Mathematical Model and Algorithm for the Task Allocation Problem of Robots in the Smart Warehouse

Transcription

1 Aerican Journal of Operations Research, 205, 5, Published Online Noveber 205 in SciRes. Matheatical Model and Algorith for the Tas Allocation Proble of Robots in the Sart Warehouse Zhenping Li, Wenyu Li School of Inforation, Beijing Wuzi University, Beijing, China Received 0 Septeber 205; accepted 6 Noveber 205; published 9 Noveber 205 Copyright 205 by authors and Scientific Research Publishing Inc. This wor is licensed under the Creative Coons Attribution International License (CC BY). Abstract In the sart warehousing syste adopting cargo-to-person ode, all the ites are stored in the ovable shelves. There are soe warehouse robots transporting the shelves to the woring platfors for copleting order picing or ites replenishent tass. When the nuber of robots is insufficient, the tas allocation proble of robots is an iportant issue in designing the warehousing syste. In this paper, the tas allocation proble of insufficient warehouse robots (TAPIR) is investigated. Firstly, the TAPIR proble is decoposed into three sub-probles: tas grouping proble, tas scheduling proble and tas balanced allocation proble. Then three sub-probles are respectively forulated into integer prograing odels, and the corresponding heuristic algoriths for solving three sub-probles are designed. Finally, the siulation and analysis are done on the real data of online boostore. Siulation results show that the atheatical odels and algoriths of this paper can provide a theoretical basis for solving the TAPIR proble. Keywords Sart Warehouse, Warehouse Robot, Tas Allocation Proble, Matheatical Model, Heuristic Algorith. Introduction In the sart warehousing syste adopting cargo-to-person ode, all the ites are stored in the ovable shelves neatly arranged in the warehouse, and picing worers stand in front of stationary picing platfors. There are soe isoorphic warehouse robots running along the ared line to coplete order picing or ites replenishent tass []. When starting order picing or ites replenishent tass, the coputer syste controls one How to cite this paper: Li, Z.P. and Li, W.Y. (205) Matheatical Model and Algorith for the Tas Allocation Proble of Robots in the Sart Warehouse. Aerican Journal of Operations Research, 5,

2 or ore warehouse robots to the sites of assigned ovable shelves; the warehouse robots lift up their assigned shelves and transport the to the corresponding picing platfors. After the picing worers coplete the order picing or ites replenishent tass, the warehouse robots transport the ovable shelves bac to their own original locations. In the warehousing anageent syste, coplex transportation wors are done by the warehouse robots instead of huan; the woring efficiency is iproved in soe ways [2]. Since different ites of one order ight be laid in different shelves. When picing an order, the robots usually need to transport ore than one shelf to the picing platfors. In another words, the warehouse robots need to coplete ultiple tass for picing one order. In order to iprove the picing efficiency, it needs the ultirobot cooperation to coplete ultiple tass. The tas allocation proble of warehouse robots greatly affects the woring efficiency of sart warehousing syste. Soe researchers investigated the tass allocation proble of abundant robots (TAPAR) where the nuber of robots was larger than the nuber of tass to be copleted in picing one order. Guo adopts the aret auction ethod to solve the tas allocation proble of abundant obile robots (TAPAR) in a sart logistic centre [3]. Li uses the genetic algorith and ant colony algorith to study the TAPAR proble [4]. Christopher et al. propose the resource allocation proble of ulti-vehicle syste in the sart warehouse by using Alphabet Soup ethod [5]. And Li designs the heuristic algorith for solving the TAPAR proble [6]. In practical, the nuber of robots is often saller than the nuber of tass to be copleted in picing one order, which eans the robots are insufficient. By now, little researches focused on the tas allocation proble of insufficient obile robots (TAPIR). In this paper, the TAPIR proble in sart warehousing syste based on cargo-to-person ode is investigated. We partition the tas allocation proble of insufficient robots into three sub-probles: tas grouping proble, tas scheduling proble and tas balanced allocation proble. The atheatical odels of three sub-probles are built, and the corresponding heuristic algoriths are designed to solve three sub-probles. Finally, the siulation and analysis are done on the real data of online boostore S. 2. Proble Description and Matheatical Models 2.. Proble Description In a sart warehousing syste based on cargo-to-person ode, there are available warehouse robots, ovable shelves, and n picing platfors. Each shelf and picing platfor is arranged in a fixed site. Each shelf stores several types of ites. The unit-distance cost of a robot waling loaded with a shelf is c, the unit-distance cost of a robot waling unloaded is c 2, and the fixed cost of every warehouse robot running one tie is r. Let d ij represents the distance of a robot running fro shelf s i to shelf s j, and u j represents the distance of a robot transporting shelf s j to its corresponding picing platfor. The operation cost of a warehouse robot copleting a given tas is coposed of three parts: the related cost, the self-cost and the fixed cost. Related cost refers to the cost that the warehouse robot needs to ove unloaded fro current position to the position of assigned tas. Self-cost refers to the cost that the warehouse robot copleting a tas. The fix cost refers to the cost running a warehouse robot one tie. Suppose the current position of robot F l is in the position of shelf s i, if we assign robot F l to finish tas s j, then the operation cost of robot F l to finish tas s j can be expressed as follows: c = dc + uc+ r lj ij 2 j where d ij is the distance of robot F l running fro current position s i to the position of assigned tas s j, u j is the double distance fro position of tas s j to picing platfor, r is the fixed cost for controlling robot F l copleting one tas. Given an order to be piced in the sart warehouse, there are p tass to be copleted for picing this order S = s, s2,, sp. ( < p). Assue that the set of tass for picing this order is { } Suppose there are e orders to be piced in a period of tie which are listed in a set: O { O O O } =, 2,, e, and the orders are piced one by one, in order to iniize the total operation cost, how to assign all tass for each warehouse robot and give its woring schedule? The tas allocation proble of insufficient warehouse robots (TAPIR) can be decoposed into three subprobles: tas grouping proble, tas scheduling proble, and tas balanced allocation proble. Firstly, all tass of picing one order are partitioned into several groups, where the nuber of groups equals to the nuber 494

3 of robots. Secondly, the scheduling schee of each group tass is given according to every robot. Finally, assign each warehouse robot to coplete one group of tass. In the following subsections, we will construct the atheatical odels for these three sub-probles Matheatical Model for the Tas Grouping Proble For the tas allocation proble of insufficient warehouse robots (TAPIR), since the nuber of robots is saller than the nuber of tass for picing an order, so soe warehouse robots need to coplete ore than one tass when picing one order. Before assign the tass to robots, we firstly partition all tass into several groups, where the nuber of group equals to the nuber of robots. In order to reduce the total operation cost and ensure the tas allocation of warehouse robots balanced, we define the siilarity r ij between tas s i and tas s j. The siilarity r ij can be calculated by forula (), where d ij is the distance for a robot to ove fro the site of tas s i to the site of tas s j, d ax is the axiu distance be- d = d ). tween any pair of tass ( ax ax{ ij} dax dij rij = () d It is easy to verify that 0 r ij. If we assign two tass with larger siilarity value to one robot, the related cost will be lower. On the contrary, if we assign two tass with saller siilarity value to one robot, the related cost will be higher. Based on this idea, we can partition the tass into several groups according to the siilarity value. To ensure the tas allocation balanced aong all robots, we restrict the nuber of tass in each group is p no ore than a given value D (for exaple D = + ). Define the decision variables:, tas si is assigned in the vth gro up; xiv = 0, otherwise i=, 2,, pv ; =, 2,,. The atheatical odel of tas grouping proble for picing one order can be forulated as an integer prograing odel. ax p p rx ij ivxjv i= j= ax z = (2) p 2 v= x iv i= xiv =, i = 2,, p (3) v= p xiv, v=, 2,, (4) st.. i= p xiv D, v=, 2,, (5) i= x = iv 0,, i =, 2,, pv ; =, 2,, (6) Objective function (2) axiizes su of the average siilarity of tass assigned to the sae group; Constraints (3) ensure that each tas is just assigned into one group; Constraints (4) and Constraints (5) ensure that each group contains at least one tas and no ore than D tass; Constraints (6) indicate that the variables are binary Matheatical Model for the Tas Scheduling Proble Because the self-cost of transporting a tas to its corresponding picing platfor is a constant no atter which warehouse robot copletes this tas, the fix cost of each robot is also a constant no atter which tas it co- 495

4 pletes, so the total operation cost of warehouse robots depends ainly on the total related cost that the warehouse robots copleting all of the tass. If a warehouse robot located in s 0 is assigned to coplete a group of tass { s, s2,, sq}, the total related cost depends on the scheduling order of the tass. For exaple, if the scheduling order of the tass is s s2 sq, then the total related cost is ( d0, + d,2 + + dq, q) c2. When the scheduling order of tass changed, the total related cost will change. Given a warehouse robot and a group of tass, the tas scheduling proble is to find the optial scheduling order of the tass, so that the total related cost for the robot copleting all tass is inial. Since the related cost depends on the distance between two successive tass, so the tas scheduling proble is siilar to Hailtonian path proble, where the robot should start its routing fro initial position s 0, and visit every site of the tas in { s, s2,, sq} one tie. After copleting all tass, the robot will stay in the site of the last tas, which eans that it needs not return to its initial position s 0. Suppose that warehouse robot F l (locating in site s 0 ) is assigned to coplete the tass { s, s2,, sq} of group. Suppose that s q + is a duy end location, which corresponding to the last location of tass{ s, s2,, sq}. Define d = iq, 0 ( i = 0,, +, q). To find the optial scheduling order of tass, we defined the variables as follows: f : the flow that run fro tas s i to tas s j, i = 0,,, q; j =, 2,, q+ x ij ij, after warehouse robot Fl finishing tas si, it directly begin tas s j; i = 0,, 2,, q; j =, 2,, q, q + = 0, Otherwise., fij > 0, We define that xij = 0, fij = 0. The tas scheduling proble can be forulated into the following integer prograing odel. q q+ in w c dx (7) = l 2 ij ij i= 0 j= q+ xij =, i = 0,, 2,, q (8) j= q xij =, j =, 2,, q+ (9) i= 0 q q+ st. fi fij =, i =, 2,, q (0) = 0 j= 0 q f0 j = q, () j= xij fij q xij i = 0,,, q; j =, 2,, q+ (2) xij { 0, } i = 0,,, q; j =, 2,, q+ (3) Objective function (7) iniizes the total related cost that the warehouse robot F l copletes all tass in group ; Constraints (8) (9) ensure that each tas ust be copleted by robot F l ; Constraint (0) denote that when a tas is copleted, the total flow run into it will be absorbed by the tas. Constraint () ensures the su of flows coe out fro location s 0 equals q. Constraint (2) represents the relationship between x ij and f ij ; Constraint (3) indicates that the variables x ij are binary Matheatical Model for the Tas Balanced Allocation Proble After calculating the related cost l l = to coplete the tass in group ( =, 2,, ), we obtained a related cost atrix W = ( w ij ). Since we have partitioned the tass into groups, each robot needs to coplete one and only one group of tass. The tass balanced allocation proble is to assign each robot copleting only one group of tass so that the total related costs is inial. w of warehouse robot F ( l, 2,, ) 496

5 Define the decision variables: y l =, assign warehouse robot Fl to coplete the th group of tass; 0, otherwise l =, 2,, ; =, 2,, The atheatical odel of tass balanced allocation proble can be forulated into an integer prograing odel as follows: in z w y (4) = l= = l l yl =, =, 2,, (5) l= st.. yl =, l=, 2,, (6) = yl ( 0, ), l=, 2,,, =, 2,, (7) Objective function (4) iniizes the total related cost that warehouse robots coplete all groups of tass; Constraints (5) ensure that each group of tass can only be assigned to one warehouse robot; Constraints (6) represent that each warehouse robot can only coplete one group of tass; Constraints (7) indicate that the variables are binary. 3. Algoriths for Solving Three Sub-Probles In Section 2, we have decoposed the TAPIR proble into three sub-probles and forulate the as integer prograing odels. For sall size of sub-probles, we can solve the integer prograing odel by Lingo software. Since soe sub-proble (such as the second sub-proble) is NP-hard, for larger size of the proble, it is not practical to solve the by Lingo software. In this section, we will respectively design heuristic algoriths for solving each sub-proble of Section Algorith for Solving the Tas Grouping Proble need to be copleted in picing one order. The heuristic algorith for grouping all tass into groups can be described as follows: Initialization: T = { s, s2,, sp}, group = 0, where T is the set of ungrouped tass, group denotes the nuber of non-epty groups. Step : Calculate the siilarity atrix R= ( rij ), where r ij is the siilarity between tas s i and tas s j, p p which can be calculated by Equation (). Given the nuber of robots and the position of each robot, the set of tass { s, s2,, sp} Step 2: Find the sallest value of atrix R, ri, j = in{ rij}, let G = { s }, G { s } { i j } T = T \ s, s i, j i 0 = ; group = 2 ; 2 j 0 Step 3: If group =, go to step 4, else. For each si T, and =,, group, calculate the average siilarity between the ungrouped tas s i and all tass in group as follows ( ) s G j AR i, =. G The su of average siilarity between tas s i and all non-epty groups SAR i group r ( ) AR ( i ) = = ij,. 497

6 Find the tas with iniu su of average siilarity ( ) iin = arg in SAR i. i Update the nuber of non-epty group, add the tass in the new non-epty group, and update the set of ungrouped tass. group : = group + ; G group { si } in { i } : = ; T : = T \ s ; Go to step 3. Step 4: If T = Φ, go to step 5. Else add the ungrouped tass to the group, with which it has the axiu average siilarity. For each si T, and =,,, where G < D. Calculate the average siilarity between s i and each tas in group. ( ) in s G j AR i, =. G Find the group with axiu average siilarity value. Update the set T and G * ( ) = ( ) AR i*, * arg ax AR i,. i, = { }, G : G { s } T : T \ si* Go to Step 4. Step 5: Output the tass in each group: G, G2,, G. r ij =. * * i* 3.2. Algorith for Solving the Tas Scheduling Proble To solve the tas scheduling proble, we design a heuristic algorith: Define the following variables: route( l, ) represents the scheduling route, whose initial value is epty; location represents the location of robot F l, whose intial position is s l0 ; w l represents the total related cost for robot F l copleting tass in G = { s s s 2 g } Step : For each robot ( ) and each tas group ( ) l,,, ; Input: The locations of robots and the sites of each tas in groups. F l G l, suppose the location of robot F l is s l0, the nuber of tass in group is Let TN {, 2,, g } g, G = { s, s,, s 2 g }. = be the index set of tass. route( l, ) = Φ ; 0 Calculate the distance atrix Dl (, ) ( dij ) ( g + ) ( g + ) the distance fro s i to Step 2: If TN = Φ, go to step 3. s, ( i j g ) j location = ; w = 0. l =, where d 0 j is the distance fro s l0 to, =, 2,,. Else, find the inial eleent of atrix (, ) = d, route( l, ): = route( l, ) { j* }, TN : TN \{ j* } {, } j* argin location j j T go to step 2. s j, d ij is Dl in the location row and coluns of TN, =, location: = j*, wl : = wl + cd 2 location, j*, Step 3: Output the total related cost w l, the optial scheduling order of tass (, ) route l, and the final loca- 498

7 tion of robot F l Algorith for Solving the Tas Balanced Allocation Proble After scheduling the tass of each group, we obtain the related cost atrix ( ) W = w l, where w l represents the related cost of robot F l copleting the tass of groups. Then the balance allocation proble is converted into an assignent proble, which can be solved by Hungarian Algorith [7]. We can also obtain the optial solution by solving the integer linear prograing odel directly. 4. Siulations Analysis In this section, we do siulation on a sart warehouse of online boostore S. At present, online boostore S uses the sart ware housing syste based on cargo-to-person ode. In the warehouse, 96 classes of boos are stored in 24 ovable shelves. Each shelf stores 4 classes of boos. There are 3 picing platfors in the sart warehouse. Currently, the orders are piced one by one. Figure depicts the layout of the warehouse, the square boxes represent 24 shelves and their positions, three rectangular boxes represent three picing platfors; three cycles with arrow represent the warehouse robots. The coordinates of shelves and picing platfors are listed in Table. The initial positions of three robots are the sae as their nearest shelves. There are 0 orders to be piced. The tass sets of 0 orders are listed in Table 2. The unit cost of a robot waling loaded with a shelf is 3 dollars, the unit cost of a robot waling unloaded is.9 dollars, and the fixed cost of controlling each warehouse robot copleting a tas is 0 dollars. In order to iniize the total cost in the process of picing 0 orders, how to allocate and schedule the tass to each warehouse robot? Figure. The layout of the warehouse, the square boxes represent shelves, three rectangular boxes represent three picing platfors three cycles with arrow represent the warehouse robots. 8 tass of the first order depicted in red squares are partitioned into three groups. The tass in the sae group are surrounded by a closed curve (Coordinate Unit: 2 ). 499

8 Table. Coordinates of each shelf and each picing platfor. Shelf s s 2 s 3 s 4 Coordinate (3, 7) (5, 7) (7, 7) (9, 7) Shelf s 5 s 6 s 7 s 8 Coordinate (, 7) (3, 7) (3, 9) (, 9) Shelf s 9 s 0 s s 2 Coordinate (9, 9) (7, 9) (5, 9) (3, 9) Shelf s 3 s 4 s 5 s 6 Coordinate (3, ) (5, ) (7, ) (9, ) Shelf s 7 s 8 s 9 s 20 Coordinate (, ) (3, ) (3, 3) (, 3) Shelf s 2 s 22 s 23 s 24 Coordinate (9, 3) (7, 3) (5, 3) (3, 3) Picing platfor t t 2 t 3 Coordinate (4, 3) (8, 3) (2, 3) Table 2. The tass of picing each order. Order Tass of picing the order { s, s, s, s, s, s, s, s } { s, s, s, s } { s, s, s, s, s, s, s } { s, s, s, s, s, s, s } { s, s, s, s, s, s, s, s } { s, s, s, s, s, s, s } { s, s, s, s, s, s } { s, s, s, s, s } { s, s, s, s, s, s, s, s } { s, s, s, s, s, s } Since the orders are piced one by one, for each order to be piced, we can calculate the distance between any pair of tass, and obtain the siilarity atrix by Equation (). For the uth ( u =, 2,,0) order, we first partition the tass into three groups by the heuristic algorith in section 3.. Then we obtain the scheduling order for each group tass and every robot by the heuristic algorith in section 3.2. Using the Hungarian Algorith, we get the tass assignent and scheduling results, and the total operation cost. After assigning the tass of the uth order to the robots, we update the initial position of each robot, and begin to assign the tass of the ( u + ) th order. Repeat the steps until all tass of 0 orders are assigned to robots. The siulation results of the first order is depicted in Figure, where 8 tass are partitioned into three groups, the tass in the sae group are surrounded by a closed curve. The optial tass assignent and the scheduling routes are F s24 s23 s2, F2 s6 s2, F3 s0 s8 s7. Which ean that robot F need to coplete tass s 24, s 23, s 2 in sequence, then stop at position s 2. Robot F 2 need to coplete tass s 6, s 2 in sequence, 500

9 then stop at position s 2. Robot F 3 need to coplete tass s 0, s 8, s 7 in sequence, then stop at position s 7. The tas grouping results of all orders are listed in Table 3. The tas allocation and optial scheduling routes of all orders are shown in Table 4, the total operation costs of each order are listed in Table 4. According to the tas assignent and scheduling routes, and the operation cost in Table 4, the total operation cost that the warehouse robots coplete all tass of 0 orders is dollars. If the nuber of warehouse robots is larger than the axiu nuber of tass for picing one order, the proble becoes TAPAR. The tass need not be grouped, and the total operation cost for picing these 0 orders will be dollars [2]. Fro the tas allocation results of 0 orders, we can see that the total operation cost in TAPIR is 39.8 dollars ore than the total operation cost in TAPAR [2]. However, the nuber of warehouse robots in the TAPIR is 7 less than that in TAPAR. Since the cost of increasing a warehouse robot is very large, in practice, the nuber of warehouse robots is often insufficient. If we consider holding cost of robots in the TAPIR, we can find that the warehouse had better hold fewer robots. Table 3. Tas grouping results of 0 orders. Order Group Group 2 Group 3 { s, s, s } { s, s, s } {, } { s 2 } { s, s 5 6} { 0} 3 { s, s, s } { s } {,, } Su of average siilarity s s s s s s { s, s, s } { s } {,, } s s s { s, s, s } { s, s, s } {, } s s { s, s, s } { s } {,, } s s s { s, s } { s, s } {, } s s { s, s } { s } {, } s s { s, s, s } { s, s, s } {, } s s { s, s, s } { s, s } { } s Table 4. The tas assignent and optial scheduling routes, and the operation cost of each order. Order Tas assignent and optial scheduling routes Operation cost F s s s ; F s s ; F s s s F s s ; F s ; F s F s s s ; F s s s ; F s F s s s ; F s s s ; F s F s s s ; F s s ; F s s s F s s s ; F s s s ; F s F s s ; F s s ; F s s F s s ; F s s ; F s F s s s ; F s s ; F s s s F s ; F s s ; F s s s

10 5. Conclusions The tas allocation proble of insufficient warehouse robots (TAPIR) in the sart warehouse is an iportant issue in the sart warehousing syste designing. By now, few researches focus on the TAPIR proble. In this paper, we investigate the TAPIR proble, and divide it into three sub-probles: tas grouping proble, tas scheduling proble and tas balanced allocation proble. We forulate each sub-proble into an integer prograing odel and design the heuristic algorith for solving each sub-proble. The atheatical odels and algoriths can be used in designing the sart warehousing syste. In this paper, we assue that the orders are piced one by one. When the nuber of orders is large, the orders can be batched firstly [8], and then consider the orders in each batch as one cobined order. Using the atheatical odels and heuristic algoriths of this paper, we can find the optial tas allocation and scheduling routes for picing each cobined order. In this paper, we have not considered the cost of holding a robot. If we consider this cost, we can analyze the relation between the total cost and the nuber of warehouse robots and find the optial nuber of robots a warehouse should hold. This is the proble we are researching on. Acnowledgeents This wor was supported by the National Natural Science Foundation of China (3009, F02408), the Funding Project for Acadeic Huan Resources Developent in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (CIT&TCD ), and Major Research Project of Beijing Wuzi University. Funding Project for Technology Key Project of Municipal Education Coission of Beijing (ID: TSJHG ); Funding Project for Beijing ey laboratory of intelligent logistics syste; Funding Project of Construction of Innovative Teas and Teacher Career Developent for Universities and Colleges Under Beijing Municipality (ID: IDHT203057); Funding Project for Beijing philosophy and social science research base specially coissioned project planning (ID: 3JDJGD03). References [] Li, Z. and Li, W. (204) Study on Optiization of Storage Bays of Sart Warehouses of Online Boostores. Logistics Technology, 2, (In Chinese) [2] Zou, S. (203) The Present and Future of Warehouse Robot. Logistics Engineering and Manageent, 6, (In Chinese) [3] Guo, Y. (200) Auction-Based Multi-Agent Tas Allocation in Sart Logistic Center. Ph.D. Thesis, Harbin Institute of Technology, Shenzhen. (In Chinese) [4] Li, G. (202) Tas Allocation of Warehouse Robots Based on Intelligence Optiization Algorith. Ph.D. Thesis, Harbin Institute of Technology, Shenzhen. (In Chinese) [5] Hazard, C.J., Wuran, P.R. and D Andrea, R. (2006) Alphabet Soup: A Tested for Studying Resource Allocation in Multi-Vehicle Systes. Aerican Association for Artificial Intelligence. [6] Li, Z. and Li, W. (205) Research on the Tas Allocation Proble of Warehouse Robots in the Sart Warehouse. Proceedings of the 2th International Syposiu on Operations Research and Its Applications, Luoyang, 2-24 August 205, [7] Kuhn, H.W. (2005) The Hungarian Method for the Assignent Proble. Naval Research Logistics,, [8] Li, Z. and Li, W. (204) Study on Supplier Warehouse Location Cobination Proble of Chain Superarets. Logistics Technology, 9, (In Chinese) 502