Cost optimization method using Monte Carlo simulation

Abstract

This invention provides a cost optimization method that quickly determines the optimal number of nodes while avoiding over-investment and under-investment under conditions of demand and price uncertainty. [Solution] A cost optimization method implemented by a management terminal 100, in which the daily cost of the target area is defined as the sum of at least transportation costs, energy costs, and depreciation costs of the bases, based on a geometric relationship in which the average transportation distance decreases as the number of bases increases, and when adding one base, the difference in profit before and after the addition is evaluated based on the unit selling price, the cost composition, and the distance reduction effect, and the addition is permitted when it is determined that the difference is not unfavorable, uncertain inputs such as demand and unit selling price are generated probabilistically and repeated trials are performed, the number of bases that maximizes the expected profit is selected based on the results of each trial, and the selection result is output.

Description

Technical Field

[0001] The present invention relates to a method for optimizing the total cost or profit by Monte Carlo simulation in a system where there is a trade-off between the number of facilities (nodes), transportation distance, energy consumption, and amortization cost.

Background Art

[0002] [[ID=1--2]]The optimization process is usually extremely important in situations where multiple factors conflict and balance is required. Optimization is widely applied in various industries, but usually, it is difficult for individuals or organizations to make optimization decisions. The optimization process requires comprehensive algorithms and calculations, and in many cases, individuals and organizations do not have access to them. Therefore, in most cases, people make blind decisions and are in a situation where there is no benchmark in sight. Thus, the optimization process becomes very important in daily life. On the other hand, the lack of algorithms and computing facilities is a serious problem. For example, a company with a "supply chain management system" that is in high demand in the industry may not be able to determine how many warehouses should be located in a specific country or region. As a specific example, the optimization decision-making for Amazon's transportation and delivery in Japan (with an area of 380,000 square kilometers) can consist of the following elements: 1) Predictive route optimization: Mapping the optimal delivery route in consideration of traffic conditions, weather, road closures, etc. in real time. 2) Inventory intelligence: Predicting demand patterns and preventing stock shortages and overstocking. 3) Real-time supply chain visibility: Monitoring the entire journey of products from the manufacturer to the customer's doorstep.

Prior Art Documents

Patent Documents

[0003] ​​​​​​​​​​

[0004] While methods have been proposed to analytically balance the effect of reducing average transport distance and increasing depreciation costs associated with increasing the number of nodes, generalized methods that simultaneously handle uncertainty in unit price and demand, and search for the optimal number of nodes while determining the feasibility of expansion using the differential gain inequality, are not yet well-developed.

[0005] Therefore, the present invention provides a cost optimization method that can quickly determine the optimal number of nodes while avoiding over-investment and under-investment under uncertain demand and price conditions. [Means for solving the problem]

[0006] A cost optimization method implemented by a management terminal, which defines the daily cost of a target area as the sum of transportation costs, energy costs, and depreciation costs of the bases, based on a geometric relationship in which the average transportation distance decreases as the number of bases increases; evaluates the difference in profit before and after the addition of one base based on the unit selling price, the cost composition, and the distance reduction effect; permits the addition when it is determined that the difference is not unfavorable; probabilistically generates uncertain inputs such as demand and unit selling price, performs repeated trials, selects the number of bases that maximizes the expected profit based on the results of each trial, and outputs the selection result. [Effects of the Invention]

[0007] This method enables a cost optimization technique that can quickly determine the optimal number of nodes while avoiding over- and under-investment. More specifically, it allows for the mathematical formulation of expansion decisions while incorporating variations in input parameters, enabling the presentation of the optimal number of nodes in a short time. It also enables the detection of areas of excessive expansion in a two-dimensional analysis under fixed-price conditions, and the identification of the point of maximum gain through a three-dimensional analysis of price range × number of nodes. [Brief explanation of the drawing]

[0008] [Figure 1] This is a diagram showing the configuration of an optimization analysis system. [Figure 2]This is a functional block diagram of the management terminal 100. [Figure 3] This is a functional block diagram of the carrier terminal 200. [Figure 4] This is a conceptual graph used to explain the concept of cost optimization. [Figure 5] This diagram shows an example of a material recovery facility (MRF) and a recycling process. [Figure 6] This diagram illustrates the relationship between the number of bases and the average transport distance. [Figure 7] This figure shows the analysis procedure (example code) under fixed-price conditions. [Figure 8] This figure shows an example of the number of locations and profit trends under fixed-price conditions. [Figure 9] This figure shows an example of a three-dimensional analysis where the price range and the number of locations are changed simultaneously. [Figure 10] This diagram shows examples of applications in other fields (e.g., an application to communication networks). [Figure 11] This figure shows an example of code for three-dimensional analysis. [Modes for carrying out the invention]

[0009] This embodiment simultaneously handles the uncertainty of the number of locations and the unit selling price through a process involving a relationship based on principles, determination based on differences, and Monte Carlo search.

[0010] The system configuration of this embodiment is as shown in Figure 1, in which a management terminal 100 and multiple business terminals 200A and 200B are interconnected via a network NW. The management terminal 100 is equipped with a simulation engine, receives input from the business terminals, and returns the result of calculating the optimal number of locations.

[0011] As shown in Figure 2, the management terminal 100 comprises a communication unit 110, a storage unit 120, and a control unit 130 (which includes simulation, optimization, and report generation functions). The storage unit 120 stores prior distributions related to the number of locations and unit sales prices, coefficients for operating costs and depreciation costs, and the history of Monte Carlo trials, while the control unit 130 performs profit difference calculations and determines whether expansion is feasible.

[0012] After the optimization process ends, the control unit 130 automatically generates report data including the recommended score, determination of the feasibility of additional construction, summary of the sensitivity analysis for the assumed price range and fluctuations in demand, and identification results of the range where there is a risk of excessive additional construction. Statistical summary values based on the input conditions (target area, cost coefficient, setting of prior distribution, etc.) and sample history are attached to the report data, which is stored in the storage unit 120 and distributed to the business operator terminal 200 via the communication unit 110. The display operation unit 220 of the business operator terminal 200 presents the report data in a dashboard format. Specifically, the recommended score and its basis, the trend of profit transition when price conditions are fixed, the maximization candidates when the price range and score change simultaneously, and precautions near the boundary of the additional construction judgment are displayed as cards or tabs. Also, as needed, output in PDF or CSV format, email transmission, and instructions for recalculation are accepted from the same screen. The above report generation function operates in cooperation with a series of processes from aggregation, formatting, distribution, browsing, and storage on the management terminal 100 side. That is, the control unit 130 generates report data based on the determination and optimization results, the communication unit 110 transmits the data to the business operator terminal 200 via the network NW, and the control unit 240 and display operation unit 220 of the business operator terminal 200 present this to the user.

[0013] As shown in FIG. 3, the business operator terminal 200 includes a communication unit 210, a display operation unit 220, a storage unit 230, and a control unit 240, and inputs the target area, price conditions, cost coefficient, etc. via the user interface, and displays the optimal score and sensitivity analysis results received from the management terminal 100.

[0014] Next, the concept of cost optimization will be explained. A configuration with a large number of points (e.g., 100 points) has a large initial cost while the transportation distance is short and a reduction in operating costs can be expected. A configuration with a small number of points (e.g., 10 points) has a small initial cost while the transportation distance is long and the operating costs increase. The present invention derives the number of points at which the total cost or profit is minimized or maximized at the boundary of these conflicting relationships.

[0015] As shown in the conceptual diagram of FIG. 4, the total cost consists of the initial cost (such as depreciation of the base) and the continuous cost (such as transportation cost, energy cost, etc.). Depending on the delivery volume and price conditions, the relationship curves of the two intersect, and after the intersection point, a configuration with a larger number of bases may be advantageous. The present invention parameterizes this intersection relationship and mechanically determines whether to expand and the optimal number of bases.

[0016] FIG. 5 is an example of a recycling process using a material recovery facility (MRF), showing the flow from collection, sorting, recycling, to shipping. The initial cost is converted into the daily depreciation amount based on the useful life of the equipment, and the continuous cost is organized as the cost of transportation, energy, etc. per day.

[0017] As shown in FIG. 6, within a service range of a certain area, as the number of bases increases, the area covered by each base becomes smaller, and the average transportation distance becomes shorter. That is, there is a regular relationship between the increase in the number of bases and the decrease in the average transportation distance, and the present invention models this relationship and reflects it in the cost estimate.

[0018] The unit cost (transportation cost, energy cost, equipment depreciation, processing capacity, transportation distance, price of recycled materials, etc.) can be set based on existing known materials. For the details of the calculation method, refer to the disclosure scope of the relevant literature. The present invention does not depend on the calculation formula itself, and is characterized in that it performs differential gain determination and optimization using the cost coefficients and price conditions given as inputs.

[0019] The cost in daily operation is defined as the sum of the daily amount related to the depreciation of the base and the variable costs such as transportation and energy. In the present invention, the "difference before and after" when adding one more base is evaluated from the perspective of profit, and the range where the difference does not result in a loss is treated as the expansion allowable range.

[0020] In Monte Carlo simulation, input values with variations such as unit selling price, demand volume, cost coefficients, etc. are generated probabilistically, and the expected profit is estimated through multiple trials. The expansion allowable range is determined in each trial, and the number of bases that maximizes the expected profit is selected. Examples of the results of two-dimensional analysis with only the number of bases changed under fixed price conditions are shown in FIGS. 7 and 8.

[0021] Figure 9 shows an example of a three-dimensional analysis that simultaneously changes the price range and the number of locations. This allows for the identification of the point where the gain per unit from expansion is maximized (maximum gain point) among the combinations of price range and number of locations. Figure 11 shows an example of the code used for the analysis.

[0022] The optimization method of the present invention can be applied not only to the logistics and recycling fields but also to other fields such as the node placement of communication networks. For example, as shown in Figure 10, the balance between performance improvement due to increased density of wireless units and increased equipment costs can be evaluated within the framework of the present invention, thereby avoiding over- or under-investment.

[0023] The cost optimization analysis method of this embodiment will be described in more detail below. First, in this example, when considering cost optimization, there are two options as follows. 1) Option A (100 nodes) The more inventory facilities (warehouses or storage locations) you allocate, the more options you have for the routing process, and the greater the cost savings. However, allocating 100 nodes will result in high initial costs. 2. Option B (10 nodes) To save on initial costs, reduce the number of facilities allocated. This will lower initial costs but increase daily operating costs. At this stage, it's impossible to say definitively which option (A or B) is better, as it depends on the initial costs and ongoing daily costs.

[0024] The graph in Figure 4 compares options A and B. Here, the initial cost (yen / day) is calculated by dividing the initial setup cost of the inventory facility by the facility's useful life. For example, if the initial cost of the facility is 3 million yen and the useful life is 10 years (approximately 3,000 days excluding holidays), the depreciation expense would be 1,000 yen / day. Following the initial setup cost, ongoing costs (yen / day) are calculated after the supply chain begins daily operations. This includes transportation costs, maintenance costs, labor costs, and miscellaneous expenses. The total cost is the sum of the initial cost and the ongoing cost. The purpose of this example is to compare the total costs of options A and B. From the graph, we can see that both curves intersect at the point of tangency. If the delivery volume (x-axis) is greater than the point of tangency (to the right of the point of tangency), option A can achieve a lower (total) cost than option B. Supply chains and transportation are extremely important to many companies, but currently, such calculation methods are not found in the industry. Most companies lack the tools to calculate the initial number of nodes (storage facilities) to allocate based on daily shipping volume estimates, preventing them from optimizing for cost reduction. Furthermore, even if initial estimates are possible, they cannot accommodate future increases in shipping volume and the resulting increase in the number of nodes.

[0025] Next, to explain the aforementioned method for calculating contact points, we will give an example of a plastic recycling process. Generally, used plastics are recycled using material recovery facilities (MRFs). Figure 5 shows an MRF facility and the recycling process within the MRF.

[0026] As mentioned above, total costs can be classified into two categories (initial costs and ongoing costs). Initial costs (A = depreciation of AMRF): If the MRF facility operates for 10 years with an annual processing capacity of 20,000 tons, it is estimated to cost $15,000,000 per unit. Based on the breakdown of depreciation costs, the constants A and the daily recycling capacity (the amount of waste that can be processed in one day) are derived as follows. The initial setup costs for n MRFs (including labor and other fixed costs) will be amortized over 10 years. A = $15,000,000 / (10 × 365) = $4,110. This is the daily depreciation cost for one unit. The daily cost for an MRF with n units is 4110 × n. Daily recycling capacity: M = 20,000 / 365 = 54.8 tons Operating costs (t = transportation costs and E = energy costs of MRF) EIN is proportional to the number of MRFs and does not depend on the transport distance (from the collection site to the MRF). On the other hand, ti and j depend on the distance between the collection site and the MRF. In other words, the more MRFs allocated to a fixed area, the higher the initial cost and MRF energy cost, but the lower the transport cost. The optimization process begins with estimating the number of MRFs to allocate to the fixed area, and in this example, the fixed area is assumed to be 1000 square kilometers (km2).

[0027] Next, we create an equation that expresses the relationship between the number of MRFs and the transport distance, specifically the relationship between the transport distance (d) and the number of MRFs (n) within 1000 km². As shown in Figure 6, at any given location Li, the circle L has a radius r0, and r0 = 1,000 km. There are n MRFs installed at location Li, and the radius of the small circle covered by each MRF is denoted by rn. Theoretically, by adding n small circles with radius rn covered by the MRFs, the entire area of ​​r0 can be completely covered. Therefore, rn is the radius of the MRF, and dn is the average distance between each MRF.

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[0028] The relationship between transport distance and MRF number reveals an important correlation: within a certain radius area, transport distance is inversely proportional to the square root of the MRF number.

[0029] For the unit costs of the input parameters (transportation costs, energy costs, number of MRFs, MRF depreciation, MRF capacity, transport distance, and price of recycled plastics), the calculation method can be found in the paper (Additive manufacturing of recycled plastics: Strategies towards a more sustainable future), Haishang Wu, Faculty of Technology, University of Sunderland, Sunderland, SR6 0DD, UK, Journal of Cleaner Production 335 (2022) 130236). The important points here are the formulas and calculation methods; the unit costs are merely for illustrative purposes.

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[0030] Cost calculation for daily operations:

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[0031] Next, we use Monte Carlo simulation (the deviation between the nth and (n+1)th MRFs) to derive the optimal value of n for the target price Pprice and the random number n. If the number of MRFs is n: Revenue is,

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[0032] Next, we perform a two-dimensional Monte Carlo simulation (number of MRFs and revenue). At a specific selling price of recycled plastic, the relationship between revenue per MRF and the number of MRFs is given by n

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[0033] Next, we perform a three-dimensional Monte Carlo simulation (number of MRFs, sales price, and revenue). Because extending the Monte Carlo simulation to a three-dimensional analysis (between the number of MRFs, sales price, and revenue) complicates the conditions, we provide an example of Python code in Figure 11. This three-dimensional analysis allows us to simulate and calculate the optimization between three-dimensional elements using the method described in this example. As shown in the graph in Figure 9, the optimized solution is obtained when the number of MRFs is set to 118 and the sales price to $294 / ton. In this case, increasing the MRF from 117 to 118 yields the highest revenue of $2.277 per ton of MRF.

[0034] The optimization process described above began with plastic recycling, used Monte Carlo simulations to calculate the number of MRFs, optimized costs, and achieved maximum profitability. Furthermore, this method can be applied to various use cases, such as in the telecommunications sector. For example, it is particularly important when promoting the expansion of communication bandwidth from 5G to 6G. An example of the broad industry application of this example is that, through this optimization approach, AI-powered technology provides intelligent signal management, optimizing network performance and routing signals while avoiding obstacles and congestion. Predictive maintenance improves reliability, reduces downtime, and facilitates the addition of nodes. As shown in Figure 10, AI-driven operations supported by RF-on-Glass and a network mesh create seamless and adaptable subnets, connecting end-user devices to the millimeter-wave world. In the service industry, customer satisfaction is paramount. As shown in Figure 10, when numerous RF-on-Glass unit sets (e.g., 100 unit sets within a 10km2 circle) are installed in one location, higher unit set density results in higher costs but improved performance. Therefore, optimizing ROI (Return on Investment) through this process is recommended. Our innovations in the optimization process help many service industries achieve customer satisfaction at minimal cost. 1) Monte Carlo simulation for initial optimization and static configuration, and 2) our innovations, supported by AI machine learning, enable dynamic configuration, allowing some device sets to be used in common areas and dynamically allocated to adjacent areas where data transfer is concentrated and load balancing is required. This Monte Carlo optimization model can serve as the foundation for future technologies. [Explanation of Symbols]

[0035] 100...Management terminal / server 110...Communication unit 120...Storage unit 130...Control unit 200...Operator terminal 210...Communication unit 220...Display / operation unit 230...Storage unit 240...Control unit

Claims

1. A cost optimization method implemented by a management terminal, Using a distance reduction effect based on a geometric relationship in which the average transport distance decreases as the number of bases increases, The daily cost for the target area is defined as the sum of at least transportation costs, energy costs, and depreciation costs of the base, and the cost structure, including coefficients for calculating said costs, can be entered. When adding a base, the difference in profit before and after the addition will be evaluated based on the unit selling price, demand volume, the aforementioned cost structure, and the aforementioned distance reduction effect, and the addition will be permitted only when it is determined that the difference will not result in a disadvantage. Based on a prior distribution stored in the storage unit of the management terminal, samples are probabilistically generated and repeated trials are conducted for the unit selling price, demand quantity, and cost coefficients included in the cost structure. Based on the results of each of the repeated trials, the number of locations that maximizes expected profit is selected. After the selection, report data is generated and output that includes at least the recommended number of locations and its rationale, a summary of the sensitivity analysis to fluctuations in demand and the unit selling price, and the identification results of the range of locations where there is a risk of excessive expansion, and is accompanied by statistical summary values ​​based on the input conditions and the sample history of the repeated trials. A method characterized by the following:

2. In the method according to claim 1, The parameters used to evaluate the difference in profit before and after the expansion are set based on modeled operating costs, depreciation costs, and processing capacity.

3. The method according to claim 1, wherein the unit selling price is fixed to a predetermined value, and only the number of locations is changed to perform an analysis, and an area of ​​excessive expansion is detected where profits begin to decrease when the number of locations exceeds a certain range, and the optimal number of locations is constrained to avoid that area.

4. The method according to claim 1, wherein the unit selling price and the number of locations are changed simultaneously and an analysis is performed to identify the combination of price range and number of locations that maximizes profit.

5. In the method according to claim 1, A method in which the operator's terminal presents the report data in a dashboard format, and this presentation includes, as cards or tabs, the recommended number of locations and its rationale, the trend of profit changes under fixed pricing conditions, the candidate for maximizing when the price range and number of locations are changed simultaneously, and precautions near the boundary for deciding on expansion, and accepts instructions for output in PDF or CSV format, email transmission, or recalculation as needed.

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