Combination of pricing and inventory policies for deteriorating products with sustainability considerations

Abstract

Economic, environmental, and social criteria are all being taken into consideration simultaneously when determining pricing policies or inventory levels in sustainable production management. The combination of pricing and inventory policies is an important source of leverage for the efficient management of perishable products. This paper, among the first studies, proposes the problem of devising optimal pricing and inventory management decisions simultaneously where the environmental and social criteria are contributed for perishable complementary products replenished and sold by the same company. This study considers two interrelated price-sensitive linear demand functions to consider the possibility of shortage with both budget and warehouse capacity constraints. Another contribution of the proposed model is to consider an upper bound for environmental pollution and a lower bound for job opportunities as the constraints to the model. As a complex optimization model, the challenge of complexity is addressed by a heuristic algorithm for finding an optimal solution. After an extensive analysis using numerical examples, some managerial insights are concluded from the results. One finding from these analyses confirms that the total capacity of the warehouse, the total available budget, carbon emissions, and variable job opportunities have a high impact on the optimal solution to find a balance between sustainability criteria for making pricing and inventory policies.

Introduction

Sustainability encompasses a wide spectrum of subjects ranging from electricity consumption (Abbasi et al., 2021a, b, c), tourism (Aman et al., 2019; Asad et al., 2017; Yu et al., 2022) supply chains (Edalatpour et al., 2018; Moosavi et al., 2022; Zhuang et al., 2022), financial systems (Li et al., 2022; Zhang et al., 2022), health (Schmidt et al., 2022), and manufacturing systems (Fathollahi-Fard et al., 2021; Song & Moon, 2019). Sustainable manufacturing is a recent concept in production management where pricing, inventory, or scheduling decisions should be made with regard to the economic, environmental, and social dimensions (Fathollahi-Fard et al., 2020; Sepehri & Gholamian, 2022; Wang et al., 2019). Since most of the models mainly consider economic criteria like the total cost or profit, limiting environmental pollution like carbon emissions, and considering social justice in the manufacturing environment like the job opportunities are well-studied in different supply chain and transportation models (Soleimani et al., 2022). However, they are rarely contributed to manufacturing systems based on inventory or pricing decisions (Asghari et al., 2022a, b). For perishable products, the inventory system needs modern refrigerators where which have a large amount of energy consumption. More products consume more energy, resulting in more carbon emissions as an environmental factor. As such, the inventory systems usually need some workers to control and check the quality of perishable products and update the mode of refrigerators based on the number of perishable products. In the same way, more perishable products in the inventory system need more workers as a social factor. Based on these needs, this paper develops an integrated inventory and pricing policy-based optimization approach with sustainability considerations for instantaneous deteriorating products which are perishable.

In various business sectors and supply chains, perishable products play a significant role (Farghadani-Chaharsooghi, et al., 2021; Majidi et al., 2022). Perishables such as meat, fish, and dairy produce accounted for 53.67% of approximately $485.5 billion in sales revenues at US supermarkets in 2014.1 Despite the tremendous efforts made for a better understanding and managing inventory systems of perishable products over the past decades (Bakker et al., 2012; Goyal & Giri, 2001; Karaesmen et al., 2011; Nahmias, 1982; Raafat, 1991; Sazvar et al., 2016; Shah et al., 2013), there are still a lot of opportunities to gain through matching supply and demand for this type of products. One-third of the food produced in the world—approximately 1.3 billion tons each year—is lost or wasted (FAO, 2011). If food waste is considered a single country, it comes in third place among the world's CO₂ emitters (Abbasi et al., 2021a, b, c). This represents a cost of $161 billion in the USA, for example. A large portion of food losses reported relate to food deterioration or not being sold/consumed before its expiration date. These losses occur during the process of distribution, including losses at wholesale and retail stores (12% of overall food losses in the world), or the business for the consumer (including restaurants/ caterers) and at home (35% of overall food losses in the world) (Lipinski et al., 2013). The total amount of food waste generated by the US retail and wholesale sectors in 2011 was estimated at 3.8 billion pounds.2 This represents an average of 11.4% for fresh fruit, 9.7% for fresh vegetables, and 4.5% for fresh meat, poultry, and seafood, according to data compiled for 2005 and 2006 (Buzby et al., 2009). Even though perishability attracts the attention of researchers and practitioners, complementary perishable products are being neglected due to their complexity. As a way to shed some light on the importance of complementary perishable goods, we examine catering businesses. A hotel's breakfast, lunch, and dinner menu include various foodstuffs most commonly classified as perishable complementary foods. Milk and cereal, coffee and milk, cake and tea, toast and butter for breakfast, salads and main courses for lunch or dinner, cake, and coffee as a middle-day meal, and numerous other products are perishable complementary products that can be found in daily life. As Fig. 1 shows, complementary products constitute the lion's share of food waste in the hotel industry.

Fig. 1.

Food waste types in hotels (the waste and resources action programs < https://www.ers.usda.gov/ >)

In a research conducted by the United States Department of Agriculture Economic Research Service (ERS)3 (USDA-ERS, 2013) for improving American diets through some pricing policies, the impacts of pricing policies on dietary quality are reviewed, and the importance of substitute and complementary has been assessed. ERS researchers investigated the price change effects on the consumption of 43 products, including 38 at-home and 3 food-away-from-home foodstuffs, also alcoholic beverages, and some non-food goods and services.

In light of Table 1, we can conclude that complementary and substitute relationships have a significant impact on consumer responsiveness in a meaningful way. As mentioned above, the importance of this topic stems from the fact that financial and vital benefits can be obtained by minimizing the volume of waste of perishable products in the distribution and retail sectors, as well as this topic's importance in the hospitality industry (hotels, restaurants, etc.), equally as most businesses produce or sell complementary products, which has become a widely accepted practice among corporations. It is actually a common practice in businesses for them to sell complementary products and to try to take advantage of the simultaneous consumption/use of these products by their clients, as well as the tendency of someone to buy several products in order to take advantage of place and time and other advantages offered by these combinations.

Table 1.

Considering a 10-percent subsidy on fruit and vegetables

	Percent change
	Only fruit and vegetable purchases	All food purchases
Food at home
Fruit and vegetables
Apples	7.7	6.2
Bananas	12.4	9.8
Citrus	14.1	11.1
Other fruit	10.3	8.2
Potatoes	7.5	5.9
Lettuce	10.1	7.9
Tomatoes	14.3	11.1
Other vegetables	10.1	7.9
Processed	7.7	6.1
Cereals and bakery*	NA	3.4
Meat and eggs*	NA	− 1.0
Dairy*	NA	0.2
Non-alcoholic beverages*	NA	− 0.4
Other food at home*	NA	− 3.2
Food and alcohol away from home
Limited service	NA	− 0.3
Full service	NA	− 0.3
Other	NA	− 0.4
Alcohol	NA	− 0.6
Non-food	NA	0.3

NA not applicable

*Average for the products within the food group

In an e-commerce environment, Diehl et al. (2015) estimated that 85% of the top 50 online retailers offered a variety of options organized into complementary sets. For this reason, there is a strong need to address the problem of simultaneously designing optimal inventory management and pricing policies for complementary perishable products to achieve optimal effectiveness in the management of these products. By definition, complementary products positively correlate with their demand for one and the other. Therefore, no optimal pricing or inventory management policy can be devised for one product without taking into account its complementary. In fact, the cross elasticity of demand for two complementary products implies that any pricing and/or inventory decisions made for one of them are likely to influence the demand and inventory status for the second.

In conclusion, this study improves the state-of-the-art literature by developing a mathematical model for the joint design of optimal pricing and inventory policies simultaneously for perishable complementary products over an infinite time horizon under various assumptions and sustainability considerations to highlight the role of carbon emissions and job opportunities. Our work offers a better representation of the inventory systems of complementary perishable products than a model that ignores this relationship as well as the sustainability considerations. Hence, it yields more valuable insights and a better pricing and inventory control policy for perishable products while optimizing the sustainability criteria.

The remainder of this paper is organized as follows. Section 2 contains a review of the literature related to our study. Section 3 presents the problem formulation, the various assumptions we have used, and the resulting models. In Sect. 4, we provide a solution method as a heuristic algorithm for finding the optimal values of the decision variables included in our model. In Sect. 5, we present the results of a numerical example, sensitivity analysis, and managerial insights of our model, along with the related managerial implications. Finally, in Sect. 6, we state our conclusion with limitations and findings and indicate possible avenues for future research.

Literature review

In the field of perishable products, research on optimal inventory management and pricing policies covers a wide body of literature, which has been reviewed comprehensively in various publications (Chan et al., 2004; Chen & Chen, 2015; Elmaghraby & Keskinocak, 2003; Petruzzi & Dada, 1999; Yano & Gilbert, 2004). The earliest studies and a major part of the research in this area treated the case of a single perishable product. Elion and Mallaya (1966) were among the earliest works to develop a joint pricing and inventory policy for a single deteriorating product, followed by (Cohen, 1977), who developed the joint pricing and order decisions for an exponentially decaying product under a deterministic price-dependent demand function by integrating continuous inventory review and their joint inventory decisions. Optimal pricing and production policy for deteriorating products within the context of a continuous review policy and deterministic demand function has been developed by Kang and Kim (1983). Lazear (1986) demonstrated that if two different prices are used to sell a product in two different periods, future profits can be increased. Also, he was able to prove that obsolescence of a product reduces the initial optimal price of a product and increases the rate at which the price of that product falls as a function of time on the shelf due to obsolescence. Aggarwal and Jaggi (1989) have developed an optimal pricing and production model for the case of a product with exponential decay.

Shiue (1990) has developed a model that is capable of determining the optimum lot-size for a single perishable product when the assumptions of a deterministic constant demand rate, partial backlog, partial loss of sales, and various quantity discount schemes are applied. Rajan and Steinberg (1992) investigated the pricing and ordering decisions of a retailer selling a single item facing a known demand function in the context of an inventory cycle in which the product, over the lifetime of the inventory cycle, may undergo physical decay or a decrease in market value over time. Gallego and van Ryzin (1994) studied the pricing decisions of a firm selling a given amount of initial inventory of a product over a finite horizon, using a stochastic price-sensitive demand function and considering the holding cost and backlogging penalty cost. Based on a model of dynamic pricing with partially backordered inventory, Abad (1996) developed a solution procedure for solving a generalized model of dynamic pricing. Wee (1999) presented a joint pricing and inventory management model for the management of an item that deteriorates over time, where the demand rate is a linear function of the selling price, and the time to deterioration is a Weibull distribution. Based on this model, unsatisfied demand is assumed to be both partial and fixed, and economic factors such as fixed ordering cost, unit holding cost, unit shortage cost, and unit lost sale cost are taken into consideration. Chatwin (2000) attempted to solve the problem of how to price a finite amount of inventory of a perishable product prior to the time before it is likely to perish. He showed that the maximum expected revenue function is non-decreasing and concave in the remaining inventory. Abad (2003) addressed the problem of determining the prices for a perishable product, as well as the lot sizes that should be used for a perishable product under the conditions of a finite horizon, finite production volume, exponential decay, and partial backordering. Chen and Chen (2004) developed a system for systematically reviewing inventory levels and pricing problems for a single product subject to continuous decay, as well as a demand function that is time- and price-dependent, as well as shortages that are completely backlogged in their work.

The pricing problem has been a research topic of intense interest, and with the advent of the last decade, a number of models, both deterministic and uncertain, have been formulated to address this issue. Dye et al. (2007) considered the same model as Dye et al. (2007), but with an exponential backlogging rate, and provided an algorithm to find the optimal selling price and replenishment schedule. Abad (2008) provides a model of and a procedure to solve the problem of simultaneous pricing and lot-sizing associated with in-stock items and partial backorders with post-order loss and backorder costs for products subject to a general rate of deterioration and partial backordering. Hsieh and Dye (2010) document a lot-sizing model for a single deteriorating item at inflation over a finite planning horizon, which incorporates a multivariate, price- and time-sensitive demand function, as well as partial backlogging. Thereafter, they provided a complete search procedure that could be used to find the optimal selling price for the system, as well as the number and timing of replenishments that would maximize its profit net present value. Maihami and Abadi (2012) presented a model of the problem of joint control of inventory and prices for a single non-instantaneously deteriorating product. Avinadav et al. (2013) examined the issue of determining the optimal price level, the order quantity, and the replenishment period for a perishable product. Chen et al. (2014) developed a model for pricing and inventory management that incorporates the possibilities of having both continuous and discrete demand distributions of the product and for a fixed lifetime of the product. This model addresses the problem of pricing and inventory management for a perishable product with a fixed lifetime over a finite period of time. Considering a positive lead time, a linear ordering cost, an inventory holding cost, a backlogging/lost-sales penalty cost, and a disposal cost, they presented an optimal order-up-to-levels heuristic policy in order to maximize the total anticipated discounted profit over the planning horizon for a perishable product facing a demand composed of a deterministic price-dependent component as well as an additive random term, over the planning horizon.

In contrast to the case of a single perishable product, multiple perishable product pricing and inventory management is a relatively new topic, and a quite few studies addressed it (Chen & Chen, 2015). According to existing studies that have addressed this topic, they can be divided into three types depending on how they have considered the relationships between the products under consideration: those that take substitute products into account, those that assume independence between products, and those that assume a general relationship among products. Accordingly, we study the simultaneous joint design of replenishment and pricing policies for two perishable products that are assumed to be complementary in our present research. The research we conducted bears some resemblance to some of the studies that focused on multiple products, which we review in some detail following this study. Gallego and Van Ryzin (1997) studied a case where multiple perishable products were produced with a limited amount of resources and were intended to be sold over a finite period of time, using a finite quantity of resources. They propose that a deterministic solution to the problem would allow the authors to set an upper bound on the expected optimal revenue based on the assumption that the demands for each product are stochastic point processes governed by the time and prices of each product. Following that, an asymptotically optimal heuristic for solving the stochastic problem was presented based on the deterministic model that was solved. Bitran et al. (2006) carried out an investigation into the optimal pricing policies for a set of substitute perishable products sold over a finite period of time, based on an inventory of a finite quantity. This study models the demand for replacement products by assuming that the potential buyers arrive according to the Poisson process, which is a process in which buyers can decide whether to purchase a product from the family of substitutes or not, depending on the prices and availability of the product. This problem is formulated as a stochastic control program by the authors, and they suggest two asymptotic approximations that are used to solve it to find a solution, including scenarios where there could be an infinite supply, as well as models where demand can be simulated in a fluid-like deterministic way. Maity and Maiti (2009) investigated optimal inventory policies over a single and finite period of time with the goal of providing optimal inventory levels for multiple deteriorating products, consisting of complementary and/or substitute types when resources were constrained. Specifically, shortages in this study are not allowed, and demand for a given product is assumed to be stock-dependent. It is also claimed that the customers of a given product may be influenced to choose substitutes for the given product based on the stock level of that substitute, while at the same time, they may be influenced to purchase complementary items to take advantage of the locational advantage. As the authors describe, this problem has been formulated as an optimal control problem and has been solved by using optimal control theory in order to obtain the optimal quantity of production for both the steady case and the transitory case. Bulut et al. (2009) studied optimal pricing and bundling policies for two perishable products that are sold individually as well as a bundle, both of which are sold under conditions of no replenishment opportunity, a situation where an unsatisfied demand would otherwise be lost. In this study, the arrival of customers is represented by a Poisson distribution with a fixed rate, and the preferences of the customers are represented by their reservation prices which have been distributed normally with a given coefficient of correlation. By utilizing numerical examples, the authors were able to demonstrate that the performance of any policy given by the author depends heavily upon the parameters that make up the demand function, and the initial level of inventory in addition to the correlation coefficient between the reservation prices of the two products. Akcay et al. (2010) considered a dynamic pricing problem involving multiple substitutable perishable products and their initial inventories, which are to be replenished over a finite time horizon without replenishment. According to this model, the demand for any given product will depend on both the price and non-price characteristics of the product itself as well as all the other products in the assortment. Shavandi et al., 2012 investigated joint pricing and lot-sizing decisions in the context of a set of perishable products that can be categorized according to their function (substitutes, complementary, and independent) over a finite planning horizon. Taking into account for each product a deterministic exponential demand function of the prices, the authors found that they were unable to solve the model analytically, so they developed a genetic algorithm for this purpose. Dobson et al. (2016) presented an Economic Order Quantity (EOQ) model for perishable goods with an age-dependent demand rate. In their study, the opportunity cost of lost sales was analyzed with respect to the aging of perishable items. Eventually, they investigated the conventional EOQ model and presented the retailer’s optimal cycle length. An article published by Herbon and Khmelnitsky (2017) reviewed the dynamic pricing of perishable goods. They represented the optimal replenishment plan and non-static price over time. A study was conducted on the nonlinear impacts of price and time on demand, and some insights were offered regarding optimal pricing as a result. Moreover, the advantages of the dynamic pricing policy were also examined by the researchers. Chan et al. (2017) presented a joint inventory and production model for the management of deteriorating products. The study also examines the effects of a production rate on the total costs of the whole system. The authors investigated a single-vendor single-buyer single-vendor study in which the product was exponentially deteriorating over time. Furthermore, the model that they propose also takes into account the possibility of deterioration during the delivery process.

Recently, Kazemi (2019) provided a case study of a dairy company to analyze and plan for the demand meeting and supply limitations, while almost the peak of demand is on the less supply of milk. This case urges an exact tradeoff between lost sales and wasted products. In another study, Lin (2019) studied an Economic Production Quantity (EPQ) inventory model dealing with an imperfect production process under a backlogged scenario and uncertain demand. Mishra et al. (2019) investigated an inventory problem for non-instantons deteriorating items which considers various inventory system costs such as preservation technology, environmental, and ordering. Their proposed model aim was to maximize the total retailer's profit while calculating optimal replenishment cycle time. Using a multi-product inventory system, Edalatpour and Mirzapour Al-e-Hashem (2019) conducted a study on joint pricing and inventory decisions for complementary and substitute items. The system that they have proposed relies on a nonlinear holding costs concept to present perishability. Orozco et al. (2020) studied an inventory control system in a three-echelon fruit supply chain. They presented a case study in a citrus supply chain and modeled the behavior of the product’s useful life to design the optimal inventory policy. Contributing to the wastewater assessment, Fathollahi-Fard et al. (2020) considered the environmental emissions of production, recycling, and disposal as well as the job opportunities. A multi-objective stochastic optimization was developed accordingly and solved by an improved social engineering optimizer as a recent powerful metaheuristic. Poursoltan et al. (2020) proposed an EPQ with deteriorated products, random machine breakdown, and stochastic repair time. An approximated method called Newton-Rawson was applied to solve their proposed EPQ. In another study, Poursoltan et al. (2021) proposed an EPQ based on the vendor-managed inventory contract, considering the learning effect for workers. To solve this complex model, a novel hybrid algorithm is proposed as a combination of red deer and Keshtel algorithms. Mashud et al. (2021) presented an inventory model for non-instantiated deteriorating items, in which preservation technology and suitable green technology investment are incorporated for reducing product deterioration and carbon emission. Advanced payment and partial back ordering strategies are also taken into account in their model. Lu et al. (2021) proposed sustainable scheduling of distributed manufacturing systems. In addition to the makespan and energy consumption, they considered the negative impact of the long scheduling process for operators. A novel multi-objective memetic algorithm was also developed to address their proposed problem. Abdul Hakim et al. (2022) presented an inventory model for the pricing policy deteriorating. Their proposed model considers linear selling prices and nonlinear green level-dependent demand. Abid et al. (2022) investigated the role of financial development and green innovation role in reducing environmental challenges. They conclude that green innovation possesses a significant impact on sustainability and can be employed as a powerful tool for pollution lowering.

Generally speaking, while the tradeoffs in any inventory management system essentially involve ordering costs, holding costs, and shortage costs, the perishability characteristics of a product introduces additional tradeoffs related to the prices and the losses resulting from deterioration. Hence, a wide body of research on the joint design of optimal inventory management and pricing policies for perishable products has been developing over the last few decades (Chan et al., 2004; Chen & Chen, 2015; Elmaghraby & Keskinocak, 2003; Farghadani-Chaharsooghi et al., 2021; Petruzzi & Dada, 1999; Sazvar et al., 2013; Yano & Gilbert, 2004). It is pertinent to note that the majority of this study was focused on the study of systems that deal with single-product inventory management, while there is still a lack of work that addresses the case of multiple products (Chen & Chen, 2015). It is also noteworthy that the majority of studies neglect many situations in which the interrelationship between the demand for a product and the demand for a competing product can be considered. A product's demand, and its inventory system, are not only affected by the price of the product itself but also by the price of its complementary products, which in turn affect its inventory system as well.

Our present study expands the previous research on perishable multiproduct inventory management and pricing policies considering sustainability dimensions that are rarely studied in the literature review (Fathollahi-Fard et al., 2021; Soleymanfar et al., 2022; Taleizadeh et al., 2018; Wang et al., 2019). This paper develops a mathematical model for the simultaneous and joint design of pricing and inventory management policies for perishable complementary products while considering sustainability criteria. This study considers the amount of energy consumption resulting in the generation of carbon emissions for holding perishable products to achieve environmental sustainability. As such, the variable job opportunities, which are related to the number of perishable products in the inventory system, are considered in our optimization model. To highlight the pricing decisions of our optimization model, two interrelated price-sensitive demand functions for complementary products are proposed. Taking backordered demand into account and obtaining a closed-form formula for optimum replenishment cycle as well as respected products' prices with both budget and warehouse capacity constraints constitute a significant departure from the current state of knowledge in this area collectively. To the best of our knowledge, this is the first attempt to formulate a sustainable inventory-pricing problem for perishable complimentary products with shortages allowed by using two generic interrelated price-sensitive demand functions and presenting a closed-form formula for optimum replenishment cycle as well as optimum product prices.

Problem definition and model formulation

As a framework for addressing this problem, the proposed solution consists of an inventory system for two perishable, complementary products that are replenished, marketed, and sold by one single company. This study seeks to determine the pricing policies as well as the inventory decisions associated with each of the two products so that the firm can maximize the profit that can be achieved by replenishing and selling them in order to maximize its overall profits. Following are some of the assumptions that can be used to support the proposed problem:

1. The planning horizon is infinite.

2. The lead time is zero.

3. The inventory management policy is a periodic review with a common replenishment cycle.

4. The shortage is allowed, and all unsatisfied demand is backlogged.

5. The cost of shortage per unit of product per unit of time is constant.

6. The on-hand inventory of each product deteriorates over time at a constant and known rate, and there is no salvage value for deteriorated units.

7. The deterioration cost (cost of dealing with a deteriorated unit of a product: separation, disposal, and replacement) is constant.

8. Perishable products need to be stored in modern refrigerators, where they are generally powered by non-renewable energy sources, resulting in the emission of carbon dioxide into the atmosphere. To achieve environmental sustainability, this study limits the amount of carbon emission by a predefined upper bound determined by the sustainable development guidelines.

9. Such modern inventory systems need engineers to check the quality of products and update the inventory mode of refrigerators. More perishable products in a period need more workers and engineers in the warehouse center. To achieve social sustainability, we have defined a lower bound for the minimum variable job opportunities created by the inventory system in the warehouse center.

10. In view of the perishable nature of the products, the inventory system is complex, and we require an environment that can be controlled by modern refrigerators that are designed to hold products of this kind. To address this issue, it is proposed that the carbon pollutants resulting from the energy consumption of such refrigerators be considered in the proposed solution.

11. The demand per unit of time for each one of the two complementary products is a linear decreasing function of the price of the product itself, as well as the complementary product’s price. Assuming the base level of the market demand for product j is ($a_j$), the price sensitivity of the demand is (s), and the degree of complementarity between the two products is (θ), the general demand functions will be:

    $$D_i\left(p_i,,,\cdots ,,,p_j,,,\cdots ,,,p_n\right)=a_i-s{(p}_i+\sum _{j\ne i}{\theta }_j{p}_j)$$ 1
    And for a two-product system can be written as follows:

    $$D_1\left(p_1,,,p_2\right)=a_1-s{p}_1-s\theta p_2$$

    $$D_2\left(p_1,,,p_2\right)=a_2-s\theta p_1-s{p}_2$$ 2

    Price sensitive linear demand functions can be referred to the literature (Gupta & Loulou, 1998; Martin, 1999; Mukhopadhyay et al., 2011; Raju & Roy, 2000; Wee, 1999; Yue et al., 2006; Zhang, 2002).

12. All the parameters of the demand function and the inventory system are constant and known.

13. The inventory system deals with both budget and warehouse capacity constraints.

To establish our mathematical model, the following notation will be used throughout the paper:

The two products are indexed using j (j = 1, 2).

Parameters

$A_j$

The fixed ordering cost for product j.

$a_j$

The base demand for product j (demand for product j when prices are set to zero).

$s$

The price sensitivity of the demand;

$h_j$

The holding cost per unit of product j per unit of time;

${\mathit{ce}}_j$

The generated carbon emissions for holding per unit of product j;

CAP

Total capacity of warehouse;

BDG

Total available budget;

UBE

The upper bound for the limitation of carob emissions in the system;

LBJ

The lower bound for the limitation of variable job opportunities;

$f_j$

Unit area/volume occupied by product j;

$c_j$

The purchase price of product j;

ć_j

The deterioration cost per unit of product j;

${\mathit{vj}}_j$

The number of created variable jobs per unit of product j;

${\pi }_j$

The shortage cost per unit of product j per unit of time;

$g_j$

The deterioration rate of product j per unit of time;

$\theta$

The complementarity coefficient of the two complementary products ($0\le \theta \le 1)$

Variables

$T$

The length of the replenishment cycle;

$t_{1j}$

Instant at which the inventory level of product j reaches zero in a replenishment cycle;

$p_j$

The sales price of product j;

$Q_j$

The order quantity of product j;

$b_j$

The shortage level of product j;

${\overline{B}}_j$

The average shortage of product j during a replenishment cycle;

$I_j(t)$

The inventory level of product j at the time t;

${I_{\max }}_j$

The maximum inventory level of product j during a replenishment cycle;

${\overline{I}}_j$

The average inventory of product j during a replenishment cycle;

${\mathrm{TC}}_{\mathrm{o}}$

The total cost of ordering per unit of time;

${\mathrm{TC}}_{\mathrm{h}}$

The total cost of holding inventory per unit of time;

${\mathrm{TC}}_{\mathrm{D}}$

The total cost of product deterioration per unit of time;

${\mathrm{TC}}_{\mathrm{s}}$

The total cost of shortage per unit of time;

$\mathrm{TC}$

The total cost of the inventory system per unit of time;

R

The profit function of the inventory system per unit of time;

In the proposed problem, there are two scenarios where the first case does not allow the possibility of a shortage, while the second scenario does allow for a shortage in the inventory system, and the unsatisfied demand is completely backlogged.

The shortage does not exist in the inventory system

In this case, the inventory level $I\left(t\right)$ of any one of the two complementary products follows the pattern depicted in Fig. 2.

Fig. 2.

Graphical representation of the inventory levels of the two products in the inventory system for the case when the shortage is not allowed

During the time interval [0, $T$], the inventory level of each product (j) of the two complementary products declines due to the demand and to the deterioration process of the product according to the equation:

$$\frac{d{I}_j(t)}{\mathit{dt}}=-g_j{I}_j\left(t\right)-(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i)$$ 3

So, the inventory level of the product at any moment between 0 and T is given by:

$$I_j\left(t\right)=\frac{\left(a_j,-,s,p_j,-,s,{\sum }_{i\ne j},{\theta }_j,p_i\right)}{g_j}\left(e^{g_j\left(T,-,t\right)},-,1\right),$$ 4

and,

$$Q_j=I_{\max j}=\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}(e^{g_{j}T}-1)$$ 5

To find the optimal values of the decision variables that maximize the profit of the firm from replenishing and selling the two complementary products under consideration, we should find the expression of the profit function. In general, this is equal to the difference between the companies' revenue from selling their products and their total costs, which includes the costs associated with buying the products, as well as the costs associated with running their inventory systems. Taking these factors into consideration, we will calculate the cost per unit of time of the inventory, which is the sum of the ordering cost, the holding cost, and the deterioration cost for each unit of time. The ordering cost per unit of time (${\mathrm{TC}}_{\mathrm{O}}$) is equal to $\frac{{\sum }_j{A}_j}{T}$, and other costs are calculated as follows:

The holding cost per unit of time is:

$${\mathit{TC}}_H=\sum _j\frac{1}{T}{h}_j{\int }_0^T{I}_j\left(t\right)=\frac{1}{T}{h}_j{\int }_0^T\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}\left(e^{g_j\left(T,-,t\right)},-,1\right)\mathrm{d}t=\frac{h_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{T{g}_j^2}(e^{g_{j}T}-g_{j}T-1)$$ 6

In this paper, the deterioration function is assumed as an exponentially decaying function, so the cost of deteriorated units of the product per unit of time is:

$${\mathit{TC}}_D=\frac{1}{T}\sum _j{\acute{c}}_j{\int }_0^T{g}_j{I}_j\left(t\right)dt=\frac{1}{T}\sum _j{\acute{c}}_j{\int }_0^T{g}_j\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}\left[e^{g_j\left(T,-,t\right)},-,1\right]dt=\frac{1}{T}{\acute{c}}_j(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i){\int }_0^T\left[e^{g_j\left(T,-,t\right)},-,1\right]dt=\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{T}\left[-,\frac{1}{g_j},e^{g_j\left(T,-,t\right)},-,t\right]\genfrac{}{}{0.0pt}{}{T}{0}=\frac{{\acute{c}}_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{T}[-\frac{1}{g_j}-T\mp \frac{1}{g_j}{e}^{g_{j}T}]=\frac{{\acute{c}}_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{T{g}_j}\left(e^{g_{j}T},-,1\right)-{\acute{c}}_j(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i)$$ 7

Assuming the value of the deterioration rate of each product j $(g_j)$ is small (${0\le g}_j\le 1$), we use the Taylor series to approximate the exponential function $e^{g_{j}T}$ by the expression $1+g_{j}T+\frac{1}{2}{(g_{j}T)}^2$. It should be noted that the considered ordering cycle ($T$) is defined per annum, so the multiplication of $(g_j)$ and ($T$) will result in a small value.

So, the total holding cost and the total cost of deteriorated units per unit of time could be, respectively, expressed as follows:

$${\mathit{TC}}_H=\sum _j\frac{h_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)T}{2}$$ 8

$${\mathit{TC}}_D=\sum _j\frac{{\acute{c}}_j{g}_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)T}{2}$$ 9

Finally, if we have two products in the proposed inventory system, the profit function that we should maximize could be written as follows:

$$R\left(T,,,p_1,,,p_2\right)=-\frac{A_1}{T}-\frac{A_2}{T}-\frac{1}{2}T({\acute{c}}_2{g}_2+h_2)(a_2-s(\theta p_1+p_2))+(c_2-p_2)(-a_2+s(\theta p_1+p_2))-\frac{1}{2}T({\acute{c}}_1{g}_1+h_1)(a_1-s(p_1+\theta p_2))+(c_1-p_1)(-a_1+s(p_1+\theta p_2))$$ 10

In Eq. 10, the deterioration cost (cost of dealing with a deteriorated unit of a product: separation, disposal, and replacement) is taken to account by using ${\acute{c}}_j$. That is why, the deterioration rate was disregarded when we expressed the purchasing cost.

In order to show the concavity of the profit function (R), let us consider the following theorem:

Theorem 1

Equation 10is concave if and only if $sT((T({\acute{c}}_1{g}_1+\theta {\acute{c}}_2{g}_2+h_1+\theta h_2)-2p_1)p_1+(T(\theta {\acute{c}}_1{g}_1+{\acute{c}}_2{g}_2+\theta h_1+h_2)-4\theta p_1)p_2-2p_2^2))<2{(A}_1+A_2)$

Proof

(Please see Appendix A).

So, to obtain the optimum values of $p_1$, $p_2$ and $T$, we take the partial derivatives of the profit function with respect to each one of these variables. Then, we put these derivatives to zero and solve the resulting system of 3 equations and 3 unknowns and find the expressions of the optimum values of the selling prices and replenishment cycle $p_1^{\ast }$, $p_2^{\ast }$ and $T^{\ast }$.

Finally, taking the partial derivatives of the resulting profit function (Eq. 10) with respect to $p_1$, $p_2$ and $T$, we get:

$$\frac{\partial R(T,p_1,p_2)}{\partial T}=\frac{A_1+A_2}{T^2}+\frac{{(h}_1+{\acute{c}}_1{g}_1)(-a_1+s{p}_1+s\theta p_2)}{2}+\frac{{(h}_2+{\acute{c}}_2{g}_2)(-a_2+s{p}_2+s\theta p_1)}{2}$$ 11

$$\frac{\partial R(T,p_1,p_2)}{\partial p_1}=\frac{Ts(\left(h_1,+,{\acute{c}}_1,g_1\right)+\theta (h_2+{\acute{c}}_2{g}_2))}{2}+a_1-2s\left(p_1,+,\theta ,p_2\right)+s(c_1+\theta c_2)$$ 12

$$\frac{\partial R(T,p_1,p_2)}{\partial p_2}=\frac{Ts(\left(h_2,+,{\acute{c}}_2,g_2\right)+\theta (h_1+{\acute{c}}_1{g}_1))}{2}+a_2-2s\left(p_2,+,\theta ,p_1\right)+s(c_2+\theta c_1)$$ 13

So,

$$p_1^{\ast }=\frac{2a_1+s\left(1,+,\theta \right)\left(2,c_1,+,T,h_1,+,T,{\acute{c}}_1,g_1\right)}{4s\left(1,+,\theta \right)}$$ 14

$$p_2^{\ast }=\frac{2a_2+s(1+\theta )(2c_2+T{h}_2+T{\acute{c}}_2{g}_2)}{4s(1+\theta )}$$ 15

After substituting $p_1^{\ast }$ and $p_2^{\ast }$ by their expressions from Eqs. 14 and 15 into Eq. 11, a function of $T$ $\left(F,\left(T\right)\right)$ is obtained as follows:

$$F\left(T\right)=\frac{(A_1+A_2)}{T^2}+\frac{(g_1{\acute{c}}_1+h_1)(2\theta a_2-2(1+2\theta )a_1+s(1+\theta )(2\theta c_2+2c_1+T(g_2\theta {\acute{c}}_2+g_1{\acute{c}}_1+\theta h_2+h_1)))}{8(1+\theta )}+\frac{(g_2{\acute{c}}_2+h_2)(2\theta a_1-2(1+2\theta )a_2+s(1+\theta )(2\theta c_1+2c_2+T(g_1\theta {\acute{c}}_1+g_2{\acute{c}}_2+\theta h_1+h_2)))}{8(1+\theta )}$$ 16

Since Eq. 16 has one single unknown $(T)$, the optimum value of the replenishment cycle $(T^{\ast })$ is the root of the function $F\left(T\right)$ (the solution of the equation $F\left(T\right)=0$).

For example, in a special state where the ordering cost is negligible, $T^{\ast }$ can be obtained by the following closed-form formula:

$$T^{\ast }=-\frac{\left(g_1,{\acute{c}}_1,+,h_1\right)\left(2,\theta ,a_2,-,2,\left(1,+,2,\theta \right),a_1,+,s,\left(1,+,\theta \right),\left(2,\theta ,c_2,+,2,c_1\right)\right)+\left(g_2,{\acute{c}}_2,+,h_2\right)\left(2,\theta ,a_1,-,2,\left(1,+,2,\theta \right),a_2,+,s,\left(1,+,\theta \right),\left(2,\theta ,c_1,+,2,c_2\right)\right)}{\left(g_1,{\acute{c}}_1,+,h_1\right)s\left(1,+,\theta \right)\left(g_2,\theta ,{\acute{c}}_2,+,g_1,{\acute{c}}_1,+,\theta ,h_2,+,h_1\right)+\left(g_2,{\acute{c}}_2,+,h_2\right)s\left(1,+,\theta \right)\left(g_1,\theta ,{\acute{c}}_1,+,g_2,{\acute{c}}_2,+,\theta ,h_1,+,h_2\right)}$$ 17

In addition to the budget and capacity constraints described above, the above model is limited by the upper and lower bounds of carbon emissions and job opportunities. In one category of challenges, a budget limitation is a real-world issue that every enterprise has to deal with on a daily basis. In order to calculate the optimal value of order quantity ($Q_j^{\ast })$ with respect to the stock capacity, the constraint.

${\sum }_j{f}_j{·I_{\max }}_j\le \mathrm{CAP}$ should be satisfied. We have:${\sum }_j\frac{f_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)(e^{g_{j}T}-1)}{g_j}\le \mathrm{CAP}$. Using the first two terms of the Taylor series, we can rewrite the capacity constraint as${\sum }_j\frac{f_j(T+\frac{1}{2}{g_{j}T}^2)\left(a_j,-,s,p_j,-,s,{\sum }_{i\ne j},{\theta }_j,p_i\right)}{g_j}\le \mathrm{CAP}$. In a similar way, we can compute the environmental constraint as ${\sum }_j{\mathit{ce}}_j.{I_{\max }}_j\le \mathrm{UBE}$ where it can be rewritten as${\sum }_j\frac{{\mathit{ce}}_j(T+\frac{1}{2}{g_{j}T}^2)\left(a_j,-,s,p_j,-,s,{\sum }_{i\ne j},{\theta }_j,p_i\right)}{g_j}\le \mathrm{UBE}$. Lastly, for social sustainability to have a limitation on the lower bound of created job opportunities, we can define the constraint set as ${\sum }_j{\mathit{vj}}_j.{I_{\max }}_j\ge \mathrm{LBJ}$ where it can be written as ${\sum }_j\frac{{\mathit{vj}}_j(T+\frac{1}{2}{g_{j}T}^2)\left(a_j,-,s,p_j,-,s,{\sum }_{i\ne j},{\theta }_j,p_i\right)}{g_j}\ge \mathrm{LBJ}$. Finally, if the budget constraint is also taken to account, the constraint ${\sum }_j{c}_j.Q_j\le \mathrm{BDG}$ should be satisfied.

Based on the selected set of constraints, if we relax each set of constraints, we can then be able to identify an optimal solution that has the lowest optimality gap when we apply the selected set of constraints. In this regard, if the solution after relaxing the capacity constraint is $T_1^{\max }$. As such, $T_2^{\max }$, $T_3^{\max }$ and $T_4^{\max }$ are the solutions for relaxing environmental, social, and budget constraints. The best solution is the maximum one as the following formula:

$$T^{\max }=\max (T_1^{\max },T_2^{\max },T_3^{\max },T_4^{\max })$$ 18

Among this relaxed solution ($T^{\max }$) and the optimal solution found by Eq. (17) as ($T^{\ast }$), the minimum solution is found by the following formula:

$$T=\min (T^{\max },T^{\ast })$$ 19

In conclusion, for the proposed model in the first case, when the shortage does not exist in the inventory system, the optimal solution is found, as illustrated above.

The inventory system in case of allowable shortage

With the assumption that inventory shortage is allowed and completely backordered, the inventory level of each product (j) of the two complementary products follows the pattern depicted in Fig. 3.

Fig. 3.

Graphical representation of the inventory and shortage levels of the two products in the inventory system

During the time interval [0,$t_{1j}$], the inventory level declines due to the demand and the deterioration process of the product. At time ${t=t}_{1j}$ the inventory level attains zero, and shortages occur during the time interval [$t_{1j},T$] solely under the effect of the demand. At $t=T,$ the inventory level attains $b_j$. Then, the inventory is replenished with a lot size $Q_j$ which raises the inventory level up, and a new inventory cycle starts again.

Hence, $I_j\left(t\right)$ the inventory level of the product j (j could be equal to 1 or 2) is governed by the two following differential equations:

$$\left\{\begin{array}{c}\frac{\mathrm{d}{I}_j\left(t\right)}{\mathrm{d}t}=-g_j{I}_j\left(t\right)-(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)0<t<t_{1j}\\ \frac{\mathrm{d}{I}_j(t)}{\mathrm{d}t}=-(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)t_{1j}\le t\le T\end{array}\right)$$ 20

Therefore, we have:

$$I_j\left(t\right)=\left\{\begin{array}{c}{e}^{-\int g_{j}dt}\times [\int -(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i){·e}^{\int g_j\mathrm{d}t}\mathrm{d}t]0<t<t_{1j}\\ \int -(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)\mathrm{d}t{t}_{1j}\le t\le T\end{array}\right)$$ 21

Solving these two differential equations, we express the inventory level for product j as follows:

$$I_j\left(t\right)=\left\{\begin{array}{c}\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}\left[e^{g_j\left(t_{1j},-,t\right)},-,1\right]0<t<t_{1j}\\ -(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)\left(t,-,t_{1j}\right)t_{1j}\le t\le T\end{array}\right)$$ 22

With the boundary condition $I_j(t_{1j})=0$ and $I_j(T)=b_j$, we get the following expressions for the maximum inventory level ${I_{\max }}_j$ and the shortage during a cycle $b_j$:

$${I_{\max }}_j=I_j\left(0\right)=\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}[e^{g_j{t}_{1j}}-1]$$ 23

$$b_j={-I}_j\left(T\right)=-(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i)(T-t_{1j})$$ 24

And the order quantity ($Q_j$) can be written as:

$$Q_j={I_{\max }}_j+b_j$$ 25

Therefore, replacing ${I_{\max }}_j$ and $b_j$ in Eq. 25, by their expressions from Eqs. 23 and 24, we get:

$$Q_j=\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}\left[e^{g_j{t}_{1j}},-,1\right]+(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i)(T-t_{1j})$$ 26

As the inventory level is positive only on the interval $\left[0,,,t_{1j}\right],$ the average amount of inventory of product j carried in the system during a cycle is given by:

$${\overline{I}}_j={\int }_0^{t_{1j}}{I}_j\left(t\right)dt={\int }_0^{t_{1j}}\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}\left[e^{g_j\left(t_{1j},-,T\right)},-,1\right]\mathrm{d}t=\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j^2}(e^{g_j{t}_{1j}}-g_j{t}_{1j}-1)$$ 27

Similarly, as the shortage occurs during the time interval [$t_{1j},T$], the average shortage for product $j$ in a cycle is given by:

$${\overline{B}}_j=-{\int }_{t_{1j}}^T{I}_j\left(t\right)dt=-{\int }_{t_{1j}}^T-(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i)\left(t,-,t_{1j}\right)\mathrm{d}t=\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i){\left(T,-,t_{1j}\right)}^2}{2}$$ 28

In addition, the average number of replenishments is equal to $\frac{1}{T}$.

Finally, as on-hand inventory deteriorates through time at the rate $(g_j)$, the total number of deteriorated units of product j during a cycle is given by:

$${\int }_0^{t_{1j}}{g}_j{I}_j\left(t\right)dt={\int }_0^{t_{1j}}{g}_j\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}\left[e^{g_j\left(t_{1j},-,t\right)},-,1\right]dt=(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i){\int }_0^{t_{1j}}\left[e^{g_j\left(t_{1j},-,t\right)},-,1\right]\mathrm{d}t=(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i)\left[-,\frac{1}{g_j},e^{g_j\left(t_{1j},-,t\right)},-,t\right]\genfrac{}{}{0.0pt}{}{t_{1j}}{0}=(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i)\left[-,\frac{1}{g_j},-,t_{1j},-,\frac{1}{g_j},e^{g_j{t}_{1j}}\right]=\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}{e}^{g_j{t}_{1j}}-(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i)t_{1j}-\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}$$ 29

It is important to find out the expression of the function in order to find the optimal values of the decision variables that will maximize the profit of the firm. Basically, this is the difference between the revenue earned by the firm from selling its products and the total cost incurred by the firm, which includes both the price of the products and the cost of maintaining the inventory system. Thus, our first step will be to calculate the total cost of the inventory system for a unit of time. This is determined by the ordering cost, the holding cost, the backordering cost, and the deterioration cost associated with both products. Like the previous model, ordering cost (${\mathit{TC}}_O$) is calculated through $\frac{{\sum }_j{A}_j}{T}$ and other costs are as follows:

The holding cost per unit of time is:

$${\mathit{TC}}_h=\sum _j\frac{h_j{\overline{I}}_j}{T}={\sum }_j\frac{h_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{T{g}_j^2}(e^{g_j{t}_{1j}}-g_j{t}_{1j}-1)$$ 30

Assuming the value of the deterioration rate of each product j $(g_j)$ is small (${0\le g}_j\le 1$), we use the Taylor series to approximate the exponential function $e^{g_j{t}_{1j}}$ by the expression $1+g_j{t}_{1j}+\frac{1}{2}{(g_j{t}_{1j})}^2,$ so we can write:

$${\mathit{TC}}_h=\sum _j\frac{h_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{T{g}_j^2}\left(1,+,g_j,t_{1j},+,\frac{1}{2},g_j^2,t_{1j}^2,-,g_j,t_{1j},-,1\right)=\sum _j\frac{h_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)t_{1j}^2}{2T}$$ 31

The backlogging cost per unit of time is:

$${\mathit{TC}}_S=\sum _j\frac{{\overline{B}}_j{\pi }_j}{T}=\sum _j\frac{{\pi }_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i){(T-t_{1j})}^2}{2T}$$ 32

The cost of deteriorated units of the product per unit of time is:

$${\mathit{TC}}_D={\sum }_j\frac{[\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}{e}^{g_j{t}_{1j}}-(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)t_{1j}-\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}]\times {\acute{c}}_j}{T}$$ 33

As discussed before, the deterioration rates have small values (${0\le g}_j\le 1$), using the Taylor series in order to approximate the exponential function $e^{g_j{t}_{1j}}$ by the expression $1+g_j{t}_{1j}+\frac{1}{2}{(g_j{t}_{1j})}^2,$ the cost of deterioration per unit of time could be written as follows:

$${\mathit{TC}}_D=\sum _j\frac{\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}[1+g_j{t}_{1j}+\frac{1}{2}{(g_j{t}_{1j})}^2{e}^{g_j{t}_{1j}}-(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)t_{1j}-1]\times {\acute{c}}_j}{T}=\sum _j\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)g_j{\acute{c}}_j{t}_{1j}^2}{2T}$$ 34

Finally, the profit function of the firm that we should maximize is obtained by subtracting the purchasing cost, the inventory holding cost, the shortage cost, and the deterioration cost calculated above from the revenue earned from selling the two complementary products. Doing so, we obtain the following expression for the firm’s profit:

$$R\left(T,,,p_j,,,b_j\right)=\sum _j[\left(p_j,-,c_j\right)\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{T}\left(\frac{g_j{t}_{1j}^2}{2},+,T\right)-\frac{A_j}{T}-\frac{h_j(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)t_{1j}^2}{2T}-\frac{{\pi }_j\left(a_j,-,s,p_j,-,s,{\sum }_{i\ne j},{\theta }_j,p_i\right){\left(T,-,t_{1j}\right)}^2}{2T}-\frac{\left(a_j,-,s,p_j,-,s,{\sum }_{i\ne j},{\theta }_j,p_i\right)g_j{\acute{c}}_j{t}_{1j}^2}{2T}]$$ 35

Substituting $t_{1j}$ in Eq. (35) by T $-\frac{b_j}{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}$, the profit function for our 2-product system could be written as follows:

$$\begin{array}{c}\hfill R\left(T,,,p_1,,,p_2,,,b_1,{,b}_2\right)=-\frac{1}{2T}(2(A_1+A_2)+\frac{{\acute{c}}_2{g}_2{(-T{a}_2+b_2+sT(\theta p_1+p_2))}^2}{a_2-s(\theta p_1+p_2)}+\frac{h_2{(-T{a}_2+b_2+sT(\theta p_1+p_2))}^2}{a_2-s(\theta p_1+p_2)}+\frac{{\acute{c}}_1{g}_1{(-T{a}_1+b_1+sT(p_1+\theta p_2))}^2}{a_1-s(p_1+\theta p_2)}+\frac{h_1{(-T{a}_1+b_1+sT(p_1+\theta p_2))}^2}{a_1-s(p_1+\theta p_2)}-2(-C_2+p_2)(a_2-s(\theta p_1+p_2))\left(T,+,\frac{1}{2},g_2,{\left(T,+,\frac{b_2}{-a_2+s(\theta p_1+p_2)}\right)}^2\right)-2(-C_1+p_1)(a_1-s(p_1+\theta p_2))\left(T,+,\frac{1}{2},g_1,{\left(T,+,\frac{b_1}{-a_1+s(p_1+\theta p_2)}\right)}^2\right)+\frac{b_1^2{\pi }_1}{a_1-s(p_1+\theta p_2)}+\frac{b_2^2{\pi }_2}{a_2-s(\theta p_1+p_2)})\end{array}$$ 36

The partial derivatives of the profit function are as follows:

$$\frac{\partial R(T,p_1,p_2,b_1,b_2)}{\partial T}=\frac{1}{2T^2}(2(A_1+A_2)+\frac{{\acute{c}}_2{g}_2{(-T{a}_2+b_2+sT(\theta p_1+p_2))}^2}{a_2-s(\theta p_1+p_2)}+\frac{h_2{(-T{a}_2+b_2+sT(\theta p_1+p_2))}^2}{a_2-s(\theta p_1+p_2)}+\frac{{\acute{c}}_1{g}_1{(-T{a}_1+b_1+sT(p_1+\theta p_2))}^2}{a_1-s(p_1+\theta p_2)}+\frac{h_1{(-T{a}_1+b_1+sT(p_1+\theta p_2))}^2}{a_1-s(p_1+\theta p_2)}-2(-C_2+p_2)(a_2-s(\theta p_1+p_2))(T+\frac{1}{2}{g}_2{(T+\frac{b_2}{-a_2+s(\theta p_1+p_2)})}^2)-2(-C_1+p_1)(a_1-s(p_1+\theta p_2))(T+\frac{1}{2}{g}_1{(T+\frac{b_1}{-a_1+s(p_1+\theta p_2)})}^2)-2T({\acute{c}}_2{g}_2(T{a}_2-b_2-sT(\theta p_1+p_2))+h_2(T{a}_2-b_2-sT(\theta p_1+p_2))+{\acute{c}}_1{g}_1(T{a}_1-b_1-sT(p_1+\theta p_2))+h_1(T{a}_1-b_1-sT(p_1+\theta p_2))-(-C_2+p_2)(a_2-s(\theta p_1+p_2))(1+g_2(T+\frac{b_2}{-a_2+s\theta p_1+s{p}_2}))-(-C_1+p_1)(a_1-s(p_1+\theta p_2))(1+g_1(T+\frac{b_1}{-a_1+s{p}_1+s\theta p_2})))+\frac{b_1^2{\pi }_1}{a_1-s(p_1+\theta p_2)}+\frac{b_2^2{\pi }_2}{a_2-s(\theta p_1+p_2)})$$ 37

$$\frac{\partial R(T,p_1,p_2,b_1,b_2)}{\partial p_1}=-\frac{1}{2T}(\frac{2s\theta b_2{g}_2(C_2-p_2)(-T{a}_2+b_2+sT(\theta p_1+p_2))}{{(a_2-s(\theta p_1+p_2))}^2}+\frac{s\theta {\acute{c}}_2{g}_2{(-T{a}_2+b_2+sT(\theta p_1+p_2))}^2}{{(a_2-s(\theta p_1+p_2))}^2}+\frac{s\theta h_2{(-T{a}_2+b_2+sT(\theta p_1+p_2))}^2}{{(a_2-s(\theta p_1+p_2))}^2}+\frac{2s{b}_1{g}_1(C_1-p_1)(-T{a}_1+b_1+sT(p_1+\theta p_2))}{{(a_1-s(p_1+\theta p_2))}^2}+\frac{s{\acute{c}}_1{g}_1{(-T{a}_1+b_1+sT(p_1+\theta p_2))}^2}{{(a_1-s(p_1+\theta p_2))}^2}+\frac{s{h}_1{(-T{a}_1+b_1+sT(p_1+\theta p_2))}^2}{{(a_1-s(p_1+\theta p_2))}^2}+2sT\theta {\acute{c}}_2{g}_2(-T+\frac{b_2}{a_2-s(\theta p_1+p_2)})+2sT\theta h_2(-T+\frac{b_2}{a_2-s(\theta p_1+p_2)})+2sT{\acute{c}}_1{g}_1(-T+\frac{b_1}{a_1-s(p_1+\theta p_2)})+2sT{h}_1(-T+\frac{b_1}{a_1-s(p_1+\theta p_2)})+2s\theta (-C_2+p_2)(T+\frac{1}{2}{g}_2{(T+\frac{b_2}{-a_2+s(\theta p_1+p_2)})}^2)+2s(-C_1+p_1)(T+\frac{1}{2}{g}_1{(T+\frac{b_1}{-a_1+s(p_1+\theta p_2)})}^2)-2(a_1-s(p_1+\theta p_2))(T+\frac{1}{2}{g}_1{(T+\frac{b_1}{-a_1+s(p_1+\theta p_2)})}^2)+\frac{s{b}_1^2{\pi }_1}{{(a_1-s(p_1+\theta p_2))}^2}+\frac{s\theta b_2^2{\pi }_2}{{(a_2-s(\theta p_1+p_2))}^2})$$ 38

$$\frac{\partial R(T,p_1,p_2,b_1,b_2)}{\partial p_2}=-\frac{1}{2T}(\frac{2s{b}_2{g}_2(C_2-p_2)(-T{a}_2+b_2+sT(\theta p_1+p_2))}{{(a_2-s(\theta p_1+p_2))}^2}+\frac{s{\acute{c}}_2{g}_2{(-T{a}_2+b_2+sT(\theta p_1+p_2))}^2}{{(a_2-s(\theta p_1+p_2))}^2}+\frac{s{h}_2{(-T{a}_2+b_2+sT(\theta p_1+p_2))}^2}{{(a_2-s(\theta p_1+p_2))}^2}+\frac{2s\theta b_1{g}_1(C_1-p_1)(-T{a}_1+b_1+sT(p_1+\theta p_2))}{{(a_1-s(p_1+\theta p_2))}^2}+\frac{s\theta {\acute{c}}_1{g}_1{(-T{a}_1+b_1+sT(p_1+\theta p_2))}^2}{{(a_1-s(p_1+\theta p_2))}^2}+\frac{s\theta h_1{(-T{a}_1+b_1+sT(p_1+\theta p_2))}^2}{{(a_1-s(p_1+\theta p_2))}^2}+2sT{\acute{c}}_2{g}_2(-T+\frac{b_2}{a_2-s(\theta p_1+p_2)})+2sT{h}_2(-T+\frac{b_2}{a_2-s(\theta p_1+p_2)})+2sT\theta {\acute{c}}_1{g}_1(-T+\frac{b_1}{a_1-s(p_1+\theta p_2)})+2sT\theta h_1(-T+\frac{b_1}{a_1-s(p_1+\theta p_2)})+2s(-C_2+p_2)(T+\frac{1}{2}{g}_2{(T+\frac{b_2}{-a_2+s(\theta p_1+p_2)})}^2)-2(a_2-s(\theta p_1+p_2))(T+\frac{1}{2}{g}_2{(T+\frac{b_2}{-a_2+s(\theta p_1+p_2)})}^2)+2s\theta (-C_1+p_1)(T+\frac{1}{2}{g}_1{(T+\frac{b_1}{-a_1+s(p_1+\theta p_2)})}^2)+\frac{s\theta b_1^2{\pi }_1}{{(a_1-s(p_1+\theta p_2))}^2}+\frac{s{b}_2^2{\pi }_2}{{(a_2-s(\theta p_1+p_2))}^2})$$ 39

$$\frac{\partial R(T,p_1,p_2,b_1,b_2)}{\partial b_1}=-\frac{(h_1+g_1(C_1+{\acute{c}}_1-p_1))(-T{a}_1+b_1+sT(p_1+\theta p_2))+b_1{\pi }_1}{T(a_1-s(p_1+\theta p_2))})$$ 40

$$\frac{\partial R(T,p_1,p_2,b_1,b_2)}{\partial b_2}=\frac{(h_2+g_2(C_2+{\acute{c}}_2-p_2))(T{a}_2-b_2-sT(\theta p_1+p_2))-b_2{\pi }_2}{T(a_2-s(\theta p_1+p_2))}$$ 41

Now, the constraints, including the capacity and budget constraints as well as the environmental and social limitations, have been applied to the model. As mentioned before, considering limited warehouse capacity is very important for perishable products, especially when we have more than one single product. In this case, we need to satisfy the inequality ${\sum }_j{f}_j\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}(e^{g_j{t}_{1j}}-1)\le CAP$.

Since these constraints are the same, for example, for the capacity constraint set, by substituting $t_{1j}=T-\frac{b_j}{D_j}$ and applying the first two terms of the Taylor series, we can rewrite it as the following second degree equation: $T^2{\sum }_j\frac{1}{2}{f}_j{D}_j{g}_j+T{\sum }_j{f}_j{(D}_j-b_j{g}_j)=CAP+{\sum }_j{f}_j{b}_j(1-\frac{b_j{g}_j}{2D_j})$. Solving this equation for T, the upper bound of the optimal replenishment cycle ($T_1^{\max }$) for the capacitated joint pricing and inventory decisions problem for perishable complementary products with shortages allowed is obtained as:

$$\begin{array}{c}\hfill T_1^{\max }=\frac{-D_1{f}_1-D_2{f}_2+b_1{f}_1{g}_1+b_2{f}_2{g}_2+\sqrt{{(f_1(D_1-b_1{g}_1)+f_2(D_2-b_2{g}_2))}^2+(g_1{D}_1{f}_1+g_2{D}_2{f}_2)(2CAp+b_1{f}_1(2-b_1{D}_1{g}_1)+b_2{f}_2(2-b_2{D}_2{g}_2))}}{b_1{D}_1{f}_1+b_2{D}_2{f}_2}\end{array}$$ 42

where ${\mathrm{D}}_1=a_1-s{p}_1-s\theta p_2$, $D_2=a_2-s\theta p_1-s{p}_2$ and $p_i,b_i$ will be calculated by the algorithm presented in the next section.

Similarly, if we have a budget limitation (BDG), then it is absolutely essential to meet the constraint:${\sum }_j{c}_j{Q}_j\le BDG$. By substituting $Q_j$ from Eq. 26, we need to satisfy inequality:

$$\sum _j\left(\frac{c_j\left(a_j,-,s,p_j,-,s,{\sum }_{i\ne j},{\theta }_j,p_i\right)(e^{g_j{t}_{1j}}-1)}{g_j},+,c,j,(a_j-s{p}_j-s\sum _{i\ne j}{\theta }_j{p}_i),(T-t_{1j})\right)\le BDG$$ 43

By substituting $t_{1j}=T-\frac{b_j}{D_j}$ and applying the first two terms of the Taylor series, we can rewrite it as the following second-degree equation:

$$T^2\sum _j\frac{1}{2}{c}_j{D}_j{g}_j+T\sum _j{c}_j{(D}_j-b_j{g}_j)=BDG-\sum _j\frac{c_j{g}_j{b}_j^2}{2D_j}$$ 44

Solving Eq. 44 for T, another upper bound of the optimal replenishment cycle ($T_2^{\max }$) for the joint pricing and inventory decisions problem for perishable complementary products with shortages allowed and under budget constraint is obtained as:

$$T_2^{\max }=\frac{-c_1{D}_1-c_2{D}_2+b_1{c}_1{g}_1+b_2{c}_2{g}_2+\sqrt{-(c_1{D}_1{g}_1+c_2{D}_2{g}_2)(-2\mathrm{BDG}+b_1^2{c}_1^2{D}_1{g}_1^2+b_2^2{c}_2^2{D}_2{g}_2^2)+{(c_1(D_1-b_1{g}_1)+c_2(D_2-b_2{g}_2))}^2}}{c_1{D}_1{g}_1+c_2{D}_2{g}_2}$$ 45

where ${\mathrm{D}}_1=a_1-s{p}_1-s\theta p_2$, ${\mathrm{D}}_2=a_2-s\theta p_1-s{p}_2$ and $p_i,b_i$ will be calculated by the algorithm presented in the next section.

In a similar way, this inequality for the environmental constraint is written as ${\sum }_j{\mathit{ce}}_j\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}(e^{g_j{t}_{1j}}-1)\le \mathrm{UBE}$. As such, the social constraints are calculated as ${\sum }_j{\mathit{vj}}_j\frac{(a_j-s{p}_j-s{\sum }_{i\ne j}{\theta }_j{p}_i)}{g_j}(e^{g_j{t}_{1j}}-1)\ge \mathrm{LBJ}.$

In a similar way, we can define $T_3^{\max }$ and $T_4^{\max }$ as the solutions from relaxing the environmental and social constraints, respectively. Without using approximate methods such as the Lagrangian approach, the optimum replenishment cycle ($T^{\ast \ast }$) for the problem with shortages allowed and under budget and warehouse capacity in addition to environmental and social constraints is therefore equal to $T^{\ast \ast }=\text{min}\{T^{\ast },T_1^{\max },T_2^{\max }{T}_3^{\max },T_4^{\max }\}$, where $T^{\ast }$ is the optimal replenishment cycle such that warehouse capacity, environmental, and social constraints are relaxed and will be calculated by the heuristic algorithm presented in Sect. 4.

Solution method

This study proposes a heuristic algorithm for finding an approximate optimal solution for the proposed inventory model based on the Lagrangian relaxation theory (Soleimani et al., 2022; Asghari et al., 2022a, b). The main idea is to reduce the complexity to have fewer computations when a commercial software is used for solving the proposed models for both cases of shortage. The main steps of our solution method are:

Step 1 (Initialization)

Step 1.1

Take derivatives from the revenue function with respect to decision variables; ${T,p}_i,and{b}_i$($i=1,2$), such that a system of 5 equations, 5 unknowns ($5\times 5$) is built.

Step 1.2

Decompose the created equations system to two $(4\times 4)$ and ($1\times 1$) subsystems.

Step 1.3

Solve the reduced $(4\times 4)$ subsystem to find the optimum functions of decision variables; $p_i,b_i,$ in terms of T, named $p_i(T),and{B}_i(T)$, respectively.

Step 1.4

Solve the first-degree equation ($1\times 1$) for T in terms of $p_i,b_i$, named $T(p_i,b_i).$

Step 1.5

Set $k=0$ and $T^{(k)}=T_0$.

Step 2 (Iteration)

Step 2.1

Compute the values of $p_i(T),B_i(T)\mathrm{for}{T}^{(k)}\to$ ($p_i^{(k)},B_i^{(k)}$).

Step 2.2

Compute the value of $T(p_i,b_i)$ for $p_i^{(k)},B_i^{(k)}\to T^{(k+1)}$

Step 2.3

Compute the absolute deviation $\left|T^{(k+1)},-,T^{(k)}\right|$

Step 2.4

If $\left|T^{(k+1)},-,T^{(k)}\right|\le \epsilon$, stop and $T^{\ast }=T^{(k+1)}$; Otherwise, $k=k+1,$ and go to step 2.1.

Where $\epsilon$ is an arbitrary small number that reflects the precision of the optimal solution and is defined by the user.

NB₁

as we are taking derivatives from the revenue function in order to obtain the optimum value of the decision variables, we need to prove that the revenue function is concave (Please see Appendix B).

NB₂

$T_0$is an initial point for the proposed heuristic algorithm. It can be set by an arbitrary number or estimated through classical inventory models.

Numerical examples and sensitivity analysis

The purpose of this section is to provide managerial insights and define numerical examples to demonstrate the optimal decisions with respect to inventory and pricing policies that can be made in light of sustainability dimensions. As an example, let us consider a company that manages the inventory of complementary products under the conditions of perishability and backlog of demand. In order to determine the optimal pricing and inventory management decisions, we employ the mathematical model we have developed along with the heuristic method based on data from a case study conducted in the Iranian food industry. Then, we do a sensitivity analysis of the optimal inventory and pricing policy in order to understand how this policy is affected by changes in the coefficient of complementarity between the two products $(\theta )$ and in the deterioration rates $(g_j)$.

Numerical example

Our case study focuses on the dairy manufacturing system for milk and cereal products. This table presents the values of different parameters associated with the inventory management system used by the company for these two products, as well as the units in which those parameters are measured. As a matter of fact, in this case, the first product (milk) has a short shelf-life and can be considered perishable, whereas the second product (cereal) is classified as non-perishable because of its long shelf-life. The important point to note is that product 1 (milk) is subject to shortages at the moment, whereas the cereal has always been available when demanded, so there would be no shortages for product 2.

For this particular case (one product is perishable and subject to shortage, while the other is not), the closed form expressions for the optimal values of the decision variables $T^{\ast },p_1^{\ast }{,p}_2^{\ast },b_1^{\ast }$ and for the optimal profit $(R^{\ast })$ are obtained (please see Appendix C).

Applying the heuristic algorithm presented in Sect. 4 on the data contained in Table 2 while assuming $s=0.5$, $g_1=0.7$ and $\theta =0.5$ and setting$\epsilon =0.0001$, we reach the optimal value of the inventory cycle ($T^{\ast })$ in 13 iterations. Table 3 reports the values of the decision variables obtained in each one of the iterations. To apply the algorithm, the initial point, $T_0$ is estimated by the following formula, taken from classical multi-product inventory models for shortage-allowed condition.

$$T_0=\frac{\sqrt{{2(A}_1+A_2)\left(c_1,+,{\acute{c}}_1\right)g_1+h_1+{\pi }_1}}{\sqrt{a_1\left(\left(c_1,+,{\acute{c}}_1\right),g_1,+,h_1\right){\pi }_1+a_2{h}_2\left(\left(c_1,+,{\acute{c}}_1\right),g_1,+,h_1,+,{\pi }_1\right)}}$$ 46

Table 2.

The case input data

$A_1$	$A_2$	$a_1$	$a_2$	$h_1$	$h_2$	$c_1$	$c_2$	${\acute{c}}_1$	${\pi }_1$	${\mathit{vj}}_1$	${\mathit{vj}}_2$	${\mathit{ce}}_1$	${\mathit{ce}}_2$
200	100	200	180	5	2	6	18	8	7	1	2	0.0065	0.0073

Table 3.

The results of applying the heuristic algorithm on the case data in Table 2

Iteration	Concavity	$T$	$p_1$	$p_2$	$b_1$	$\mathrm{\Delta }T\%$
1	✓	0.66766	150.37984	116.00499	39.04475	34.84
2	✓	0.86200	150.57520	116.09767	49.67736	15.88
3	✓	0.94390	150.66152	116.13861	54.36697	7.89
4	✓	0.98367	150.70343	116.15850	56.64219	4.01
5	✓	1.00366	150.72450	116.16850	57.78521	2
6	✓	1.01386	150.73525	116.17360	58.36843	1.06
7	✓	1.01910	150.74078	116.17622	58.66826	0.55
8	✓	1.02321	150.74510	116.17827	58.90296	0.15
9	✓	1.02393	150.74587	116.17863	58.94436	0.08
10	✓	1.02430	150.74626	116.17882	58.96579	0.04
11	✓	1.02450	150.74647	116.17892	58.97689	0.02
12	✓	1.02460	150.74657	116.17897	58.98264	0.01
13	✓	1.02465	Algorithm halted			0.00

Sensitivity analysis

In the following set of experiments, we study the variations of the optimal values of decision variables in response to variations in the coefficient of complementarity between the two products $(\theta )$ and of the deterioration rate of product 1 $(g_1$). The results of these experiments are then commented, and managerial insights are illustrated.

Sensitivity analysis for the coefficient of complementarity $(\mathit{\theta })$

In this experiment, we let the ratio of complementarity between the two products $(\theta )$ vary from 0 (the two products are independent) to $1$ (the two products are perfectly complementary) with a step equal to 0.1, while the other parameters of the model are fixed. Then, we report and comment the response of the optimal values of the decision variables. For $s=0.5\mathrm{and}{g}_1=0.7,$ our model gives the results shown in Table 4 and in Fig. 4 for the optimal values of $T^{\ast },p_1^{\ast }{,p}_2^{\ast },b_1^{\ast }$ and $R^{\ast }$, respectively.

Table 4.

Results of sensitivity analysis for the coefficient of complementarity between the two products $(\theta )$

$\theta$	Concavity	$T^{\ast }$	$p_1^{\ast }$	$p_2^{\ast }$	$b_1^{\ast }$	$R^{\ast }$
0	✓	1.01403	204.069	189.507	59.8298	33,432.6
0.1	✓	1.01614	187.909	171.124	59.6627	30,165.8
0.2	✓	1.01826	174.907	155.342	59.4949	27,448.7
0.3	✓	1.0204	164.515	141.378	59.3269	25,156.3
0.4	✓	1.02255	156.459	128.559	59.1582	23,200.5
0.5	✓	1.02471	150.747	116.179	58.9888	21,518.6
0.6	✓	1.02689	147.832	103.263	58.8193	20,068.4
0.7	✓	1.02908	149.183	87.9459	58.6498	18,828.4
0.8	✓	1.03129	159.643	65.0712	58.4784	17,819.2
0.9	✓	1.03351	204.089	9.51675	58.3071	17,267.5
1	✓	1.31722	Infinity	Infinity	55.489	1.31722

Fig. 4.

Changes of the $T^{\ast }$, $p_1^{\ast }$, $p_2^{\ast }$,$b_1^{\ast }$,$R^{\ast }$ with respect to changes in the coefficient of complementarity between the two products $(\theta )$

Results show that the optimal inventory cycle ${(T}^{\ast })$ has an increasing trend with $\theta$ (Fig. 4a). So, a higher degree of complementarity between the two products implies a lower ordering frequency. As for the optimal prices of the two products (Fig. 4b, c), they have a declining trend in the first (lower) range of values of $\theta$. Then, the optimal price of the perishable product $p_1^{\ast }$ increases with $\theta ,$ while the optimal price of its complementary non-perishable product $p_2^{\ast }$ decreases. The optimal shortage level also shows a decreasing trend in $\theta$ (Fig. 4d). This last result could be interpreted as follows: the shortage of one product will result in a higher reduction in the sales of its complementary product when the ratio of complementarity between them is higher. Therefore, when the coefficient of complementarity between the two products increases, the cost related to the shortage increases. Hence, in order to minimize this cost increase, the model tends to lower the shortage level. Finally, the optimal profit decreases when $\theta$ increases (Fig. 4e). This could be interpreted by the fact that a higher degree of complementarity between the two products reduces the demand for each of them, and consequently, this has led to a reduced profit from selling the products.

Sensitivity analysis for the deterioration rate $({\mathit{g}}_1)$

In this experiment, we study how the deterioration rate of the perishable product $(g_1)$ influences the optimal values of the decision variables. To do so, we set $s=0.5\mathrm{and}\theta =0.5$ and vary $g_1$ from 0 to 1 with a step equal to 0.1. Results obtained from this experiment are shown in Table 5 and in Fig. 5 for the optimal values of T,$p_1,p_2,b_1$ and R, respectively.

Table 5.

Results of sensitivity analysis for the deterioration rate of product 1 $(g_1)$

$g_1$	Concavity	$T^{\ast }$	$p_1^{\ast }$	$p_2^{\ast }$	$b_1^{\ast }$	$R^{\ast }$
0	✓	1.15782	150.511	116.246	46.1599	21,586.2
0.1	✓	1.12757	150.561	116.23	48.8766	21,572.2
0.2	✓	1.10276	150.603	116.218	51.1851	21,560.2
0.3	✓	1.08204	150.64	116.208	53.1739	21,549.8
0.4	✓	1.06446	150.672	116.199	54.9064	21,540.6
0.5	✓	1.04935	150.7	116.191	56.4302	21,532.4
0.6	✓	1.03622	150.724	116.185	57.7814	21,525.2
0.7	✓	1.02471	150.747	116.179	58.9888	21,518.6
0.8	✓	1.01453	150.767	116.174	60.0742	21,512.7
0.9	✓	1.00546	150.785	116.169	61.0553	21,507.4
1	✓	0.997325	150.801	116.165	61.9466	21,502.5

Fig. 5.

Changes of $T^{\ast }$,$p_1^{\ast }$, $p_2^{\ast }$,$b_1^{\ast }$,$R^{\ast }$ with respect to changes in the deterioration rate of product 1 ${(g}_1)$

Results of this experiment show that the more the product is perishable, the shorter the optimal inventory cycle $(T^{\ast })$ is (Fig. 5a). The intuitive explanation for this result is as follows. In order to limit the deterioration cost, the more perishable a product is, the less time it should spend as an inventory, so the inventory cycle should be shorter. As for the optimal prices of the two products, we observe an increase in the optimal price of the perishable product ($p_1^{\ast })$ (Fig. 5b) = with $g_1$, while the optimal price of the non-perishable product decreases with $g_1$ (Fig. 5c). A similar interpretation could be given to the increasing trend of the optimal value of shortage ${(b}_1^{\ast })$ with $g_1$ (Fig. 5d): in response to a higher deterioration cost due to a higher deterioration rate, the trade-off between shortage cost and deterioration cost incorporated in the model will result in a higher shortage cost, leading, by consequence, to a higher optimal shortage level. Finally, we observe that the higher the deterioration rate is, the lower the firm’s profit is (Fig. 5e). This is not surprising concerning the precedent results. In fact, this experiment shows that when the deterioration rate of the product increases, the costs of the inventory system increase due to a higher orders' frequency, a higher number of deteriorated units, and a higher shortage level, so the firm gets less profit.

Sensitivity analysis for budget, capacity, environmental, and social constraints

In Sect. 3, we explain how, if we implement all four mentioned constraints in the model, the ordering cycle should be recalculated, and a minimum value will be found among all calculation values ($T^{\ast \ast }=\text{min}\{T^{\ast },T_1^{\max },T_2^{\max },T_3^{\max },T_4^{\max }\}$). As part of this section, we will examine how each constraint affects the optimal ordering cycle. We will do calculations for $s=0.5,g_1=0.7\text{and}\theta =0.5$ which $T_0=0.67665$ taken from the presented heuristic in Sect. 4.

Figure 6 shows that all of the variables increase with larger warehouse capacities. As mentioned before, the ordering interval is determined with regard to the $I_{\max }$. By having more storage capacity, we are able to set longer intervals between orders. Aside from that, we are able to sell our products at a higher price because we have access to more facilities within the system that enable us to do so. According to another perspective, in the event that there is a limited amount of storage space, the model is designed to ensure that orders are adjusted in a shorter period of time, that prices are decreased, and that demand is stimulated in order to maintain a reasonable overall profit of the system.

Fig. 6.

Changes of,$T_1^{\max },p_1$, $p_2,b_1,b_2$ with respect to changes in the total capacity of the warehouse (CAP)

In Fig. 7, we see that all variables increase as a result of a higher budget. A larger budget will allow us to extend the interval between orders by a longer amount of time. Our products can also be priced higher than our competitors because we have more facilities within the system, allowing us to charge a higher price for our products. In the event of a lower budget level, the model tends to shorten the ordering interval in order to achieve a greater turnover rate of the storage in order to maintain a higher level of liquidity from the storage to compensate for the lower budget level.

Fig. 7.

Changes of,$T_2^{\max },p_1$, $p_2,b_1$ with respect to changes in the total available budget (BDG)

Figure 8 illustrates that as the carbon emission constraint is loosened, all variables increase. In an environment where environmental limitations are less stringent, it is evident that a system can act more freely in terms of both ordering and pricing. According to the model, prices are gradually decreased, followed by relevant demand to maintain the economic viability of the system. So, what is to be concluded from the above is that the improvement in environmental factors is being responded to by charging the shortage costs and reducing the prices (in order to increase the demand and keep the profit for a short period of time). It has been shown that when ordering intervals are smaller, inventory can be reduced at the same time as meeting demand, and as a result, environmental factors can be improved by reducing inventory levels and managing backorders effectively.

Fig. 8.

Changes of,$T_3^{\max },p_1$, $p_2,b_1$ with respect to changes in the upper bound for the limitation of carbon emissions in the system (UBE)

Figure 9 illustrates that as a result of the fact that variable job opportunities are attributed to inventories, the model attempts to increase the ordering interval in an attempt to improve both the storage inventory and warehouse handling. According to another viewpoint, increased inventory levels would result in higher final prices, resulting in the need to increase the price of the products, which can be done by increasing the price of the products.

Fig. 9.

Changes of,$T_4^{\max },p_1$, $p_2,b_1$ with respect to changes in the lower bound for the limitation of variable job opportunities (LBJ)

Conclusion, implications, limitations, and future research directions

In this paper, an inventory management system for a pair of perishable complementary products is presented in this study under the assumption that the demand function is deterministic, with a possibility of shortages or unsatisfied orders completely backordered, in the process. The proposed model contributes to sustainability by an upper bound for the number of carbon emissions and the number of job opportunities. The proposed model aimed to find simultaneously the optimal values of the inventory cycle and each product’s price and shortage level over the cycle. We first treated the case when the shortage is not allowed, solved the resulting model, and expressed the optimal values of the two prices as functions of the replenishment cycle with the budget, warehouse capacity, environmental, and social constraints. Then, we modeled the general case when the shortage is allowed and presented a solution method and a heuristic algorithm based on the Lagrangian relaxation to calculate the optimal values of the replenishment cycle, the two products’ prices, and the shortage levels with and without the aforementioned constraints. To conclude, we applied the solution method and the algorithm to solve a numerical example and completed a sensitivity analysis of the model based on the results.

In the following, we explain our implications and managerial insights from our results (Sect. 6.1). Some limitations of this study are illustrated accordingly (Sect. 6.2). Finally, potential future research directions are recommended (Sect. 6.3).

Implications and managerial insights

In light of the findings of the study, some managerial insights can be made as follows: The purpose of this study is firstly to conceptually transform a traditional inventory system into a pricing-inventory problem that is accompanied by a concern for sustainability. As a result, the effects of shortages on the optimal cycle of replenishment were particularly high when budget, capacity, and environmental and social constraints all had the same impact. As perishable products need to be kept in refrigeration units to be kept in the inventory systems, they consume a great deal of energy. Therefore, our model finds a solution for this dilemma while considering an upper bound to limit the environmental emissions during the planning horizon. Although modern inventory systems consume fewer carbon emissions, they may need more workers. However, a traditional inventory system usually has more workers in comparison with modern inventory systems. Thus, job opportunities as an important social factor have been contributed to the proposed model, and the proposed solution considers a lower bound to guarantee a minimum number of created job opportunities during the planning horizon in the proposed optimization model. As a result of our findings from an industrial example of a dairy manufacturing system, we have developed models that can be used to generate accessible guidelines for making decisions about inventory and pricing in an industry of dairy products concerning some factors like sustainability and cost of ownership.

Limitations

Since 2020, the COVID-19 pandemic recently has limited all the business networks, supply chains, and manufacturing systems concerning perishable products (Al Halbusi et al., 2022; Liu et al., 2022; Mamirkulova et al., 2022; Moosavi et al., 2022). This study did not consider the limitations of the COVID-19 pandemic resulting in several disruptions in our inventory system. Another important limitation was that one popular tool for analyzing the sustainability criteria is multi-objective decision-making which this study did not utilize it. At last but not least, in a real-life inventory system, there are several dynamic issues like demand and prices of products which are difficult to model using certain mathematics. Hence, machine learning tools can help us with the formulation and prediction of them.

Recommendation for future studies

Using the knowledge gained from this study, researchers could improve the inventory system by focusing on other aspects, such as applying discounts for complementary products in the future. While the parameters in this study were deterministic, it goes without saying that there is a significant amount of uncertainty when it comes to pricing decisions and demand functions of perishable goods. As a result, one suggestion would be to predict such data through a machine learning method to improve the accuracy of our optimal cycles by predicting the data. In the case of the demand function, this can be assumed to be a probabilistic function, and stochastic optimization methods can be applied to solve the resulting model. It is possible to include lead times in both deterministic and stochastic models of the inventory system. A constrained replenishment rate also can be incorporated into the proposed model. As a future extension of the presented model, considering the relationship between products, including substitutions or competitions, is another future approach that could be considered. Lastly, but certainly not least, another potential direction for further research would be to consider the dynamics of parameters about time using dynamic multi-period methods.

Appendix A

To prove Theorem 1, let us introduce Theorem 2, which is already proved in Bertsekas (2009).

Theorem 2

$F(X)$ is strictly concave if and only if $X\times H\times X^{\mathrm{T}}<0$, where $H$ is the Hessian matrix of F(X).

By applying the above theorem to the revenue function of the first proposed model (Eq. 10), we have: $F\left(X\right)=R\left(T,,,p_1,,,p_2,,,b_1,,,b_2\right),$

$$H=\left[\begin{array}{ccc}\frac{{\partial }^{2}R}{\partial T^2}& \frac{{\partial }^{2}R}{\partial T\partial p_1}& \frac{{\partial }^{2}R}{\partial T\partial p_2}\\ \frac{{\partial }^{2}R}{\partial p_1\partial T}& \frac{{\partial }^{2}R}{\partial p_1^2}& \frac{{\partial }^{2}R}{\partial p_1\partial p_2}\\ \frac{{\partial }^{2}R}{\partial p_2\partial T}& \frac{{\partial }^{2}R}{\partial p_2\partial p_1}& \frac{{\partial }^{2}R}{\partial p_2^2}\end{array}\right],X=\left(\begin{array}{ccc}T& p_1& p_2\end{array}\right),\mathit{and}{X}^{\mathrm{T}}=\left(\begin{array}{c}T\\ p_1\\ p_2\end{array}\right)$$

So, we need to show $\left[T,,,p_1,,,p_2,,,b_1,,,b_2\right]\times H\times {[T,p_1,p_2,b_1,b_2]}^{\mathrm{T}}<0$, and consequently, the following inequality should be met:

$$p_2\left(\frac{1}{2},s,T,(\theta {\acute{c}}_1{g}_1+{\acute{c}}_2{g}_2+\theta h_1+h_2),-,2,s,\theta ,p_1,-,2,s,p_2\right)+p_1(\frac{1}{2}sT({\acute{c}}_1{g}_1+\theta {\acute{c}}_2{g}_2+h_1+\theta h_2)-2s{p}_1-2s\theta p_2)+T(-\frac{2(A_1+A_2)}{T^2}+\frac{1}{2}s({\acute{c}}_1{g}_1+\theta {\acute{c}}_2{g}_2+h_1+\theta h_2)p_1+\frac{1}{2}s\left(\theta ,{\acute{c}}_1,g_1,+,{\acute{c}}_2,g_2,+,\theta ,h_1,+,h_2\right)p_2)\le 0$$ 47

After simplifying the arithmetic expressions, we have:

$$\frac{1}{T}(-2A_1-2A_2+sT((T(e_1{g}_1+\theta e_2{g}_2+h_1+\theta h_2)-2p_1)p_1+(T(\theta e_1{g}_1+e_2{g}_2+\theta h_1+h_2)-4\theta p_1)p_2-2p_2^2))\le 0$$ 48

Since $T>0$, inequality $sT((T({\acute{c}}_1{g}_1+\theta {\acute{c}}_2{g}_2+h_1+\theta h_2)-2p_1)p_1+(T(\theta {\acute{c}}_1{g}_1+{\acute{c}}_2{g}_2+\theta h_1+h_2)-4\theta p_1)p_2-2p_2^2))<2{(A}_1+A_2)$ is equivalent to (48). It means that $X\times H{\times X}^{\mathrm{T}}<0,$ and the revenue function (Eq. 10) is strictly concave if and only if the above inequality is satisfied. Therefore, the proof of Theorem 1is complete.

The necessary and sufficient conditions for the concavity of the second type of the proposed model can be obtained in the same way.

Appendix B

To show the concavity of the profit function for the numerical example (R), we apply Theorem 2:

Let H be the Hessian matrix for the example discussed in Sect. 5.1:

$$H=\left|\begin{array}{cccc}\underset{1}{\underbrace{\frac{{\partial }^{2}R}{\partial T^2}}}\hfill & \underset{2}{\underbrace{\frac{{\partial }^{2}R}{\partial T\partial p_1}}}& \underset{3}{\underbrace{\frac{{\partial }^{2}R}{\partial T\partial p_2}}}& \underset{4}{\underbrace{\frac{{\partial }^{2}R}{\partial T\partial b_1}}}\\ \underset{5}{\underbrace{\frac{{\partial }^{2}R}{\partial p_1\partial T}}}\hfill & \underset{6}{\underbrace{\frac{{\partial }^{2}R}{\partial p_1^2}}}& \underset{7}{\underbrace{\frac{{\partial }^{2}R}{\partial p_1\partial p_2}}}& \underset{8}{\underbrace{\frac{{\partial }^{2}R}{\partial p_1\partial b_1}}}\\ \underset{9}{\underbrace{\frac{{\partial }^{2}R}{\partial p_2\partial T}}}\hfill & \underset{10}{\underbrace{\frac{{\partial }^{2}R}{\partial p_2\partial p_1}}}& \underset{11}{\underbrace{\frac{{\partial }^{2}R}{\partial p_2^2}}}& \underset{12}{\underbrace{\frac{{\partial }^{2}R}{\partial p_2\partial b_1}}}\\ \underset{13}{\underbrace{\frac{{\partial }^{2}R}{\partial b_1\partial T}}}\hfill & \underset{14}{\underbrace{\frac{{\partial }^{2}R}{\partial b_1\partial p_1}}}& \underset{15}{\underbrace{\frac{{\partial }^{2}R}{\partial b_1\partial p_2}}}& \underset{16}{\underbrace{\frac{{\partial }^{2}R}{\partial b_1^2}}}\end{array}\right|$$ 49

where

$$1=\frac{2(A_1+A_2)(-a_1+s(p_1+\theta p_2))-b_1^2({\acute{c}}_1{g}_1+h_1+{\pi }_1)}{T^3(a_1-s(p_1+\theta p_2))}$$

$$2=\frac{s(T^2{a}_1^2({\acute{c}}_1{g}_1+h_1+\theta h_2)-2s{T}^2{a}_1({\acute{c}}_1{g}_1+h_1+\theta h_2)(p_1+\theta p_2)+s^2{T}^2({\acute{c}}_1{g}_1+h_1+\theta h_2){(p_1+\theta p_2)}^2+b_1^2({\acute{c}}_1{g}_1+h_1+{\pi }_1))}{2T^2{(a_1-s(p_1+\theta p_2))}^2}$$

$$3=\frac{s(T^2{a}_1^2(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2)-2s{T}^2{a}_1(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2)(p_1+\theta p_2)+s^2{T}^2(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2){(p_1+\theta p_2)}^2+\theta b_1^2({\acute{c}}_1{g}_1+h_1+{\pi }_1))}{2T^2{(a_1-s(p_1+\theta p_2))}^2}$$

$$4=\frac{b_1({\acute{c}}_1{g}_1+h_1+{\pi }_1)}{T^2(a_1-s(p_1+\theta p_2))}$$

$$5=\frac{s(T^2{a}_1^2({\acute{c}}_1{g}_1+h_1+\theta h_2)-2s{T}^2{a}_1({\acute{c}}_1{g}_1+h_1+\theta h_2)(p_1+\theta p_2)+s^2{T}^2({\acute{c}}_1{g}_1+h_1+\theta h_2){(p_1+\theta p_2)}^2+b_1^2({\acute{c}}_1{g}_1+h_1+{\pi }_1))}{2T^2{(a_1-s(p_1+\theta p_2))}^2}$$

$$6=\frac{s(-s{b}_1^2({\acute{c}}_1{g}_1+h_1)-2T{(a_1-s(p_1+\theta p_2))}^3-s{b}_1^2{\pi }_1)}{T{(a_1-s(p_1+\theta p_2))}^3}$$

$$7=\frac{s\theta (-s{b}_1^2({\acute{c}}_1{g}_1+h_1)-2T{(a_1-s(p_1+\theta p_2))}^3-s{b}_1^2{\pi }_1)}{T{(a_1-s(p_1+\theta p_2))}^3}$$

$$8=-\frac{s{b}_1({\acute{c}}_1{g}_1+h_1+{\pi }_1)}{T{(a_1-s(p_1+\theta p_2))}^2}$$

$$9=\frac{s(T^2{a}_1^2(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2)-2s{T}^2{a}_1(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2)(p_1+\theta p_2)+s^2{T}^2(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2){(p_1+\theta p_2)}^2+\theta b_1^2({\acute{c}}_1{g}_1+h_1+{\pi }_1))}{2T^2{(a_1-s(p_1+\theta p_2))}^2}$$

$$10=\frac{s\theta (-s{b}_1^2({\acute{c}}_1{g}_1+h_1)-2T{(a_1-s(p_1+\theta p_2))}^3-s{b}_1^2{\pi }_1)}{T{(a_1-s(p_1+\theta p_2))}^3}$$

$$11=\frac{s(-s{\theta }^2{b}_1^2({\acute{c}}_1{g}_1+h_1)-2T{(a_1-s(p_1+\theta p_2))}^3-s{\theta }^2{b}_1^2{\pi }_1)}{T{(a_1-s(p_1+\theta p_2))}^3}$$

$$12=-\frac{s\theta b_1({\acute{c}}_1{g}_1+h_1+{\pi }_1)}{T{(a_1-s(p_1+\theta p_2))}^2}$$

$$13=\frac{b_1({\acute{c}}_1{g}_1+h_1+{\pi }_1)}{T^2(a_1-s(p_1+\theta p_2))}$$

$$14=-\frac{s{b}_1({\acute{c}}_1{g}_1+h_1+{\pi }_1)}{T{(a_1-s(p_1+\theta p_2))}^2}$$

$$15=-\frac{s\theta b_1({\acute{c}}_1{g}_1+h_1+{\pi }_1)}{T{(a_1-s(p_1+\theta p_2))}^2}$$

$$16=-\frac{{\acute{c}}_1{g}_1+h_1+{\pi }_1}{T(a_1-s(p_1+\theta p_2))}$$

Computing $X\times H\times X^{\mathrm{T}}$ and after simplifying the arithmetic expressions, we have:

$$\frac{1}{T{(a_1-s(p_1+\theta p_2))}^3}(a_1^3(-2A_1-2A_2+sT((T({\acute{c}}_1{g}_1+h_1+\theta h_2)-2p_1)p_1+(T(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2)-4\theta p_1)p_2-2p_2^2))+3s{a}_1^2(p_1+\theta p_2)(2A_1+2A_2+sT(p_1(-T({\acute{c}}_1{g}_1+h_1+\theta h_2)+2p_1)-(T(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2)-4\theta p_1)p_2+2p_2^2))+s^3{(p_1+\theta p_2)}^3(2A_1+2A_2+sT(p_1(-T({\acute{c}}_1{g}_1+h_1+\theta h_2)+2p_1)-(T(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2)-4\theta p_1)p_2+2p_2^2))-s{a}_1(p_1+\theta p_2)(3s(p_1+\theta p_2)(2A_1+2A_2+sT(p_1(-T({\acute{c}}_1{g}_1+h_1+\theta h_2)+2p_1)-(T(\theta {\acute{c}}_1{g}_1+\theta h_1+h_2)-4\theta p_1)p_2+2p_2^2))+b_1^2({\acute{c}}_1{g}_1+h_1+{\pi }_1)))$$

By substituting the parameters, it is straightforward to show that $X\times H\times X^{\mathrm{T}}<0,$ the associated profit function (R) is therefore concave.

Appendix C

The optimum closed formulas for decision variables of the example presented in Sect. 5 are as follows:

$${F(T}^{\ast })=\frac{1}{8T^2{({\acute{c}}_1{g}_1+h_1+{\pi }_1)}^2}(8A_1{({\acute{c}}_1{g}_1+h_1+{\pi }_1)}^2+8A_2{({\acute{c}}_1{g}_1+h_1+{\pi }_1)}^2+T^2({({\acute{c}}_1{g}_1+h_1)}^2{h}_2(-2a_2+s(2\theta C_1+2C_2+T{h}_2))+2({\acute{c}}_1{g}_1+h_1)(-(a_1-s(C_1+\theta C_2))({\acute{c}}_1{g}_1+h_1)+(-2a_2+s(2\theta C_1+2C_2+T\theta ({\acute{c}}_1{g}_1+h_1)))h_2+sT{h}_2^2){\pi }_1+(({\acute{c}}_1{g}_1+h_1)(-2a_1+s(2C_1+2\theta C_2+T{\acute{c}}_1{g}_1+T{h}_1))+2(-a_2+s(C_2+\theta (C_1+T{\acute{c}}_1{g}_1+T{h}_1)))h_2+sT{h}_2^2){\pi }_1^2))$$ 50

$$p_1^{\ast }=\frac{-2\left(a_1,+,s,C_1,-,\theta ,\left(a_2,+,s,\theta ,C_1\right)\right)\left({\acute{c}}_1,g_1,+,h_1\right)+\left(-,2,a_1,+,2,\theta ,a_2,+,s,\left(-,1,+,{\theta }^2\right),\left(2,C_1,+,T,{\acute{c}}_1,g_1,+,T,h_1\right)\right){\pi }_1}{4s\left(-,1,+,{\theta }^2\right)\left({\acute{c}}_1,g_1,+,h_1,+,{\pi }_1\right)}$$ 51

$$p_2^{\ast }=\frac{2\theta a_1-2a_2+s\left(-,1,+,{\theta }^2\right)\left(2,C_2,+,T,h_2\right)}{4s\left(-,1,+,{\theta }^2\right)}$$ 52

$$b_1^{\ast }=\frac{T\left({\acute{c}}_1,g_1,+,h_1\right)\left(-,\left({\acute{c}}_1,g_1,+,h_1\right),\left(-,2,a_1,+,s,\left(2,C_1,+,2,\theta ,C_2,+,T,\theta ,h_2\right)\right),-,\left(-,2,a_1,+,s,\left(2,C_1,+,2,\theta ,C_2,+,T,\left({\acute{c}}_1,g_1,+,h_1,+,\theta ,h_2\right)\right)\right),{\pi }_1\right)}{4{\left({\acute{c}}_1,g_1,+,h_1,+,{\pi }_1\right)}^2}$$ 53

And the expression of the company’s optimal profit is as follows:

$$R^{\ast }=\frac{1}{2}(-\frac{2\left(A_1,+,A_2\right)}{T}+T{h}_2\left(-,a_2,+,s,\left(\theta ,p_1,+,p_2\right)\right)+2\left(C_2,-,p_2\right)\left(-,a_2,+,s,\left(\theta ,p_1,+,p_2\right)\right)+2\left(C_1,-,p_1\right)\left(-,a_1,+,s,\left(p_1,+,\theta ,p_2\right)\right)-\frac{{\acute{c}}_1{g}_1{\left(-,T,a_1,+,b_1,+,s,T,\left(p_1,+,\theta ,p_2\right)\right)}^2}{T\left(a_1,-,s,\left(p_1,+,\theta ,p_2\right)\right)}-\frac{h_1{\left(-,T,a_1,+,b_1,+,s,T,\left(p_1,+,\theta ,p_2\right)\right)}^2}{T\left(a_1,-,s,\left(p_1,+,\theta ,p_2\right)\right)}-\frac{b_1^2{\pi }_1}{T\left(a_1,-,s,\left(p_1,+,\theta ,p_2\right)\right)})$$ 54

Authors contributions

MAE was involved in writing, original draft, and mathematical modeling and performed the experiments and data analysis. SMJMA contributed to the conceptualization, supervision, validating, review and editing. AMF-F assisted in the visualization, investigation, and data analysis.

Funding

Not applicable.

Data availability

The authors confirm that the data supporting the findings of this study are available within the article.

Availability of data and materials

The data generated in the current study are available from the first author upon a reasonable request.

Declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Ethical approval

As authors, we declare the manuscript is our original work. Thus, the arguments, ideas, points of view, innovations, and results presented in this manuscript are entirely ours, unless otherwise stated in the text.

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The other researchers’ papers if used in our work are cited.

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