Optimal Pricing Strategies and Inventory Management for Fresh Food Products in Sustainable Cold Chain: Analytical Modeling with Korean Market Validation

Abstract

With rising consumer concerns regarding food safety, cold chain management—which preserves product freshness through low-temperature distribution—has emerged as a critical competitive factor for retailers. This study examines how retail firms can manage quality deterioration over time to maximize profits, with a focus on pricing strategies and discard rates. Through game-theoretic modeling and empirical data analysis of milk products, we find that while individual items exhibit no consistent pattern, bundled fresh food items demonstrate an inverted U-shaped relationship between discount rates and profits, indicating an optimal discount level. Furthermore, we identify a U-shaped relationship between order quantity and disposal rate, highlighting the importance of determining optimal inventory levels to minimize waste and maximize efficiency for a sustainable competitiveness.

1. Introduction

As countries’ economic development advances, individual consumption levels rise correspondingly. Consequently, unlike in the past, a culture of enjoying diverse cuisines has become prevalent, and the delivery of fresher products to consumers has emerged as a critical operational concern for businesses. As consumers increasingly demand higher quality food, companies throughout the food supply chain have adopted cold chain systems to maintain freshness, which serve to attract consumers who are particularly sensitive to food safety.

Cold chains can be broadly defined as the management of food distribution processes at low temperatures to maintain product freshness. As food safety regulations have strengthened globally, the cold chain market was valued at USD 248.4 billion in 2020 and is projected to reach USD 410 billion by 2028 [1]. Despite the importance of cold chains, many firms hesitate to adopt this technology due to the substantial capital requirements for refrigerated trucks and storage facilities. For retailers, cold chain management transcends technical considerations—it represents a key competitive differentiator. Products that maintain visible freshness and extended shelf life generate greater consumer trust and drive increased sales. Moreover, with the expansion of online grocery shopping and rapid delivery services, the ability to ensure freshness throughout the distribution network has become a core capability, particularly for perishable categories including produce, meat, dairy, and seafood. Retailers that fail to invest in cold chain infrastructure face both revenue loss and reputational damage from spoilage, stockouts, and food safety incidents.

From the perspective of retail companies managing product quality deterioration over time within cold chain systems, this study seeks to identify profit-maximizing strategies through two approaches: price discounting and discard rate optimization. We examine this relationship using game-theoretic modeling, and following model development, we assess actual cold chain operations and validate various model parameters through empirical analysis. As integrated approaches that combine theoretical and empirical analyses yield more nuanced insights, such methodologies have gained increasing prominence in recent literature and represent a significant contribution to the research stream [2,3,4].

In addition to collecting quantitative data for empirical research, this study conducts qualitative research through in-depth interviews with industry experts to comprehensively understand the current state of the Korean cold chain and capture field-level perspectives. When developing models that integrate analytical frameworks with quantitative data, expert interviews provide crucial insights into practical accessibility and implementation feasibility, thereby enhancing the meaningfulness of both theoretical and empirical findings. By incorporating qualitative methods that capture nuanced aspects of Korean cold chain operations often overlooked in quantitative research, this study achieves methodological rigor while contributing to the balanced advancement of both quantitative and qualitative research paradigms in the field.

This study examines innovation strategies and optimal pricing within the Korean cold chain context. The findings provide a foundation for future research that integrates strategic challenges and economic dynamics currently faced by industry practitioners.

This study addresses the following research questions: How can retail firms optimize pricing strategies and inventory management for fresh food products in cold chain environments to maximize profits while minimizing waste? Specifically, (1) what is the optimal relationship between discount rates and profitability for fresh food products, and how does product bundling moderate this relationship? (2) What is the optimal order quantity that minimizes disposal rates for perishable products? While the existing literature has examined pricing strategies and inventory management separately, limited research integrates both dimensions within the specific context of cold chain management for fresh foods, particularly with empirical validation using actual retail data. This gap motivates our integrated analytical and empirical approach.

This paper is organized as follows. Section 2 provides the theoretical background and reviews relevant literature. Section 3 presents the analytical model. Section 4 discusses the empirical analysis and results. Finally, Section 5 concludes with managerial insights, limitations, and future research directions.

2. Theoretical Background

2.1. Time-Based Quality Deterioration Theory

Time-dependent quality deterioration represents one of the fundamental challenges in fresh food distribution management. While traditional quality deterioration research originated from the exponential decay model proposed by Ghare & Schrader [5], recent studies have demonstrated the applicability of linear quality deterioration models under specific conditions.

Fu & Labuza [6] presented various approaches to food quality deterioration, emphasizing that deterioration patterns vary depending on storage conditions and duration. They suggested that linear quality deterioration models may be more appropriate for relatively short-term and controlled cold chain environments. This is expressed as the initial quality level decreasing at a constant rate over time, indicating that quality deterioration follows a predictable pattern. Moshtagh et al. [7] addressed the challenges of perishable product markets by developing a joint model for optimizing issuing, quality disclosure, production, and markdown pricing. Their research evaluates the effectiveness of different markdown strategies, finding that while concealing quality information can be optimal, the benefits of complex multi-stage or dynamic markdowns are often marginal compared to simpler single-stage approaches.

While exponential quality deterioration models exhibit gradual changes initially followed by rapid deterioration over time; linear models demonstrate quality reduction at a constant rate. In cold chain environments, where temperature is maintained relatively consistently, quality deterioration often follows uniform patterns. Due to these characteristics, linear models can provide more practical and accurate predictions when analyzing quality changes over short periods.

Quality deterioration manifests differently depending on product types and storage conditions. In fresh foods, quality deterioration occurs through biochemical processes including microbial proliferation, enzymatic activity, and oxidation, which affect appearance, flavor, nutritional value, and safety simultaneously. Critically, quality deterioration in most fresh foods is irreversible and cannot be restored through subsequent treatment. Therefore, quality management of fresh foods must emphasize preventive measures rather than remedial interventions, underscoring the importance of time-based quality deterioration modeling. Recently, Syed et al. [8] demonstrated that dynamic pricing is vital for short-shelf-life products, consistent with our approach. Their study proposes a three-phase data-driven digital transformation (DD-DT) model for dynamic pricing of perishables: initiation (data collection and analytics), facilitation (algorithm development and real-time integration), and strategic adaptation (ongoing refinement) in supermarkets. Similarly, Kayikci et al. [9] note that fresh products are wasted in low- and middle-income countries due to inappropriate pricing and volatile demand. Their study addresses retail food waste by proposing a four-stage data-driven dynamic pricing model for bulk produce, utilizing real-time IoT sensor data to determine freshness scores and optimize pricing. Through empirical validation, the model evaluates the impact of pricing, replenishment, discounts, and freshness on profit and waste, demonstrating the significant potential of hyperspectral imaging for reducing waste and increasing sales.

2.2. Price–Quality–Demand Relationship

Consumer demand is influenced simultaneously by product price and quality level. The linear demand function subtracts the price effect from total market size while incorporating the quality effect, demonstrating a clear relationship between price and quality in monopolistic markets. Price sensitivity and quality sensitivity represent crucial parameters that characterize fundamental aspects of consumer behavior.

According to Bernstein & Federgruen [10], in competitive markets, when one firm enhances its quality level, competing firms tend to respond by lowering prices. This occurs because consumers respond more sensitively to price than to quality. Consequently, since price exerts greater influence on demand than quality does, quality sensitivity typically exhibits lower levels relative to price sensitivity. To address this, Motlagh & Nasiri [11] develop static and dynamic pricing strategies for retail and chain stores, incorporating discounts and revenue-sharing under uncertain, price-dependent demand with stock-out considerations.

As quality deterioration progresses, consumer purchase intention declines, necessitating price discounts to compensate for this reduction. This price–quality–demand interaction directly affects retailers’ strategic decision-making, and for perishable products, the temporal dimension further complicates this relationship. As quality deteriorates over time, demand decreases even at constant price points, making price adjustments essential for maintaining demand levels.

2.3. Optimal Discount Rate Existence Theory

For an optimal solution to exist in the relationship between discount rates and profitability, the profit function must exhibit strictly concave characteristics with respect to the discount rate. This implies that the rate of profit change demonstrates diminishing returns when the discount rate varies, suggesting that maximum profit can be achieved at a specific discount rate.

The research by Lal & Staelin [12] theoretically demonstrated that optimal values exist for price discounting, indicating that both low and high discount rates can negatively impact profitability. When discount rates are low, the inventory clearance effect remains limited, resulting in increased disposal costs. If quality-deteriorated products are not processed appropriately, they eventually reach a point where disposal becomes necessary, leading to direct losses.

Conversely, when discount rates are excessively high, unit profit margins decrease, potentially reducing overall profits. While sales volume may increase through discounting, if the profit per unit diminishes significantly, total profit may decrease. This dual characteristic demonstrates that the relationship between discount rates and profits follows an inverted U-shaped curve, supporting the concavity of the profit function. Therefore, identifying the appropriate discount rate becomes a fundamental challenge for profit optimization.

2.4. Order Quantity Optimization and Inventory Management

In inventory management of perishable products, order quantity determination constitutes a critical factor. While traditional economic order quantity models did not consider quality deterioration, perishable products require order quantity optimization that reflects time-dependent deterioration and corresponding demand changes. Excessive order quantities result in increased disposal losses due to quality deterioration, while insufficient quantities lead to opportunity losses and higher costs from increased ordering frequency.

Richter [13,14] confirmed the close relationship between spoilage rates and order quantities in economic order quantity models. To achieve optimal spoilage rates from a total cost perspective, order quantities must be adjusted, suggesting that perishable product management requires not only preventing quality deterioration but also minimizing disposal costs through order quantity optimization.

Herbon & Khmelnitsky [15] emphasized that profit maximization for quality-deteriorating products necessitates optimal order quantity calculation, while Wang et al. [16] demonstrated the significance of optimal order quantities for enhancing corporate profits in food supply chains.

An interesting observation is that spoilage rates tend to decrease as order quantities increase. This occurs because the sales volume increase effect exceeds the loss increase due to quality deterioration. While larger order quantities result in higher absolute losses from quality deterioration, the loss ratio relative to total order quantity tends to decrease. This phenomenon resembles economies of scale, indicating that appropriate increases in order quantity can enhance overall efficiency.

2.5. Disposal Costs and Disposal Management Strategies

Inventory disposal due to quality deterioration is an inevitable phenomenon in the distribution of perishable products. Qin et al. [17] systematically analyzed the costs associated with inventory disposal and presented a methodology for calculating disposal processing costs by linking them to the quantity of disposed products. Furthermore, retailers must set expiration dates that balance the risk of discarding still-saleable products against unknowingly selling expired ones, accounting for random product lifetimes while ensuring reasonable shelf life for customers [18]. This represents a critical concept for quantifying the processing costs of products affected by time-dependent quality deterioration.

The disposal rate is defined as the ratio of residual inventory exceeding market demand plus losses due to quality deterioration to the initial order quantity. This metric serves as a key indicator for measuring inventory management efficiency, where higher disposal rates indicate inefficient inventory management. Disposal rate management not only contributes to loss reduction but also directly enhances overall profitability.

In practice, two types of product disposal occur. The first involves disposal due to quality deterioration over time, corresponding to cases where fresh food products exceed their expiration dates and can no longer be sold. The second involves disposal of products that remain unsold within a certain timeframe to create space for new inventory. While this occurs in relatively limited circumstances, it represents a phenomenon observed in actual distribution settings.

This categorization of disposal types reflects actual distribution practices and enhances the accuracy of disposal cost calculations. It also suggests that different management strategies are required for each type, highlighting the necessity for integrated disposal management strategies. According to expert interviews, companies are making efforts to minimize related costs by outsourcing disposal processing operations, which demonstrates the significance of the impact that disposal costs have on profitability.

2.6. Characteristics of Fresh Food Distribution in Cold Chain Environments

Fresh food distribution in cold chain environments possesses distinctive characteristics that differentiate it from general product distribution. A cold chain is a system that maintains consistent low temperatures from production to final consumers to ensure product quality and safety, representing an essential element in fresh food distribution. This study assumes linear quality deterioration over time, focusing on quality changes within retail establishments where the cold chain is relatively short. Akram et al. [19] reveals FCC management (FCCM) as an interdisciplinary field and highlights that integrating various transportation modes, advanced planning, optimization techniques, and ICT can significantly improve cold supply chain efficiency and effectiveness.

First, consumers exhibit high sensitivity to freshness. Consumers maintain particularly high quality standards for fresh foods distributed through cold chains, placing significant emphasis on expiration dates and freshness. They tend to demand substantial discounts for products approaching their expiration dates, which directly influences pricing policies and marketing strategies. Milk represents one of the products for which consumers prioritize checking expiration dates most frequently, and such consumer behavior provides empirical evidence for the concepts of price sensitivity and quality sensitivity considered in analytical models. And further, milk is a representative dairy product, and both milk and milk-based products play a critical role in cold chain logistics [20]. Second, there is significant seasonal and temporal demand variability. Fresh food demand exhibits considerable fluctuations according to season, day of the week, and time of day, and this variability complicates inventory management. In the case of milk, high demand patterns in summer and relatively low demand in winter reflect changes in overall market size, which also influences optimal discount rate determination. During periods of high demand, the necessity for discounting is relatively low, while periods of low demand require more aggressive discounting strategies.

Third, there is complexity in inventory management due to short shelf lives. Cold chain products generally have considerably shorter shelf lives compared to other processed foods, which presents the dual challenge of increasing inventory turnover while preventing stockouts. Quality deterioration in milk occurs through microbial proliferation, enzymatic activity, and lipid oxidation, and once quality has deteriorated, restoration to the original state is impossible. These characteristics highlight the importance of quality deterioration rate and disposal rate concepts, emphasizing the necessity for time-dependent discounting strategies.

These characteristics of fresh food distribution in cold chain environments support the empirical validity of concepts established in analytical models and provide important background for analyzing the integrated impact of discount rates and disposal rates on profitability.

3. Analytic Research Model

3.1. Model Formulation

In this study, game theory will be used to construct an analytical model. From the perspective of the retailing company constituting the supply chain, the purpose of this study is to find an optimal strategy in the cold chain situation when sales can be increased by implementing a price discount policy. Assuming that there are manufacturers and distributors that provide fresh food, the demand function is judged to have a linear relationship between product price and product quality level in an exclusive situation.

This study will conduct an analysis based on the following scenario.

Scenario 1. A retailer implements the price discount policy to recover demand due to lessened quality.

For the development of scenario 1, assuming the overall market size as $\alpha$, the case of price discounts to cope with the decrease in demand due to quality degradation without efforts to improve quality is defined as follows:

$$\begin{gathered} D_{1a}={\int }_0^t\left. [\alpha -\beta p+\gamma q_0\left(1-\lambda t\right)\right]dt,0\le t\le t_1 \\ {D}_{2a}={\int }_{t_1}^t\left. [\alpha -\beta p\left(1-\theta \right)+\gamma q_0\left(1-\lambda t\right)\right]dt,t_1\le t\le t_2 \end{gathered}$$

(1)

$D_{1a}$ and $D_{2a}$ are the expected demands for each and for the period $0\le t\le t_1$ and $t_1\le t\le t_2$, respectively. $\beta$ is the sensitivity of a change in demand for price, and $\lambda$ is the sensitivity of a change in demand for quality level. In the first scenario, it is set that retailing companies perform a price discount as much as possible to compensate for the decrease in demand due to quality degradation during $t_1\le t\le t_2$. In addition, in this study, the range of time is normalized to 1, that is, the value of $t_2$ is judged as 1.

In particular, the deterioration of quality over time is a major factor influencing demand and there are various approaches [21]. In this study, quality deterioration is considered linearly because the cold chain is not long and focuses on quality changes within retailers. Therefore, it is assumed that the quality level $q_0$ at the time of warehousing decreases at a rate of $\lambda$ over time. The value is a value that can be estimated by the product storage capacity and condition, and acts as an important factor when developing into an empirical study.

The retailer’s sales and processing costs are assumed to be normalized to zero to consider only quality costs according to quality, and for convenience of calculation, the overall market size $\alpha$ is assumed to be 1, the sensitivity of demand changes to price $\beta$ is assumed to be 1, and the following profit function is obtained:

$${\mathrm{\pi }}_{ra}=D_{1a}\times p+D_{2a}\times p\left(1-\theta \right)$$

(2)

$$={\displaystyle \frac{1}{2}}p\left(\begin{array}{c}\left(-1+\theta \right)\left(-2-2p\left(-1+\theta \right)+\gamma \left(-2+\lambda \right)q_0\right)\\ +2\theta \left(1+p\left(-2+\theta \right)+\gamma q_0\right)t_1-\gamma \theta \lambda q_0{t}_1^2\end{array}\right)$$

From Equation (2), the following price is obtained:

$$p*={\displaystyle \frac{-2+2\theta -2\theta t_1+\gamma q_0\left(\left(-1+\theta \right)\left(2-\lambda \right)-\theta t_1\left(2-{\lambda t}_1\right)\right)}{-4{\left(-1+\theta \right)}^2-4\left(2-\theta \right)\theta t_1}}$$

(3)

Based on Equations (2) and (3), the following optimal profit can be derived:

$${\pi }_{ra}*={\displaystyle \frac{{\left(-2+2\theta -2\theta t_1+\gamma q_0\left(\left(-1+\theta \right)\left(2-\lambda \right)-\theta t_1\left(2-{\lambda t}_1\right)\right)\right)}^2}{16{\left(-1+\theta \right)}^2+16\left(2-\theta \right)\theta t_1}}$$

(4)

3.2. Existence of Optimal Discount Rate

Suppose Equation (4) is a function of the discount rate. In order for an optimal discount rate that maximizes retailer’s profit to exist, the following condition must be satisfied:

$${\pi }_{ra}^{*}\left(\left(1-a\right)x+ay\right)>\left(1-a\right){\pi }_{ra}^{*}\left(x\right)+a{\pi }_{ra}^{*}\left(y\right)$$

(5)

If this condition is satisfied under $a\in \left(\mathrm{0,1}\right),x\ne y$, Equation (5) can be considered strictly concave. Since the range of the discount rate is $\left. [\mathrm{0,1}\right]$, we substitute 0 and 1 into $x,y$, respectively, and simplify the expression. In the case of market competition, when one firm raises its quality level, competing firms tend to lower their prices in response. This is because consumers are often more sensitive to price [10].

In other words, since price has a greater effect on demand than quality, it is reasonable to assume that the coefficient of quality is less than 1. In this study, we assume $\gamma$ to be 0.5.

In addition, both price and quality are standardized in this study and as in previous studies, where the initial quality level was often set close to 1 to analyze variations in other parameters, we conveniently set $q_0$ to 1 [21]. While the quality degradation coefficient in Wang & Li [22] is given per hour and does not exactly match the unit-time coefficient in this study, we similarly assume $\lambda$ to be 0.05.

Under these assumptions, we obtain the following proposition:

Proposition 1.

The profit function is strictly concave with respect to the discount rate under certain conditions.

Assuming $\gamma$, $q_0$, and $\lambda$ are 0.5, 1, and 0.05, respectively,

$${\pi }_{ra}^{*}\left(a\right)-\left(\left(1-a\right){\pi }_{ra}^{*}\left(0\right)+a{\pi }_{ra}^{*}\left(1\right)\right)$$

is always greater than 0. Therefore, the profit function exhibits strict concavity under specific conditions, and thus Proposition 1 is proven.

Proposition 1 is related to Hypothesis 1.1 described below. When profit is considered as a function of the discount rate, this implies that an optimal discount rate exists. This suggests that deriving and applying an appropriate discount rate can contribute to profit maximization. It is expected that the above theorem may also hold under other reasonable conditions beyond the specific assumptions on $\gamma$, $q_0$, and $\lambda$. The study by Rong et al. [23] served as a motivation for this research. They suggested that if a firm can minimize quality degradation over time and maintain product value, more consumers may be induced to purchase, thereby increasing profits. In fact, Hsu et al. [24] and Dye [25] analytically explored how investments in preservation technologies for maintaining quality levels could generate profits for actual retailers.

3.3. Model Including Order Quantity and Disposal Costs

In the study by Qin et al. [17], the cost associated with inventory disposal is denoted as $c_h$. Multiplying this cost by the amount of discarded products can be seen as the cost of handling items affected by quality degradation over time.

According to expert interviews, separate costs should be considered for handling discarded products. Companies are attempting to minimize related costs by outsourcing the disposal of materials—such as waste paper and miscellaneous items—that could otherwise yield some revenue. Moreover, because the cost of disposal may differ between the early sales stage and later periods, the value of $c_h$ could also be set in intervals. In this model, however, we initially treat it as a single cost.

If the initial order quantity is denoted as $Q_o$, the remaining inventory at time $t_2$ can be expressed as follows:

$$Q_1=Q_0-\left(Q_{d0}+Q_{d1}+D_{1a}+D_{2a}\right)$$

(6)

Accordingly, the disposal rate is defined as follows:

$$disposal (discarded) rate={\displaystyle \frac{Q_0-\left(D_{1a}+D_{2a}\right)}{Q_0}}={\displaystyle \frac{Q_1+Q_{d0}+Q_{d1}}{Q_0}}$$

(7)

In other words, the disposal rate is defined as the ratio of the remaining inventory that exceeds market demand and the quantity degraded due to quality deterioration, divided by the initial order quantity. While the quantities degraded in the interval $\left. [0,t_1\right]$ and $\left. [t_1,t_2\right]$, respectively, are $Q_{d0}=0.5\gamma \lambda q_0{t_1}^2$, and $Q_{d1}=0.5\gamma \lambda q_0\left({t_2}^2-{t_1}^2\right)$.

From Equation (7), it can be observed that as the order quantity increases, the disposal rate tends to decrease. Although $Q_{d0},Q_{d1}$ may also increase as $Q_0$ increases, it is empirically known that the quantity lost due to quality deterioration is typically smaller than the quantity sold. Moreover, it is expected that the increase in order quantity due to demand fluctuations will be greater than the increase in the amount of products discarded. This observation relates to Hypothesis 2.

In this model, two distinct types of product disposal are addressed, in an effort to closely reflect actual distribution practices. The first refers to products that have deteriorated in quality over time—empirically, this corresponds to items such as milk that has passed its expiration date in the real-world analysis.

The second refers to products that are not sold within a certain time frame and are discarded to make room for new products (either competing brands or new stock of the same product). Although this second category tends to occur in more limited cases, such as with unpopular items, it is a phenomenon observed in practice and thus has been included. Given these two types of disposal, the disposal rate due to quality degradation has been newly defined, and this forms the basis for calculating disposal-related costs. For convenience, $Q_1$ is defined as 0 in the following model. When the first derivative has a negative slope and the second derivative has a positive slope, the function is convex and there is a possibility of a local minimum. Since the current function is an inverse function of the order quantity, the following interpretation can be made:

Proposition 2.

The disposal rate function is strictly convex with respect to the order quantity under certain conditions.

Accordingly, by incorporating Equations (6) and (7), the profit function is modified as follows:

$${\pi }_{ra}=D_{1a}\times p+D_{2a}\times p\left(1-\theta \right)-c_h\times \left(Q_1+Q_{d0}+Q_{d1}\right)$$

(8)

$${{\pi }_{ra}}^{*}={\displaystyle \frac{\left(\begin{array}{c}4\gamma q_0\left(\begin{array}{c}\left(1-\theta +\theta t_1\right)\left(\left(-1+\theta \right)\left(2-\lambda \right)+\theta t_1\left(2-\lambda t_1\right)\right)\\ -2\lambda c_h\left({\left(-1+\theta \right)}^2+\left(-2+\theta \right)\theta t_1\right)\end{array}\right)\\ -4{\left(1-\theta +\theta t_1\right)}^2-q_0^2{\left(\left(-1+\theta \right)\left(-2+\lambda \right)+\theta t_1\left(2-\lambda t_1\right)\right)}^2\end{array}\right)}{16\left({\left(-1+\theta \right)}^2+\left(-2+\theta \right)\theta t_1\right)}}$$

(9)

The following flowchart (Figure 1) visually summarizes the Analytical Research Model. It begins with product arrival and tracks linear quality deterioration over time. As quality drops, retailers decide whether to implement price discounts to stimulate demand. Profit is then calculated based on adjusted prices. If discardable inventory occurs, total profit is recalculated by factoring in disposal costs. This model is directly linked to sales strategy, as it determines optimal discount rates and order quantities that maximize profit while minimizing waste in cold chain operations.

4. Empirical Research Model

There is a growing trend toward conducting analytical research and empirical research simultaneously while validating results from both approaches [2,3,4]. Validation utilizing accumulated data based on analytical models can provide multifaceted implications. Furthermore, this study conducted not only empirical research analysis using accumulated quantitative data but also qualitative research through interviews with industry experts. Through interviews with industry experts, we explored the processes and contexts of the current competitive environment in Korea’s cold chain industry more deeply, capturing insights and understanding stakeholder relationships that might be overlooked in quantitative research. Additionally, when establishing models for analytical research and empirical research utilizing quantitative data, we aimed to enhance the qualitative level of research by conducting experiential integration through interviews with industry experts regarding the trade-off relationship between fidelity and feasibility, thereby presenting more meaningful implications from both analytical and empirical research.

We have expanded the empirical research model introduction section to clearly present the components as follows: Our empirical research model consists of three core components: (1) the relationship between price discount strategies and profits, (2) the moderating effect of bundling strategies, and (3) the relationship between order quantity and disposal rates. First, we predict a non-linear relationship (inverted U-shape) between discount rates and profits. Second, we expect this relationship to be moderated by whether products are offered in bundled form. Third, we hypothesize a U-shaped relationship between order quantity and disposal rates.

4.1. Data Collection and Data Refinement Process

Following four in-depth interviews with fresh food team managers at Company A, a major domestic retailer, monthly fresh food discount rates and sales revenue data spanning 12 months were obtained for research and analytical purposes. Company A is a major Korean supermarket chain established in 2001, operating as a corporate supermarket (SSM) under one of Korea’s largest retail conglomerates. The company operates 352 stores nationwide and achieved an operating profit of KRW 29.3 billion in FY2024, making it one of the leading players in Korea’s retail food distribution market. As a large-scale retailer specializing in fresh food products with extensive cold chain infrastructure, Company A is considered to provide a suitable empirical context for analyzing pricing strategies and inventory management in perishable goods. The company’s substantial market presence and sophisticated supply chain management systems are expected to offer a robust empirical foundation for testing our theoretical models in real-world retail environments.

Table 1. Number of SKUs by Major Category of Fresh Foods by Month.

	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Fruits	1212 (13%)	1206 (13%)	1239 (13%)	1462 (14%)	1383 (14%)	1322 (13%)	1331 (14%)	1422 (14%)	1331 (13%)	1430 (13%)	1330 (13%)	1433 (13%)
Vegetables	1991 (21%)	2000 (21%)	1995 (21%)	2282 (22%)	2197 (22%)	2099 (21%)	2166 (22%)	2245 (22%)	2232 (23%)	2505 (23%)	2567 (24%)	2597 (23%)
Seafood	1836 (19%)	1886 (20%)	1894 (20%)	1971 (19%)	1881 (19%)	1849 (19%)	1853 (19%)	1837 (18%)	1693 (17%)	1987 (19%)	1947 (18%)	2002 (18%)
Meat	1199 (13%)	1199 (13%)	1131 (12%)	1176 (11%)	1205 (12%)	1209 (12%)	1233 (13%)	1240 (12%)	1219 (12%)	1315 (12%)	1270 (12%)	1278 (11%)
Dairy	1141 (12%)	1112 (12%)	1119 (12%)	1281 (12%)	1174 (12%)	1158 (12%)	1115 (11%)	1179 (12%)	1153 (12%)	1160 (11%)	1159 (11%)	1260 (11%)
Frozen	709 (7%)	715 (7%)	693 (17%)	776 (7%)	723 (7%)	704 (7%)	697 (7%)	734 (7%)	716 (7%)	714 (7%)	724 (7%)	772 (7%)
MS	1417 (15%)	1441 (15%)	1403 (15%)	1631 (15%)	1595 (16%)	1509 (15%)	1415 (14%)	1552 (15%)	1529 (15%)	1595 (15%)	1534 (15%)	1830 (16%)
Total	9505 (100%)	9559 (100%)	9474 (100%)	10,579 (100%)	10,158 (100%)	9850 (100%)	9810 (100%)	10,209 (100%)	9873 (100%)	10,706 (100%)	10,531 (100%)	11,172 (100%)

presents the number of SKUs by major category of fresh food products across the months.

As income levels rise, consumption of dairy products is increasing, and the importance of dairy cold chain systems for providing high-quality dairy products to consumers is growing [26]. Dairy products fundamentally refer to all processed foods made from animal milk, which are typically manufactured through processing facilities and distributed to retailers as finished products, with milk (plain and flavored) being the most representative form. In Table 1 above, the dairy category includes not only milk but also cheese, yogurt, soy milk, and lactic acid bacteria products. The reason for selecting milk as the subject of analysis is that it is the most frequently chosen dairy product, price discounts—which are the main focus of this study—occur frequently, and quality deterioration over time is more pronounced compared to other dairy products. The researchers classified milk products, including chocolate, strawberry, and coffee milk, separately from other dairy products through internet searches on a monthly basis. This process was reviewed multiple times to complete the first data cleaning procedure.

Table 2. Number of SKUs for Milk Products by Month.

	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Milk	255	241	235	268	237	228	217	213	212	207	207	208

below shows the number of SKUs for milk products by month.

As can be observed in Table 2, there are differences in the number of SKUs for milk products across months. To enable monthly comparisons, a second refinement process was conducted to standardize the SKU count across the entire period based on January as the reference month. Products were refined based on product names to ensure identical products across each month, resulting in 207 SKUs per month. Subsequently, data lacking information on sales revenue, discount rate, and disposal rate were removed for each month. Using January as the baseline, 33 SKUs were found to be included across all 12 months of the entire period. To secure a sufficient sample size for analysis, this study also included SKUs that were commonly present for 11, 10, and 9 months based on January as the reference, comprising 20, 15, and 12 SKUs, respectively. Ultimately, a total of 80 SKUs were utilized for analysis across the entire period (January to December).

4.2. Empirical Research Model 1

The first empirical research model is presented in Figure 2. This model analyzes the impact of the independent variable, discount rate, on the dependent variable, profit. Based on Proposition 1 derived from the analytical model, an inverted U-shaped relationship is expected to exist between discount rate and profit. To verify this, a nonlinear model including the quadratic term of discount rate was established, incorporating the moderating effect of bundling on the relationship between discount rate and profit.

Price discounting is not only a preferred strategy for revenue improvement in distribution but also enables various discount methods from a cost-effectiveness perspective [27]. The existence of optimal values for such price discounts has been confirmed in previous studies [12], and it is meaningful to examine patterns observed in milk, which is consumed by many people.

Two important mechanisms operate in price discount strategies for fresh foods including milk. First, discount rates create conflicting effects of demand increase and unit profit decrease. At low discount rates, the demand increase effect dominates, leading to increased total profits, but when discount rates exceed a certain level, the unit profit decrease effect becomes larger, resulting in decreased total profits. Second, consumer value perception and price sensitivity operate differently between bundled and individual products. Bundling can change consumers’ reference prices and evaluation criteria, making their response to discounts different from individual products.

Based on these mechanisms, this study goes beyond empirically validating the theoretical results of analytical models to additionally consider the moderating effect of bundling to reflect the complexity of actual distribution environments. Since differences in consumer perceived value caused by bundling change consumption patterns [28], understanding the impact of bundling strategies widely used in actual distribution stages on optimal discount rates is important for enhancing the practical applicability of theoretical results. From a retailer’s perspective, optimizing brand-specific discount strategies has already become a core concern [29], and verification of such moderating effects is expected to provide more realistic implications. Therefore, we establish the following hypotheses:

Hypothesis 1.1.

As the magnitude of the discount rate increases, profit will exhibit an inverted U-shaped pattern.

Hypothesis 1.2.

The inverted U-shaped relationship between discount rate magnitude and profit will vary depending on whether products are bundled.

4.3. Analysis Results: Empirical Research Model 1

4.3.1. Impact of Discount Rate on Profit

To test Hypothesis 1.1, this study conducted a multiple regression analysis model.

Table 3. Analysis Results for Hypothesis 1.1.

	B	Beta	t-Value
Constant	799.182	-	8.270
Discount Rate	−326.592	−0.042	−0.738
Discount Rate2	−1342.420	−0.067	−1.197
$\mathit{A}\mathit{d}\mathit{j}.{\mathit{R}}^{\mathbf{2}}$		0.008
${\mathit{R}}^{\mathbf{2}}$		0.104
$\mathit{F}$		4.617 **

Dependent Variable: Profit (Unit: KRW 10,000). ** $p<0.05$.

analysis reveals important discount rate–profit relationship implications. The R² of 0.104 indicates that independent variables explain 10.4% of variance in the dependent variable. The F-statistic, which tests the overall significance of the regression model, is 4.617, showing statistically significant results at the 0.05 significance level (p < 0.05).

The first-order coefficient of discount rate is −326.592, showing a negative direction, which indicates a tendency for profit to decrease as the discount rate increases. However, with a t-value of −0.738, the absolute value is less than 1.96, making it statistically non-significant at the 0.05 significance level. The standardized coefficient (Beta) is −0.042, showing a very small effect size. The second-order coefficient of discount rate is −1342.420, showing an even larger negative value, with a t-value of −1.197 and a standardized coefficient of −0.067. The negative second-order coefficient indicates that the rate of profit decrease accelerates with increasing discount rates, but this is also statistically non-significant (p > 0.1).

Consequently, contrary to the theoretical predictions of Proposition 1 derived from the analytical model, the empirical data failed to confirm an inverted U-shaped relationship between discount rate and profit.

Figure 3 presents a scatter plot and estimated regression line that visually depicts the relationship between discount rate and profit. Unlike the theoretically expected inverted U-shaped curve, the actual data distribution does not exhibit a clear pattern. A detailed examination of the graph reveals that profit variability is considerably large in the low discount rate range, and it is difficult to observe a clear trend of consistent profit decline as the discount rate increases. This may be because the effectiveness of discount strategies in actual markets is complexly influenced by various exogenous factors (seasonality, competitive conditions, changes in consumer preferences, etc.). Particularly noteworthy is that the data points are not evenly distributed around the regression line but show considerable dispersion. This suggests the existence of unobserved variables that influence profit beyond the discount rate. Furthermore, the theoretically expected optimal point or inflection point cannot be clearly identified in the intermediate range of discount rates, failing to support the existence of an inverted U-shaped relationship.

4.3.2. Analysis of Differential Impact of Discount Rate on Profit According to Bundled Product Status

Table 4. Analysis Results for Hypothesis 1.2.

	Bundle (n = 174)		Unbundle (n = 680)		Fisher Z-Test
	B	Beta	B	Beta
Constant	1601.686	-	610.497	-	-
Discount Rate	2845.651	0.283 **	−1029.732	−0.144 **	5.09 ***
Discount Rate2	−7159.762	−0.304 **	93.317	0.005	−3.73 ***
$\mathit{A}\mathit{d}\mathit{j}.{\mathit{R}}^{\mathbf{2}}$	0.024		0.017
${\mathit{R}}^{\mathbf{2}}$	0.035		0.020
$\mathit{F}$	3.124 **		6.75 ***

Dependent Variable: Profit (Unit: KRW 10,000). ** $p<0.05$, *** $p<0.01$.

presents the results of subgroup analysis divided into bundle products and individual products. The subgroup analysis results for testing Hypothesis 1.2 clearly demonstrate differential patterns between bundle products and individual products [23,24,25].

For the bundle product group (n = 174), the coefficient of determination (R²) representing the model’s explanatory power is 0.035, which is low, but the F-value of 3.124 (p < 0.05) is statistically significant, indicating that the model itself is appropriate. While the model shows relatively low explanatory power, this reflects the inherent complexity of fresh food distribution environments with multiple influencing factors. The first-order coefficient of discount rate is 2845.651 (Beta = 0.283, t-value not presented but p < 0.05), showing a significant positive effect, which means that initial increases in discount rate contribute to profit improvement. The second-order coefficient of discount rate is −7159.762 (Beta = −0.304, p < 0.05), showing a significant negative value. This satisfies the mathematical conditions for a typical inverted U-shaped relationship, meaning that Proposition 1 derived from the analytical model is empirically supported for bundle products. Specifically, the positive first-order coefficient of discount rate for bundle products indicates that marginal revenue has a positive value in the initial discount range, while the negative second-order coefficient indicates that marginal revenue diminishes as the discount rate increases. This pattern suggests that consumer price perception and purchasing behavior for bundled products are fundamentally different from those for individual products.

In contrast, the individual product group (n = 680) exhibits a considerably different pattern. The model’s F-value is 6.75 (p < 0.001), showing high significance, but the pattern of individual coefficients contrasts with that of bundle products. The first-order coefficient of discount rate is −1029.732 (Beta = −0.144, p < 0.05), showing a significant negative effect, suggesting that increases in discount rate actually lead to profit decreases. The second-order coefficient is 93.317 (Beta = 0.005), which is close to zero and statistically non-significant, indicating that no inverted U-shaped relationship exists.

An important finding is the Fisher Z-test result. The Fisher Z-test for inter-group differences showed significance at the p < 0.01 level, providing strong statistical evidence that the regression coefficients of the two groups represent structurally different patterns rather than differences due to chance. These results empirically confirm that bundling is not simply about selling products together, but rather a strategic tool that changes the consumer price response function itself.

The scatter plot and regression curves presented in Figure 4 provide important visual evidence supporting the core assertion of Hypothesis 1.2. The curve shape observed in the graph displays a typical inverted U-shape, clearly revealing that the effectiveness of discount strategies for bundle products is nonlinear.

In the ascending portion of the curve, it can be confirmed that increases in discount rate improve total profit through increased demand. This aligns with the core proposition of bundling theory that bundle discounts provide consumers with greater value perception than individual product discounts.

In the low discount rate range (0–30%), the pattern shows profit increasing as the discount rate increases, suggesting that appropriate levels of discounting can enhance the attractiveness of bundle products, simultaneously improving both sales volume and profitability. The peak of the curve appears at approximately the 40–50% discount rate range, where profit maximization can be confirmed. Subsequently, as the discount rate increases further, profit shows a declining pattern. This pattern empirically refutes the common misconception among practitioners that “more discounting always means more sales.” In other words, it indicates that excessive discounting can undermine the value of bundle products or lead to deteriorating profitability due to margin reduction.

Additionally, examining the distribution of data points reveals considerable variability around the regression curve. This indicates that various factors beyond discount rate—such as seasonality, competitive conditions, and inventory levels—influence profit, but the overall inverted U-shaped pattern is consistently observed despite this noise. This supports that the existence of optimal discount rates for bundle products is a robust phenomenon.

The relationship between discount rate and profit for individual products presented in Figure 5 shows a contrasting pattern to bundle products. The most important characteristic observable in the graph is that no clear inverted U-shaped curve appears. This means that the relationship between discount rate and profit for individual products shows a relatively linear or monotonic pattern. Examining the distribution of data points, the changes in profit as discount rate increases do not exhibit the same systematic nonlinearity observed in bundle products.

This phenomenon offers important theoretical implications. First, individual products’ high price transparency enables consumers to accurately perceive discount value, resulting in predictable discount responses. Second, reference price effects operate more strongly for individual products. Because consumers clearly recognize the regular prices of individual products, purchase decisions according to changes in discount rates are relatively rational and calculated. This explains why extreme discount ranges do not show sudden changes in demand or abrupt deterioration in profitability.

The relatively gentle slope of the regression line in the graph is also noteworthy. This suggests that the effectiveness of discount strategies for individual products is limited compared to bundle products. In particular, the fact that optimal discount points are not clearly identified means that discount rate determination for individual products is more tactical choice based on market conditions and competitive environment rather than an optimization problem.

Additionally, the relatively large data dispersion suggests that discount effects for individual products may be more greatly influenced by other marketing variables (display position, advertising, competitor prices, etc.). This means that discount strategies for individual products function as part of the overall marketing mix rather than independently determining profitability.

A comparative analysis of the graphs for bundle products and individual products clearly reveals the core assertion of Hypothesis 1.2 that bundling fundamentally changes the effectiveness of discount strategies. The contrast between the distinct inverted U-shaped curve observed in bundle products and the relatively flat pattern of individual products empirically confirms that bundling is a strategic tool that changes consumer behavior and market dynamics beyond simple product composition methods.

This contrasting pattern suggests that discount mechanisms operate through different pathways in the two product types. While bundle products have an optimal point where discounts maximize consumer perceived value, individual products show limited optimization effects. This means that in practice, differentiated pricing strategies must be established for bundle products and individual products within the product portfolio. The rejection of Hypothesis 1.1 in the full sample while H1.2 was supported for bundled products can be explained by several factors. External market conditions such as competitive pressures and seasonal variations, consumer heterogeneity, and concurrent marketing effects may obscure the theoretical inverted U-shaped relationship in aggregated data. Our findings align with strategic pricing literature, particularly bundling theory, which suggests that bundled products create a more controlled environment by reducing consumer heterogeneity and external noise effects, thereby revealing the underlying optimization pattern.

4.4. Empirical Research Model 2

The second empirical research model is presented in Figure 6. This model analyzes the impact of the independent variable, order quantity, on the dependent variable, disposal rate. Based on Proposition 2 derived from the analytical model, a U-shaped relationship is expected to exist between order quantity and disposal rate. To verify this, a nonlinear model including the quadratic term of order quantity was established.

According to Richter [13,14], it was confirmed that new product setup and repair are related to disposal rates in inventory models considering economic order quantities, and showed that order quantities must be adjusted to derive optimal disposal rates when considering various economic environments from a total cost perspective. Herbon & Khmelnitsky [15] mentioned that calculating optimal order quantities is necessary to maximize profits for products experiencing quality deterioration, and Wang et al. [16] also showed that optimal order quantities are necessary for companies to increase profits in food supply chains. That is, we can understand that inventory must be introduced with optimal order quantities to reduce disposal rates.

In fresh food inventory management, order quantity has a dual effect on disposal rates. When order quantities are too small, operational costs increase due to increased ordering frequency, along with opportunity costs from inventory shortages. Conversely, when order quantities are too large, quality deterioration and disposal increase due to extended inventory holding periods. Therefore, considering this trade-off relationship, disposal rates are expected to be minimized at optimal order quantities.

Proposition 2 derived earlier confirmed that the disposal rate function for order quantities is strictly convex under specific conditions and theoretically proved the existence of optimal order quantities that minimize disposal rates. Based on this theoretical background and the characteristics of fresh foods, we establish the following hypothesis.

Hypothesis 2.

As the magnitude of order quantity increases, disposal rate will exhibit a U-shaped pattern.

4.5. Analysis Results: Empirical Research Model 2

The analysis results in

Table 5. Analysis Results for Hypothesis 2.

	B	Beta	t-Value
Constant	0.018	-	12.296
Order Quantity	−0.000	−0.605 ***	−5.512
Order Quantity2	0.000	0.417 ***	3.804
$\mathit{A}\mathit{d}\mathit{j}.{\mathit{R}}^{\mathbf{2}}$		0.08
${\mathit{R}}^{\mathbf{2}}$		0.08
$\mathit{F}$		19.739 ***

Dependent Variable: Disposal Rate. *** $p<0.01$.

provide strong empirical evidence for the U-shaped relationship between order quantity and disposal rate. Consistent with the theoretical predictions of Proposition 2 derived from the analytical model, the empirical data confirms that the disposal rate function exhibits convex function characteristics with respect to order quantity.

The regression model’s F-value of 19.739, which shows statistical significance at the 0.01 significance level, confirms the overall significance of the model. Both the coefficient of determination (R²) and adjusted coefficient of determination (Adj. R²) appearing as 0.08 are considered meaningful levels when considering the various exogenous variables that influence disposal rates in actual milk distribution environments.

The first-order beta coefficient of order quantity showing a significant negative value of −0.605 at the 0.01 significance level means that disposal rates decrease in the initial order quantity increase range. This can be explained economically by economies of scale effects and inventory turnover improvement effects. While small orders result in high disposal rates due to relatively high per-unit fixed costs and inefficient inventory management, increasing order quantities to appropriate levels improves operational efficiency and reduces disposal rates.

The squared term beta coefficient of order quantity showing a significant positive value of 0.417 at the 0.01 significance level means that disposal rates increase again once order quantities exceed the optimal point. This can be explained by insufficient storage space due to excessive inventory, increased complexity in expiration date management, and expansion of demand forecasting errors. Particularly for perishable products like milk, excessive order quantities directly lead to quality deterioration due to inventory accumulation, causing increased disposal rates. Given milk’s short shelf life and essential cold storage requirements, the relationship between order quantity and disposal rate may appear more sensitive than other product categories.

Multicollinearity diagnostic results show VIF values of 6.015 for both order quantity and order quantity squared, satisfying the general criterion of below 10, making the model estimates reliable. The combination of the two coefficients (negative first-order, positive second-order) mathematically represents a complete U-shaped relationship, empirically confirming the existence of an optimal order quantity.

The relationship between order quantity and disposal rate presented in Figure 7 shows the estimated U-shaped regression curve along with the scatter plot of raw data. Examining the distribution of data points, a tendency is observed where relatively high disposal rates appear in both low and high order quantity ranges, while relatively low disposal rates are shown in the middle range. The estimated U-shaped curve precisely reflects this data pattern, providing clear visual support for the core assertion of Hypothesis 2.

In the low order quantity range, a clear pattern can be confirmed where increases in order quantity lead to decreases in disposal rate. In this range, order quantity increases bring positive effects such as reduced per-unit ordering costs, improved delivery efficiency, and optimized inventory turnover rates. Particularly from the perspective of cold chain management, which is important in milk distribution, appropriately scaled orders are interpreted as minimizing temperature management costs and contributing to quality maintenance.

The lowest point of the curve represents the optimal order quantity that minimizes disposal rate. At this point, marginal benefits and marginal costs from order quantity increases are balanced, serving as a core reference point for inventory management strategies in practice. The clear identification of this optimal point in the graph strongly supports the practical applicability of the theoretical model.

After the optimal point, the phenomenon where excessive order quantities actually increase disposal rates is demonstrated. This is explained by complex factors such as deterioration of storage environments due to physical limitations of storage space, difficulties in expiration date management due to inventory accumulation, and reduced flexibility in responding to demand fluctuations. Considering milk’s short shelf life and cold storage requirements, excessive inventory directly leads to quality deterioration, causing rapid increases in disposal rates.

Examining the scatter plot, certain variability exists around the U-shaped curve, but the overall pattern is consistently maintained. While this variability reflects the influence of various exogenous factors such as seasonal demand changes, promotional effects, and supplier changes, the clear observation of the U-shaped relationship means that the relationship between order quantity and disposal rate has structural and predictable characteristics.

5. Conclusions and Limitations

5.1. Conclusions

This study delves into strategies for maximizing corporate profits within the context of cold chain management for fresh food products, specifically focusing on the interplay of price discounts and discard rates. This research employed a multi-faceted methodology, integrating analytical models based on game theory, empirical studies utilizing quantitative data from a major Korean retailer, and qualitative research through in-depth interviews with industry experts. This comprehensive approach aimed to provide both theoretical insights and practical implications for the Korean cold chain environment.

A central focus of this study was to analyze the relationship between discount rates and profit, and between order quantity and disposal rates, particularly for perishable items like milk, which served as the primary subject of empirical analysis due to its frequent discounting and clear quality deterioration over time. This study operated under the assumption of linear quality deterioration within the retailer’s environment due to the relatively short and controlled cold chain segment under consideration.

Key findings from empirical analysis provide valuable insights. While Hypothesis 1.1, which posited a general inverted U-shaped relationship between discount rate and profit, was rejected based on the overall data, Hypothesis 1.2 yielded a more nuanced understanding. It was confirmed that the presence of bundled products significantly moderates this relationship: for bundled fresh food items, an inverted U-shaped curvilinear relationship between discount rate and profit was observed. This indicates that profit initially increases with a discount but declines if the discount becomes excessively high, suggesting the existence of an optimal discount threshold for bundled products. Conversely, this inverted U-shaped relationship was not identified for individual (non-bundled) products. This highlights the strategic importance of bundling in influencing consumer purchasing behaviors and pricing effectiveness.

Furthermore, Hypothesis 2, which examined the relationship between order quantity and disposal rate, was adopted. This study found a U-shaped relationship between the magnitude of order quantity and the disposal rate. This crucial finding implies that an optimal order quantity exists that can effectively minimize the disposal rate. This aligns with previous research emphasizing the importance of optimal order quantity in maximizing profits and minimizing waste for deteriorating products. It also provided an empirical resolution to a conceptual point in the mathematical model’s initial definition regarding the correlation between disposed volumes and order quantities.

In conclusion, this research confirms that for fresh food products in a cold chain environment, strategic price discounting (especially for bundled items) and optimized order quantities are critical levers for maximizing profitability and minimizing waste. This study contributes to a deeper understanding of the complex interactions between pricing, quality deterioration, demand, and inventory management in the perishable food supply chain, offering meaningful insights for retailers navigating these challenges.

5.2. Academic Implication

This study offers several significant academic implications, primarily through its integrated research methodology and empirical validation of theoretical constructs in a real-world cold chain context. Firstly, it exemplifies the growing trend of conducting both analytical and empirical research simultaneously to validate findings from multiple perspectives. By combining game theory-based analytical models with quantitative empirical data and qualitative insights from industry experts, this study provides a multifaceted and robust approach to understanding complex phenomena. This blend enhances the depth and richness of the implications, addressing the trade-off between theoretical fidelity and practical feasibility.

Secondly, this research contributes to the existing body of literature on perishable product management by empirically testing and refining key theoretical relationships. The analytical model proposed the existence of an optimal discount rate based on profit function concavity. While the general Hypothesis 1.1 regarding an inverted U-shaped relationship between discount rate and profit was not supported broadly, this study’s confirmation of Hypothesis 1.2 for bundled products significantly advances this argument. It introduces product bundling as a critical moderating variable that influences the optimal pricing strategy for perishable goods, demonstrating that the inverted U-shaped relationship, and thus an optimal discount, is context-dependent. This refines the application of optimal pricing theories, suggesting that bundling strategies need to be explicitly considered in pricing models for deteriorating products.

Thirdly, this study validates the theoretical importance of optimal order quantity in minimizing disposal rates for perishable products. By empirically confirming the U-shaped relationship between order quantity and disposal rate (Hypothesis 2), this research provides strong support for the notion that efficient inventory management is crucial not only for controlling costs but also for reducing waste in the cold chain. This reinforces and builds upon the works of Richter [9,10] and Herbon & Khmelnitsky [11] regarding the close link between order quantity, disposal rates, and profit maximization in deteriorating product contexts. This empirical evidence also resolves a conceptual nuance in this study’s mathematical model regarding the correlation between disposed volumes and order quantities.

Finally, by focusing on the Korean cold chain and the characteristics of fresh food distribution, this study provides contextualized empirical evidence that can serve as a foundation for comparative studies in different markets or for different product categories. This contributes to a more universal understanding of cold chain dynamics and perishable inventory management.

5.3. Managerial Implication

This study offers several crucial managerial implications for retailers and businesses operating within the cold chain for fresh food products.

Firstly, cold chain management should be recognized as a strategic imperative, not just an operational cost. Investing in robust cold chain infrastructure is essential for building and maintaining consumer trust, enhancing sales, and safeguarding brand reputation. Failure to do so carries significant risks of revenue loss due to spoilage and potential food safety incidents.

Secondly, our empirical analysis demonstrates that bundled products exhibit a statistically significant inverted U-shaped relationship between discount rates and profits (first-order coefficient: 2845.651, p < 0.05; second-order coefficient: −7159.762, p < 0.05). This empirical evidence indicates the existence of an optimal discount rate that maximizes profitability for bundled fresh food items. For bundled fresh food products, retailers should actively seek to identify and apply an optimal discount rate. The finding of an inverted U-shaped relationship implies that while discounts can boost sales and profits initially, excessive discounting can erode profit margins and lead to overall profit reduction. This means blindly applying high discounts might be detrimental; instead, a data-driven approach to finding the peak of this “U-curve” for bundled items is advised. Conversely, our findings show that individual products demonstrate a negative linear relationship (coefficient: −1029.732, p < 0.05), suggesting that discounting strategies should be fundamentally different for bundled versus individual products. In contrast, for individual products, this study did not identify the same inverted U-shaped relationship, suggesting that different pricing models or more nuanced strategies might be required for single items. Our statistical analysis provides strong evidence (p < 0.01) that bundling fundamentally changes the discount–profit relationship structure, indicating that bundled products can benefit from strategic discounting within an optimal range, while individual products require alternative approaches such as improved positioning, enhanced marketing, or non-price promotions. Furthermore, the linear quality deterioration model (λ = 0.05) validated in our study suggests that price adjustments are necessary as product quality deteriorates over time. Retailers should also consider seasonal and temporal demand fluctuations; more aggressive discounting may be needed during periods of low demand to move inventory and prevent spoilage.

Thirdly, optimizing order quantity is paramount for minimizing waste and maximizing profitability. Our analysis confirms a U-shaped relationship between order quantity and disposal rate (first-order coefficient: −0.605, p < 0.01; second-order coefficient: 0.417, p < 0.01). This relationship provides strong empirical evidence that both under-ordering and over-ordering increase disposal rates, with an optimal order quantity that minimizes waste. The negative first-order coefficient indicates that initial increases in order quantity reduce disposal rates through economies of scale, while the positive second-order coefficient shows that excessive orders lead to increased disposal due to quality deterioration and storage limitations. The confirmed U-shaped relationship between order quantity and disposal rate means that there is an optimal order quantity that leads to the lowest disposal rate. Retailers should invest in analytical tools and data to precisely calculate these optimal quantities. Ordering too much leads to increased discard costs due to quality deterioration, while ordering too little results in lost sales opportunities and higher reordering frequencies. Effective inventory management, therefore, directly contributes to cost reduction and profit enhancement.

Finally, this study underscores the importance of proactive disposal management strategies. The confirmed relationships in our study demonstrate that systematic optimization of order quantities and discount strategies can significantly reduce disposal rates while maintaining profitability. The U-shaped disposal rate function indicates that precise demand forecasting and order optimization are critical for both economic and environmental outcomes in fresh food retailing. Recognizing that disposal costs exist and can be significant, companies should explore methods to minimize these expenses, such as outsourcing disposal activities. Understanding the two types of product disposal—due to time-based quality deterioration and clearing space for new products—can help in tailoring more effective management strategies for each. This holistic approach to managing the entire product lifecycle, from ordering to potential disposal, is key for long-term profitability in the fresh food sector.

5.4. Limitations and Future Research

This study, while offering valuable insights into cold chain management for fresh food, is subject to certain limitations that provide clear avenues for future research.

One significant limitation lies in the assumptions made within the analytical model. For simplification and calculation convenience, several parameters were normalized or set to specific values, such as the overall market size and price sensitivity assumed as 1, quality sensitivity as 0.5, initial quality as 1, and quality degradation coefficient as 0.05. Furthermore, sales and processing costs were normalized to zero, and the disposal cost was initially treated as a single cost, despite the possibility of variation across different intervals. While these simplifications enable model tractability, they might limit the direct generalizability of the quantitative results to real-world scenarios where these parameters fluctuate significantly. Future research could explore the model’s robustness by relaxing these rigid assumptions and investigating scenarios with varying parameter values and more complex cost structures, including interval-based disposal costs.

The scope of this empirical study also presents limitations. The data for the empirical analysis was obtained from a single major domestic retailer in Korea (Company A). This naturally raises questions about the generalizability of the findings to other retailers with different operational models, market positions, or to international markets with distinct cold chain infrastructures and consumer behaviors. Moreover, the empirical analysis primarily focused on milk products. While milk was chosen due to its frequent price discounting and clear quality deterioration, its characteristics may not be fully representative of other fresh food categories such as fruits, vegetables, meat, or seafood, which exhibit different spoilage patterns, shelf lives, and consumer preferences. Future studies should aim to incorporate data from a broader range of retailers, diverse geographical markets, and multiple fresh food categories to enhance the external validity and applicability of the findings.

Another area for future exploration stems from the rejection of Hypothesis 1.1 (the general inverted U-shaped relationship between discount rate and profit) for the overall dataset. While Hypothesis 1.2 successfully identified product bundling as a crucial moderator for this relationship, it suggests that the initial theoretical premise might be too simplistic without accounting for other complex moderating factors. Future research could investigate additional moderating variables such as brand type (e.g., national brands vs. private brands, as briefly mentioned in the source), competitive intensity, seasonal demand shifts, or even specific marketing efforts that might influence how discount rates impact profitability.

Finally, while this study effectively integrated analytical and empirical methods, there is always room for further methodological advancements. For instance, developing more dynamic and adaptive models that can respond in real-time to quality degradation, fluctuating demand, and inventory levels could provide more practical decision-making tools for retailers. Additionally, extending the game theory model to encompass multi-echelon supply chain interactions—involving manufacturers, distributors, and other intermediaries—would provide a more holistic understanding of profit optimization across the entire cold supply chain, rather than primarily from the retailer’s perspective. Given the overarching theme of sustainability, future research could also integrate explicit sustainability objectives, such as food waste reduction targets, directly into the optimization models, exploring the trade-offs between profit maximization and environmental impact.