Optimization device and optimization method

Abstract

To eliminate the need for hyperparameter tuning regarding the QUBO formula.SOLUTION: An optimization device comprises a transmission unit that transmits a Quadratic Unconstrained Binary Optimization (QUBO) formula to a predetermined quantum computer, where the QUBO formula includes multiple cost functions, and cost functions other than one of the multiple cost functions are transformed into inequality constraints with upper bounds set, and a receiving unit that receives a solution for the transmitted QUBO equation from the quantum computer.SELECTED DRAWING: Figure 26

Description

[Technical Field]

[0001] The present disclosure relates to an optimization device and an optimization method. [Background technology]

[0002] Methods for solving optimization problems using quantum computers are being studied (for example, Patent Document 1, Non-Patent Documents 1 and 2). [Prior art documents] [Patent documents]

[0003] [Patent Document 1] Japanese Patent Publication No. 2022-118555 [Non-patent literature]

[0004] [Non-Patent Document 1] Kofler, M., Beham, A., Wagner, S., Affenzeller, M., & Achleitner, W. (2011, August). Re-warehousing vs. healing: Strategies for warehouse storage location assignment. In 3rd IEEE international symposium on logistics and industrial informatics (pp. 77-82). IEEE. [Non-patent document 2] Khullar, C. (2021). Development of a warehouse slotting model to improve picking performance (Doctoral dissertation, Concordia University). Summary of the Invention [Problem to be solved by the invention]

[0005] Regarding the Quadratic Unconstrained Binary Optimization (QUBO) formula, which performs multi-objective optimization with two or more different cost functions, for example, a hyperparameter tuning method is performed by searching for hyperparameter values ​​at multiple points to obtain various Pareto solutions, as disclosed in Patent Document 1. Thus, conventionally, a large number of searches for hyperparameter values ​​are required to obtain a solution desired by the optimization user.

[0006] An object of the present disclosure is to provide a technology that eliminates the need to adjust hyperparameter values ​​for QUBO formulas. [Means for solving the problem]

[0007] One aspect of the present disclosure provides an optimization device comprising: a transmitter that transmits a Quadratic Unconstrained Binary Optimization (QUBO) formula including a plurality of cost functions, in which all but one of the plurality of cost functions are inequality constrained and have upper limits set, to a predetermined quantum computer; and a receiver that receives a solution to the transmitted QUBO formula from the quantum computer.

[0008] One aspect of the present disclosure provides an optimization method that transmits a Quadratic Unconstrained Binary Optimization (QUBO) formula including multiple cost functions, in which all but one of the multiple cost functions are inequality constrained and have upper limits set, to a predetermined quantum computer, and receives a solution to the transmitted QUBO formula from the quantum computer.

[0009] These comprehensive or specific aspects may be realized as a system, an apparatus, a method, an integrated circuit, a computer program, or a recording medium, or may be realized as any combination of a system, an apparatus, a method, an integrated circuit, a computer program, and a recording medium. [Effects of the Invention]

[0010] According to the present disclosure, it is possible to eliminate the need to adjust hyperparameter values ​​for QUBO formulas. [Brief explanation of the drawings]

[0011] [Figure 1] FIG. 1 is a schematic diagram showing an example of item placement before and after optimization in a logistics warehouse according to the first embodiment; [Figure 2] Schematic diagram for explaining the setting of the slotting problem in the QUBO formulation according to the first embodiment. [Figure 3] Schematic diagram for explaining constants and variables for setting a slotting problem in QUBO formulation according to the first embodiment. [Figure 4] FIG. 1 is a diagram showing a QUBO formulation of the slotting problem according to the first embodiment. [Figure 5] FIG. 1 is a diagram illustrating an equation for cost term 1 according to the first embodiment and cost term 1. [Figure 6] FIG. 1 is a diagram illustrating an equation for cost term 2 according to the first embodiment and cost term 2. [Figure 7] 1 is a diagram illustrating an equation for the path constraint F0(p) of the constraint term 5 according to the first embodiment and the path constraint F0(p) of the constraint term 5. [Figure 8] FIG. 10 is a diagram showing an equation for the path constraint F1(p) of the constraint term 5 according to the first embodiment. [Figure 9] FIG. 10 is a diagram for explaining the first term of the path constraint F1(p) of the constraint term 5 according to the first embodiment. [Figure 10] FIG. 10 is a diagram for explaining the second term of the path constraint F1(p) of the constraint term 5 according to the first embodiment. [Figure 11] FIG. 10 is a diagram showing an equation for the path constraint F2(p) of the constraint term 5 according to the first embodiment. [Figure 12] FIG. 10 is a diagram for explaining the first term of the path constraint F2(p) of the constraint term 5 according to the first embodiment. [Figure 13] FIG. 10 is a diagram for explaining the second and third terms of the path constraint F2(p) of the constraint term 5 according to the first embodiment. [Figure 14]FIG. 10 is a diagram showing an equation for the path constraint F3(p) of the constraint term 5 according to the first embodiment. [Figure 15] FIG. 10 is a diagram for explaining the first term of the path constraint F3(p) of the constraint term 5 according to the first embodiment. [Figure 16] FIG. 10 is a diagram for explaining the second and third terms of the path constraint F3(p) of the constraint term 5 according to the first embodiment. [Figure 17] FIG. 10 is a diagram showing an equation for the path constraint F4(p) of the constraint term 5 according to the first embodiment. [Figure 18] FIG. 10 is a diagram for explaining the first term of the path constraint F4(p) of the constraint term 5 according to the first embodiment. [Figure 19] FIG. 10 is a diagram for explaining the second and third terms of the path constraint F4(p) of the constraint term 5 according to the first embodiment. [Figure 20] FIG. 10 is a diagram showing an equation for the path constraint F5(p) of the constraint term 5 according to the first embodiment. [Figure 21] FIG. 10 is a diagram for explaining the first term of the path constraint F5(p) of the constraint term 5 according to the first embodiment. [Figure 22] FIG. 10 is a diagram for explaining the second and third terms of the path constraint F5(p) of the constraint term 5 according to the first embodiment. [Figure 23] 1 is a diagram illustrating an equation of a path constraint F6(p) of the constraint term 5 according to the first embodiment and F6(p) of the constraint term 5. [Figure 24] FIG. 1 shows the formulas for PF and PA according to Non-Patent Document 1. [Figure 25] FIG. 10 is a diagram showing a QUBO formulation of PF and PA according to the second embodiment. [Figure 26] FIG. 1 shows a QUBO formula C1 of the prior art and a QUBO formula C2 according to the third embodiment. [Figure 27] FIG. 10 shows a QUBO formula obtained by transforming the cost term of the first embodiment using the method of the third embodiment. [Figure 28] FIG. 1 is a block diagram illustrating an example configuration of an optimization system according to the present disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0012] Hereinafter, embodiments of the present disclosure will be described in detail with appropriate reference to the drawings. However, more detailed description than necessary may be omitted. For example, detailed descriptions of well-known matters and redundant descriptions of substantially identical configurations may be omitted. This is to avoid unnecessary redundancy in the following description and to facilitate understanding by those skilled in the art. Note that the accompanying drawings and the following description are provided to enable those skilled in the art to fully understand the present disclosure, and are not intended to limit the subject matter described in the claims.

[0013] The functions of one configuration shown in this embodiment may be realized by two or more physical configurations, or the functions of two or more configurations may be realized by, for example, one physical configuration.

[0014] (Embodiment 1) In this embodiment, in order to improve the efficiency of large-scale item picking processes in warehouses and logistics centers, a Quadratic Unconstrained Binary Optimization (QUBO) equation is formulated for slotting, i.e., for changing item placement, to minimize picking movement costs, and solution is made possible using quantum annealing.

[0015] Conventional methods for optimizing slotting based on the item quantity and frequency of appearance in a picklist generated from an item order, as disclosed in Non-Patent Documents 1 and 2, are known. However, these conventional methods are solutions to overcome the long calculation time required when calculating picking movement costs based on movement distance. While they reduce calculation time, they do not directly reduce movement distance. Therefore, this embodiment proposes a method for directly calculating movement distance using a quantum computer capable of high-speed calculations. Since whether significant changes from the initial item placement before slotting are possible depends on the on-site situation, a method for controlling the item placement change cost (item movement cost) is also required. Furthermore, while quantum computers are theoretically capable of high-speed calculations, there are currently limitations in hardware performance, which may make them impractical depending on the problem scale. Therefore, we also propose a method for implementing this method using a conventional quantum computer.

[0016] That is, in this embodiment, a QUBO equation for slotting optimization that minimizes the picking movement cost (movement distance) while controlling the item movement cost based on the initial item placement is formulated.

[0017] According to this embodiment, by making it possible to control the item movement cost from the initial item placement, slotting optimization can be realized, minimizing the picking distance based on the item movement cost acceptable to users in a real environment. Furthermore, the QUBO formulation enables the solution of optimization problems using quantum computers. For example, quantum annealing has the characteristic of outputting a solution in a constant time regardless of the problem scale, making it possible to solve problems within a realistic time frame even when the problem becomes large. Furthermore, this embodiment can also be applied to quantum gate methods, and by using the Grover Adaptive Search algorithm, it is possible to obtain an optimal solution while achieving quadratic speedup compared to a full search. This will be explained in detail below.

[0018] FIG. 1 is a schematic diagram showing an example of item placement in a logistics warehouse before and after optimization according to the first embodiment.

[0019] From a start point, the picker moves through the aisles of the warehouse, picking items from shelves to fill a picklist.

[0020] For example, in a logistics warehouse, if items 0, 1, and 2 are arranged as shown before optimization in Figure 1(a), the picker will need to travel a relatively long distance to pick all items 0, 1, and 2 registered in the pick list, as shown by arrow 11 before optimization in Figure 1(a).

[0021] Therefore, by optimizing item placement (slotting) as described in this embodiment, we propose placing items 0, 1, and 2 near the start point as shown after optimization in Figure 1(b), and we place items 0, 1, and 2 in this manner in advance.

[0022] This allows the picker to pick all items 0, 1, and 2 registered in the pick list in a relatively short movement, as shown by arrow 12 after optimization in Figure 1(b).

[0023] In other words, by optimizing item placement (slotting) to best meet demand, it is possible to improve the efficiency of order picking operations in logistics warehouses (reducing the distance that pickers have to travel).

[0024] Non-Patent Document 1 discloses a method for optimizing slotting based on the item quantity and frequency of appearance in a pick list generated from item orders. However, Non-Patent Document 1 has the following three issues: (Problem 1) Non-Patent Document 1 is a heuristic method and does not aim to minimize the picking distance. (Problem 2) Non-Patent Document 1 has a problem that, depending on the solution algorithm, it may not be able to solve the problem in a realistic time when the problem becomes large-scale (it does not support large-scale problems). (Problem 3) Non-Patent Document 1 does not take into consideration initial item placement, and the cost of moving items to optimize slotting may become very large (not dealing with initial item placement).

[0025] This embodiment proposes an optimization device 1100 (see FIG. 28) and an optimization method for minimizing the picking distance cost while taking into account the initial item placement for Problems 1 and 3. In this case, a quantum computer 1200 (see FIG. 28) (e.g., quantum annealing) is used for Problem 2, and the necessary QUBO formulation is performed for this purpose.

[0026] Quantum Annealing (QA) is a computational technology specialized for solving combinatorial optimization problems, and has the characteristic that the speed of solving the problem remains constant regardless of the problem scale. Quantum Annealing sets the problem as a model formula called the Ising model or QUBO formula.

[0027] The QUBO equation is composed of a binary variable x and the interaction J between adjacent variables as shown in (Equation 1) below. ij and the externally applied local magnetic field h i The state changes according to the formula, and the pattern of x that results in the optimal solution is formulated so that the energy H is minimized. The pattern of 0 and 1 in this formula is the solution to the combinatorial optimization problem that we are looking for.

[0028]

number

[0029] Below, we will show the slotting problem setting and QUBO formulation for slotting optimization.

[0030] FIG. 2 is a schematic diagram for explaining the setting of the slotting problem in the QUBO formulation according to the first embodiment.

[0031] <Problem and constant settings> As shown in FIG. 2, in the slotting environment, aisles 20 with movement directions indicated by down arrows 21, up arrows 22, and right arrows 23, and shelves 50 indicated by solid-line rectangular frames, exist in a regular pattern. As shown in the dotted square frame in FIG. 2, the shelf 50 has a plurality of slots 51, and each slot 51 can accommodate one item. · The passage 20, whose movement direction is indicated by the downward arrow 21, the upward arrow 22, and the rightward arrow 23, has a distance. As shown in Figure 2, two shelves 50 are arranged between the upper and lower aisles, and the picker can pick items from the shelves 50 only when moving up and down the aisles 20. In other words, the picker cannot pick items from the shelves 50 when moving left and right in the aisles 20. There are multiple picklists, which are lists of items to be picked, and a picking route is generated for each picklist. There is an initial item layout that shows what items are on each shelf.

[0032] <Purpose> Minimize the total distance cost of the picking routes corresponding to each picklist. -Consider the distance cost of moving items from the initial item placement.

[0033] <Restrictions> All items included in each picklist can be collected by following the picking route corresponding to each picklist. In other words, a picking route that satisfies each picklist can be generated. The item must be present on one of the shelves. The number of items that can be stored on Shelf 30 must be less than the maximum number. Each item has weights of "large (H)", "medium (M)", and "small (L)", and there is a slot 51 that can store only "small (L)" items. In other words, the number of items of weight "large (H)" and "medium (M)" that can be stored on the shelf 30 is below the upper limit. The route during picking must be a single route from "start" to "end" following the movement directions of the down arrow 21, up arrow 22, and right arrow 23 shown on the aisle 20 (see, for example, the thick dotted arrows in Figure 2).

[0034] FIG. 3 is a schematic diagram for explaining constants and variables for setting a slotting problem in QUBO formulation according to the first embodiment.

[0035] Next, the constants and variables to be set will be explained. <constant> X T : The total number of right-direction passages (i.e., right arrows 23) (16 in the problem setting in Figure 3) Y T : The total number of vertical passages (i.e., down arrow 21 and up arrow 22) (15 in the problem setting in Figure 3) Y M : The total number of vertical passages (i.e., dotted down arrows 31 consisting of consecutive down arrows 21 and dotted up arrows 32 consisting of consecutive up arrows 22) (5 in the problem setting in Figure 3) X S : The total number of right-direction passages (i.e., right arrows 23) that exist between two passages that combine vertical passages (for example, between the dotted down arrow 31 and the dotted up arrow 32) (in the problem setting in Figure 3, there are four) Y S : The total number of vertical passages (e.g., the down arrow 21 included in the dotted down arrow 31) that exist in a single passage that combines vertical passages (e.g., the dotted down arrow 31) (3 in the problem setting in Figure 3) ·S: Number of shelves I: Number of item types P: Total number of picklists PL: Picklist array (array of items to be picked) ·W x : Vertical aisle distance array (X T ×1) ·W y : Right direction aisle distance array (Y T ×1) ·S Pick [b]:Aisle b(∈Y T) Shelf arrangement from which items can be picked X Ind : The index assigned to each right-hand passage is two-dimensionalized ((Y S +1)×(X S )) array (x[0]~x

[15] in Figure 3) Y Ind : The index assigned to each vertical aisle is two-dimensional ((Y S )×(X S +1)) array (y[0] to y

[14] in Figure 3) I HM :Arrangement of items with weight "large (H)" and "medium (M)" ·S HM : An array (S x 1) that contains values ​​indicating how many "large (H)" and "medium (M)" weight items can be stored on each shelf. ·S Max : An array (S x 1) that contains values ​​indicating how many items can be stored on each shelf. Init I : An array (I x 1 binary) that indicates which shelf each item was originally stored on Dist ss : Shelf distance arrangement (S × S) λ: Hyperparameter

[0036] <variable> x:P×X T A binary variable. The picking path generated for PL[p] is α(∈X T ) passage, x[p,a]=1. y:P×Y T A binary variable. When the picking route generated for PL[p] passes through aisle b, y[p,b]=1. · shelf: S × I binary variable. When item i (∈I) exists on shelf s (∈S), shelf[s,i] = 1.

[0037] The final solution is defined by the output of shelf[s,i]. For example, shelf[0,1]=1 indicates that item number 1 should be placed on shelf number 0, and shelf[1,10]=1 indicates that item number 10 should be placed on shelf number 1.

[0038] FIG. 4 is a diagram illustrating a QUBO formulation of the slotting problem according to the first embodiment.

[0039] As shown in FIG. 4, the QUBO formula for energy H is composed of the sum of cost term 1, cost term 2, constraint term 1, constraint term 2, constraint term 3, constraint term 4, and constraint term 5.

[0040] Cost term 1 indicates the total distance cost of the picking route corresponding to each picklist.

[0041] Cost term 2 indicates the total distance cost of moving an item from the initial item placement.

[0042] Constraint term 1 indicates a constraint that all items included in each picklist must be collected by following the picking route corresponding to each picklist.

[0043] Constraint 2 indicates that the item must be present on one of the shelves.

[0044] Constraint 3 indicates that the number of items that can be stored on the shelf must be equal to or less than the upper limit.

[0045] Constraint item 4 indicates that the number of items of weight "large (H)" and "medium (M)" that can be stored on the shelf must be equal to or less than the upper limit.

[0046] Constraint 5 indicates that the route during picking must be a single route from start to end, following the direction of the aisle (i.e., the direction of the arrow).

[0047] The goal is to find the variables x[p,a], y[p,b], and shelf[s,i] that minimize H in the QUBO formula shown in Figure 4. Note that constraint terms 1 to 5 are expressions that are 0 when the constraint is satisfied, and are greater than 0 when the constraint is not satisfied. Therefore, if constraint terms 1 to 5 are not satisfied, H is unlikely to be minimized.

[0048] Next, cost terms 1 and 2 and constraint terms 1 to 5 will be explained in detail.

[0049] <Cost Item 1> FIG. 5 is a diagram for explaining the equation for cost term 1 and cost term 1 according to the first embodiment.

[0050] Cost term 1, which indicates the total distance cost of the picking route, can be expressed as the sum of (binary variables x and y representing the aisle) × (each aisle Wx, Wy), as shown in Figure 5.

[0051] For example, in the case of the route shown in Figure 5, the path distance W is assigned to x[1], x[4], y[0], y[3], y[6], y[7], and y[8]. x ,W y The sum of these multiplications is the total distance cost of the picking route (i.e., the value of cost term 1).

[0052] <Cost item 2> FIG. 6 is a diagram for explaining the equation for cost term 2 and cost term 2 according to the first embodiment.

[0053] Cost term 2, which indicates the total item movement cost, can be expressed as the sum of the differences between the initial item placement and the optimized item placement, as shown in FIG.

[0054] In the formula shown in Figure 6, when item i is moved to shelf s, the initial placement shelf of item i is Init I Dist[s,Init I[i]] is added. The influence of the item movement cost is controlled by the hyperparameter λ. Note that the hyperparameter λ may be adjusted separately (externally determined). For example, the hyperparameter λ may be set by the user.

[0055] <Restriction 1> The equation for constraint term 1 shown in Figure 4 indicates the constraint that all items included in each picklist must be collected by following the picking route corresponding to each picklist.

[0056] The equation for constraint term 1 shown in Figure 4 is an equation that determines whether item i in the picklist PL[p] is stored on a shelf sy that can be picked from aisle variable y. The equation for constraint term 1 shown in Figure 4 is 0 if each item is stored. In other words, if the equation for constraint term 1 shown in Figure 4 is 0, it can be determined that the constraint related to constraint term 1 is satisfied, and if the equation for constraint term 1 shown in Figure 4 is greater than 0 (non-zero), it can be determined that the constraint related to constraint term 1 is not satisfied.

[0057] <Restriction 2> The equation for constraint term 2 shown in Figure 4 indicates the constraint that the item must be present on one of the shelves.

[0058] The equation for constraint term 2 shown in Figure 4 is an equation that determines whether or not a certain item i is stored on a shelf. In the equation for constraint term 2 shown in Figure 4, the shelf variable shelf is a variable that becomes 1 when added up for all s for a certain item i. Therefore, if the equation for constraint term 2 shown in Figure 4 is 0, it can be determined that the constraint related to constraint term 2 is satisfied, and if the equation for constraint term 2 shown in Figure 4 is greater than 0 (non-zero), it can be determined that the constraint related to constraint term 2 is not satisfied.

[0059] <Restriction 3> The equation for constraint term 3 shown in FIG. 4 indicates a constraint that the number of items that can be stored on a shelf must be equal to or less than the upper limit number.

[0060] The equation for constraint term 3 shown in Figure 4 is an equation that determines whether the number of items that can be stored on a shelf is equal to or less than the upper limit. The equation for constraint term 3 shown in Figure 4 uses slack variables because it is an inequality constraint.

[0061] The equation for constraint term 3 shown in Figure 4 is an equation that determines whether the difference between (the number of items stored on the shelf + the slack variable) and the upper limit is 0. If the number of items exceeds the upper limit, even if the slack variable is set to the minimum value of 0, the equation for constraint term 3 shown in Figure 4 will not be 0, and it can be determined that a constraint violation has occurred.

[0062] <Restriction 4> The equation for constraint term 4 shown in Figure 4 determines whether the number of "large (H)" and "medium (M)" items that can be stored on the shelf is equal to or less than the upper limit. The equation for constraint term 4 shown in Figure 4 uses a slack variable because it is an inequality constraint.

[0063] The equation for constraint term 4 shown in Figure 4 is an equation that determines whether the difference between (the number of items stored on the shelf + the slack variable) and the upper limit is 0. If the number of items exceeds the upper limit, even if the slack variable is set to the minimum value of 0, the equation shown in Figure 12 will not be 0, and it can be determined that a constraint violation has occurred.

[0064] <Restriction 5> 7 is a diagram for explaining the equation of the path constraint F0(p) of constraint term 5 and the path constraint F0(p) of constraint term 5 according to embodiment 1. Next, F0(p) in the equation of constraint term 5 shown in FIG. 4 will be explained with reference to FIG. In the following explanation of constraint term 5, p is omitted because it is an index of the pick list.

[0065] The formula for F0 in constraint term 5 shown in FIG. 7 indicates the constraint that the route should be one (the route should not be interrupted midway), as shown in the lower part of FIG.

[0066] For example, if we write the formula for F0 according to the path shown in the bottom of Figure 7, it becomes as follows: y[0]-y[1]-aux[0]=0 y[1]-y[2]-aux[1]=0 Ensure that y[1]=1 and y[0]=0 do not occur at the same time. In this case, y[0]-y[1]-aux[0] becomes 0-1-aux[0], and since aux[0] can only be 0 or 1, 0-1-aux[0] cannot be 0. Therefore, this is a constraint violation (non-zero).

[0067] Note that aux is an auxiliary bit for the path constraint, and aux can be either 0 or 1. If F0 becomes zero when aux is either 0 or 1, it can be determined that the constraint F0 is satisfied. The reason for using aux in this way is to comprehensively express the constraints for both cases where the path ends and where it continues. Specifically, when the path ends, it passes through y[0] and does not pass through y[1] (it ends at y[0]). In this case, aux[0]=1 satisfies the constraint. When the path continues, it passes through y[0] and passes through y[1] (the path continues beyond y[1]). In this case, aux[0]=0 satisfies the constraint.

[0068] If a path suddenly appears (if it is strange as a path), the value of the constraint term will not be zero, even if the value of aux is 0 or 1. Specifically, if a path appears along the way, it will not pass through y[0], but through y[1]. In this case, y[0]-y[1]=-1, and since aux[0] is further subtracted, the value of the constraint term will not be zero whether aux[0] is 1 or 0 (i.e., it will be -2 or -1).

[0069] Fig. 8 is a diagram showing the equation of the path constraint F1(p) of the constraint term 5 according to the first embodiment. Fig. 9 is a diagram for explaining a first term 101 of the path constraint F1(p) of the constraint term 5 according to the first embodiment. Fig. 10 is a diagram for explaining a second term 102 of the path constraint F1(p) of the constraint term 5 according to the first embodiment. Next, F1(p) in the equation of the constraint term 5 shown in Fig. 4 will be described with reference to Figs. 8, 9 and 10.

[0070] The formula of the first term 101 of F1 in constraint term 5 shown in Fig. 8 indicates a no-branch constraint (a constraint to keep the path to one) as shown in Fig. 9. For example, when the formula of the first term 101 of F1 is written in accordance with the path shown in Fig. 9, it becomes as follows: x[0]*y[0]=0 x[1]*y[1]=0 x[2]*y[2]=0

[0071] The value of the first term 101 of F1 is non-zero if the paths x and y are chosen simultaneously, resulting in a constraint violation.

[0072] The formula of the second term 102 of F1 in constraint term 5 shown in Fig. 8 indicates the constraint that the route must be continuous (that is, the route must not be interrupted midway), as shown in Fig. 10. For example, when the formula of the second term 102 of F1 is written in accordance with the route shown in Fig. 10, it becomes as follows: y[0]-x[1]-aux[2]=0 y[1]-x[2]-aux[3]=0 y[2]-x[3]-aux[4]=0

[0073] Fig. 11 is a diagram showing the equation of the path constraint F2(p) of the constraint term 5 according to the first embodiment. Fig. 12 is a diagram for explaining the first term 111 of the path constraint F2(p) of the constraint term 5 according to the first embodiment. Fig. 13 is a diagram for explaining the second term 112 and the third term 113 of the path constraint F2(p) of the constraint term 5 according to the first embodiment. Next, F2(p) in the equation of the constraint term 5 shown in Fig. 4 will be described with reference to Figs. 11, 12 and 13.

[0074] The first term 111 in the equation for F2 of constraint term 5 shown in Fig. 11 indicates a constraint that reverse driving is prohibited, as shown in Fig. 12. For example, when the equation for the first term 111 in the equation for F2 is written in accordance with the route surrounded by the dotted line in Fig. 12, it becomes as follows. x[0]*y[3]=0 x[1]*y[4]=0 x[2]*y[5]=0

[0075] The same is true for the other passages, so the same pattern appears halfway through the first term 111 (k=2, 6, 10, . . . ).

[0076] The second term 112 and the third term 113 of F2 of the constraint term 5 shown in FIG. 11 indicate the constraint that the route be one, as shown in FIG.

[0077] The second term 112 corresponds to the case where there are two relationships, as shown in the dotted line box 114 in FIG. 13, and when written in accordance with the dotted line box 114 in FIG. 13, it becomes as follows. x[1]+y[4]-y[3]-aux[6]=0 x[2]+y[5]-y[4]-aux[7]=0

[0078] The third term 113 corresponds to the case where there is one relationship, as shown in the dotted line box 115 in FIG. 13, and when written in accordance with the dotted line box 115 in FIG. 13, it becomes as follows. x[3]-y[5]-aux[8]=0

[0079] The same applies to the other passages, so the same pattern appears halfway through the second term 112 and the third term 113 (k=2, 6, 10, ...).

[0080] Fig. 14 is a diagram showing the equation of the path constraint F3(p) of the constraint term 5 according to the first embodiment. Fig. 15 is a diagram for explaining the first term 121 of the path constraint F3(p) of the constraint term 5 according to the first embodiment. Fig. 16 is a diagram for explaining the second term 122 and the third term 123 of the path constraint F3(p) of the constraint term 5 according to the first embodiment. Next, F3(p) in the equation of the constraint term 5 shown in Fig. 4 will be described with reference to Figs. 14, 15 and 16.

[0081] The first term 121 of the formula for F3 in constraint term 5 shown in Fig. 14 indicates a constraint that prohibits branching, as shown in Fig. 15. For example, when the formula for the first term 121 of the formula for F3 is written to fit the path surrounded by the dotted line in Fig. 15, it becomes as follows. x[3]×y[5]=0 x[4]×y[6]=0 x[5]×y[7]=0

[0082] The same is true for the other passages, so the same pattern appears halfway through the first term 121 (k=3, 7, 11, ...).

[0083] The second term 122 and the third term 123 of F2 of the constraint term 5 shown in FIG. 14 indicate the constraint that the route be a single line, as shown in FIG.

[0084] The second term 122 corresponds to the case where there are two relationships, as shown in the dotted line box 124 in FIG. 16, and when written in accordance with the dotted line box 124 in FIG. 16, it is as follows: x[1]+y[3]-x[4]-aux[9]=0 x[1]+y[4]-x[5]-aux

[10] =0 x[2]+y[5]-x[6]-aux

[11] =0

[0085] The third term 123 corresponds to the case where there is one relationship, as shown in the dotted line box 125 in FIG. 16, and when written in accordance with the dotted line box 125 in FIG. 16, it becomes as follows. x[3]-y[7]-aux

[12] =0

[0086] The same applies to the other passages, so the same pattern appears halfway through the second term 122 and the third term 123 (k=3, 7, 11, ...).

[0087] Fig. 17 is a diagram showing the equation of the path constraint F4(p) of the constraint term 5 according to the first embodiment. Fig. 18 is a diagram for explaining the first term 131 of the path constraint F4(p) of the constraint term 5 according to the first embodiment. Fig. 19 is a diagram for explaining the second term 132 and the third term 133 of the path constraint F4(p) of the constraint term 5 according to the first embodiment. Next, F4(p) in the equation of the constraint term 5 shown in Fig. 4 will be described with reference to Figs. 17, 18 and 19.

[0088] The first term 131 in the equation for F4 in constraint term 5 shown in Fig. 17 indicates a constraint that reverse driving is prohibited, as shown in Fig. 18. For example, when the equation for the first term 131 in the equation for F4 is written in accordance with the route surrounded by the dotted line in Fig. 18, it becomes as follows. x[5]*y[6]=0 x[6]*y[7]=0 x[7]*y[8]=0

[0089] The same is true for the other passages, so the same pattern appears halfway through the first term 131 (k=4, 8, 12, ...).

[0090] The second term 132 and the third term 133 of F4 of the constraint term 5 shown in FIG. 17 indicate the constraint that the route be one, as shown in FIG.

[0091] The second term 132 corresponds to the case where there are two relationships, as shown in the dotted line box 134 in FIG. 16, and when written in accordance with the dotted line box 134 in FIG. 19, it becomes as follows. x[5]+y[6]-y[7]-aux

[13] =0 x[6]+y[7]-y[8]-aux

[14] =0

[0092] The third term 133 corresponds to the case where there is one relationship, as shown in the dotted line box 135 in FIG. 19, and when written in accordance with the dotted line box 135 in FIG. 19, it becomes as follows. x[4]-y[6]-aux

[15] =0

[0093] The same applies to the other passages, so the same pattern appears halfway through the second term 132 and the third term 133 (k=4, 8, 12, ...).

[0094] Fig. 20 is a diagram showing the equation of the path constraint F5(p) of the constraint term 5 according to the first embodiment. Fig. 21 is a diagram for explaining the first term 141 of the path constraint F5(p) of the constraint term 5 according to the first embodiment. Fig. 22 is a diagram for explaining the second term 142 and the third term 143 of the path constraint F5(p) of the constraint term 5 according to the first embodiment. Next, F5(p) in the equation of the constraint term 5 shown in Fig. 4 will be described with reference to Figs. 20, 21 and 22.

[0095] The first term 141 of the equation for constraint term 5 F5 shown in Fig. 20 indicates a constraint that reverse driving is prohibited, as shown in Fig. 21. For example, when the equation for the first term 141 of the equation for F5 is written in accordance with the route shown in Fig. 21, it becomes as follows: y[6]*x[8]=0 y[7]*x[9]=0 y[8]*x

[10] =0

[0096] The same is true for the other passages, so the same pattern appears halfway through the first term 141 (k=5, 9, 13, ...).

[0097] The second term 142 and the third term 143 of the constraint term F5 shown in FIG. 20 indicate the constraint that the route be one, as shown in FIG.

[0098] The second term 142 corresponds to the case where there are two relationships, as shown in the dotted line box 145 in FIG. 22, and when written in accordance with the dotted line box 145 in FIG. 22, it becomes as follows. x[5]+y[6]-y[9]-aux

[16] =0 x[6]+y[7]-y

[10] -aux

[17] =0 x[7]+y[8]-y

[11] -aux

[18] =0

[0099] The third term 143 corresponds to the case where there is one relationship, as shown in the dotted line box 144 in FIG. 22, and when written in accordance with the dotted line box 144 in FIG. 22, it becomes as follows. x[4]-x[8]-aux

[19] =0

[0100] The same is true for the other passages, so the same pattern appears halfway through the first term 141 (k=5, 9, 13, ...).

[0101] 23 is a diagram for explaining the equation of the path constraint F6(p) of the constraint term 5 and F6(p) of the constraint term 5 according to the first embodiment. Next, F6(p) in the equation of the constraint term 5 shown in FIG. 4 will be described with reference to FIG.

[0102] The formula F6 in Figure 23 indicates the constraint that the end point is a passage that leads downward. The formula F6 in Figure 23 adds up all the "passages going downward" in the bottom row and subtracts all the "passages going upward" in the bottom row, which results in 1.

[0103] In the formula for F6 shown in Figure 23, the path that always goes downward (y[2], y[8], y

[14] in the example of Figure 23) will be 1. When multiple paths are 1, the path always goes upward (y[5], y

[11] in the example of Figure 35), so subtract the value of this path.

[0104] For example, if the formula for F6 is written according to the path shown in the lower part of Figure 23, it becomes as follows. (y[2]+y[8]+y

[14] )-(y[5]+y

[11] )=1 Here, (y[2]+y[8]+y

[14] ) is the sum of the lowest passage going down, and (y[5]+y

[11] ) is the sum of the lowest passage going up.

[0105] (Embodiment 2) FIG. 24 is a diagram showing the formulas for PF and PA according to Non-Patent Document 1.

[0106] Non-Patent Document 1 defines PF and PA shown in FIG. 24 as equations relating to slotting optimization.

[0107] The variables in the equations shown in FIG. 24 are defined as follows: orders(p i ):Item p i The number of orders in which |L(p i )|:Item p i Total number of shelves containing ·dist(s,origin): Distance between the origin and shelf s dist(s k ,s l ):shelf k and s l Distance from dffinity(p i ,p j ,):Item p i and p j The number of orders that appear simultaneously α: Hyperparameter for score calculation

[0108] PF stands for total Pick Frequency Score, and its goal is to place items that frequently appear on the pick list on shelves as close to the start point as possible.

[0109] PA stands for total Part Affinity Score, and its goal is to place items that appear together on the picklist as close to each other as possible on shelves.

[0110] Non-Patent Document 1 defines the cost by adding the functions of PF and PA as shown in the following equation 2.

[0111] Cost = PF + αPA ... (Equation 2)

[0112] Non-Patent Document 1 performs slotting optimization by calculating shelf s and item p that minimize the above-mentioned cost equation 2.

[0113] FIG. 25 is a diagram illustrating a QUBO formulation of PF and PA according to the second embodiment.

[0114] The QUBO formula according to the second embodiment shown in FIG. 25 is a simplification of the PF and PA of Non-Patent Document 1 shown in FIG. 24 by utilizing the constraints of the target slotting optimization.

[0115] The variables in the equations shown in FIG. 25 are defined as follows: ·S: Number of shelves I: Number of items (= number of item types) ·shelf: A binary variable of X × I. If item i exists on a shelf s, then shelf·[s,i]=1, otherwise shelf[s,i]=0. Dist[s,origin]: Distance from shelf s to the origin Dist[s1,s2]: Distance from shelf s1 to shelf s2 Order[i]: The number of times item i appears in the picklist. This is a change from the definition in the prior art, as it assumes the existence of a picklist. Affinity[i1,i2]: The number of times that two items i1 and i2 appear simultaneously across all picklists. This definition has been changed from the conventional definition because it assumes the existence of picklists. α: Hyperparameter

[0116] As shown in FIG. 24, the PF of Non-Patent Document 1 is i is the total number of shelves containing |L(p i In contrast, the PF of the second embodiment includes division by item p i Since there is a constraint that there is only one shelf that contains |L(p i )| is omitted and the QUBO formulation is shown in PF of Figure 25.

[0117] Furthermore, the PA of Non-Patent Document 1 is, as shown in FIG. 24, i )|*|L(p j In contrast, in the PA of the second embodiment, the slotting optimization involves division by item p i Since there is a constraint that there is only one shelf that contains |L(p i )|*|L(pj )| is omitted and the QUBO formulation is shown in PA in Figure 25.

[0118] In the second embodiment, the cost is defined by adding the functions of PF and PA shown in FIG. 25 as shown in the following equation 3.

[0119] Cost = PF + αPA ... (Equation 3)

[0120] Constraint terms 1 to 5 may be the same as those described in the first embodiment, and therefore will not be described in the second embodiment. In other words, the QUBO equation to be solved in the second embodiment is the above-described (Equation 3) plus the constraint terms described in the first embodiment. However, when using the QUBO equation of Equation 3, constraint terms 1 and 5 described in the first embodiment are unnecessary. Furthermore, the hyperparameter α may be adjusted separately (externally determined). For example, the hyperparameter α may be set by the user.

[0121] In addition, |L(p i If |L(p i )| is treated as a constant, and both the PF and PA equations are simplified due to the constraints of the slotting optimization in question.

[0122] In this way, by defining the QUBO formulas for PF and PA that eliminate division, it is possible to properly calculate the solution using a quantum computer.

[0123] (Embodiment 3) In the third embodiment, a method is described for eliminating the need to adjust hyperparameters (balance of the influence of each cost) to obtain a user's desired solution for a QUBO formula that performs multi-objective optimization with two or more different cost functions.

[0124] Regarding the QUBO formula, which performs multi-objective optimization with two or more different cost functions, for example, the hyperparameter tuning method is performed by performing a multi-point search for hyperparameter values ​​to obtain various Pareto solutions, as shown in Patent Document 1. However, a large number of searches are required to obtain the solution desired by the optimization user. Furthermore, current quantum annealing machines and QUBO solvers do not have a function for automatically tuning hyperparameters for cost functions.

[0125] Therefore, in the third embodiment, for a QUBO formula that performs multi-objective optimization with two or more different cost functions, all costs except one are soft-constrained (inequality constraints), thereby eliminating the need for parameter adjustment between cost functions. Here, soft-constraints (inequality constraints) are applied so that the cost before soft-constraint is equal to or less than the allowable upper limit.

[0126] This allows the user of the optimization to obtain a desired solution by simply setting an upper limit value that the user can tolerate, without having to adjust many parameters. This will be explained in detail below.

[0127] In this embodiment, when optimizing multiple cost functions simultaneously, the conventional method of multiplying each cost function by a hyperparameter and adding them up is changed to soft constraints (inequality constraints) for all cost functions except one, eliminating the need to adjust the hyperparameters.

[0128] The user obtains the desired solution by setting an acceptable value for the upper limit in the soft constraint.

[0129] FIG. 26 is a diagram showing a QUBO formula C1 of the prior art and a QUBO formula C2 according to the third embodiment.

[0130] When a QUBO equation consists of two different cost functions F0 and F1, as shown in equation C1 in Figure 26, in order to obtain the desired solution, it is necessary to multiply one of them by the hyperparameter λ to adjust its influence.

[0131] In contrast, in the proposed method according to this embodiment, as shown in equation C2 in FIG. 26, one of the cost functions (F1 in this case) is soft-constrained, and an upper limit is set for the soft-constrained equation, thereby obtaining the desired solution.

[0132] In the formula C2 in FIG. 26, the constant L is an upper limit value and may be determined arbitrarily by the user.

[0133] In equation C2 in Fig. 26, the part in box 201 including the constant D and variable b is used to absorb the difference (<0) of (F1(x)-L) in binary representation and make it 0. This may be a common method used when setting inequality constraints.

[0134] In formula C2, when the minimum value of F1(x) is 0, the value of D (the number of bits of variable b) is given by ceil(log2L).

[0135] The part in box 201 of formula C2 in FIG. 26 will be explained using a specific example.

[0136] When L=30, D=ceil(log2(30))=5, so the number of bits of variable b is 5.

[0137] When F1(x)=20, the pattern of variable b is b[4]=0, b[3]=1, b[2]=0, b[1]=1, b[0]=0, and (20-30+2 0 *0+2 1 *1+2 2 *0+2 3 *1+2 4 *0) 2 =0, and the inequality constraint is satisfied.

[0138] Similarly, when L=30, if F1(x)=100, the difference (=70) exceeds the range of values ​​that can be expressed in a 5-bit binary number, so (···) in formula C2 2 cannot be set to 0, which violates the inequality constraint.

[0139] The upper limit (acceptable value) can also be determined by performing optimization with all soft-constrained equations disabled, and using the cost function value evaluated for the solution obtained as a guideline.

[0140] FIG. 27 is a diagram showing a QUBO formula obtained by transforming the cost term of the first embodiment using the method of the third embodiment.

[0141] Cost formula C3 of the first embodiment shown in FIG. 27 is obtained by extracting cost term 1 and cost term 2 from the QUBO formula shown in FIG.

[0142] In this cost formula C3, the larger the value of the hyperparameter λ multiplied by cost term 2, the more optimal a solution (= small item movement cost) is obtained in which the item placement is not changed significantly from the initial item placement, and the smaller the value, the more optimal a solution (= large item movement cost) is obtained in which the item placement is changed significantly from the initial item placement.

[0143] Therefore, in cost formula C3, trial and error (numerous searches) is required to determine how large λ should be to obtain a solution for the item movement cost.

[0144] In contrast to this, in the cost formula C4 according to the third embodiment shown in FIG. 27, as explained in FIG. 26, the cost term 2 (see box 211) of the cost formula C3 is made into a soft constraint (inequality constraint), and an upper limit value is set for the soft-constrained formula (see box 212).

[0145] The variables in cost formula C4 are defined as follows: L: Upper limit (upper limit for cost term 2) b: Slack variable for inequality constraints D: Number of bits of slack variable b

[0146] This allows us to obtain the same solution as when adjusting the hyperparameter λ by setting the upper limit value. In other words, there is no need for trial and error (multiple searches) to determine how large the hyperparameter λ should be to obtain a solution for the item movement cost.

[0147] The upper limit L of the added soft constraints may be determined by disabling all soft-constrained equations, performing optimization, and evaluating the cost function value (original cost term 2) for the solution obtained. The upper limit L may also be set based on the characteristics of the QUBO equation used. For example, in the case of slotting, the upper limit L may be set based on the upper limit of the work time and work cost required to move items from the initial item placement.

[0148] Note that the method for eliminating the need to adjust the hyperparameter λ described in the third embodiment is not limited to the QUBO formula for the slotting problem. In other words, the above method is applicable to various QUBO formulas with two or more optimization indices (objective functions). For example, the above method is applicable to a QUBO formula that uses distance and time in a delivery plan as optimization indices and minimizes the total cost of distance and time. For example, the above method is applicable to a QUBO formula that uses personnel shifts (determining the number of man-hours for each person) and line allocation (where to place people) in a factory as optimization indices and minimizes the total cost of man-hours and placement movements.

[0149] Furthermore, which optimization indices (objective functions) among multiple optimization indices are not soft-constrained and which are soft-constrained may be determined according to the characteristics of the QUBO formula. For example, it may be determined that the optimization indices to be optimized are not soft-constrained, and the remaining optimization indices are soft-constrained. For example, it may be determined that an optimization indices for which an upper limit can be easily set (such as on-time in the above delivery plan example) is soft-constrained. For example, in order to suppress the increase in variables when soft-constraining, if the value range is known, it may be determined that an optimization indices (objective functions) with a narrow value range are soft-constrained.

[0150] (Hardware configuration) FIG. 28 is a block diagram illustrating a configuration example of an optimization system according to the present disclosure.

[0151] As shown in FIG. 28, the optimization system 1000 includes an optimization device 1100 and a quantum computer 1200.

[0152] The optimization device 1100 includes a processor 1101, a memory 1102, a storage 1103, an input unit 1104, a display unit 1105, a transmitting unit 1106, and a receiving unit 1107. The optimization device 1100 may be interpreted as a classical computer.

[0153] The processor 1101 executes a computer program in cooperation with the memory 1102 to set parameters and generate QUBO formulas as described in the first to third embodiments.

[0154] The memory 1102 is configured by a volatile storage medium and / or a non-volatile storage medium, and stores computer programs, data, and the like.

[0155] The storage 1103 is configured by a non-volatile storage medium and stores computer programs, data, and the like.

[0156] The input unit 1104 is, for example, a keyboard, a mouse, a touchpad, a microphone, etc., and accepts input operations from the user.

[0157] The display unit 1105 is, for example, a display, and displays various information and images.

[0158] The transmitter 1106 connects to a communication network 1300 and transmits, for example, information related to the QUBO formula described in the first to third embodiments to the quantum computer 1200. Examples of the communication network 1300 include the Internet, a wired LAN, a wireless LAN, a mobile communication network, and a VPN.

[0159] The receiving unit 1107 is connected to the communication network 1300 and receives, for example, information relating to the solution of the transmitted QUBO equation from the quantum computer 1200.

[0160] The quantum computer 1200 is connected to the communication network 1300. The quantum computer 1200 receives information about the QUBO equation described in the first to third embodiments from the optimization device 1100, for example, and finds a solution to the QUBO equation using a quantum computing mechanism. The quantum computer 1200 then transmits information about the solution to the QUBO equation to the optimization device 1100. The quantum computer 1200 may be an Ising machine type computer and find a solution to the QUBO equation using quantum annealing. However, the quantum computer 1200 may also be a gate type computer.

[0161] The user can set the hyperparameter λ described in the first embodiment, the hyperparameter α described in the second embodiment, or the upper limit value L described in the third embodiment via the input unit 1104.

[0162] In addition, the processor 1101 may display the optimal item placement locations and picking routes for each picklist obtained based on information regarding the solution of the QUBO equation received from the quantum computer 1200 on a layout image showing the shelves and aisles in a warehouse, as shown in Figure 2.

[0163] The processor 1101 may also transmit, to the terminal of the picker having the picklist, information indicating the picking route corresponding to the picklist, obtained based on the solution of the QUBO equation by the quantum computer 1200. This allows the picker to pick the items registered in the picklist using the optimal picking route.

[0164] The processor 1101 may also instruct the robots in the warehouse to move each item to the optimal item placement position obtained based on the solution of the QUBO equation received from the quantum computer 1200. This automatically optimizes the item placement in the warehouse.

[0165] Summary of the Disclosure

[0166] Based on the above description of the present disclosure, the following techniques are disclosed.

[0167] <Technology 1> The optimization device (1100) includes a transmitting unit (1106) that transmits a Quadratic Unconstrained Binary Optimization (QUBO) formula including a plurality of cost functions, in which all but one of the plurality of cost functions are inequality constrained and have an upper limit value set, to a predetermined quantum computer, and a receiving unit (1107) that receives a solution to the transmitted QUBO formula from the quantum computer (1200). This eliminates the need for hyperparameter tuning.

[0168] <Technology 2> In the optimization device described in Technology 1, the QUBO formula is:

[0169]

number

[0170] where F0 and F1 are cost functions, x is a binary variable, L is an upper limit, b is a slack variable for inequality constraints, and D is the number of bits of the slack variable b. This eliminates the need for hyperparameter tuning.

[0171] <Technology 3> In the optimization device described in Technique 2, the QUBO formula is a formula for warehouse shelf slotting optimization, F0(x) is a cost function related to the sum of picking path costs for each picklist, and F1(x) is a cost function related to the sum of item movement distance costs from the initial item placement. This makes it possible to determine a picking route and item placement that minimizes the total cost based on the sum of the total picking route costs and the total item movement distance costs in the warehouse.

[0172] <Technology 4> In the optimization device described in Technique 3, the QUBO formula further includes at least one of a constraint term regarding a picking path and a constraint term regarding the slotting of items onto the warehouse shelves. This makes it possible to determine a picking path and item placement that minimizes the total cost while satisfying at least one of the constraints on the picking path and the constraints on the slotting of items onto warehouse shelves.

[0173] <Technology 5> The optimization method involves sending a Quadratic Unconstrained Binary Optimization (QUBO) formula including multiple cost functions, in which all but one of the multiple cost functions are inequality constrained and have upper limits set, to a specified quantum computer, and receiving a solution to the sent QUBO formula from the quantum computer (1200). This eliminates the need for hyperparameter tuning.

[0174] Although the embodiments have been described above with reference to the accompanying drawings, the present disclosure is not limited to such examples. It is clear that a person skilled in the art can conceive of various modifications, alterations, substitutions, additions, deletions, and equivalents within the scope of the claims, and it is understood that these also fall within the technical scope of the present disclosure. Furthermore, the components in the above-described embodiments may be combined in any manner without departing from the spirit of the invention. [Industrial Applicability]

[0175] The techniques of the present disclosure are useful in using computers to solve industrial optimization problems. [Explanation of symbols]

[0176] 1000 Optimization System 1100 Optimization Device 1101 processor 1102 memory 1103 Storage 1104 Input section 1105 Display section 1106 Transmitter 1107 Receiving unit 1200 quantum computer 1300 Communication Network

Claims

1. a transmission unit that transmits, to a predetermined quantum computer, a Quadratic Unconstrained Binary Optimization (QUBO) formula including a plurality of cost functions, in which all but one of the plurality of cost functions are inequality constrained and have upper limits set; A receiving unit that receives a solution to the transmitted QUBO equation from the quantum computer. Optimizer.

2. The QUBO formula is: [Equation 1] Including F ０ and F １ is a cost function, x is a binary variable, L is an upper limit, b is a slack variable for inequality constraints, and D is the number of bits of the slack variable b. The optimization device according to claim 1 .

3. The QUBO formula is a formula for warehouse shelf slotting optimization, Said F ０ (x) is a cost function related to the sum of the picking path costs for each picklist; Said F １ (x) is a cost function related to the total cost of moving items from the initial item placement. The optimization device according to claim 2 .

4. the QUBO formula further includes at least one of a constraint term regarding a picking path and a constraint term regarding slotting of items onto the warehouse shelves; The optimization device according to claim 3 .

5. A Quadratic Unconstrained Binary Optimization (QUBO) formula including a plurality of cost functions, in which all but one of the plurality of cost functions are inequality constrained and have upper limits set, is transmitted to a predetermined quantum computer; receiving a solution to the transmitted QUBO equation from the quantum computer; Optimization methods.

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