Seasonal Inventory Management Model for Raw Materials in Steel Industry
Abstract
We developed a seasonal inventory management model for raw materials, such as iron ore and coal, for multiple suppliers and multiple mills. The Nippon Steel Corporation imports more than 100 million tons of raw material annually by vessels from Australia, Brazil, Canada, and other countries. Once these raw materials arrive in Japan, they are transported to domestic mills and stored in yards before being treated in a blast furnace. A critical problem currently facing the industry is the limited capacity of the yards, which leads to high demurrage costs while ships wait for space to open up in the yards before they can unload. To reduce the demurrage costs, the inventory levels of the raw materials must be kept as low as possible. However, inventory levels that are too low may lead to inventory shortage resulting from seasonal supply disruptions (e.g., a cyclone in Australia) that delay the supply of raw materials. Because both excess and depleted inventory levels lead to increased costs, optimal inventory levels must be determined. To solve this problem, we developed an inventory management model that considers variations on the supply side, differences that should be observable upon looking at the ship operations. The concept is to model the probability distribution of ship arrival intervals by brand groups and mills. We divided ship operations into two stages: arrival at all mills (in Japan) and arrival at individual mills. We modeled the former as a nonhomogeneous Poisson process and the latter as a nonhomogeneous Gamma process. Our proposed model enables inventory levels to be reduced by 14% in summer and 6% in winter.
Introduction
Product managers are facing increasing pressure to manufacture high-quality products at low cost. In the steelmaking industry, determining optimal inventory levels is crucial. Excess inventory slows down cash flow, whereas depleted inventory leads to unstable operation and drives up costs. Determining optimal inventory levels is a critical management issue in the steel industry.
Figure 1 shows the raw-material operations of Nippon Steel, which imports raw materials, such as iron ore and coal, from various suppliers—more than 60 million tons of iron ore and 30 million tons of coal annually. The raw materials are transported to nine steel mills. When a ship arrives at a mill, the raw materials are unloaded by machines, transported by belt conveyors to stockyards according to type, reclaimed when needed, and then mixed variously to maintain constant quality before being fed into the blast furnace.

Because yard space for the raw materials is considerably limited in the company, inventory levels must be kept as low as possible; otherwise, ships must wait out at sea at high cost until yard space is available. To reduce inventory levels, a common technique is to group together the raw materials of similar quality and manage the inventory of that group so that it does not run out (Tang et al. 2008). By treating several raw materials as a group, the inventory level can be reduced because the replenishment cycles of the raw materials are shorter by grouping and demand and supply fluctuations in the cycles are reduced. For this reason, inventory managers define the brand groups of each mill according to the nature of the mill, and shipping schedules are planned to maintain the inventory level of each brand group.
Also, the supply of raw materials varies seasonally because of sudden disruptions caused by bad weather. For example, cyclones tend to occur around February and March in Australia, and the rainy season in Brazil is from December to May, leading to supply disruptions from floods. In the past, floods caused by a cyclone have shut down the railway network in Australia, preventing the shipment of raw materials from Australia to Japan and resulting in stock running out at Nippon Steel’s mills. Therefore, inventory managers must consider the seasonal supply disruption risk in advance when determining inventory levels in the medium to long term (from three to six months ahead).
However, in the complex supply chain for raw materials, it is not apparent how much inventory to hold to minimize the management costs by giving the brand group a strategic level. Primarily, this is because it is difficult to quantify the variability in raw material lead times because the allocation plans for ships carrying raw materials change frequently. To address the problem, Agra et al. (2018) propose a formulation of a maritime inventory routing problem and its solution, but that is difficult to solve even at the operational level. Therefore, this issue does not determine inventory levels over the medium to long term.
Previous Research
There has been some previous research on inventory management for bulk materials, such as iron ore and coal. Chao et al. (1989) use stochastic dynamic programming to determine optimal inventory policies for electric utility companies that may face market disruption. Silver and Zufferey (2005) deal with the seasonal lead time for raw materials under constant and fixed demand using descent-based and Tabu search algorithms. Tang et al. (2008) introduce an optimization problem for raw material inventory management with lot size and supply interval as the decision variables; in that study, the distribution functions of demands and order lead times were given. Kim et al. (2009) optimize the placement of raw material in yards considering the yard capacity, which varies greatly depending on how the raw material pile is stacked.
There have been several studies of inventory management policies that consider supply chain disruptions and lead-time variation. Parlar and Berkin (1991) propose an economic order quantity with disruptions (EOQD) model, which assumes that the durations of a supplier’s “wet” and “dry” periods are distributed exponentially; Berk and Arreola-Risa (1994) then investigate the EOQD model further. Arreola-Risa and DeCroix (1998) propose an extension to the EOQD model with the added condition that back-ordering is stochastically possible. Mohebbi (2003) presents an analytical model for computing the stationary distribution of on-hand inventory in a continuous-review inventory system with compound Poisson demand, Erlang-distributed lead time, and lost sales. Tomlin (2006) studies a single-product setting in which a firm can source from two suppliers (one that is unreliable and one that is reliable but more expensive). That study focuses on the supply-side tactics available to a firm, namely sourcing mitigation, inventory mitigation, and contingent rerouting. Song et al. (2010) investigate the behavior of the optimal policy parameters and the long-run average cost in response to stochastically shorter or less-variable lead times in a single-item (r, q) inventory system with compound-Poisson-process demand; they find that some of the optimality of the base-stock system could be extended to their model. Schmitt and Snyder (2012) analyze the complexity of systems with uncertain supply when that supply is subject to both yield variability and disruptions and the importance of planning for future periods by considering more than a single-period model. Snyder (2014) proposes a tight approximation for the EOQD model of Berk and Arreola-Risa (1994). Chakraborty et al. (2020) investigate how coordinating contracts for supply disruptions work in a supply chain under risk and a competitive environment. Bakal et al. (2017) analyze the value of disruption information that provides an additional ordering opportunity for the buyer when the supplier is disrupted. Schmitt et al. (2017) investigate adjustments in order activity across four echelons, including assembly; embedded adaptive ordering analytics enable the relaxation of customary assumptions, such as aggregation of demand and supply data. Konstantaras et al. (2019) study a single-echelon inventory installation controlled by the (S, T) inventory policy with supply disruptions following an independent Bernoulli process. Sevgen and Sargut (2019) study an inventory model of a retailer that faced a deterministic and constant demand as a part of a two-echelon supply chain, in which both levels were subject to independent random disruptions. Saithong and Lekhavat (2020) derive a closed-form expression for optimal base-stock levels considering partial back order, deterministic demand, and stochastic supply disruption. (See Snyder et al. (2016) for a comprehensive review of supply chain disruption models and see Dolgui et al. (2013) for a comprehensive review of supply planning and inventory control under lead time uncertainty.)
Because it is difficult to determine the optimal inventory quantity analytically, various other methods have been applied. For example, Zhang and Wang (2011) consider the economic order quantity problem for a deteriorated multi-item. Roy et al. (2017) develop a probabilistic inventory model for deteriorating items. Pervin et al. (2017) formulate and solve an economic production quantity inventory model with deteriorating items. Pervin et al. (2018) develop a deterministic inventory control model with Weibull distribution deterioration. Pervin et al. (2019) consider a multi-item two-echelon inventory model for deteriorating items. Roy et al. (2020) study a deteriorating inventory model with defective products and variable demand. Pervin et al. (2020) consider an integrated vendor–buyer model for deteriorating items in which demand is a quadratic increasing function of time.
Although various inventory models are proposed in previous studies, it would be extremely difficult to apply those inventory management techniques directly to our problem. This is because ship allocation operations always involve human intervention, meaning that we cannot model the probability of how often raw materials are supplied to each steel mill.
In this paper, we propose an inventory calculation model for raw materials under the conditions of multiple suppliers, multiple mills, and supply disruptions. The novelty of this research is that we approximate the structure of the ship-scheduling operation by a two-step probability distribution, and we estimate the seasonal variation of the supply disruption from past shipping-schedule data using a nonhomogeneous Poisson process. The method enables us to determine the optimal seasonal inventory for given brand groups at each mill.
In the following sections, we begin by describing our method for calculating a safety inventory level without considering seasonal variation. We then propose an inventory calculation method with seasonal variation. Finally, we discuss the validity of the results and how they might be implemented and the applied result.
Notations and Assumptions
To formulate our inventory model, we introduce the following notations and assumptions.
Notations
λ: the average number of vessel arrivals per unit of time
x: arrival interval between vessels
I: set of ports
i: port name
T: period for inventory calculation
qi: total arrival counts for port i during a given period T
Q: total arrival counts for all mills during a given period T
Ni: ratio of total vessels assigned to port i
Z: safety inventory days for each brand group and mill
α: service level
: the solution to , meaning a point covering the gamma distribution by
D: the maximum delay,
A: the average arrival interval,
Assumptions
The brand groups for each mill are given.
The demand rate is constant, but the lead time is stochastic.
The arrival process of a ship is a nonhomogeneous Poisson process, and the arrival of ships is allocated in accordance with the arrival process to several steel mills in Japan.
Inventory Calculation Model Without Seasonal Effect
Modeling Procedure
In this section, we describe how to calculate the safety inventory from shipping records. First, in the steel industry, it is difficult to quantify the lead-time variability because the actual shipping schedule differs dramatically from the planned one, including changes to charging and/or discharging ports because of unexpected weather-related problems, such as strong wind and heavy rain. If charging and/or discharging ports are changed from those on the planned schedule, we cannot compare the arrival times between the planned schedule and the actual one, thereby making it difficult to assess the lead-time variability quantitatively.
To cope with this problem, we consider the average arrival interval of the ships for a mill and a brand group as an ideal planned schedule, and we regard the safety inventory as the variability between the ship’s average arrival interval and the maximum variation of its arrival interval. An example of the variation of the arrival interval of ships for a mill and a brand group is shown in Figure 2.

To apply this concept, we must construct a probabilistic model of the arrival interval for each mill and brand group. Figure 3 shows the actual arrival interval at each mill. As shown in Figure 3, the tendencies of the ship’s arrival interval are different by mill. To create a probabilistic model for each place and group, we must clarify why the distribution of arrival intervals differs by mill.

To clarify the distribution of arrival intervals, we construct a model that considers the structure of the ship-scheduling operation. That operational investigation shows that the operator typically decides the ship’s destination in such a way as to prevent shortage of inventory when a ship is approaching a domestic mill. In other words, we found two stages of ship operations, one in which we know when the ship will arrive in Japan and the other in which the ship is allocated to the steel mill. Based on these findings, we consider ship operations to be divided into two types: arrival at all mills (AAM) and arrival at each mill (AEM). We define AAM as the arrival of ships near domestic mills for each brand group and AEM as the arrival of ships for each brand group and at an individual mill.
In the following sections, we discuss how to model two probability distributions, AEM and AAM, based on this idea.
First, we describe the AAM modeling process. The actual AAM data of a brand group are shown in Figure 4, showing clearly that AAM follows an exponential distribution. An exponential distribution is a probability distribution of events that occur continuously and independently at a constant average rate, and the probability density function is given as

Next, we describe the AEM modeling process, the basic concept of which is shown in Figure 5. For each mill i, we define as the arrival counts for each mill during a given period T, and we define Q as the arrival counts for all mills. We assume that a given mill is scheduled regularly being allocated once every times, so under this assumption, the operator allocates the ships arriving in Japan to the mill once every Ni times. Because the arrival interval at mill i is given as a time until a ship arrives in Japan according to a Poisson process for Ni arrivals, the arrival interval of the steel mill is the time interval until the exponential distribution with intensity parameter λ occurs Ni times. Because the time until the exponential distribution occurs Ni times can be the convolution of the exponential distribution Ni times and is known to be a gamma distribution, we model AEM using the gamma distribution as follows:

Estimation Results of AEM
Figure 6 shows a comparison between the gamma distribution model and the actual histogram for a brand group. We used the real shipping scheduling data of a brand group for two years. As shown, the proposed model seems to be consistent. The point that we wish to highlight is that, even if the brand groups change, we can determine the AEM distribution from just the two parameters λ and Ni, which are calculated for each group and an individual mill. We discuss the statistical testing of this model in a later section.

Using the AEM distribution, we can calculate the safety and optimal inventories. We assume that the demand for the raw materials is fixed for each brand group. As shown in Figure 2, we calculate the safety inventory holding days from the difference between the average arrival interval and the maximum delay of the interval. The safety inventory holding days Z for a brand group and a mill is given as follows:
This is our proposed model for calculating the safety inventory for the raw materials without seasonal variation. The benefit of the method is that it is possible to calculate the safety stock even if the brand group is changed because we use only data on the number of vessel arrivals in total and the number of vessels allocated to each steel mill to calculate the lead-time variation. In a later section, we discuss the change in inventory volume when the brand group is changed.
Seasonal Safety Inventory Model
Modeling Procedure
As discussed, we propose a model for calculating the optimal inventory at a certain level during the given period. However, the manager must change the inventory level according to seasonal disruption risk because irregular weather disrupts the supply of raw materials.
As mentioned in the introduction, we know empirically that raw materials imported from Australia and Brazil have seasonal disruption risk in the Japanese winter rather than in summer. To model this feature, we consider the seasonality that appears in the frequency of ships arriving at all mills (i.e., AAM). Based on this concept, we model the seasonal variation in the AAM process.
There are various methods for modeling seasonal variation, such as Holts–Winters (Winters 1960, Holt 2004) and auto regressive integrated moving average (Box et al. 1977). In this case, AAM is an arrival process that follows a Poisson process, and the interval between events is not constant but random. Based on this premise, we model the seasonality using a nonhomogeneous Poisson process.
In this section, we convert the intensity parameter λ of the exponential distribution used for the AAM modeling to a time-dependent variable. We newly describe λ as , where parameter t denotes a particular time of year. In this way, the process considering the nonstationary of the intensity is called a nonhomogeneous Poisson process.
This is a counting process in which individual events occur independently, but the incidence of events is time dependent. For example, the occurrence of an earthquake described in Hong and Guo (1995) and the ignition of a nerve spike follow a nonhomogeneous Poisson process.
We model the seasonality of AAM for each brand group by means of a nonhomogeneous Poisson process. Let N(t) be the AAM counting process, and let the intensity function be the AAM arrival rate at time t. The probability of the occurrence of AAM during can be expressed as
To treat all the arrival processes as nonhomogeneous Poisson ones, we must model the intensity function so that it coincides with AAM. Thus, is modeled as an exponential Fourier series to provide it as a function that varies over a period of one year. The formulation is as follows:
We estimate the coefficients based on maximum likelihood estimates (MLEs), and we determine the optimal order by minimizing the value of the Akaike information criterion.
Figure 7 shows the AAM estimation results for one brand group. For each brand group, we estimate the seasonality of AAM using actual data from the past 10 years. The lower figure depicts the arrival number of ships at a certain time t. Because we assume that AAM is a nonhomogeneous Poisson process, no more than one ship arrives at the port at the same time, which is why the number of ships arriving is always one. The upper figure shows the estimated results for the AAM seasonality .

Notes. Bottom: Actual arrival times of ships. Top: Estimated seasonality results for AAM.
As shown in Figure 7, the estimated intensity suggests that, for this brand group, the arrival rate of ships from February to June is low. This brand group contains goods imported from Brazil, and we know in the Brazilian rainy season (December to April) the shipment of iron ore decreases because of flooding in the mines. It takes more than 40 days to transport the raw material to Japan, so it is reasonable that the number of ships decreases from February to June.
We conducted this analysis on other brand groups and confirmed that the nonhomogeneous Poisson process models could accurately capture the actual seasonal variability.
Next, we explain how we reflect the estimated seasonal variation in the proposed inventory model. We require long-term data to analyze seasonal variation, which is why we use actual data from the past 10 years for modeling. However, the average number of ships arriving differs from year to year because the contract composition changes in accordance with the market conditions of that year. Bearing this is mind, we assume that the tendency of seasonal effects never changes but that the average arrival rate varies with time. We define the past intensity function as and the current average intensity in the analysis period as . The seasonal intensity function is calculated by
Calculation Result of Seasonal Optimal Inventory
We calculated the optimal inventory levels using Equation (8). Figure 8 shows the calculation results for the total seasonal optimal inventory for all mills. The solid line shows the total number of optimal inventory days for all mills, which was calculated from the weighted average of each inventory level for each brand group and mill by the amount of raw material used. The results show that the raw materials should have more inventory in winter than in summer. Moreover, as shown in Figure 8, compared with the conventional inventory policy indicated by the dotted line, we can reduce the inventory level considerably by 14% in summer and 6% in winter. The validity of these results is discussed in the next section.

Discussion
Kolmogorov–Smirnov Test
In this section, we examine the validity of the proposed model by using the Kolmogorov–Smirnov (KS) test. The KS test determines whether the distributions of two data sets are the same or not. We set the significance level to 5%, and we generate data sets for the proposed model using a sampling method featuring the time-rescaling theorem Brown et al. (2002), which produces samples whose arrival interval follows the gamma distribution .
Figure 9 shows a comparison of the actual arrival interval with the proposed model generated from the time-rescaling theorem (Brown et al. 2002). The KS test results show that the p-value was 0.96. Under the null hypothesis that there is no difference between the actual value and the sample obtained from the proposed model, the p-value greatly exceeds the significance level of 5%. Therefore, the null hypothesis is not rejected, and we can assume that there is no difference between the two data sets.

Table 1 lists the KS test results for all the brands and mills. A model whose p-value exceeds 0.05 indicates that the KS test is not rejected. As shown in the table, the p-value of most stock brands exceeds 0.05; therefore, the discrepancy between actual and modeled is low. Note that brand groups showing en-dashes in Table 1 are not used at the mill and, therefore, were not tested.
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Table 1. KS Test p-Values
Mill 1 | Mill 2 | Mill 3 | Mill 4 | Mill 5 | Mill 6 | Mill 7 | Mill 8 | |
---|---|---|---|---|---|---|---|---|
Brand group 1 | 0.13 | – | 0.37 | – | 0.89 | 0.5 | – | 0.82 |
Brand group 2 | 0.46 | 0.11 | 0.18 | 0.08 | 0.46 | 0.81 | 0.29 | 0.82 |
Brand group 3 | – | 0.75 | 0.27 | – | 0.31 | 0.47 | 0.36 | 0.47 |
Brand group 4 | 0.53 | – | 0.28 | – | 0.94 | 0.6 | 0.56 | – |
Brand group 5 | 0.16 | – | 0.2 | 0.89 | 0.72 | 0.25 | 0.47 | 0.66 |
Brand group 6 | 0.38 | 0.23 | 0.1 | 0.03 | 0.09 | 0.1 | 0.06 | 0.06 |
Brand group 7 | 0 | 0 | 0 | 0.77 | 0 | 0 | 0.33 | 0.02 |
Brand group 8 | 0.46 | – | 0.08 | 0.27 | 0.67 | 0.1 | 0.08 | – |
Brand group 9 | 0.05 | – | 0.3 | – | 0.48 | 0.12 | 0.67 | 0.65 |
Brand group 10 | 0.22 | 0.06 | 0.52 | – | 0.4 | 0.96 | 0.56 | 0.35 |
Note. Bold values indicate that the p-value of the KS test is below 0.05.
One concern here is that the p-value is less than 0.05 for six of the eight mills in brand group 7. Regarding this result, note that the proposed model and the actual sample do not share the same data set in this brand group. Brand group 7 features the brands that Nippon Steel imports most frequently, so the arrival frequency of the related ships is higher and more random than that of other brands. We conclude that it might not be possible to establish the assumption that vessels arrive in Japan and have goods sent to a mill once every Ni times.
Inventory Simulation
To verify the proposed model, we also performed inventory simulations. We set the initial inventory calculated from the proposed model and then simulate the operations based on shipping and planning information at the steelworks. After calculating the inventory shortage rate, we verify the feasibility of actual operation in the proposed model by comparing the inventory shortage rate derived from the proposed model with the simulation results.
We used the actual arrival times and lot sizes for each ship along with the exact usage amount of raw material. We estimated the seasonal variation of AAM from July 2002 to June 2012 and calculated the optimal inventory level from June 2013 to July 2014. Then, we performed the inventory simulation for two months from July 2014 to September 2014. We figured the initial inventory to have the average inventory during the simulation coincide with the optimal inventory and then set the initial inventory to the simulation.
We set the service level α (as shown in Table 2) on the basis of past operation data. For example, for mill 8 and brand group 5, the service level is set to 18% because there was an 18% inventory shortage in the actual operation during the calculation period. Figure 10 shows the inventory simulation results for mill 1. In brief, there was no inventory shortage at mill 1, thus demonstrating that it is possible to operate with the calculated optimal stock at this mill.

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Table 2. Inventory Shortage Ratio Obtained by Simulation
Mill 1 | Mill 2 | Mill 3 | Mill 4 | Mill 5 | Mill 6 | Mill 7 | Mill 8 | |
---|---|---|---|---|---|---|---|---|
Brand group 1 | 0.05 | 1.00 | 0.05 | 1.00 | 0.05 | 0.05 | 1.00 | 0.05 |
Brand group 2 | 0.05 | 0.00 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 |
Brand group 3 | 1.00 | 0.05 | 0.05 | 1.00 | 0.05 | 0.05 | 0.05 | 0.05 |
Brand group 4 | 0.05 | 1.00 | 0.05 | 0.84 | 0.05 | 0.05 | 0.05 | 1.00 |
Brand group 5 | 0.05 | 0.25 | 0.21 | 0.05 | 0.05 | 0.38 | 0.05 | 0.18 |
Brand group 6 | 0.05 | 0.05 | 0.11 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 |
Brand group 7 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 |
Brand group 8 | 0.05 | 0.52 | 0.05 | 0.69 | 0.22 | 0.05 | 0.01 | 1.00 |
Brand group 9 | 0.05 | 1.00 | 0.88 | 0.79 | 0.05 | 0.38 | 0.05 | 0.05 |
Brand group 10 | 0.05 | 0.42 | 0.05 | 0.25 | 0.05 | 0.02 | 0.76 | 0.05 |
Table 3 lists the service level for all mills in the simulation. As shown, most brand groups maintain their service level (except for brand group 7).
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Table 3. Inventory Shortage Ratio Obtained from Simulation for July 2014 to September 2014
GR 1 | GR 2 | GR 3 | GR 4 | GR 5 | GR 6 | GR 7 | GR 8 | GR 9 | GR 10 | |
---|---|---|---|---|---|---|---|---|---|---|
Threshold | 0.41 | 0.05 | 0.29 | 0.39 | 0.06 | 0.33 | 0.05 | 0.15 | 0.41 | 0.21 |
Simulation | 0.41 | 0 | 0.25 | 0.38 | 0 | 0.24 | 0.08 | 0.07 | 0.26 | 0.19 |
Note. GR, brand group.
As shown in Table 1, for brand group 7, six out of eight mills did not pass the KS test, and the inventory shortage rate was above the threshold value. These results suggest that we cannot construct an appropriate inventory model for this brand group.
However, the arrival frequency of the ships related to brand group 7 is higher than that of the other groups. For this reason, the arrival times of vessels by location are random compared with the other brands. Therefore, there is a high possibility that the assumption that one vessel is shipped Ni times to each place, which is the premise of the proposed model, is not satisfied. However, because brand group 7 has a high frequency of arrival, the optimal inventory can be determined from the rate of results without using the proposed method. We estimated the arrival frequency by using an appropriate probability distribution of the exponential distribution. Therefore, by using this method, both inventory reduction and stable operation can be achieved.
Implementation and Result
Head-office raw-material managers use the inventory model to determine inventory levels with staff in charge at mills. The managers make management policies such as raw-material procurement plans and the rules to divide brands into brand groups to meet the mills’ requirements. Then, by the policy, the manager executes a calculation of the inventory model. The inventory calculation results for each mill and its brand groups are output in a spreadsheet format and shared with all mills. The managers and the staff determine the final inventory level by repeating the negotiation and the calculation. After that, to maintain the determined inventory level, the managers and the staff operate the daily ship allocation and yard management.
We show a practical example of inventory calculations for the different inventory groups. Table 4 shows original stock groups for each mill, and Table 5 shows revised brand groups of some stocks aggregated for fine ores. In the revised groups, brands 8, 7, 12, and 13 among the fine ores were mainly aggregated as a group, and the inventory was recalculated. Tables 6 and 7 show the results of the inventory holding day calculations for each stock group. We normalized the value of the number of days as 100 for the sum of all steel mills in the original stock group. As shown in Table 7, we found that the revised brand groups could reduce the inventory holding days at all steel mills by 18.1%. Based on this calculation, Nippon Steel changed brand groups at some steel mills and reduced inventory levels to an appropriate level. A noteworthy feature of this method is that, even if the group of stocks changes, we know how much inventory to hold as long as we know how often ships arrive at each mill.
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Table 4. Original Brand Groups for Each Mill
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Table 5. Revised Brand Groups for Each Mill
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Table 6. Inventory Holding Days for Original Brand Groups
Mill 1 | Mill 2 | Mill 3 | Mill 4 | Mill 5 | Mill 6 | Mill 7 | Mill 8 | All mill | |
---|---|---|---|---|---|---|---|---|---|
Pellet | – | 107.4 | 163.8 | 121.0 | – | 221.0 | 209.2 | 105.5 | 148.3 |
Lump ore | 239.1 | 50.2 | 80.4 | 43.2 | 104.8 | 142.1 | 192.3 | 108.1 | 90.0 |
Fine ore | 81.9 | 69.0 | 107.4 | 87.5 | 100.7 | 119.2 | 9.6 | 69.7 | 88.9 |
Total | 106.3 | 69.0 | 105.2 | 76.4 | 101.8 | 132.8 | 180.4 | 110.0 | 100.0 |
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Table 7. Inventory Holding Days for Revised Brand Groups
Mill 1 | Mill 2 | Mill 3 | Mill 4 | Mill 5 | Mill 6 | Mill 7 | Mill 8 | All mill | |
---|---|---|---|---|---|---|---|---|---|
Pellet | – | 107.4 | 163.8 | 121.0 | – | 221.0 | 209.2 | 105.5 | 148.3 |
Lump ore | 239.1 | 50.2 | 80.4 | 43.2 | 104.8 | 142.1 | 192.3 | 108.1 | 90.0 |
Fine ore | 81.9 | 49.4 | 62.7 | 69.7 | 76.8 | 69.7 | 9.6 | 49.4 | 62.7 |
Total | 106.3 | 57.2 | 70.1 | 65.7 | 84.5 | 98.9 | 180.4 | 96.3 | 81.9 |
There are two main benefits of using the inventory model. First, Nippon Steel has applied the seasonal inventory model and succeeded in reducing the inventory of the raw material of which they had unconsciously had too much. They have revised the quantity of the raw material procured and reduced inventory level at each steel mill by 14% in summer and 6% in winter. As a result, they have cut the cost of demurrage resulting from the limited capacity of the yards without stock-out. Second, they have been able to estimate the inventory level even if they have no operational performance. For example, when the managers plan to purchase a new brand, they use the inventory model to estimate the effect of the inventory level on the mills in advance. Also, because the managers can estimate the inventory level after reviewing the management policy, such as the brand groups, the managers and the staff can now discuss in depth the inventory model’s ideal way of managing raw-material inventory.
Figure 11 tracks inventory over three years from June 2013 to July 2016. We normalize the inventory level on a 100-unit scale. The inventory model has been implemented since April 2015, resulting in an average 11% inventory decrease after application. Also, there were no serious out-of-stock incidents. We confirm that this result has achieved a cost reduction of demurrage of at least $4.5 million per year. These results show that the model is an essential tool for determining the company’s raw-material inventory levels and is used in daily work.

Conclusion
We develop a new method for obtaining the optimal season-based inventory levels for raw materials, such as iron ore and coal. In the proposed method, we divide the shipping operations into two distinct models—arrival at all mills and arrival at each mill—by analyzing the relevant shipping operations. We find that the arrival interval of ships arriving in Japan follows an exponential distribution. Furthermore, assuming that the goods on vessels arriving in Japan are shipped equally to each mill, we model the arrival interval of ships to each mill as a gamma distribution. We can establish the optimal inventory level by using just two parameters: the frequency of ship arrival and the ratio of goods to be shipped to each mill. We model the arrival process of ships to Japan as a nonhomogeneous Poisson process and find that, compared with the current inventory level, we can reduce the inventory level by 14% in summer and 6% in winter.
We performed KS tests to evaluate the discrepancy between the proposed model and the actual data sets. Samples of the nonhomogeneous gamma process were generated using a time-rescaling theorem and then tested with real data sets. Results show that the discrepancy between the proposed model and the actual data sets was small. We also carried out an inventory simulation based on the exact shipping and operation plans of mills. Except for one brand group, we confirm that the inventory shortages were lower than the given threshold level, which means that we can operate without any inventory shortage.
We recalculated the optimal inventory level whenever there were changes to either the composition of the raw materials or the fleet. By comparing the obtained optimal inventory level with the capabilities of the facility (such as the yard capacity), we can judge whether it is possible to operate with the current brand group or fleet compositions. This way, we can promptly determine the optimal inventory level and change the brand group and fleet composition in advance. The proposed model contributes dramatically to reducing inventory levels by 11% and demurrage by at least $4.5 million per year in the mills and maintaining stable operation.
In summary, in comparison with previous studies, we have established a method for estimating the seasonal variation of the lead time of ships by assuming that ships arrive according to a nonstationary Poisson process and are allocated regularly at a mill. This enables us to determine the optimal seasonal inventory for all brand groups and all mills.
Many tasks remain for future research. In this study, we assume that the lot size is constant, but in practice the lot size varies greatly from ship to ship. Demand can also vary greatly depending on economic fluctuations and other factors. These variables will be the subject of our future research. We also calculated the safety stock based on a 5% out-of-stock rate. In practice, however, the amount of safety stock should be determined by taking into account the trade-off between inventory holding costs and out-of-stock costs.
Because the impact of demurrage resulting from yard waiting is the most important factor in inventory holding costs, in future work, we intend to study the trade-off between demurrage cost from yard waiting and out-of-stock costs.
The present method is confirmed to be effective at Nippon Steel. Other steel companies have similar operations, and we expect our way to be useful, but verification is a subject for future study. We can also apply the way to companies that operate with similar systems, such as liquefied natural gas (LNG) procurement. However, in LNG inventory management, it is more important to consider the capacity of the LNG tanks, which may cause problems that cannot be solved by this method alone.
Finally, herein, we calculated only the stock volume for a given group of stocks. However, in the future, we would like to investigate how to find the optimal grouping strategy for a given group of stocks with the lowest management costs and the most moderate seasonal variation effects.
The authors acknowledge the Intelligent Algorithm Research Center and the Measurement and Instrumentation & Control Research Lab in Process Research Laboratory for their excellent guidance. The authors would also like to express their deepest gratitude to the Raw Materials Division-1 and Raw Materials Division-1 of Nippon Steel Corporation for providing valuable data and for their cooperation in the practical application of this analysis. Also, they acknowledge the work of the anonymous peer reviewers and associate editor, whose timely and helpful comments improved the final product.
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Verification Letter
Naoto Horishita, Senior Manager, Head of Department, Coal & Coke Management Department, Raw Materials Division-1, Nippon Steel Corporation, 6-1, Marunouchi 2-chome, Chiyoda-ku, Tokyo, 100-8071, Japan, writes:
“This letter is to confirm that this study has achieved an average inventory reduction of 11% per year and a cost reduction of at least $4.5 million per year. The result of the study is one of the most important tools for determining the company’s raw material inventory levels and is used in daily works. Please contact us if you should require any further information.”
Kosuke Kawakami is a PhD student of the department of industrial engineering and economics at Tokyo Institute of Technology and chief digital officer at Negocia, Inc. His research experience areas are supply chain management in the steel industry and optimizing bidding policy in online advertising.
Hirokazu Kobayashi is the section chief of the scheduling section of the Intelligent Algorithm Research Center of Nippon Steel Corporation’s Process Research Laboratories. He is engaged in research on optimization of production planning, scheduling, and logistics.
Kazuhide Nakata is associate professor in the department of industrial engineering and economics at Tokyo Institute of Technology. His main research interests are continuous optimization and counterfactual machine learning.