the classified storage strategy

The bibliography [3] proves that under the random storage strategy d=(n+1)/2, namely, when the entrance is in the middle position of the front channel, the average expected walking distance is shortest. Based on this conclusion, under the classified storage strategy, it supposes that the position of the entrance of the layout model is sure to be located in the middle of the front channel, and in order to be advantageous for the analysis, its channel number is bilateral symmetry as the Figure 1. It respectively adopts different path strategies to calculate the average expected walking distance according to the different classification layout situation

It carries on the ordering for the goods according to COI principle; storing the goods which have low COI value near the entrance is the common method of the classified storage strategy. The ABC classified storage curve which is based on the COI principle can be expressed by the function as follows[2，8]:

In the formula, x expresses the ratio between the needed storage space and the total storage space; s is the shape factor and expresses the deflection extent of the curve. The greater the s value is, the more gently the ABC curve is, for example, the 50/20, 60/20, 70/20, 80/20 ABC curve under the COI storage principle, the values of s are respectively 0.33, 0.20, 0.12 and 0.07.

4.2.1 The return strategy of the cross laneway mode

The expected walking total distance of picking process likewise is composed of the walking distance of picking channel and the walking distance of front and rear channel two parts. In the mode of cross laneway storage, each picking laneway has the same access frequency; the ABC curve which is based on COI principle in each laneway is same. Therefore, the number of expected picking laneway is the same as that of the random storage strategy, namely E[A]= n1-()m. Thus the expected picking number in each laneway is q = . The distance of the return part depends on the position of the item which is the farthest from the front channel in the laneway, and these q items can be regarded as the random variable according to F(x) distribution, so the ration R that the expected value of the farthest picking position in a laneway accounts for the length of laneway can be expressed [2]:

4.2.2 The traversing strategy of the mode within laneway [8]

Under the classification storage mode within the laneway, the products’ distributions of the storage area in the left and right two sides of I/O are symmetrical, the probability of having a picking position for the two laneways which are in the symmetrical position which is determined by F (x) is same. It should distinguish the odd number and even number of the laneways’ number, when n is the odd number, the probability of a laneway k which has a picking product is:

The laneway j has the same probability with the laneway which is symmetric with k.

When n is the even number, the expression for the probability of laneway k which has a picking product is:

Obviously, the probability of no picking goods in the laneway is (1-Pk)m. Therefore, there is at least a picking product in the laneway, the probability for the pickers to traverse these laneways is 1-(1-Pk)m. It can calculate the expected walking distance in the picking laneways after knowing the probability of different circumstances.

V. The numerical examples of layout optimization for

order picking area

According to the total length of different shelves, it uses the layout model to determine the best layout structure of the picking area. In the situation of certain total length of shelf, with the increase in the number of laneways, the length of the laneway will be gradually reduced. Thus the length of shelf y in each laneway is equal to 1/2 of the ratio of the shelf’s total length L and the number of the laneways. Because it has been demonstrated that the entrance is in the middle of the front channel to be the best position, therefore the example sets the entrance position in the middle of the front channel. It can get the best layout of picking area which relates to the walking distance through considering the number of laneways under the minimum path length.

Because the minimum path length has the relationship with the storage strategy and the path strategy, here takes the traversing strategy of the random storage strategy as the example to carry on optimizing layout for the picking area that its total length of the shelf surface L is 300 meters and the laneway center distance wa is 4 meters. When the entrance is in the middle position, d= (n+1)/2, for simplicity, the walking distance of the picking laneway uses the formula which is similar to formula (1). According to the above assumption, the picking walking distance can be simplified as:

Thus the layout optimization model can be written as follows:

The layout model is the nonlinear function, in addition to asking the laneway’s number to use integers, there also has the absolute value computation which relates to the variables in the objective function, and the variable also is emerged in the upper limit of summation, at present, this model is difficult to be solved through using the authority software of MATALAB and LINGO and so on. But after careful analysis, it is not difficult to find that the first constraint is an integer, the dependent variable y can be directly replaced by the variable n to be substituted into the objective function, at this time the objective function is changed into a function of one variable which only has one variable. It can be seen from the second constraint that when L is equal to 300, the value of n is the integer between 1 and 150, the solution space of the model is very small, and the layout structure only has 150 possibilities (it arranges the shelves symmetrically on both sides of the laneway), namely from 150 meters length of a laneway to 1 meter of 150 long laneways. In the case of a small number of solutions, the exhaustive algorithm is a very competitive algorithm, and it will certainly be able to get the exact optimal solution. In this paper’s layout model for order picking area, the number of laneways of shelves’ layout is not a great number, and it uses the integers, belongs to the discrete problem and its solution space is very small, the exhaustive algorithm can draw the results on the computer less than a second after inputting the correlative data. In order to ensure the correction and effectiveness of the results, it firstly does not consider the integer requirements for n and carries on the integer processing for the upper limited variable in summation; it firstly regards the model as the real number planning, and transfers the bounded scalar nonlinear minimization procedures of METNABLE to obtain the real solutions, then carries on the integer solutions searching around the real solutions, the obtained results are consistent with that of exhaustive algorithm. The following carries on the specific analysis for the computed results.

When L is equal to 300 meters and wa is equal to 4 meters, the numerical calculations show when the size of the picking order m is from 1 to 13, the optimal laneway number is 10; when m is 14, the optimal laneway number is 6; and when m is above 15, the optimal laneway number is 3. When it changes the total length of the shelf and the laneway center distance wa, the value domain range of the optimal laneway number has a little change, but when the picking items reach a certain value (this value is different along with different L and wa, it is 15 in Figure 3), the optimal laneway number is also 3. Here chooses the representative picking order to carry on the comparative analysis within three levels of the value domain range, when the size of the picking order m are respectively 5, 14 and 40, the optimal laneway number are respectively 10, 6 and 3 as the Figure 3. It can be seen from amount of data’s analysis, once m is equal to or above 15, the expected path lengths of one laneway layout and two laneways layout are respectively 225 meters and 195.5 meters, they are the fixed values and have no relationship with the size of the picking order. When m is equal to or above 26, the expected picking path length which corresponds to the minimum laneway is 191 meters, and it does not change along with the changes of m. When L is equal to 300 meters and wa is equal to 3 meters, the distribution of the optimal laneway number has 6 kinds of situations; it is as the Table 1.

It can draw the conclusion from this that when the total length of the shelf is certain, in the picking system layout of narrow laneway, the smaller the picking order is, the greater the number of optimal laneway is; the greater the picking order is, the small the number of optimal laneway is. In the case of the same number of laneway, the more the number of items to be picked is, the longer the expected walking path is. When the size of the picking order reaches to a certain extent, the optimal number of laneway is no longer to change. But in the layout of the greater picking order system, along with the increase of the items in the picking order, in the range of a small number of laneways, the laneway’s number―the expected path length curve can present the jumping or fluctuation phenomena in varying degrees. This phenomenon can be explained that if the number of laneways in picking area is odd number and all the laneways must be visited, thus there is a part of distance which needs to return the entrance and walk repetitively in the last laneway to be visited. If the quantity of the goods to be picked is great, the probability of all the laneways to be visited can be increased, it leads to the increase of the distance which needs to walk repetitively for the last laneway. Along with the increase of the number of laneways, this phenomenon can be reduced. Because when the quantity of the items to be picked is certain, if the quantity of the laneways’ number can be increased, so the probability of all the laneways to be visited can be reduced. The numerical calculations also show that when the total length of the shelf is certain, if it hopes to arrange the least number of laneways, it must meet that the items of the picking order reach certain quantity; when the number of picking goods’ items exceeds this value, the value of the optimal laneway’s number is no longer to change and keeps a fixed value. It can be seen from the situation of the jumping of the smallest number of laneways, the differences between the total lengths of the shelf and the shortest path lengths which are corresponded to the values of two alternate laneways are smaller. Besides, according to these characteristics, when it makes the specific layout decision, it also can be according to the size of the area of the picking area and the actual shape to select a layout of laneway which is the most fitting the actual.

VI. Conclusion

It has established the picking area’s layout model for man-to-part order picking system, it has carried on the specific analysis to the average expected walk distance under the various storage strategies and path strategies through using some of the conclusions of previous studies, and it uses examples to show the solving method of the model. The numerical calculations show: when the total length of the shelf is certain, in the picking system layout of narrow laneway, the smaller the picking order is, the greater the number of optimal laneway is; the greater the picking order is, the small the number of optimal laneway is. In the case of the same number of laneway, the more the number of items to be picked is, the longer the expected walking path is. When the size of the picking order reaches to a certain extent, the optimal number of laneway is no longer to change. But in the layout of the greater picking order system, along with the increase of the items in the picking order, in the range of a small number of laneways, the laneway’s number―the expected path length curve can present the jumping or fluctuation phenomena in varying degrees. Along with the increase of the number of laneways, this phenomenon can be reduced. When the total length of the shelf is certain, if it hopes to arrange the least number of laneways, it must meet that the items of the picking order reach certain quantity; when the number of picking goods’ items exceeds this value, the value of the optimal laneway’s number is no longer to change and keeps a fixed value. It is worth noting that the layout model has universality, the different average expected walking distances can generate different layout models, when the distribution centers make layout decisions, they can use different layout model to design and optimize according to their actual situation.

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