BUS606: Operations and Supply Chain Management

The Plant Location Problem

Cabbage Production Quantities and Supply Failure Data

The problem in hand is to select the best location for a new semiprocessed kimchi plant by using the imprecise information provided by our research partner, World Institute of Kimchi. This information includes food production conditions, plant operation data, and average supply failure data as shown in Figure 1.

Figure 1 The given information for plant location selection.

The proposed model for plant location selection is aimed at supporting decision makers when they have difficulty in estimating a suitable distribution form related to supply failures for simulation modeling due to the lack of information. In other words, we could not estimate a probability density function using the conventional distribution fitting because there was no detailed supplier failure information such as date and duration of each failure in a specific region. We approximated distribution forms by a fuzzy supply vulnerability analysis based on food production conditions and average supply failure data as shown in Figure 1. It is recommended for decision makers to choose normal, exponential, and gamma distributions that have been widely used for failure occurrence modeling in the literature.

Figure 2 illustrates the food production conditions for the eight prospective locations and the four seasons which include the production quantities in locations, the area of the production region, the demand for food raw materials, the number of customers, and the annual mean temperature. These conditions play critical roles in determining the seasonal production quantities.

Figure 2 The given information of the production conditions for eight plant location candidates.

It is assumed that the delivery distance for raw materials is related to the area of the production and, hence, the delivery time in a relatively large area is longer than that in a small area. For the sake of simplicity, other decisive factors, such as delivery cost per mile, taxation, plant construction, operation cost, and local government policies, are assumed to be the same for all locations.

Plant Location Selection Procedure

This section presents the basic ideas of plant location selection considering the unstable supply of food raw materials. In the food industry, unexpected conditions, such as natural disasters, abnormal climate, or the abandonment of cultivation, sometimes lead to shortfalls in the supply of raw materials. In this case, a food manufacturer should search for alternative sources for food raw materials in other regions. We take this supply shortage situation into account for a supply vulnerability factor, that is, the possibility of replacing raw material feedstock using alternative sources. It can be said that if the possibility is small, the supply vulnerability is high. The plant location selection procedure is illustrated by a case study of semiprocessed kimchi production. The case study was chosen upon the request of our research partner, World Institute of Kimchi, who are aiming to construct a new kimchi processing plant in a suitable location.

Kimchi is a traditional Korean side dish made by combining cabbage and other fermented vegetables in a salted brine. Recently, there has been a strong customer demand for semiprocessed kimchi and, hence, many food manufacturers have focused their attention on building new semiprocessed kimchi plants that can automatically produce salted cabbage on a large scale. The main raw materials for this process include a considerable amount of cabbage, salt, and water, of which the stable supply of cabbage is the most important, irrespective of seasonal and regional variations.

As illustrated in Figure 3, the plant location selection procedure involves the supply vulnerability analysis by estimating supply failure rates and failure durations and stochastic simulation as follows:

Figure 3 The simulation-based selection procedure of a food plant location according to supply vulnerability.

Step 1 (vulnerability analysis). Quantify the fuzzy supply vulnerability from the standpoint of regional and seasonal instability in the supply of raw materials.

Step 2 (simulation modeling).

(i) Convert the quantified supply vulnerability scores (the instability level of raw material supply in a specific region) into raw material supply variations.

(ii) Estimate supply failure rates (the number of supply failure occurrences per season) from the supply vulnerability scores.

(iii) Adjust the probability density functions for the supply failure durations (inter-supply failure time).

(iv) Specify production process parameters (e.g., malfunction rate, processing time).

Step 3 (simulation-based location selection).

(i) Adjust the daily utilization of a production system.

(ii) Calculate a target production volume and an estimated production volume using the adjusted daily utilization.

(iii) Determine the best plant location.

Table 1 summarizes the variables used in the production volume estimation for the proposed method.

Fuzzy Vulnerability Analysis

For supply failure estimation, the supply vulnerability of food raw materials is incorporated into the simulation model in which three main vulnerability factors are involved: raw material availability, production efficiency of raw material, and possibility of replacing raw material feedstock using alternative sources.

(1) Raw material availability assesses whether the amount of raw materials meets the market demand, including the current consumption by competitors in a prospective plant location. It can be linguistically assessed by considering the ratio of production quantity (location, season) to demand (location, season). production quantity (location, season) is the total amount of raw material growing in a location during a particular season, while demand (location, season) is the market demand in a location during a particular season.

(2) Production efficiency of raw materials represents the proportion of the production quantity, relative to the food-growing area in a certain location. It can be said that the higher the production efficiency, the smaller the supply vulnerability.

(3) Possibility of replacing raw material feedstock using alternative sources represents easy accessibility of raw materials from the neighboring region, based on the fact that insufficient raw materials in a certain location can be supplemented from other locations, and it can be assessed (imprecisely) by the normalized ratio $\text { (1/n) } \sum_{\mathrm{i}=1}^{\mathrm{n}} \text { (surplus }_{\mathrm{i}} / \text { distance }_{\mathrm{i}} \text { ) }$. Furthermore,  $n$ is the number of other locations that can support the insufficient raw materials for the location being assessed, $\text { (surplus }_{\mathrm{i}}$ is the surplus of raw materials in the $i ^{th}$ location, and $\text { distance }_{\mathrm{i}}$ is the average distance between the prospective location and the $i ^{th}$ location that will affect delivery efficiency.

Imprecise linguistic assessments of prospective locations with respect to each factor make it difficult to do a direct quantitative evaluation of supply vulnerabilities. Fuzzy quantification is normally performed by clustering and aggregation. Experts often describe their assessment results by using linguistic descriptors such as high, medium, and small. Further, to consider the effects of unknown exogenous factors, a fuzzy aggregation method can be employed. For example, this study divided the levels of vulnerability factor values into two subgroups (i.e., low and high) using a fuzzy c-means clustering method with a conventional triangular-shaped membership function. We used Xie-Beni index S, compactness and separation function, for data clustering to define a membership function, and it is efficient for easy calculation.

$\mathrm{S}=\frac{\sum_{i=1}^{c} \sum_{k=1}^{n} u_{i, k}^{2}\left\|V_{i}-X_{k}\right\|^{2}}{n \min _{i, j}\left\|V_{i}-V_{j}\right\|^{2}}$

where $X_k$ is the $k^{th}$ data point, $V_i$ and $V_j$ are cluster centroids, $u_{i, k}$ is the membership value of data $X_k$, and $min_{i,j} || V_i - V_j ||$ is the minimum distance between cluster centroids.

To find the optimal cluster number for fuzzy rules, it is necessary to find the minimum S. Figure 4 illustrates the clustering result of raw material availability, and the clustering number 2 that minimizes S will be selected as the level of vulnerability factor. Table 2 summarizes the fuzzy input data for the supply vulnerability of raw material. The conventional Mamdani method, which uses minimum implication and maximum aggregation, was employed, after which defuzzification was performed to derive an aggregated vulnerability score by means of finding the center of gravity as follows:

$\text { Center of gravity }=\frac{\sum_{x=a}^{b} \mu_{A}(x) x}{\sum_{x=a}^{b} \mu_{A}(x)}$

where $a \leq x \leq b, a, b \in \mathbf{R}, \mu_{A}(x)$ is a membership function.

Table 2 Fuzzy input data for the supply vulnerability of raw material.

Figure 4 Fuzzy c-means clustering of raw material availability (S: Xie-Beni index).

The score was normalized to have a range from 0 to 1, with 1 meaning highly vulnerable. Table 3 summarizes the supply vulnerability evaluation. Note that if food raw materials are not cultivated in a particular season and region, such that the production quantity is zero, the values of raw material availability and production efficiency of raw materials are calculated to be zero as shown in Table 3.

Table 3 Summary of the supply vulnerability evaluation by fuzzy quantification.

Estimation of Supply Failure Rate and Duration

This subsection describes how to derive the number of supply failure occurrences per season and the inter-supply failure time from the supply vulnerability scores. This information will provide the foundation for estimating the supply failure rate and the duration of each supply failure.

From the estimated probability distribution of failure occurrences, $f_N$, it is possible to identify an empirical relationship between the supply vulnerability scores and the failure occurrences. As shown in Figure 5, supply vulnerability is proportional to the cumulative probability of the supply failure occurrences. There are minimum and maximum numbers of failure occurrences during a particular season in the historical data. Thus, the random variable should be restricted for failure occurrences within a specific range. To do this, we employed truncated distribution models for supply failure occurrence and duration.

Figure 5 Supply failure rate estimation for stochastic simulation.

The supply vulnerability score obtained in the previous subsection determined the parameters of gamma distribution (e.g., the shape parameter and the scale parameter), as shown in Figure 6. In addition, the parameters of gamma distribution were adjusted according to the failure duration information obtained by the supply vulnerability analysis. This was achieved by multiplying the average failure duration by the vulnerability score. The shape parameter $\alpha$ and scale parameter $\beta$ of the truncated gamma distribution $f_G$ were estimated using the moments method; i.e., $\widehat{\alpha}=(\bar{\mu} / \bar{\sigma})^{2}, \beta=\bar{\sigma}^{2} / \bar{\mu}$, $\bar{\mu}\$, where is the mean value of the historical data for failure durations and $\bar{\alpha}\$ is its standard deviation. To include the vulnerability score in the gamma distribution, this study mapped the peak point of gamma distribution to the vulnerability score in the center of the fuzzy membership function. In this case, 0.5 was set as the reference value for which the peak point of gamma distribution correlated with supply failures, moved either to the left (more stable) or to the right (more vulnerable) (see Figure 6). The shape and scale parameters can be adjusted by modifying the mean value as follows.

$\bar{\mu}^{*}=\bar{\mu} \times(0.5+\text { Vulnerability Score (location, season) })$

Figure 6 Supply failure durations in the form of gamma distributions.

Finally, the obtained probabilistic distribution of supply failure duration simulates the daily utilization of a food production system in a location for a particular season, i.e.,$\text { adj_daily_util(location, season) }$ (see step 3 in Figure 3).

Simulation Model of Semiprocessed Kimchi Production

Initial conditions and process information for the simulation of semiprocessed kimchi production are given in Figure 7. The target production volume of a new plant is 2,000 tons/year. The initial inter-arrival time (IAT) of raw material supply is two days. The amount of raw material per order is set as twelve tons. Eight workers handle the entire production process and they work eight hours per day. The semiprocessed kimchi production process consists of nine processes: loading, cutting, first treatment, washing, salting, cleaning, second treatment, rinsing, and dewatering. The operation time information of each process is given in Figure 7. The vulnerability analysis was performed to provide the supply vulnerability scores, then the supply failure rates and durations are estimated in order to adjust IAT of raw material supply during simulation.

Figure 7 Initial conditions and process information for the simulation of semiprocessed kimchi production.

In the simulation, raw material supply continues to produce a demand quantity. In other words, twelve tons of raw material will be supplied every two days until the simulated production volume meets the demand quantity. For this reason, there is no oversupply of raw material in the simulation. On the other hand, in case of supply failure in the simulation, the production volume cannot meet the demand quantity in the required production time, and therefore an estimated production volume is always less than a planned target production volume.

We used a commercial software, Delmia QUEST™, to simulate semiprocessed kimchi production. The QUEST model consists of six main simulation elements: part (cabbage), source (part input), sink (processed part output), machine, labor, and buffer. Refer to Figure 7 for the detailed process information. The average simulation run time for one year production without 3D animation was 39 minutes (CPU: Intel Core i7-7700 3.6GHz, RAM: 16GB).

In summary, we conducted food production simulations by considering seasonal supply variations for more detailed evaluation. However, the proposed plant location selection model aims to rank order of prospective plant locations with respect to decision attributes such as production quantity of raw materials, demand, and food-growing area in a certain location. Therefore, the rank-ordering is still possible even in the case that there is no significant difference in the seasonal supply variations.

The Best Location Selection of New Kimchi Plant

In this study, the best plant location is the one where a planned target production volume can be steadily produced, despite the unstable supply of raw materials. It is formulated as follows:

$\text { Location* } \left.=\underset{\text { location}}{\arg \min } \sum {season} \:{P_T}(\text { season })-P_{E}(\text { location, } \text { season })\right\}$

where $P_T$ is the target production volume for the new plant $P_F$ and is the estimated production volume, considering the regional and seasonal supply vulnerability of food raw materials for the prospective location. The target production volume for a particular season,$P_T (season)$, is determined by:

$P_{T}(\text { season })=w{\text {_days }}(\text { season }) \times w{\text {_hours }} \times \text { daily_ } u t i l(\text { season }) \times \text { prod_vol }$

where w_days (season) is the total number of work days during a particular season, w_hours denotes the maximum work hours per day, and daily_util (season) indicates the daily utilization of the production system with respect to daily demand and production quantities. In addition, the daily utilization is given by the ratio of the scheduled work hours per day (scheduled_w_hours) to w_hours, and prod_vol is the production volume per hour (tons/hour). In general, daily_util (season) is assumed to be sensitive to seasonal demand to maximize the utilization of the production system.

Conversely, the estimated production volume in a location for a particular season,$P_E (location, season)$ , is obtained by the following.

$\text P_{E}( { location, season })= w_{-} \text {days }(\text { season }) \times w{\text {_hours }} \times \text { adj_daily_util }(\text { location, season }) \times \text { prod_vol }$

The adjusted daily utilization of the production system, adj_daily_util (location, season), is determined by the following simulation analysis:

$\text{adj_daily_util (location, season)} =\dfrac{\text { total utilization of a production system in the face of supply failures }}{\text { total utilization of a production system per season }} \times \text{daily_util (season)}$

where

(i) total utilization of a production system per season = $w_{-} \text {days }(\text { season }) \times w{\text {_hours }} \times \text{daily_ util(season)}$ ,

(ii) total utilization of a production system in the face of supply failures = $w_{-} \text {days }(\text { season}) \times \text {w_hours} \times  \text {daily_util (season) - supply_failure_time (location, season)}$

The supply failure time, $\text {supply_failure_time (location, season)}$, represents the total interruption time due to supply failures during which normal food processing is impossible for a particular season at a certain location.

Results and Discussion

Figure 8 illustrates the simulation results of supply failure durations in each prospective location for one year. There are more frequent supply failures for a relatively long duration in location 8, particularly during spring, summer, and winter, whereas it can be said that the supply of food raw materials in location 5 is relatively stable, owing to less failure occurrences and shorter failure durations.

Figure 8 The simulation results of supply failure in locations for a year; repetition of simulation: 1,000 times; supply failure rate (occurrences/season): average 1.5, min. 0, max. 5; duration of a supply failure (day): average 2.5, min. 1, max. 20.

Table 4 summarizes the estimated supply failure duration for the four seasons, the simulation results of $\text{adj_daily_util (location, season)}$, the estimated production volume $P_E$ in one year, and the gap between the target production volume  $P_T$ and $P_E$ in one year. In addition, it is assumed that the total number of work days during a particular season  $\text{w_days(season)}$ is 55 days, the maximum work hours per day $\text {w_hours}$ is 8 hours, the daily utilization of the production system with respect to daily demand and production quantities $\text{daily_util (season)}$ is full (namely, 1), the production volume per hour $\text{prod_vol}$ is 10 tons/hour, and the target production volume in one year is 17,600 tons. The simulation results show that location 5 is the best prospective location in which the planned target production volume can be steadily achieved, despite the unstable supply of raw materials. As summarized in Table 4, the estimated supply failure duration in autumn is relatively short compared to the other seasons. This is because autumn is the harvest season, and thus raw material supply is relatively stable. Table 5 shows an example of supply failure occurrences in location 8 according to the result of one year simulation by using Delmia QUEST™, and the simulation result also shows that the supply failures in autumn are relatively shorter than the other seasons.

Table 4 The simulated daily utilization of a new plant in each location and the gap between the target production volume and the simulated production volume.

Table 5 An example of the supply failure occurrences in location 8 (one year simulation).