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ScienceDirect Procedia Engineering 174 (2017) 579 – 587

13th Global Congress on Manufacturing and Management, GCMM 2016

The Optimization and Scheduling Research of Shuttle Combined Vehicles in Automated automatic Three-dimensional Warehouse Wujun Caoa, Mengzhuo Zhangb,* a

School of Management Engineering, Zhengzhou University, NO.100 Kexue Road High-Tech Development Zone, Zhengzhou,450001, P.R.China School of Management Engineering, Zhengzhou University, NO.100 Kexue Road High-Tech Development Zone, Zhengzhou,450001, P.R.China

b

Abstract This paper studies the scheduling problem about the Automatic Vehicle Storage and Retrieval System controlling of the shuttle combined vehicles and the elevator to complete the storing and picking-up operations on the order in automatic threedimensional warehouse. For the order with the pure storing operations or pure picking-up operations, it directly solve them similar to Traveling Salesman Problem(TSP), and for storing and picking-up compound operations,it will respectively cooperate and schedule the elevator with one or two sets of shuttle combined vehicles to complete the task in the shortest time. Pertinent mathematical models has been established, and have been simulated and optimized with MATLAB.Compared with the conventional random access strategy, the current method can greatly save the travel time of the elevator and shuttle bus. The lower time consumption and higher efficiency indicates that the method presented is of significant value in both theory and practical usage. ©2017 2016The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license © Published by Elsevier Ltd. This Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management Keywords: Automated three-dimensional warehouse, shuttle combined vehicles, optimization and scheduling, genetic algorithm ;

1. Introduction With the automated three-dimensional warehouses springing up and the concept of "intensive storage" being proposed, the requirements of enterprises storage system about delivering into and out of warehouse become everchanging. RGV(Rail Guided Vehicle) is playing an increasingly important role in modern manufacturing, logistics and other industries with its higher speed, lower cost and stability. However, its optimization and scheduling management is still many deficiencies. The optimal scheduling of shuttle vehicles is to seamlessly link the shuttle vehicle system and the automated three-dimensional warehousee system into the AVS / RS (Automatic Vehicle * Corresponding author. Tel.: +86 13137154110. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management

doi:10.1016/j.proeng.2017.01.190

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Storage and Retrieval System) in the automated three-dimensional warehouses, and find the optimal storing / picking-up solution through certain models, so that shuttle vehicles complete high-quality storing / picking-up operations in the shortest possible time, and avoid the conflict of tasks. On the AVS / RS, the first study had been done by Malmborg [1]. Later, Malmborg [2] proposed a state equation model for predicting the proportion of Double Command Cycle (DCC) in AVS / RS. Fukunari and Malmborg [3] developed an efficient cycle time model for AVS / RS and compared their performance with the AS / RS with crane. Their model was based on an iterative computational scheme that took into account the assumption of random storage and it was be similar to queuing models. However, the most relevant articles are done by Carlo and Vis [4].a conveyor, two non-passing lifts that share a mast, multiple transfer shuttles, and a storage rack were included in the AS / RS they studied. They focused on the scheduling problem, where two piecewise linear functions were used to evaluate the alternative solution. Chung and Lee (2008) [5] used the GA and random storage strategy to develop a double-loop mode to order the tasks for the AS / RS with a single shuttle. Speaking specifically, the GA model proposed simultaneously determines the positions and order of tasks. And it also showed that the GA model was superior to two greedy heuristic algorithms. Based on the characteristics of the shuttle vehicles and its combination with the elevator, this paper will use the mathematical modeling knowledge to optimize the travel time of shuttle combined vehicles system (one or two sets of shuttle combined vehicles combined with a elevator) completing the order tasks. And reasonable mathematical models will be established, them contain: (1) pure storing operations; (2) pure picking-up operations; (3) storing and picking-up compound operations. Then based on the genetic algorithm, the mathematic models will be programmed by MATLAB software, and the problem of order picking route will be solved. At the same time, the results will be compared in several examples. 2. Problem description and symbols 2.1. Problem Description The automated three-dimensional warehouse with shuttle combined vehicles make the use of WMS(Warehouse Management System) to analyze orders, with AVS / RS controling the shuttle combined vehicles and elevator. When the orders come, WMS will analyze them, and AVS / RS will control elevator and shuttle combined vehicles to the corresponding layer, then the shuttle mother car will move in the roadway and stay in the end of the cargo aisle, at the same time it will release the sub car, sub-car will inventory or pick up the cargo and then return. For example, the physical map of automated three-dimensional warehouse with shuttle combined vehicles, transforming it into the model showed in figure 1. Elevator raise the cargo in the direction of the z axis, the shuttle mother car send the sub car with cargo or not in the x axis direction, then the sub car will be shipped in y axis direction. The study of the number of shuttle combined vehicles is still relatively lacking, this paper focuses on the optimization and Scheduling of a set of shuttle combined vehicle and two sets of shuttle combined vehicle per elevator, and then compared the results with the results of stochastic allocation. The study is divided into three types: pure storing operation, simple picking-up operation, storing and picking-up compound operations.The optimization ideas of this paper are as follows: x pure storing operations: The task on the order are all inventory items. According to the position coordinates of cargo on the order , the system will sort tasks by the genetic algorithm .And then the shuttle combined vehicles will pick up the cargoes from the starting point to a specified location for inventory, and return to the starting point for picking, and then to another designated location for inventory, so back and forth, until all the cargoes on the order are stored.

Wujun Cao and Mengzhuo Zhang / Procedia Engineering 174 (2017) 579 – 587

x:Roadway of the mother car y: Roadway of the sub car z: Roadway of the elevator

z

y x Fig. 1 Model diagram of automated three-dimensional warehouse with shuttle combined vehicles

x pure picking-up operations: Pure picking-up operations are similar to pure storing operations. The difference is that the tasks on the order are all pick-up operations. x storing and picking-up compound operations: The task on the order are both storing operations and picking operations.The system will coordinate the adjacent storing operation and pick-up operation. It will pick up the cargo after inventory, and then return to the starting point. It will reduce the empty travel time, and sort the pairs of storing operations and picking operations, at last it will find the route with the shortest time. 2.2. symbolic description and hypotheses x x x x x x x x x x x

(xi,yi,zi) vs vm vm’ vz’ vz’ tz tm tm’ ts ts’

location coordinates on the shelf speed of Elevator speed of the no-load shuttle mother car speed of the full-load shuttle mother car speed of the no-load shuttle sub car speed of the full-load shuttle sub car time of the shuttle sub car picks up or unloads cargoes time of the shuttle mother car picks up or unload the no-load shuttle sub car time of the shuttle mother car picks up or unload the full-load shuttle sub car time of the elevator picks up or unload the no-load shuttle mother car time of the elevator picks up or unload the full-load shuttle mother car

In this paper, we must assume the following conditions: (1)If the system has a set of shuttle combined vehicles, after the car finish storing operation,it will still stay at the position of storing operation; (2)If the system has two sets of shuttle combined vehicles, when a set of shuttle combined vehicles(A )are doing the storing operation , the elevator will control another set of shuttle combined vehicles(B). (3)After A(or B) finish the storing operation, the sub car of A(or B) will go back to the position of its mother car,which is the end of the column ,at where the cargo need be stored is. At the same time,ignore the time of sub car return. (4)The initial position of the elevator and the shuttle combined vehicles is at the in / out station. (5)If the system has two sets of shuttle combined vehicles, two adjacent operations are must be not st the same floor.

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3. Optimization and scheduling model of the shuttle combined vehicles 3.1. Models for each elevator with one set of shuttle combined vehicles in automatic three-dimensional warehouse x pure storing operations: Since the initial position of the shuttle combined vehicles and the elevator is at point O and the shuttle combined vehicles return to point O after each stocking, the travel time for each whole storing operation Tc1 is as follows: Tc1

2 zi x x y y  i  i  i  i  t s  t s 'tm  tm '2t z vs vm vm ' vz vz '

(1)

x pure picking-up operations: Similar to pure storing operations, the travel time for each whole picking-up operation Tq1 is as follows: Tq1

2 zi x x y y  i  i  i  i  t s  t s 't m  t m '2t z vs vm vm ' v z v z '

(2)

x storing and picking-up compound operations: Pair the storing operations and picking operations one by one,as several storing and picking-up pairs.That means a storing operation and a picking up operation are a unit.For each storing and picking pair, there are two cases: (a)The cargo space of storing and picking up operations are at the same layer(showed by Tt.). (b) The cargo space of storing and picking up operations are at the different layer(showed by Tb.). Then the time of finishing a pair of storing operation and picking up operation(i means for storing operation, j means for picking up operation) could be showed by Tt and Tb, which are as follows: Tt

Tb

xi  x j xi  x j yi  y j yi  y j 2 zi      4t z  2t m  2t m '2t s ' vs vm ' vz ' vz vm

(3)

zi  zi  z j  z j

(4)

vs



xi  x j vm '



yi  y j vz '



yi  y j vz



xi  x j vm

 4t z  2t m  2t m '2t s  2t s '

Set a 0-1 variate mij for the cargo space of storing operation and picking up operation are at the same layer or not: mij

­1 ˈThe cargo space of inventory and picking up operations are at the same layer ® ¯0 ˈThe cargo space of inventory and picking up operations are at the different layer

(5)

The total time of the storing and picking-up compound operations Td is: Td

¦ >m

ij

Tt  (1  mij ) Tb @

(6)

Then the overall time model is:

T

­ °Tc ° ° ®Tq ° °T ° d ¯

2 zi

¦( v

2 zi

¦( v ¦ >m



xi x y y  i  i  i  t s  t s 't m  t m '2t z ) , Simple inventory operations vm vm ' v z v z '



xi x y y  i  i  i  t s  t s 't m  t m '2t z ) , Simply picking up operations vm vm ' v z v z '

s

s

ij

Tt  (1  mij ) Tb @

, Inventory and picking - up compound operations

(7)

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3.2. Model for each elevator with two sets of shuttle combined vehicles in automatic three-dimensional warehouse The study of each elevator with two sets of shuttle combined vehicles is mainly about how the elevator to schedule and choose the two sets of shuttle combined vehicles x pure storing operations Operational explanation diagram of pure storing operations is as shown in Figure 2. So the elevator will screan car A or B for each storing operation and choose a car that could finish the next storing operation in the shortest time to do the task.The travel time of single storing operation Tc1 :

Tc1

ª z j  zi x j º z j ½ ­x zi y y x z ° , » ¾  t s  t s 't z  min ® i  i  i  i  i  t m  t m 't z , max « vs vm » vs ° «¬ v s ¯ vm ' v z ' v z vm v s ¼ ¿

(8)

x pure picking-up operations Operational explanation diagram of pure picking-up operations is as shown in Fig. 4. The time of single picking-up operation Tq1: Tq1

§x § zi  z j x j · z j · zi y y x z ¸ ¸  t s  t s 't z  min ¨ i  i  i  i  i  t m  t z  t m ' , max ¨ , ¨ vm v z v z ' vm ' v s ¨ vs vs vm ' ¸ v s ¸ © ¹ © ¹

(9)

Roadway of the mother car Roadway of the sub car Roadway of the elevator k (Location of next storing operation )

4

j (Stored by car B, Location of the last storing operation)

3

i (Stored by car A, 2 Location of storing operation which is underway) 1

Fig.2 Plane explanation diagram of the pure storing operations

x storing and picking-up compound operations: Operational explanation diagram of storing and picking-up compound operations is as shown in Fig. 4. Also adopt the way of pairing the storing operations and picking-up operations here. The time of single pair of the storing and picking-up operations Td1 is showed as follows. Roadway of the mother car

Roadway of the sub car

Roadway of the elevator k (Location of next picking-up operation ) 4 The route of elevator if it chooses car A to pick up the cargo that is at point k.

j (Picked up by car A) i (Picked up by car B)

The route of elevator if it chooses car B to pick up the cargo that is at point k.

3 2 1

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Wujun Cao and Mengzhuo Zhang / Procedia Engineering 174 (2017) 579 – 587 Fig.3 Plane explanation diagram of the pure picking-up operations

If i,j,k are at the same layer: Td 1

zi  zk x y y x  k  k  k  k  tm  tm '3t z  2t s  2t s ' vs vm vz vz ' vm '

­x ª z j  zi x j º zk  z j z  zi y y x °  min ® i  i  i  i  k  tm  tm 't z , max « , » v v v v v vs vm » vs ' ' z z m s ° ¬« ¼ ¯ m

½ ° ¾ ° ¿

Roadway of the sub car Roadway of the mother car Roadway of the elevator k (Location of next picking-up operation ) The route of elevator if it chooses car A to pick up j (Store by car A) the cargo that is at point k. i (Store by car B) The route of elevator if it chooses car A to pick up the cargo that is at point k.

(10)

4 3 2 1

Fig.4 Plane explanation diagram of the storing and picking-up compound operations

If only i,k are at the same layer: Td 1

2

x  xk z i xi  x k y i  y k y i  y k     i  4t z  2t s '2t m  2t m ' vs vm ' vz ' vz vm

(11) If only j,k are at the same layer: Td 1

½ ­ y y x z zi ° ° z i  z j x j  xk ,  t z  t s ' max ®  k  k  k  k  t m  t m 't s '2t z ¾ v v v v v v vs ' ' s m z z m s ° ° ¿ ¯

(12) Min{}shows choosing the shortest time from sending car A to point k and sending car B to point k. Then the overall time model is:

T

­Tc' ° ' ° ®Tq ° ' ° ¯Td

¦T ¦T ¦T

c1

, Simple inventory operations

q1

, Simply picking up operations

d 1 , Inventory and picking - up compound operations

(13)

4. Solutions of the models with computational example Genetic algorithm cannot obtain highly accurate results as accurate algorithm, but its results are approximate value, also can get satisfactory solution.The following will analysis and calculate the optimization and scheduling problem about both each elevator with one set of shuttle combined vehicles and two sets of them. The parameters are listed in table 1 as follows :

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Wujun Cao and Mengzhuo Zhang / Procedia Engineering 174 (2017) 579 – 587 Tab.1 Parameters table Parameter name Speed of the no-load shuttle sub car(m/s) Speed of the full-load shuttle sub car(m/s) Speed of the no-load shuttle mother car(m/s) Speed of the full-load shuttle mother car(m/s) Speed of Elevator(m/s) Time of the shuttle sub car picks up or unloads cargoes (s) Time of the shuttle mother car picks up or unload the no-load shuttle sub car(s) Time of the shuttle mother car picks up or unload the full-load shuttle sub car(s)

numerical value vz=1 vz’=0.8 vm=2.5 vm’=2 vs=2/3 tz=1.2 tm=0.8 tm’=1.2

Time of the elevator picks up or unload the no-load shuttle mother car(s)

ts=1

Time of the elevator picks up or unload the full-load shuttle mother car(s) Other time(s)

ts’=2 Tr=1

4.1. Solution for models of each elevator with one set of shuttle combined vehicles Code all the locations of the cargoes that need be store or pick up ,and the code depends on the traversal order.Use Randperm(N) to bring a N*N matrix as the random route,and make use of n*N matrix to store the initial population getting from n random groups. In this paper, the positions of the cargoes in the calculation examples is specified before compilation, but it can also be randomly generated.The data of time is stored in a N*N matrix which is named Tz ,Tz(i,j) is on behalf of the whole time that this system operates for a cargo whose location is point j after operating for a cargo whose location is point i. Here ,several examples with different quantities tasks will be used to verify the models of each elevator with one set of shuttle combined vehicles. Use genetic algorithm to optimize them,and the results are showed in table 2: Tab.2 The results of models of each elevator with one set of shuttle combined vehicles. Item

Task quantity

Travel time before optimization(s)

Travel time after optimization(s)

Optimize efficiency(%)

The number of storing operations=the number of picking-up operations

30

1015.2

977.75

3.7

54

2217.5

2047.1

7.7

The number of storing operations >the number of picking-up operations

46

1234.8

1183.4

4.2

16

680.95

646.15

5.1

The number of storing operations the number of picking-up operations>0 The number of picking-up operations > the number of storing operations>0

Travel time before optimization(s) 642.35 1404.7 1082.9 452.05 1208.2 645.4

Travel time after optimization(s) 589.25 1281.6 1008.7 408.05 1141.7 592.6

Optimize efficiency(%) 8.3 8.8 6.9 9.7 5.5 8.2

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Wujun Cao and Mengzhuo Zhang / Procedia Engineering 174 (2017) 579 – 587

15 50 15 50

Pure storing operations Pure picking-up operations

445.9 1578.4 466.4 1680.9

427.6 1544.2 456.9 1647.4

4.1 2.2 2.0 1.99

4.3. The analysis optimization results In this paper.every Numerical example are running 50 times to Prove the stability and accuracy,here it only shows the result of 50 times for 30 cargo coordinates with one set of shuttle combined vehicles(the number of storing operations =the number of picking-up operations).And get the figure 5 and table 4:

Fig. 5 Results of running 50 times for one set of shuttle combined vehicles with 30 position coordinates. Tab.4 Analysis of results of running 50 times for one set of shuttle combined vehicles with 30 position coordinates. Travel time of random route (s)

Travel time of optimized route (s)

Optimize efficiency(%)

The average

1054.568

974.81

7.54%

The variance

252.1438531

10.82653061

—

With the order of thr same cargo coordinates, table5 shows the comparison of one set of shuttle combined vehicles and the two sets of them: Tab.5 The comparison of one set of shuttle combined vehicles and the two sets of them. Category The number of storing operations = the number of picking-up operations The number of storing operations> the number of picking-up operations The number of storing operations < the number of picking-up operations Pure storing operations Pure picking-up operations

Number of tasks

Travel time of random route (s)

Travel time of optimized route (s)

Optimize efficiency(%)

One set

Two sets

One set

Two sets

One set

Two sets

30

1015.2

642.35

977.75

589.25

3.7

8.3

46

1234.8

1082.9

1183.4

1008.7

4.2

6.9

19

790.35

645.4

746.35

592.6

5.5

8.2

15 15

— —

445.9 466.4

— —

427.6 456.9

— —

4.1 2.0

It can be seen from the table that the order of the routes for accessing is optimized by the models of this paper, and the travel time of combined vehicles and elevator can be shortened obviously, which can save the time cost.This paper analyzes the orders on the whole, and makes the combination of storing and picking-up, which can greatly reduce the journey with no-load , not only shorten the time cost, but also save energy.What is more important is it could improve efficiency .This further proved the feasibility of the model and algorithm. In addition, you can find the travel time of two sets of shuttle combined vehicles is less than it of one set of shuttle combined vehicles whether it is optimized or not. Can be learned that although increasing in one set of

Wujun Cao and Mengzhuo Zhang / Procedia Engineering 174 (2017) 579 – 587

shuttle combined vehicles will bring higher equipment costs, in the long run, the addition of one set of shuttle combined vehicles can also reduce much cost and improve efficiency. 5. Conclusions A reasonable mathematical model for the schedule and optimization of a shuttle combined vehicles has been presented. Particularly, the current project focuses on the problem that each elevator cooperated with one or two set of shuttle combined vehicles. Simulated by the models, a global optimization based on the genetic algorithm has been investigated. The results indicate that the travel time could be greatly shortened by the optimization. Additionally, the elevator with two sets of shuttle combined vehicles could save more travel time than that with only one set. But in some aspects, we still need to continue to make further research. Pickup and Delivery Problem (PDP)with various constraints, such as the constraint of a variety of products is not considered in this paper. Most of them are complex that have not been effectively solved. In addition, this research is just for one and two sets of shuttle combined vehicles. The research in future will also involve more sets of shuttle combined vehicles, which may find the most suitable number of the stes of shuttle combined vehicles to achieve the highest efficiency and lowest cost .

Acknowledgements I would like to express my gratitude to all those who helped me during the writing of this thesis. My deepest gratitude goes first and foremost to Professor Wujun Cao, my supervisor, for his constant encouragement and guidance. Without his consistent and illuminating instruction, this thesis could not have reached its present form.

References [1] Malmborg CJ (2002) ,Conceptualizing tools for autonomous vehicle storage and retrieval systems. Int J Prod Res 40(8):1807–1822 [2] Malmborg CJ (2003), Interleaving rule dynamics in autonomous vehicle storage and retrieval systems. Int J Prod Res 41(5):1057–1069 [3] Fukunari M, Malmborg CJ (2008) ,An efficient cycle time model for autonomous vehicle storage and retrieval systems. Int J Prod Res 46(12):3167–3184 [4] Carlo HJ, Vis IFA (2012) ,Sequencing dynamic storage systems with multiple lifts and shuttles. Int J Prod Econ 140:844–853 [5] Chung E,Lee HF (2008) ,Agenetic algorithm for the generalized sequencing problem for automated storage and retrieval systems. Int J Serv Oper Inf 3(1):90–106

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