Design and implementation of optimal algorithm for shelves handling based on the change of return position of movable shelves

A shelf‐to‐person picking system based on automatic guided vehicles (AGVs) is developed in the mobile shelf warehouse, which can meet the needs of multi batches, small volume and multi variety picking. In this system, the efficiency of picking is affected by Storage location allocation, optimal combination of picking orders, optimization of handling shelves and AGVs scheduling. This article establishes a mathematical model according to the shortest walking distance of AGV in view of the situation that the return position of the shelf can be changed. In this article, a hybrid intelligent algorithm is designed, and its effectiveness is verified by numerical examples. At the same time, the heuristic algorithm and intelligent algorithm were compared in two cases, and the range of intelligent algorithm parameters was analyzed to verify the stability and effectiveness of the algorithm.


INTRODUCTION
Market pressure under consumption upgrading, massive stock keeping unit (SKU) and uncontrollable labor cost have become the common problems in e-commerce, retail and other industries.The commonly manual picking used in the past has low efficiency and many errors, and the labor cost increases year by year.The traditional automatic equipment overcomes the defects of manual picking, but large investment with a slow payback and insufficient flexibility make it not adapt to multi batches, small volume and multi variety picking in the new retail.In recent years, a picking system of " shelf-to-person" developed from logistics enterprises not only meets the needs of multi batches, small volume and multi variety picking, but also saves a lot of labor.It has become the main picking method in the picking industry at present.The shelf-to-person picking system is composed of picking stations (seeding wall), mobile stock shelves and AGVs.In order to improve work efficiency, there are generally multiple picking stations.A picking station which is composed of multiple storage spaces can pick multiple orders at the same time.The mobile stock shelves consist of multiple lanes, whose ground is pasted the two-dimensional code for positioning and navigation.Each shelf has multiple storage spaces and can store a variety of goods.The number of AGVs is configured according to the operation frequency, and multiple AGVs can work at the same time.Firstly, after all of orders are combined and optimized, they are released to the picking station.According to the orders, the optimized shelves combination to be transported is formed.Secondly, AGVs carries the shelves to the picking stations in turn for picking operation.After the picking is finished, the shelves are moved back to the storage area by AGVs.
In the shelf-to-person picking system, the research on improving picking efficiency focuses on storage location allocation, group batch of orders, optimization of handling shelves, AGV real-time scheduling and so on.In References 1 and 2, K-mean clustering method is applied to order batching in a shelf-to-person picking system.In Reference 3, a mathematical model was constructed with the goal of minimizing the sum of manual picking costs and AGV handling costs in order batching, and an improved adaptive genetic algorithm was designed to solve the problem.In Reference 4, based on the order frequency of items, Chaoqun et al. use the order volume index method to solve the storage location optimization problem, and design a tabu search algorithm to solve the storage location optimization model.In Reference 5, the robotic mobile fulfillment system can improve the picking efficiency by 3-4 times, and develop a two-stage stochastic model.To analyze robot assignment strategies for multiple storage zones.In Reference 6, items were clustered based on the correlation between orders and items, and a greedy algorithm was designed to solve the model.In References 7 and 8, a path planning model was established for shelf-to-person picking system, and the problem was solved by improving the A* algorithm.
In Reference 9, Fangyuan et al. design a fast and effective algorithm to solve the storage allocation problem by using the basic idea of community structure division of complex binary network.Wang10 carry out grid clustering optimization on orders.Li11 construct a 0-1 integer model to determine which shelves should be carried and optimized these shelves with a heuristic algorithm.The model takes the distance to and from the picking station as the goal, which is not applicable to the change of return position.At the same time, the algorithm will increase the amount of calculation with the increase of the number of shelves in the warehouse.The randomness is strong in shelf exchange, so many feasible solutions will be excluded.
Boysen et al.'s 12 study the arrival picking system with only one picking station and establish a mixed-integer programming model.The sequence of shelves arriving at the picking station is optimized and orders are executed by minimizing the number of shelves arriving at the picking station.However, it is only suitable for the situation where there are few varieties of goods, and it is inefficient for the situation where there are many varieties of goods.
The shelf-to-person picking system is very important because it promotes machine to person substitution, reduces labor costs, reduces errors, and improves picking efficiency.In the task optimization of the shelf-to-person picking system, most of the current research focuses on the situation where the shelves return to their original position, and there has been no research on the situation where the return position can change.This article mainly studies the optimization problem of the handling task when the shelf return position is not fixed after the order is released to the picking station of the shelf-to-person picking system.Aimed at the shortest total path of AGV, a 0-1 programming mathematical model is established, and an intelligent algorithm is designed.Additionally, the simulation is carried out with practical case data, and the sensitivity of the algorithm parameters is analyzed.For both fixed and variable shelf return positions, numerical simulations were performed using the heuristic presented in Reference 11 and the genetic algorithm presented in this article.The results of this study will provide valuable insights into relevant engineering fields.The rest of this article is organized as follows.The mathematical model is presented in Section 2. The algorithm is designed in Section 3. The calculation example is performed in Section 4. Two case and two algorithms are compared and analyzed in Section 5.

Problem description
Suppose there are K kinds of goods and M shelves in the warehouse, and each shelf has Q storage spaces.Each storage space can store no more than one kind of goods.In order to ensure continuous operation, the picking station is equipped with multiple shelf parking positions, one for operation and the rest for waiting operation.Several AGVs in the warehouse are parked under the shelves or other storage areas.The AGV initial position is at the charging position.For example, Figure 1 is a warehouse layout diagram with six rows and eight columns, with three charging stations, one picking station, and three shelf parking spaces.The position of the shelves in the warehouse is known, and the distance from each shelf to the picking station can be calculated.In a picking operation, AGV travel distance can be divided into the distance from the initial position to the position of the shelf to be moved, the distance from the warehouse to the picking station, and the distance from the picking station to the warehouse.Assume that R ′ is the set of shelves to be moved in the warehouse, and the total number is N m ; L is the set of shelf locations to be moved in the warehouse; L m is the m-th location, and its coordinates are(x m , y m ); The set RL represents the relationship between the shelf to be moved and the location of the shelf; There are w positions on the picking station where shelves can be parked; and there are w ′ robots is used for handling in the warehouse.When the handling task is completed, the total traveling distance f of AGVs as in Equation ( 1): where z is the sum of shelves handling distance from warehouse to the picking station, z r is the sum of shelves handling distance from picking station to warehouse, z a is the sum of AGVs' distance from the starting position to the position of the shelves to be moved.

Theorem 1.
In the shelf-to-person system, for a task set of shelves to be handled , the sum of shelves handling distance from warehouse to picking station is equal to the sum of shelves handling distance from picking station to warehouse.That is, z = z r .Proof: Let z t is the distance from a shelf to the picking station, we have ( Let z r t is the distance of a shelf from picking station to warehouse, we have There are two cases according to the position of the shelf from the picking station back to the warehouse: 1.When the return position of the shelf is consistent with the starting position, the distance from each shelf to the picking station is equal to the distance from the picking station to the warehouse, that is, Therefore That is as: 2. When the return position of the shelf is not consistent with the starting position, the distance from each shelf to the picking station is not equal to the distance from the picking station to the warehouse, that is, But For ∀z t ∃z r i which satisfy so that: that is, Theorem 2. In the shelf-to-person system, for given set of shelves to be handled R ′ , when z a is the minimum, the total distance for AGV to complete the handling task R ′ is the minimum.

Proof:
Let z a t is the distance from AGV parking position to handling shelf, we have Suppose the AGV initial position set is The element L ′ n represents the n-th location, and its coordinates are (x n , y n ).The set of shelf locations to be handled in the warehouse is The element L m represents the mth location, and its coordinates are (x m , y m ).
The set of shelves and the shelf positions is When the tth shelf is handled, the distance between positions L m and When the ith shelf on the storage location m is moved, represents the Manhattan distance from the initial location of AGV n to storage location m.
As shown in Figure 2, for any And then let We have Therefore The following equation holds: Remark.
1.In the shelf-to-person system, z a is a function of R ′ , and it can be defined as follows: F I G U R E 2 Schematic diagram of distance from AGV to the shelves to be handled.
The function represents the sum of the distances traveled by each AGV during the completion of the handling task set R ′ from its starting position to the position of the shelf.
2. According to the two theorems, the total AGVs' travel distance f can be written as follows: The function represents the sum of the distances traveled by the AGV after completing the handling task set R ′ .

Mathematical model
Supposed that n combined orders need to be picked over a period of time, and picking station is independent of each other.During picking, the shelves need to be transported to the picking station, therefore, it is important to study which shelves should be transported, so as to minimize the transportation cost and maximize the transportation efficiency.
On the basis of above description, the picking problem based on "shelf-to-person" can be expressed as the following 0-1 mixed integer programming model: where i is the Warehouse shelf index, i = 1, 2. … .M, j is the Item index, j = 1, 2. … .K, K is the total number of items.M is the total number of shelves.C i is the distance from the ith shelf to picking station, i = 1,2, … ,M.b j is the total number of the jth good in order, j = 1,2, … ,K. a ij is the number of the jth goods on the i-th shelf, i = 1,2, … ,M; j = 1,2, … ,K.x i is 0 or 1, 0 means the ith shelf does not need to be carried, 1 means the ith shelf needs to be carried.

Remark.
1. Objective function(23)indicates the shortest travel distance of AGVs to complete picking orders.
2. The constraint condition (24) indicates that the sum of the number of items on shelves to be transported should be greater than or equal to the number of items to be picked in a picking order.3. The constraint condition(26) represents a collection of shelves to be moved.

ALGORITHM DESIGN
0-1 programming is a special case of integer programming, which is widely used in practice.Generally, 0-1 programming solution includes exhaustive method, implicit enumeration method and goal ranking method.Although these methods can find an optimal solution, if the above methods are directly applied to problem (23), there will be a problem that calculation scale will increase exponentially with the increase of the number of warehouse shelves, which lead it will be impossible to find the solution.In addition, every picking order may have nothing to do with many shelves.Therefore, in the algorithm design, the stock shelves that have nothing to do with the order or have low correlation should be excluded and then the optimal solution is solved in the remaining small space to improve the calculation speed and accuracy.
In the view of the above analysis, this article proposes to use the hybrid intelligent algorithm to solve the problem (23), that is, first use the heuristic algorithm to generate a better initial population, and then use the genetic algorithm to solve the optimal problem.

Algorithm flow
The proposed genetic algorithm is summarized as below.
Step 1: The initial control parameters were determined such as population S, generation gap g, crossover probability P c , mutation probability P m , repeated crossover coefficient u, repeated variation coefficient v and termination algebra Ter.
Step 2: The heuristic algorithm generates S chromosomes, as described in Section 3.3, satisfying the constraints (24).
Step 3: for i = 1,2, … , Ter do Step 4: Calculate the fitness of each chromosome according to the target value and sort it from large to small.
Step 5: Select individuals participating in genetic operation through roulette and elite reservation strategy.
Step 6: Crossover and mutation operations.In order to ensure that the excellent individuals produced in each generation can be successfully inherited to the offspring, the S(1-G) parents with the best fitness among the parents directly enter the offspring.Cross and mutate the remaining chromosomes and test the feasibility of the offspring with the constraint (24).The offspring will be abandoned and the parents will be retained to ensure that the population size meeting the constraints remains unchanged, if the offspring are not feasible after repeating a certain number of times operations of crossover and mutation.
Step 7: end for Step 8: The best chromosome is given as the optimal solution.
The specific implementations of steps one to eight are explained in Figure 3.

Genetic coding scheme
Since the model in this article is 0-1 programming, the binary coding strategy is adopted.0 means that the shelf does not need to be handled, and 1 means that the shelf needs to be handled.

"Shelf heat" evaluation and ranking
If the shelves which are less relevant to picking orders are excluded, it can reduce the calculation scale.In order to evaluate the correlation between shelves and orders, the evaluation method of shelf heat is designed as follows.Assuming P i ={p i1 ,p i2 , … ,p iK } T is the column vector of goods that should be picked on the ith shelf, then, Step 1: Initialize control parameters Step 2: Heuristic algorithm to generate initial population of S GEN=Ter The termination algebra between the quantity of goods that should be picked on the shelf and the distance from the shelf to the picking station is defined as the heat of the shelf, that is, . Sort the heat of the shelf in descending order and form a shelf sequence: R = {r 1 ,r 2 , … ,r M }.

Chromosome formation
The process of generating the initial population from the flowchart in Figure 4 is as follows: 1. sub matrix of X is extracted as the chromosome matrix of the initial population If the column vectors of X are zero vectors, then it means that the shelf has low correlation with the picking order (there are few goods to be picked on the shelf or the shelf is far away from the picking station) and they should be deleted from X. Let x is submatrix of X after the deletion and x is described as follows: where x is called the initial population chromosome matrix, m is the number of chromosomes, n is the number of shelves.Each chromosome is composed of n elements and the ith chromosome is x i. ={x i1 ,x i2 , … ,x in }.The new shelf sequence corresponding to n shelves is RR ′ ={r 1 ,r 2 , … ,r n }.
According to the new shelf sequence RR ′ ={r 1 ,r 2 , … ,r n }, obtain the distance vectors from the shelf to the picking station and describe them as The storage volume vector of the corresponding n shelves is defines as

Chromosome fitness assessment
Because the problem which is solved is a minimum problem, the fitness evaluation function is defined as the reciprocal of the objective function.Considering that the reciprocal is too small to compare, the fitness evaluation is defined as follows: Calculate the fitness fit(k)(k = 1 … … m)of each individual and sort them in decreasing order, it means that chromosomes are sorted from good to bad.

Selection operation
The function of choice is survival of the fittest.The basic idea of roulette algorithm is to convert the fitness of each individual into the probability of being selected, so as to achieve better survival of the fittest.But in the concrete operation, according to each generation of the random number interval to decide whether to select individuals and participate in the genetic operation.People with low fitness also have a chance of being selected.People with high fitness are more likely to be selected, but there is also the possibility of not being selected, leading to the loss of good genes.To avoid this, roulette is combined with elite retention strategies to select the best individuals from each generation directly into next, ensuring that highly adaptable individuals are not lost.This article adopts the method of Roulette and elite reservation strategy.
In roulette strategy, the probability of individual-k being selected is If an individual's selection probability is high, then it will be selected for many times, and its genetic genes will expand in the population.If the individual's selection probability is low, the probability of eliminated will be relatively At the same time, the optimal preservation strategy is adopted so that the optimal m*(1−G) individuals directly enter the next generation.Therefore, m*G individuals need to be selected to participate in genetic operation.

Cross operation
Based the selection operation, the cross operation steps are as follows: Step 1: two adjacent individuals x i , x i+1 in the population are selected as crossover objects.
Step 2: Generate the random number z in [0,1].If z > P c , the crossover operation will not be carried out and x i , x i+1 directly enter the offspring.If z ≤ P c , then the crossover operation will be carried out and form the offspring Step 3: The crossover operation adopts single point crossover.If the individual has n elements, then an integer t within n is randomly generated as the intersection position.The two individuals exchange some gene codes at the intersection position to form two sub individuals.Assume that the chromosome is divided into two parts at point t, that is, ) .
Then, the individuals after crossing are The above cross operation case is shown in Figure 5. Step 4: Use the constraints (24) to test the feasibility of offspring x ′ i ,x ′ i+1 .If it is not feasible, repeat Step 3 at most u times.If the offspring are still not feasible, then the original individuals are retained to ensure that the population size satisfying the constraints remains unchanged.
Step 5: Repeat Step 1 to Step 4 until the cross operation of all parents is completed, and then select m*G children.

Mutation operation
Step 1: For each individual, a random number ζ in [0,1] is generated.If ζ < P m , then select the individual as the parent of a mutation, otherwise the individual will not mutate.
Step 2: For the parent generation of mutation, V = (x 1 , x 2 … x n ), generate a random integer within n as the mutation point.Take the inverse mutation, that is, 0 becomes 1 or 1 becomes 0, and get a new chromosome.
Step 3: Test whether the offspring chromosomes meet the model constraints (24).At the same time, the improvement about the mutation algorithm is made.In order to ensure that the optimal gene is not destroyed and make the mutation develop in a good direction, the fitness of the chromosome after mutation should be better than that before mutation.If the individual after mutation meets both the constraints (24) and the conditions for fitness improvement, the original chromosome is replaced by a new chromosome; Otherwise, return to step 2 and repeat mutation V times until a qualified mutant chromosome is formed.If there are no children generated which is meet both the constraints (24) and the conditions for fitness improvement, the children will be discarded and the parent generation will be retained to the next generation.

Formation of offspring
A new offspring composed of m individuals is formed by the optimal m*(1−G) individuals from the parent generation and m*G individuals formed by cross and mutation operation.

Iterative process termination
Repeat Steps 3.4-3.8until the number of iteration Ter is reached.Meanwhile, record the maximum fitness and the corresponding individual value of the population in the whole iterative process.

Description of calculation example
A warehouse layout (6 columns, 8 rows, and 48 groups of shelves) is shown in Figure 1.In order to facilitate calculation, we establish a rectangular coordinate system with the picking station as the origin and a shelf width as the length unit to

TA B L E 3
The types and quantities of goods to be picked in picking orders.

The picking order
Items amount The total 15 18 16 19 form the position coordinate system of the warehouse plane.All of shelves are arranged back-to-back and each two rows of shelves share a channel.The path length from the left and the right shelf of a channel to the picking station is the same.Table 1 shows the distance from a shelve in a row or column to the pick station.Table 2 shows AGVs' initial position coordinates and the coordinates of the picking station.Table 3 shows the types and quantities of goods to be picked in picking orders.Table 4 shows the type and quantity of goods placed on the shelf, as well as the abscissa x and ordinate y of each shelf.

Analysis of results
The proposed algorithm is implemented with MATLAB r2014a.Figure 6 shows the convergence process of the objective function, which tends to be flat and converge to the optimal value when the number of iterations is less than 100.The optimal value is fit(k) = 0.8621, the corresponding shortest path z = 115.6,and the corresponding handling scheme is the 1st, 2nd, 3rd, 9th, 10th, 25th, and 33th shelves need to be transported.

Parameter sensitivity analysis
To further verify the validity of the model and algorithm, the mean, standard deviation, and running time of fitness evaluation function were compared when the parameter generation gap g, the population size S, the iteration times Ter, the repeated crossing times u, the repeated variation times v, the crossing probability P c and the variation probability P m were taken different values.Each set of data is run 10 times with the same parameters and gets the mean of the program run time, the mean of the fitness function, and the standard deviation, as shown in Table 6.By comparing Table 6, it can be concluded that 1.In order to improve the calculation efficiency, generally a value of G = 0.95, S = 40, Ter should bigger than 150, u and v should be between 10 and 20, P c should be bigger than 0.7 and P m should be smaller than 0.3 according to the analysis of experimental results.2. Under different parameter conditions, the maximum average value of the fitness function is 0.86136 and the minimum is 0.80488.The maximum standard deviation is 0.042492 and the minimum is 0.00222.This means that the proposed hybrid intelligent algorithm is insensitive to parameter values and has good robustness.

COMPARATIVE ANALYSIS OF DIFFERENT ALGORITHMS
In order to further validate the effectiveness of the algorithm proposed in this article, numerical simulations were conducted using the heuristic algorithm proposed in Reference 11 and the genetic algorithm proposed in this article, with the return position of the shelves unchanged.And compare with the changes in the return position of the shelves.When the return position of the shelves remains unchanged, only one AGV is suitable for operation and the shelves are sequentially transported to the picking station.The distance traveled by an AGV is determined by the sum of the distance from its initial position to the shelves and the distance to and from the shelves and the picking stations.Referring to Figure 1 and Tables 1-4, assuming that AGV 3 starts from the initial charging position and completes the processing work, the statistics is shown in Table 7.
It can be seen from Table 7 that the efficiency of the genetic algorithm in this article is 21% higher than that of the algorithmic in Reference [11], and the efficiency of the intelligent algorithm in this article is 29% and 10% higher than that of the first two algorithmic.The intelligent algorithm in this article is applicable to multi AGV collaborative operations, and the efficiency of picking work will be greatly improved.

Step 8 : 3 4
Output Diagram of hybrid intelligent algorithm.Heuristic algorithm diagram for generating initial population.

F I G U R E 5
Example diagram of cross operation.

TA B L E 5 F I G U R E 6
The algorithm parameters.Convergence process of objective function.On the bases of Table3 anothermatrix b is defined as follows: b = [15 18 16 19], Distances from location of the shelves to the picking station.AGVs' initial position coordinates.
TA B L E 1
Comparison of different parameter combinations.
TA B L E 6