Optimizing Inventory and Pricing for Substitute Products with Soft Supply Constraints

Abstract

This paper presents a profit optimization model for substitute products in a competitive, time-sensitive market with scarcity and shifting user preferences. The model maximizes profit, considering production costs and inventory maintenance. It uses a discrete choice model to represent demand, sensitivity to price, availability, and changing preferences. A two-phase PSO-type metaheuristic solution tackles the nonlinear, recursive model, efficiently managing inventories and evolving consumer preferences. The model integrates production decisions, inventories, and sales prices, considering scarcity conditions and user preferences. It uses a multinomial logit for the consumers’ demand function with soft exogenous constraints, which influence utility and change consumption preferences and choices. This research offers a tool for companies to manage stock, production, and pricing in a context where goods are substitutes, providing a new perspective on business strategy.

1. Introduction

Integrating pricing strategies with logistics decisions is crucial for optimizing operations in supply chain management. Failure to align pricing with production and inventory choices may compromise supply chain efficiency. Pricing decisions are typically made in dynamic environments, influencing production and storage choices. Demand is directly linked to the period-specific price, challenging traditional assumptions of deterministic demand models, which may result in overestimating or underestimating outcomes.

To support decisions integrating prices and demand, it is imperative to develop methods aligning with utility maximization goals. This study optimizes decision-making within a lot-sizing model with a nonlinear probabilistic endogenous demand. The proposed model incorporates multiple substitute products and integrates sales prices. The model includes the consumers’ demand function using a multinomial logit (MNL) with soft exogenous constraints. Given the substitutable nature of the analyzed assets, these constraints impact the utility, prompting changes in consumption preferences and choices, thereby generating demand dynamics that differ from those derived from an unconstrained model.

The solution methodology is a heuristic optimization using the particle swarm optimization (PSO) algorithm, which is used because of the model’s complexity and nonlinearity. Given the characteristics of the multinomial logit (MNL) model with constraints, the system of equations representing the first-order conditions of optimality formed a fixed-point system of equations; this requires a recurring calculation that involves numerically estimating stock in each period. The described methodology was implemented in Python, and a series of experiments across various periods and two substitute products were successfully solved.

This research aims to optimize the management of substitute products, addressing storage, production, and pricing decisions in dynamic environments with production constraints while accounting for demand responsive to market characteristics.

In this context, the research highlights are as follows:

• The proposed approach quantifies the impact of capacity constraints and resource availability on customer choices, revealing changes in purchasing decisions when products are unavailable.
• The proposal defines optimal price schemes, accounting for resource capacity and availability, a dimension not analyzable with an unconstrained multinomial logit (MNL) model.
• The approach determines optimal production and warehousing, accommodating changes in customer decisions resulting from constraints in the discrete choice model.
• The comprehensive model integrates price, production, and storage decisions, optimizing profit in a context where customers alter choices due to product unavailability.

2. Literature Review

The literature review serves three primary objectives. First, it demonstrates the significance and diversity of works integrating production decisions, inventories, and pricing. Second, it explores the relationship between multinomial logit-type discrete choice models and pricing estimation models across different contexts. Third, it provides an overview of the mathematical modeling contributions of the constrained logit discrete choice model.

Both static and dynamic environments are considered when examining integration in production, inventory, and pricing decision-making. In a static context, Burwell et al. [1] discuss the effects on price and lot size, incorporating parameters such as quantity discounts and freight under linear demand. Chang [2] extends this work to analyze cross discounts on the same variables. Additionally, Taleizadeh et al. [3] consider the impact of carbon dioxide emissions on these decisions. In a dynamic context, seminal works by Thomas [4] and Wagner and Whitin [5] highlight the variability of optimal prices across different periods, independent of each other. These works involve mathematical models and algorithms addressing simultaneous price and production decisions for a single product with a deterministic demand function, excluding pending orders.

Zhang et al. [6] developed an analytical model informing pricing, promotion, and inventory decisions for a single item within a finite horizon, focusing on profit maximization through periodic reviews. Dong et al. [7] explored dynamic pricing policies and inventory control for substitute products in a retailer’s long-term supply with short sales periods, using a multinomial logit discrete choice model for consumer demand. Smith et al. [8] addressed joint planning optimization of inventories, production, and pricing for a single item with capacity and inventory limitations, maximizing profits in a discrete multiperiod horizon. Avinadav et al. [9] proposed and analyzed two mathematical models for optimal pricing, order quantity, and replacement period, considering demand functions separable into price and age components. Building on these works, some authors extended the analysis to incorporate reference prices in inventory and pricing dynamics, assuming positive consumer valuation of price changes over time [10,11,12,13].

A highly accepted demand estimation model used in marketing to analyze consumer purchasing preferences is a discrete choice model known as multinomial logit (MNL) and its extensions. These models, rooted in microeconomic utility maximization theory, have been integrated into various theoretical and applied studies. In the realm of logistics and production decisions, the joint consideration of sale prices for one or more products has been explored by researchers such as Lüer-Villagra and Marianov [14], and Wang et al. [15]. Yaghin et al. [16] proposed a multistage model incorporating lot-sizing, pricing, and coordination decisions, utilizing a multinomial logit demand function. Terzi et al. [17] focused on pricing and batch size decisions for a manufacturer selling a single product across different channels, aiming to design an optimal production plan and prices. Adeinat et al. [18] studied supply chain coordination using a nonlinear integer optimization model, emphasizing the importance of considering varying sales prices for geographically dispersed retailers with distinct cost structures and marketing strategies. Their findings suggest that models with variable sales prices generate higher profits than those with uniform sales prices, and they propose a simulated annealing (SA) algorithm for solution optimization.

An important assumption in classic multinomial logit (MNL) models is the compensatory behavior of decision-makers, implying a decision strategy that permits exchange or variation between attribute values to maintain a fixed level of utility [19,20]. These models, however, do not accommodate thresholds or limits on attributes, such as price, and typically overlook the impact of product shortages or competition among consumers for limited units. In practical scenarios where unbounded assumptions are unrealistic, there is a need to curtail or penalize specific alternatives to align with system and consumer constraints.

Building on the works of Swait and Adamowicz [21] and Cascetta and Papola [22], Martínez et al. [23] propose a constrained multinomial logit (MNL) heuristic model. This model incorporates the utility function of consumers and endogenous constraints of the system, including capacity or the supply of goods and services. The assumption is that the utility can be separated into compensatory and non-compensatory terms, indicating the feasibility or availability of each alternative. Cutoff or logarithmic penalty functions are utilized to achieve this, which can be imposed on individual attributes or system conditions. These penalty functions facilitate a smooth transition between the feasible compensatory space and the infeasible non-compensatory space, allowing for slight breaches of certain constraints.

Recent applications of the constrained multinomial logit (CMNL) model span various domains, including the choice of transportation mode [24], school selection [25], urban economy [26,27], parking pricing [28], location modeling [29], renewable energy development [30], and modeling path choices in urban rail transit networks [31]. While the CMNL has been employed in pricing decisions [32,33], primarily analyzing the impact of constraints on the maximum availability of payment to different consumers, none of the models incorporate the influence of exogenous constraints or the system in a dynamic or intertemporal environment of integrated inventory management with pricing.

Demand models based on CMNL become fixed-point multidimensional problems when incorporating system constraints such as capacities or supply. This is because one agent’s purchase decision influences others’ decisions and can alter the preferences of those who do not have access to a product due to its shortage.

Based on the literature review, the primary contributions of this study revolve around quantifying the impact of capacity constraints and resource availability on customer choices, thereby revealing and incentivizing changes in purchasing decisions when products are unavailable. The model establishes optimal prices within an intertemporal framework, considering resource capacity and availability, a dimension typically overlooked in consumer valuations (unconstrained multinomial logit model). The approach determines optimal production and warehousing strategies, accommodating shifts in customer decisions due to constraints in the discrete choice model. This comprehensive model integrates price, production, and storage decisions to optimize profit in a context where customer choices are influenced by product unavailability.

Moreover, due to the intertemporal nature of the model, it is solved using integrated PSO algorithms. The advantage of this approach lies in its ability to adapt the classic PSO to address subproblems within each iteration. This adaptation is necessary because estimating demand relies on a fixed-point problem, which lacks an explicit algebraic expression. Therefore, PSO serves as a suitable alternative for implementing these modifications, which are unfeasible within a traditional nonlinear solver.

3. Mathematical Model

The model represents a company that maximizes the total profit by defining prices $P_{it}$, inventory $S_{it}$, and production $X_{it}$ of $i\in I$ products in $t\in T$ periods. The demand is modeled with a multinomial discrete choice model in which consumers perceive a utility $V_{it}$ for acquiring a product to Equation (1):

$$V_{it}\left(P_{it}\right)=I_{it}-\beta ·P_{it}$$

(1)

where, $\beta$ is the marginal utility of consumer income, $P_{it}$ is the product i price in period t in T, and $I_{it}$ is product attractiveness. The attractiveness ($I_{it}$), is the value of a product, that refers to consumers’ valuation of its attributes or characteristics (quality, design, brand). It is a measure of how desirable or interesting the product or service is to the target market compared to other available alternatives.

In a multinomial logit (MNL) context, if there are $M_t$ consumers in the market in period $t \in T$ and if $V_0$ is the nonpurchase utility, then the demand $D_{it}$ can be expressed as Equation (2), which is the logit probability of acquiring product $i \in I$ multiplied by the market size $M_t$.

$$D_{it}\left(P_{1t},\dots ,P_{1t}\right)=D_{it}\left({\stackrel{⃑}{P}}_t\right)=M_t·\frac{e^{V_{it}\left(P_{it}\right)}}{V_0+{\sum }_{k=1}^I{e}^{V_{kt}\left(P_{kt}\right)}}$$

(2)

The soft constraints within the MNL are represented with cutoffs, which are ‘s’-shaped functions as in Equations (3) and (4),

$${\varphi }_{{Prod}_{it}}=\frac{1}{1+e^{{{\omega }_{Prod}}_i\left(X_{it}-m_{Cap}-{\rho }_{{Prod}_i}\right)}}$$

(3)

$${\varphi }_{{Stock}_{it}}=\frac{1}{1+e^{{{\omega }_{Stock}}_i\left({\sum }_{i=1}^I{S}_{it}-k_{Cap}-{\rho }_{{Stock}_i}\right)}}$$

(4)

$${\rho }_{{Prod}_i}=\frac{1}{{{\omega }_{Prod}}_i}·ln \left(\frac{1-{\eta }_{Prod}}{{\eta }_{Prod}}\right)$$

(5)

$${\rho }_{{Stock}_i}=\frac{1}{{{\omega }_{Stock}}_i}·ln \left(\frac{1-{\eta }_{Stock}}{{\eta }_{Stock}}\right)$$

(6)

where $m_{Cap}$ and $k_{Cap}$ represent the maximum production and storage levels, respectively. The other parameters of these cutoff functions are the scale parameters ${\omega }_{{Stock}_i}$ and ${\omega }_{{Prod}_i}$ of the quantity stored and produced, respectively. ${\eta }_{Stock}$ and ${\eta }_{prod}$ indicate the proportion of individuals not accomplishing the corresponding storage and production constraint, estimated according to Equations (5) and (6).

Cutoffs (3) and (4) are soft constraints of production and storage, aiming to ensure that the quantity produced does not exceed the maximum capacity ($X_{it}\le m_{Cap} \forall i,t$) and that the storage does not exceed a limit (${\sum }_{i=1}^I{S}_{it}\le k_{Cap}, \forall t$). Additionally, as detailed by López-Ospina et al. [29] and Pérez and López-Ospina [33], the purpose of smoothed functions is not only to allow penalizing an alternative based on a specific attribute but also to demonstrate the impact on consumers of including and valuing endogenous constraints that affect their utility function due to a shortage of products in the market.

Including these cutoffs, the MNL turns into a constrained MNL (CMNL), and the demand $D_{it}$ in (2) now is expressed as in (7). All the parameters of CMNL models can be estimated using the methodologies described in Martínez et al. [23] and Castro et al. [24], which provide guidelines and insights to perform maximum likelihood estimations specialized in CMNL.

$$D_{it}\left({\stackrel{⃑}{P}}_t,{\stackrel{⃑}{X}}_t,{\stackrel{⃑}{S}}_t\right)=M_t·\frac{{\varphi }_{{Prod}_{it}}·{\varphi }_{{Stock}_{it}}·e^{V_{it}}}{V_0+{\sum }_{k=1}^I{\varphi }_{{Prod}_{kt}}·{\varphi }_{{Stock}_{kt}}·e^{V_{kt}}}$$

(7)

If the production and storage costs are $C_{it}$ and $H_{it}$, respectively, then the complete model is developed in expressions (8)–(10).

$$\max \sum _{i\in I}{\displaystyle \sum _{t=1}^T}\left(P_{it}\cdot D_{it}\left({\stackrel{⃑}{P}}_t,{\stackrel{⃑}{X}}_t,{\stackrel{⃑}{S}}_t\right)-C_{it}\cdot X_{it}-H_{it}\cdot S_{it}\right)$$

(8)

s.t.

$$S_{it}=S_{it-1}+X_{it}-D_{it}\left({\stackrel{⃑}{P}}_t,X_t,{\stackrel{⃑}{S}}_t\right), \forall i\in I, \forall t\in T$$

(9)

$$D_{it}\left({\stackrel{⃑}{P}}_t,{\stackrel{⃑}{X}}_t,{\stackrel{⃑}{S}}_t\right)=M_t·\frac{{\varphi }_{{Prod}_{it}}·{\varphi }_{{Stock}_{it}}·e^{V_{it}}}{V_0+{\sum }_{k=1}^I{\varphi }_{{Prod}_{kt}}·{\varphi }_{{Stock}_{kt}}·e^{V_{kt}}}$$

(10)

$$P_{it}, X_{it}, S_{it}\ge 0, \forall i\in I, \forall t\in T$$

The objective function (8) is the profit represented by the sum of the total income subtracting the total production and warehousing costs. Equation (9) is the dynamic balance of the product in which storage from prior periods remains available for future ones. For the initial period, the initial stock $S_{i0}$ is known for every product. Please note that constraints are handled within the discrete choice model for users, employing soft constraints. The multinomial logit (MNL) associated with the Equation (10) model ensures that consumer demand in each period and product is probabilistically determined, considering exogenous constraints on the probability of purchase. This means that the demand, representing what will be sold, adapts to constraints through these cutoff functions.

4. Solution Method

The particle swarm optimization (PSO) algorithm is used specifically for optimizing the whole model. The reason is because it is nonlinear and requires a method capable of handling this complexity through specially tailored modifications. The PSO heuristic was adapted with a fixed point to solve the model. The focus was not on contributing to the field of heuristics but rather on using a known method that allowed for modifications, including the inclusion of the fixed point. Traditional solvers were ruled out due to the high nonlinearity and the impossibility of having an algebraic expression due to the fixed point. The solution algorithm integrates two PSO-type metaheuristics. The first metaheuristic seeks the global solution of the problem, and the second aims to maintain the balanced intertemporal condition in each period and product. This approach allows us to calculate optimal storage using cutoff capacity constraints and to estimate the probability of demand based on the utility of the goods. The PSO algorithm achieved convergence effectively, making it an appropriate choice for our optimization needs. The algorithm achieves convergence to solve fixed-point equations (see Figure 1).

Specific challenges are raised when implementing a fixed-point system for each period within the PSO scheme. Inspired by bird flock behavior, PSO integrates group dynamics to influence individual decisions. For a comprehensive understanding, refer to Shi [34], Wang et al. [15], Kilichev and Kim [35], and Zhang et al. [36]. The PSO process involves creating a particle swarm, evaluating them in the objective function, and iteratively updating positions and velocities.

$${FO}_{Fixed-point,t}={\displaystyle \sum _{i=1}^I}{{(S}_{it}-(S_{it-1}+X_{it}-D_{it}\left(S_{it}, X_{it}, P_{it}\right)))}^2$$

(11)

The stored quantity is calculated using fixed-point equations for each particle with known production and price. This calculation considers the amount produced, prices, and the demand for the current period. Therefore, the constraint of the balance of the product between periods arises (9), which is recursive. Note that the calculation of the demand also depends on production, and at the same time, the storage value depends on all the available products. Hence, to obtain the whole set of solutions $S_{it}$, the following optimization problem is solved at each period. To solve (11), one can use the PSO method again, or any other library suited for numerical optimization because these inner problems are quite easy to solve. For each solution evaluated at each iteration $k$ of the main PSO to solve the whole problem, it is necessary to find an optimal solution for inventory flow. Please see this inner methodology in Figure 2.

In this context, when a solution for iteration $k$ is ready, then a classic PSO is performed at each step to solve the main problem. Let ${\stackrel{⃑}{Y}}_m^{\left(k\right)}$ the matrix of decision variables of the whole problem at iteration $k$ for particle $m$. Then, by following the classic PSO procedure (see Equations (13) and (14)), it is possible to optimize the whole problem. See Appendix A for details of the PSO parameters.

$${\stackrel{⃑}{Y}}_m^{\left(k\right)}=\left[\begin{array}{c}{\stackrel{⃑}{X}}_{m1}^{\left(k\right)},\dots ,{\stackrel{⃑}{X}}_{mT}^{\left(k\right)}\\ {\stackrel{⃑}{P}}_{m1}^{\left(k\right)},\dots ,{\stackrel{⃑}{P}}_{mT}^{\left(k\right)}\\ {\stackrel{⃑}{S}}_{m1}^{\left(k\right)},\dots ,{\stackrel{⃑}{S}}_{mT}^{\left(k\right)}\end{array}\right]$$

(12)

$${\stackrel{⃑}{Y}}_m^{\left(k+1\right)}={\stackrel{⃑}{Y}}_m^{\left(k\right)}+{\stackrel{⃑}{Vel}}_m^{\left(k\right)}$$

(13)

$${\stackrel{⃑}{Vel}}_m^{\left(k+1\right)}=w{\stackrel{⃑}{Vel}}_m^{\left(k\right)}+c_1{r}_1\left({\stackrel{⃑}{P}}_{best}^{\left(k\right)}-{\stackrel{⃑}{Y}}^{\left(k\right)}\right)+c_2{r}_2\left({\stackrel{⃑}{G}}_{best}^{\left(k\right)}-{\stackrel{⃑}{Y}}^{\left(k\right)}\right)$$

(14)

where

${\stackrel{⃑}{Vel}}_m^{\left(k\right)}:$ velocity of particle m in dimension d at iteration (k).

$w$ is the inertia weight that controls the influence of the previous velocity.

$c_1$ and $c_2$ are acceleration coefficients that control the influence of the best-known position of the particle itself (${\stackrel{⃑}{P}}_{best}^{\left(k\right)}$) and the best-known global position, respectively, (${\stackrel{⃑}{G}}_{best}^{\left(k\right)}$).

$r_1$ and $r_2$ are random values between 0 and 1.

${\stackrel{⃑}{P}}_{best}^{\left(k\right)}$ is the best historical position of particle m.

${\stackrel{⃑}{G}}_{best}^{\left(k\right)}$ is the best historical global position in dimension.

${\stackrel{⃑}{Y}}_m^{\left(k\right)}$: position of particle m at iteration k.

${\stackrel{⃑}{Y}}_m^{\left(k+1\right)}$ is the new position of particle m at iteration k+1.

5. Results Analysis

Considering the characteristics of the proposed model, the analysis and comparison have to consider the following:

• Nonlinearity and Probabilistic Demand: The model is non-linear and incorporates a demand representation that deviates significantly from classical deterministic lot sizing and inventory management instances. Traditional models often assume deterministic demand, whereas our model addresses probabilistic, non-fixed demand, adding complexity.
• Substitution Effects Representation: The model utilizes a logit function to represent substitution effects, further complicating the problem. This approach creates a non-algebraic structure, making it unsuitable for traditional solvers.
• Fixed Point Inclusion: This approach requires a specialized solution method distinct from conventional models to include a fixed point.

Given these unique features, a validation approach is required. Instead of a comparison with the existing literature (remember, this is a novel model), a baseline scenario where no differences exist between products is used. From this symmetric and straightforward starting point, variations to explore the model’s recommendations were introduced to emulate when decision-makers face challenges in offering consumers the most attractive product or service. This methodology permits the highlighting of the novel aspects and contributions of the model while reassuring you of its practical applicability in real-world scenarios.

The objective is to analyze how the proposed model handles scarcity and overproduction when a product has higher attractiveness for consumers than the alternative. Scenarios are defined by varying values for attractiveness ($I_{it}$), maximum capacities ($k_{cap}$, ${m_{cap}}_i$), costs ($C_{it}$ and $H_{it}$), price sensitivity ($\beta$), market size ($H_t$) and nonpurchase utility ($V_0$). The model is tested in a time horizon of five periods ($n_T=5$) and for two products ($n_{PROD}=2$). The values presented are obtained for the objective function value, prices $P_{it}$ and quantities produced ($X_{it}$) and stored ($S_{it}$). Additionally, the demand values are presented for the cases described and the respective convergence graphs for the PSO heuristics.

In this study, the testing scenario considers a time horizon of five periods and two substitute products to effectively illustrate the model’s capabilities while maintaining a balance between complexity and clarity.

• Number of Products (Substitutes)—The primary objective of this research is to develop a robust modeling tool for handling substitute products. Given this focus, we found it sufficient to work with two products for several reasons:

  ○

  Illustration of Substitution Effects: With two products, one can adequately demonstrate the model’s ability to handle substitution effects and consumer preferences. This setup allows us to showcase the ranking of products based on consumer utility using the discrete choice model.

  ○

  Complexity Management: Limiting the scenario to two products helps in managing the complexity of the model. It allows us to highlight the key features and behaviors of the model without overwhelming the reader with excessive variations in product types.

  ○

  Focused Analysis: By concentrating on two products, it is possible to provide a clear and concise representation of the model’s performance, making it easier to understand and interpret the results.

• Time Horizon (Periods)—The model’s sequential nature guides inventory management decisions over time, with each period representing a discrete time step. The choice of a five-period time horizon is justified as follows:

  ○

  Dynamic Adaptation: A short time horizon allows us to demonstrate how the model adapts to changing conditions over a concise and manageable period. This approach effectively highlights the essential dynamics of inventory adjustments and decision-making processes.

  ○

  Simplified Demonstration: Analyzing a limited time horizon helps present the model’s core concepts and behaviors without introducing unnecessary complexity. It provides a clear view of how inventory levels are adjusted in response to potential availability constraints for one of the products.

  ○

  Practical Application: In practical scenarios, the model could be executed periodically in batches over a few periods, considering the overlapping nature of solutions. This periodic execution is feasible for real-world problems where updated demand and product availability information becomes available.

In summary, the choice of two products and a five-period time horizon effectively showcases the model’s capabilities while maintaining clarity and relevance. This setup ensures that the model’s essential features are highlighted, providing valuable insights into its performance.

5.1. Baseline Scenario

To validate the model, we compared it against a baseline scenario. In this scenario, we assume symmetric conditions with no differentiation between products. This baseline serves as a reference point to illustrate the impact of introducing product attributes and availability variations. By comparing the model’s performance against this controlled environment, we can effectively demonstrate its robustness and adaptability. This approach provides valuable insights into the model’s behavior and potential applications in inventory and pricing decisions for substitute products under soft supply constraints.

A case where there are two products with the same attractiveness is presented, along with costs and capacities, as shown in

Table 1. Parameters for the baseline scenario.

		Period
Parameter	Product	1	2	3	4	5
Attractiveness $I_{it}$	1 and 2	4.00	3.50	3.00	2.50	2.00
Production costs $C_{it}$	1 and 2	3.00	2.75	2.50	2.25	2.00
Storage costs $H_{it}$	1 and 2	0.05	0.04	0.03	0.02	0.01
Storage cap. $k_{cap}$	1 and 2	10.00	10.00	10.00	10.00	10.00
Production cap. $m_{cap}$	1 and 2	5.00	5.00	5.00	5.00	5.00

. Storage costs decrease as time increases. The segment size, nonpurchase utility, and marginal sensitivity to price are shown in

Table 2. MNL parameters for the baseline scenario.

MNL Parameters
Market size $M_t$	10.00
Nonpurchase utility $V_0$	4.00
Price sensitivity $\beta$	0.10

.

In the baseline scenario, the decrease in storage costs over time is due to the incorporation of a small discount rate to reflect the present value of money within the simulations. This adjustment aligns with standard financial practices, where future costs are discounted to their present value, providing a more realistic assessment of long-term expenses. By applying a discount rate, the model ensures that the economic reality of storing inventory over multiple periods is accurately represented, capturing the time value of money and its impact on storage costs.

Three trials are considered to observe the methodology’s performance in this baseline scenario.

Table 3. Objective function—firm’s profit and deviations from average for three trials.

Objective Function and Deviations
	Total
Average	83.14
Trial 1	1.67%
Trial 2	−0.57%
Trial 3	−1.10%

shows the profit and the percentual deviations from the average value.

Table 4. Objective function value for three trials and each period—the closer to zero, the lower the error.

	Objective Function Values for the Inner Problems
Trials	$\mathit{t}=1$	$\mathit{t}=2$	$\mathit{t}=3$	$\mathit{t}=4$	$\mathit{t}=5$
1	1.718 × 10−5	1.542 × 10−6	1.245 × 10−6	5.449 × 10−5	1.100 × 10−6
2	1.055 × 10−3	1.163 × 10−3	1.124 × 10−5	1.114 × 10−5	1.144 × 10−6
3	7.114 × 10−5	3.207 × 10−5	7.081 × 10−5	5.224 × 10−5	8.536 × 10−5

contains the objective function for the inner problems for each period; this problem represents the calculation of the fixed-point equation in (11); considering this quantity is modeled as an error, the closer to zero, the more accurate the PSO solution.

Figure 3 shows the convergence curves for the second trial of PSO of the main problem to the left, and on the right, they show the convergence for the inner PSO to find storage values. As one can observe, the convergence of the inner problems is quite fast, because the problem is easy to solve, while the main PSO takes more iterations to converge. The results of other trials can be observed in Appendix B.

Table 5. Decision variables and demands for trial 2 and baseline scenario.

Decision Variables and Demands for Trial 2
		$\mathit{t}=1$	$\mathit{t}=2$	$\mathit{t}=3$	$\mathit{t}=4$	$\mathit{t}=5$	Total	Average
$P_{it}$	$i=1$	$7.415$	$7.373$	$7.446$	$7.469$	$7.465$	-	$7.434$
	$i=2$	$7.446$	$7.491$	$7.448$	$7.404$	$7.373$	-	$7.432$
$X_{it}$	$i=1$	$2.178$	$2.121$	$2.301$	$2.975$	$0.479$	$10.054$	$2.011$
	$i=2$	$2.202$	$2.105$	$3.765$	$0.380$	$1.451$	$9.903$	$1.981$
$D_{it}$	$i=1$	$2.171$	$2.091$	$2.028$	$1.901$	$1.842$	$10.033$	$2.007$
	$i=2$	$2.170$	$2.088$	$1.878$	$1.935$	$1.855$	$9.926$	$1.985$
$S_{it}$	$i=1$	$0.000$	$0.000$	$0.271$	$1.363$	$0.000519$	$1.635$	$0.327$
	$i=2$	$0.000$	$0.000$	$1.889$	$0.412$	$0.00771$	$2.309$	$0.462$

,

Table 6. Attractiveness values.

$\mathbf{Attractiveness} ({\mathit{I}}_{\mathit{i}\mathit{t}}$)
Period	Product N°1	Product N°2
1	4.0	6.0
2	3.5	6.25
3	3.0	5.5
4	2.5	4.75
5	2.0	4.0

and

Table 7. Limited production capacity—Scenario No. 1.

Capabilities	Product No. 1	Product No. 2
Production capacity (${m_{cap}}_i$)	5	2

present the values of prices ($P_{it}$), production ($X_{it}$), demand ($D_{it}$), and stock ($S_{it}$) for the three trials, covering the entire time horizon.

In summary, the methodology demonstrates satisfactory performance as evidenced by nearly identical results across three trials, numerical proximity of errors for the inner PSO methodology to zero, and convergence in both the main and inner PSO. In this baseline case, the outcomes are deemed satisfactory.

5.2. Scenario 1: Analyzing Substitution by Constraining More Attractive Products through Adjustments in Costs and Capacities

This section analyzes the behavior of consumers for more attractive products to illustrate and consider the attractiveness indicated in Table 6. However, this product is limited in terms of production capacity or high costs, generating potential shortages of this good. Likewise, two markets are examined as follows: one in which consumers are considerably sensitive to a change in price and one in which they do not respond to a change in price.

The same holding costs ($H_{it}$) and inventory capacity ($k_{cap}$) as those indicated in Table 1 are used. Regarding the market, the same market size ($M_t)$ and the nonpurchase utility ($V_0$) are considered. This is indicated in Table 2.

In the following subsections, we first analyzed when a product is more attractive and when its production is limited by high costs and low capacities. Secondly, the cases with high- and low-price sensitivities are analyzed.

5.2.1. High Production Costs and Low Production Capacity in a Highly Price-Sensitive Market

For both cases, high production costs and low production capacity, the impact of consumers’ marginal utility of income (or price sensitivity) on the model results is investigated. This is achieved by increasing price sensitivity ($\mathsf{\beta }$) to 0.15:

• For the case of high production costs, an adjustment in the production costs of product 2, as outlined in

  Table 8. Production costs for Scenario 1.

  $\mathbf{Production} \mathbf{Costs} ({\mathit{C}}_{\mathit{i}\mathit{t}}$)
  Period	Product 1	Product 2
  1	3.00	5.00
  2	2.75	4.75
  3	2.50	4.50
  4	2.25	4.25
  5	2.00	4.00

  , is performed. The production capacity is the same for both products (${m_{cap}}_i=5, \forall i\in \left\{1, 2\right\}$) as in the baseline scenario.
• For the case of low production capacity, it is modified for the most attractive good (Table 7).

The results are presented when constraining the production capacity or increasing the costs of the most attractive good compared to the baseline scenario. Here, ${{ ML}_{\uparrow C_{Prod}}}_{ }$ and ${{ML}_{\downarrow m_{Cap}}}_{ }$ refer to the high-cost and low-capacity markets, respectively.

Table 9. Total profit objective function for Scenario 1—high sensitivity to price.

Setting	${\mathbf{U}}_{\mathbf{T}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$	${\mathbf{U}}_{\mathbf{P}\mathbf{r}\mathbf{o}\mathbf{d}\mathbf{u}\mathbf{c}\mathbf{t} 1}$	${\mathbf{U}}_{\mathbf{P}\mathbf{r}\mathbf{o}\mathbf{d}\mathbf{u}\mathbf{c}\mathbf{t} 2}$
${{{ML}_{\uparrow C_{Prod}}}_{ }}_{ }$	59.5005	31.887	27.614
${{ML}_{\downarrow m_{Cap}}}_{ }$	79.967	41.429	38.538

provides the utilities obtained in each setting. Limiting the production capacity yields better margins than increasing costs, as cost variations directly affect the objective function.

Let us point out that, in both cases, the less attractive goods generate more profit for the firm than the goods that are more attractive to consumers. Likewise, the scenarios present values below the baseline.

Regarding PSO SUB, there is a high level of convergence of both scenarios, as evidenced in the values obtained in

Table 10. Fixed-point storage function—Scenario No. 1: high price sensitivity market.

	$\mathit{t}=1$	$\mathit{t}=2$	$\mathit{t}=3$	$\mathit{t}=4$	$\mathit{t}=5$
${{{ML}_{\uparrow C_{Prod}}}_{ }}_{ }$	$6.081\times {10}^{-5}$	$0.000$	$1.045\times {10}^{-6}$	$9.296\times {10}^{-5}$	$1.169\times {10}^{-6}$
${{ML}_{\downarrow m_{Cap}}}_{ }$	$1.876\times {10}^{-7}$	$0.000$	$4.763\times {10}^{-6}$	$3.276\times {10}^{-5}$	$9.843\times {10}^{-7}$

.

In

Table 11. Variables and demand calculations—Scenario No. 1 (${\mathrm{M}\mathrm{L}}_{\uparrow {\mathrm{C}}_{\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{d}}}$).

	t = 1	t = 2	t = 3	t = 4	t = 5	Total	Average
$P_{ it}$
$i=1$	$7.496$	$4.774$	$7.445$	$6.376$	$7.491$	-	$6.716$
$i=2$	$7.497$	$7.496$	$7.497$	$7.443$	$7.481$	-	$7.483$
$X_{ it}$
$i=1$	$1.915$	$1.925$	$2.599$	$1.317$	$1.395$	$9.151$	$1.830$
$i=2$	$2.838$	$2.641$	$2.513$	$3.734$	$0.706$	$12.432$	$2.486$
$D_{ it}$
$i=1$	$1.915$	$1.925$	$1.804$	$1.820$	$1.697$	$9.161$	$1.832$
$i=2$	$2.830$	$2.641$	$2.513$	$2.230$	$2.224$	$12.438$	$2.488$
$S_{ it}$
$i=1$	$0.000$	$0.000$	$0.795$	$0.302$	$0.000579$	$1.098$	$0.220$
$i=2$	$0.000$	$0.000$	$0.000754$	$1.518$	$0.00131$	$1.520$	$0.304$

. the variables are observed as production costs increase. Regarding the price ($P_{it}$) of the most attractive goods, the values are very close to the maximum $P_{max}$ because there is higher demand ($D_{it}$) for the evaluated product. For the prices of the substitute good, there are periods of considerable downturn ($T_2$ and $T_4$), which leads to a lower average than that for the substitute.

For the quantities produced, the results are very interesting. For Product No. 2, in the first three periods, production is limited to that only needed to supply demand, which is a great strategy because production costs decrease over time. Likewise, this trend can be seen for Product No. 1 in $T_1$ and $T_2$.

By focusing on the quantity demanded ($D_{it}$), Product No. 2 has higher values in all periods; therefore, despite having a higher price, a large percentage of consumers continue to purchase the good above No. 1. In total, three more units of Product No. 2 are consumed for the described scenario.

In

Table 12. Variables and demand calculations—Scenario No. 1 (${\mathbf{M}\mathbf{L}}_{\downarrow {\mathbf{m}}_{\mathbf{C}\mathbf{a}\mathbf{p}}}$).

	t = 1	t = 2	t = 3	t = 4	t = 5	Total	Average
$P_{it}$
$i=1$	$7.480$	$7.463$	$7.491$	$7.492$	$7.497$	-	$7.485$
$i=2$	$7.473$	$7.493$	$7.494$	$7.488$	$7.498$	-	$7.489$
$X_{it}$
$i=1$	$2.151$	$2.040$	$2.118$	$2.991$	$0.544$	$9.844$	$1.830$
$i=2$	$1.912$	$1.868$	$1.864$	$1.976$	$1.501$	$9.121$	$1.824$
$D_{it}$
$i=1$	$2.150$	$2.040$	$1.975$	$1.948$	$1.732$	$9.845$	$1.969$
$i=2$	$1.912$	$1.868$	$1.836$	$1.420$	$2.089$	$9.125$	$1.825$
$S_{it}$
$i=1$	$0.000$	$0.000$	$0.141$	$1.194$	$0.00677$	$1.342$	$0.268$
$i=2$	$0.000$	$0.000$	$0.0292$	$0.588$	$0.000378$	$0.621$	$0.124$

, which shows the variables in the scenario, the production capacity of Product No. 2 is limited to two units per period and is maintained for Product No. 1 at a maximum of five. First, very similar values are observed in the prices ($P_{it}$) of both products, close to $P_{max}$. Furthermore, when comparing the averages, they are practically the same.

Second, the quantities produced ($X_{it}$) of Product No. 2 are close to the maximum in all periods, except the last one. Likewise, in this scenario, there is a higher percentage of consumers who substitute their consumption due to a lack of availability, which can be seen in $T_1$, $T_2$ and $T_3$, because the quantity produced equals the demand. Regarding the production of Product No. 1, the values are higher because capacity is not as limited. Nevertheless, the maximum is not obtained because the good in question is not the most attractive.

Third, the values for total quantity demanded and produced are similar for both items due to the conservation of flow, being only 0.72 units higher for the less attractive product but without production limitations.

In the present scenario, a decrease in the utility associated with the most attractive product is expected due to increased costs. In addition, an increase in the substitution of the product by consumers is expected because it represents a highly elastic market induced by the increase in the beta parameter.

5.2.2. High Production Costs and Low Production Capacity in a Market with Low Price Sensitivity

For the cases with high production costs and low production capacity, customers with less sensitivity to a change in prices ($\beta =$ 0.05) were considered.

For the case with high production costs, the costs in Table 8 are preserved, whereas Product No. 2 is more expensive. The production capacities of both goods are those of the baseline scenario. In this case, the quantities that are demanded, and the prices that are expected to be higher than those in the previous scenario, are associated with the change in elasticity.

In the case of low production capacity, the production costs are the same (Table 1). The productive capacity of Product No. 2 is limited, as indicated in Table 7. The described scenario should maximize the production of the most attractive good. In addition, when implementing a market where beta is lower, the associated costs and the quantities demanded are expected to be higher than those of the previously described market. Additionally, a higher percentage of substitution toward the less attractive good is estimated because of the scarcity generated.

The purpose of adjusting the price sensitivity is to illustrate how the model’s recommendations shift in response to discrepancies between two substitute products, especially when the more appealing product encounters challenges in customer selection. Within this framework, scenarios are examined where the more attractive product incurs higher costs and experiences capacity constraints. The distinction between Section 5.2.1 and Section 5.2.2 lies in simulating the impact on customers with varying degrees of price sensitivity.

First, the context where the production capacity of the most attractive good. ${MBI}_{\uparrow C_{Prod}}$ is limited, presents higher utilities than do those that consider higher production costs. ${{MBI}_{\downarrow m_{Cap}}}_{ }$. This is logical because the first case does not have a direct effect on utilities as an increase in costs does (

Table 13. Objective utility function—Scenario No. 1: low price sensitivity market.

Firm’s Utility Function
	${\mathit{U}}_{\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$	${\mathit{U}}_{\mathit{P}\mathit{r}\mathit{o}\mathit{d}\mathit{u}\mathit{c}\mathit{t} 1}$	${\mathit{U}}_{\mathit{P}\mathit{r}\mathit{o}\mathit{d}\mathit{u}\mathit{c}\mathit{t} 2}$
${{MBI}_{\uparrow C_{Prod}}}_{ }$	68.074	34.735	29.339
${{MBI}_{\downarrow m_{Cap}}}_{ }$	83.491	43.836	39.655

). Likewise, the less attractive good, a product that is not limited, yields higher profits.

Table 14. Fixed-point storage function—Scenario No. 1: low price sensitivity market.

Fixed-Point Storage Function Value
	t = 1	t = 2	t = 3	t = 4	t = 5
${{MBI}_{\uparrow C_{Prod}}}_{ }$	$0.000$	$1.822\times {10}^{-3}$	$6.517\times {10}^{-5}$	$7.644\times {10}^{-5}$	$2.749\times {10}^{-6}$
${{MBI}_{\downarrow m_{Cap}}}_{ }$	$1.757\times {10}^{-5}$	$7.100\times {10}^{-6}$	$6.313\times {10}^{-7}$	$1.228\times {10}^{-5}$	$3.913\times {10}^{-7}$

shows the storage PSO fixed-point convergence.

Table 15. Variable and demand calculations—Scenario No. 1 (${\mathbf{M}\mathbf{B}\mathbf{I}}_{\uparrow {\mathbf{C}}_{\mathbf{P}\mathbf{r}\mathbf{o}\mathbf{d}}}$).

	t = 1	t = 2	t = 3	t = 4	t = 5	Total	Average
$P_{it}$
$i=1$	$6.808$	$7.463$	$7.418$	$6.376$	$7.473$	-	$7.108$
$i=2$	$7.492$	$7.480$	$7.495$	$7.482$	$7.470$	-	$7.484$
$X_{it}$
$i=1$	$2.019$	$1.976$	$2.318$	$2.376$	$0.879$	$9.568$	$1.914$
$i=2$	$2.920$	$2.853$	$3.200$	$2.278$	$1.976$	$13.227$	$2.645$
$D_{it}$
$i=1$	$2.019$	$1.970$	$1.920$	$1.854$	$1.808$	$9.571$	$1.914$
$i=2$	$2.920$	$2.811$	$2.620$	$2.504$	$2.355$	$13.210$	$2.642$
$S_{it}$
$i=1$	$0.000$	$0.000$	$0.394$	$0.930$	$0.0000931$	$1.324$	$0.265$
$i=2$	$0.000$	$0.000$	$0.587$	$0.382$	$0.002985$	$0.972$	$0.194$

provides the variables for the case of high production costs in a market in which a lower beta of the baseline scenario is presented. When focusing on the prices, they are higher than those in the same case but in the market in which there is a higher beta value. This finding is consistent with the expected result because in the present scenario, individuals are less sensitive to a change in prices. In addition, the value for the most attractive good is higher than that for Product No. 1.

Then, the demand and quantity produced for both items are higher than those in the previous market, a finding that is consistent with the decrease in the beta parameter. Likewise, the amounts corresponding to Product No. 2 are higher throughout the time horizon.

For both goods, the quantities produced are equivalent to those demanded for the first two periods because production costs decrease over time. Therefore, overproducing to store goods is not advisable in these stages.

Next,

Table 16. Variables and demand calculations—Scenario. 1 (${\mathbf{M}\mathbf{B}\mathbf{I}}_{\downarrow {\mathbf{m}}_{\mathbf{C}\mathbf{a}\mathbf{p}}}$).

	$$\mathit{t}=1$$	$$\mathit{t}=2$$	$$\mathit{t}=3$$	$$\mathit{t}=4$$	$$\mathit{t}=5$$	Total	Average
$P_{it}$
$i=1$	$7.486$	$7.498$	$7.500$	$7.492$	$7.030$	-	$7.401$
$i=2$	$7.484$	$7.459$	$7.496$	$7.492$	$7.468$	-	$7.480$
$X_{it}$
$i=1$	$2.311$	$2.214$	$2.102$	$2.507$	$1.380$	$10.514$	$2.103$
$i=2$	$1.946$	$1.919$	$1.985$	$1.741$	$1.796$	$9.387$	$1.877$
$D_{it}$
$i=1$	$2.307$	$2.211$	$2.102$	$1.957$	$1.931$	$10.508$	$2.102$
$i=2$	$1.946$	$1.919$	$1.593$	$2.089$	$1.849$	$9.396$	$1.879$
$S_{it}$
$i=1$	$0.000$	$0.000$	$0.000645$	$0.555$	$0.00304$	$0.559$	$0.112$
$i=2$	$0.000$	$0.000$	$0.391$	$0.0547$	$0.00116$	$0.447$	$0.089$

provides the values for the variables in the context in which the production capacity for the most attractive item is limited. In this scenario, prices very close to the upper limit for both products are obtained, except for the last period for the less attractive good. However, for Product No. 2, a slight decrease is observed with respect to the case in which the costs are increased.

In addition, as in the previous limitation, the quantities produced and stored exceed those in the market with greater sensitivity to a change in prices. This has its origin in the need that arises in consumers to acquire the good. Likewise, there is greater substitution between products when the production capacity of the most attractive good is limited, with respect to an increase in manufacturing costs. This is seen in the total quantity demanded and has its origin in the generation of a shortage because of limitations.

Additionally, the quantities produced must be equal to those demanded in the first periods for the costs to be reduced for both products.

5.3. Scenario 2: The Limitation of a More Attractive Product through CMNL Parameters

In Scenario 2, a more attractive product is modeled. However, at present, the shortage is contextualized by modifying the parameters associated with CMNL. Specifically, the customer group ($H_t$) is increased, and the nonpurchase utility is decreased ($V_0$).

First, the attractiveness of each product is described in Table 6. The capacities and costs remain constant for both products. The maximum production capacity is modeled with the information in Table 7. The production costs are maintained as in Table 8. As in the baseline scenario, the price sensitivity ($\mathsf{\beta }$) is 0.1.

This scenario is designed to analyze variations in the CMNL parameters for a context in which Product No. 2 is more attractive than the alternative. Specifically, the size of the market segment, i.e., an increase to 15 individuals, is evaluated. Then, the nonpurchase utility decreases, first when it takes a value equivalent to two and then when it equates to the unit.

5.3.1. Large Customer Segment Market

For the present case, the number of individuals in the market is modified, and a normal nonpurchase utility is maintained (

Table 17. CMNL parameters—Scenario No. 2: high $H_t$.

CMNL Parameters
Segment size ($H_t$)	15
Nonpurchase utility ($V_0$)	4
Price sensitivity ($\beta$)	0.10

). Thus, the shortage is modeled by many customers who try to acquire the available goods, but these are limited in availability because of production capacities.

As the size of the market increases, the demand for both products is expected to increase and thus the units produced, even more for the most attractive good. In addition, an increase in prices could be seen because of the increase in demand. To represent a large market, ${ H}_t=15$ is established. The utility functions obtained when executing the algorithm with the proposed change are shown in

Table 18. Objective utility function—Scenario No. 2: increase in market size.

Firm’s Utility Function
$U_{ Total}$	$U_{ Product 1}$	$U_{ Product 2}$
137.200	59.117	78.083

.

The profits increased considerably compared with those in the baseline scenario, a finding that is consistent with the expected result because in this case, there are five additional individuals. Likewise, Product No. 2 generates the highest utilities due to its superior attractiveness.

Regarding PSO SUB, it estimates the correct amount stored for the problem because the values of the fixed-point function (

Table 19. Fixed-point storage function—Scenario No. 2: increase in market size.

Fixed-Point Storage Function
$t=1$	$t=2$	$t=3$	$t=4$	$t=5$
$2.878\times {10}^{-4}$	$4.760\times {10}^{-4}$	$4.631\times {10}^{-5}$	$5.270\times {10}^{-5}$	$1.081\times {10}^{-4}$

) are minimal.

Next, the variables in

Table 20. Variables and demand calculations—Scenario No. 2: increase in market size.

Variables and Demands for Scenario No. 2: Increase in Market Size
		$\mathit{t}=1$	$\mathit{t}=2$	$\mathit{t}=3$	$\mathit{t}=4$	$\mathit{t}=5$	Total	Average
${\mathit{P}}_{\mathit{i}\mathit{t}}$	$i=1$	$7.489$	$7.401$	$7.482$	$7.490$	$7.465$	-	$7.465$
	$i=2$	$7.498$	$7.489$	$7.451$	$7.462$	$7.458$	-	$7.472$
${\mathit{X}}_{\mathit{i}\mathit{t}}$	$i=1$	$3.025$	$2.936$	$3.295$	$2.792$	$2.027$	$14.075$	$2.815$
	$i=2$	$4.008$	$3.864$	$3.713$	$3.787$	$2.977$	$18.349$	$3.670$
${\mathit{D}}_{\mathit{i}\mathit{t}}$	$i=1$	$3.008$	$2.914$	$2.772$	$2.743$	$2.637$	$14.074$	$2.815$
	$i=2$	$4.008$	$3.864$	$3.713$	$3.443$	$3.393$	$18.421$	$3.684$
${\mathit{S}}_{\mathit{i}\mathit{t}}$	$i=1$	$0.000$	$0.000$	$0.516$	$0.610$	$0.00850$	$1.135$	$0.227$
	$i=2$	$0.000$	$0.000$	$0.000515$	$0.417$	$0.00721$	$0.425$	$0.0850$

are analyzed; price ($P_{ it}$) is quite similar for both products and very close to $P_{ max}$, and ($X_{ it}$) is superior for the most attractive product compared to its alternative throughout the time horizon. This equates to the quantity demanded in the first period and is not generated for storage because the production costs at these times are higher.

5.3.2. Low Nonpurchase Utility

Finally, scarcity is presented, limiting the utility of nonpurchases that consumers present and maintaining a $H_t$ within the baseline parameters. In this way, the probability of demand increases by reducing the nonpurchase attempt (

Table 21. CMNL parameters—Scenario No. 2: low $V_0$.

CMNL Parameters
Segment size ($H_t$)	10
Nonpurchase utility ($V_0$)	[1; 2]
Price sensitivity ($\beta$)	0.10

).

This last case study represents a market with the greatest need to purchase the product. Therefore, the quantities demanded are expected to increase as the nonpurchase utility decreases, showing a higher increase for the most attractive product. In addition, the above could show an increase in prices.

In

Table 22. Objective utility function—Scenario No. 2: nonpurchase utility variation.

Firm’s Utility Function
	${\mathit{U}}_{\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$	${\mathit{U}}_{\mathit{P}\mathit{r}\mathit{o}\mathit{d}\mathit{u}\mathit{c}\mathit{t}1}$	${\mathit{U}}_{\mathit{P}\mathit{r}\mathit{o}\mathit{d}\mathit{u}\mathit{c}\mathit{t}2}$
${E_{V_0=2}}_{ }$	127.968	53.799	74.169
${E_{V_0=1}}_{ }$	154.436	67.609	86.827

, the utilities of each context are observed. For the case where the nonpurchase utility is lower, the total profit generated for both products individually is higher. This is a logical result because a lower $V_0$ represents a market in which there is a higher need to buy, like the case of a reduced beta.

Table 23. Fixed-point storage function—Scenario No. 2: nonpurchase utility variation.

Fixed-Point Storage Function
	$\mathit{t}=1$	$\mathit{t}=2$	$\mathit{t}=3$	$\mathit{t}=4$	$\mathit{t}=5$
${E_{V_0=2}}_{ }$	$5.117\times {10}^{-6}$	$0.000\times {10}^0$	$1.285\times {10}^{-5}$	$8.264\times {10}^{-6}$	$4.210\times {10}^{-5}$
${E_{V_0=1}}_{ }$	$0.000$	$0.000\times {10}^0$	$8.934\times {10}^{-6}$	$6.956\times {10}^{-6}$	$2.554\times {10}^{-6}$

indicates a positive function of PSO SUB. In these, the algorithm’s convergence is detailed, in which the fixed-point function related to storage tends to zero in all periods and for both items.

The

Table 24. Variables and demand calculations—Scenario No. 2.

$\mathbf{Variables} \mathbf{and} \mathbf{Demands} \mathbf{for} \mathbf{Scenario} \mathbf{No}. 2 \mathsf{(}{\mathit{E}}_{{\mathit{V}}_0=2}\mathsf{)}$
		$\mathit{t}=1$	$\mathit{t}=2$	$\mathit{t}=3$	$\mathit{t}=4$	$\mathit{t}=5$	Total	Average
${\mathit{P}}_{\mathit{i}\mathit{t}}$	$i=1$	$7.480$	$7.488$	$7.490$	$7.485$	$7.437$	-	$7.476$
	$i=2$	$7.495$	$7.489$	$7.492$	$7.496$	$7.494$	-	$7.493$
${\mathit{X}}_{\mathit{i}\mathit{t}}$	$i=1$	$2.701$	$2.644$	$2.881$	$3.908$	$0.704$	$12.838$	$2.568$
	$i=2$	$3.758$	$3.612$	$3.556$	$3.526$	$3.018$	$17.470$	$3.494$
${\mathit{D}}_{\mathit{i}\mathit{t}}$	$i=1$	$2.700$	$2.644$	$2.598$	$2.379$	$2.515$	$12.836$	$2.567$
	$i=2$	$3.755$	$3.612$	$3.485$	$3.395$	$3.220$	$17.467$	$3.493$
${\mathit{S}}_{\mathit{i}\mathit{t}}$	$i=1$	$0.000$	$0.000$	$0.286$	$1.819$	$0.000259$	$2.105$	$0.421$
	$i=2$	$0.000$	$0.000$	$0.0729$	$0.208$	$0.00998$	$0.291$	$0.0582$

shows the values for the variables; price ($P_{ it}$) trends to $P_{ max}$ for both products but is always slightly increased for the most attractive good.

The quantity produced ($X_{ it}$) is like that demanded ($D_{ it}$) in the first two periods; therefore, no additional costs are incurred. Likewise, the quantities associated with Product No. 2 are higher than those of the alternative by approximately five units. Therefore, there is a tendency to consume attractive goods; however, many customers tend to substitute their purchases. This originates in representing a market of greater need, in which the nonpurchase utility is quite low.

In

Table 25. Variables and demand calculations—Scenario No. 2: decreasing the nonpurchase utility.

Variables and Demands for Scenario No. 2
		$\mathit{t}=1$	$\mathit{t}=2$	$\mathit{t}=3$	$\mathit{t}=4$	$\mathit{t}=5$	Total	Average
$P_{ it}$	$i=1$	$6.755$	$7.484$	$7.486$	$7.420$	$7.492$	-	$7.327$
	$i=2$	$7.493$	$7.495$	$7.489$	$7.484$	$7.489$	-	$7.490$
$X_{ it}$	$i=1$	$3.373$	$3.367$	$3.395$	$3.602$	$2.850$	$16.557$	$3.311$
	$i=2$	$4.244$	$4.198$	$4.285$	$3.875$	$3.860$	$20.462$	$4.092$
$D_{ it}$	$i=1$	$3.373$	$3.367$	$3.395$	$3.188$	$3.267$	$16.590$	$3.318$
	$i=2$	$4.244$	$4.198$	$3.979$	$4.128$	$3.923$	$20.472$	$4.094$
$S_{ it}$	$i=1$	$0.000$	$0.000$	$0.000588$	$0.419$	$0.000385$	$0.420$	$0.0840$
	$i=2$	$0.000$	$0.000$	$0.304$	$0.0623$	$0.000885$	$0.367$	$0.0734$

, by further decreasing the nonpurchase utility ($V_0=1$) and generating more need for the acquisition of the goods, similar prices are observed to those in the previous case, in which these tend to the maximum value. The quantities produced and stored are considerably increased for both goods. In addition, a higher percentage of customers substitute their consumption of the most attractive good (difference of four units) due to the increase in the need to purchase the product.

Notably, in the previous cases, just enough is produced to supply the demand in the first three periods due to the high associated production costs. Likewise, in the following stages, production is generated to supply because these expenses decrease in the time horizon.

6. Conclusions

This research integrates production decisions, inventories, and sales prices in competitive and intertemporal environments for multiple products in the face of scarcity conditions and user preferences. A nonlinear optimization model is designed and validated in which the objective function seeks profit maximization, which includes production costs and inventory maintenance for substitute products. Given the phenomenon of scarcity or potential competition between customers, it is assumed that the demand is sensitive to the sale price and the product’s availability. To model the behavior of demand in each period, a restricted MNL model is used that allows the inclusion within the utility function of customers’ said variation in preferences using penalty constraints known as cutoffs. These functions reduce the probability of buying a scarce product, varying the buyers’ preferences. Given the intertemporal behavior of the model, a balance condition is included. This intertemporal or dynamic nature extends the investigations of Pérez et al. [32], López-Ospina et al. [29], and Page et al. [37].

This research employs a constrained multinomial logit (MNL) model to model the demand behavior in each period. This model is not only theoretically sound but also has practical implications. It allows the inclusion of customer preference variations within the utility function through penalty constraints known as cutoffs. These cutoff functions are crucial as they reduce the probability of purchasing a scarce product, effectively varying buyer preferences and simulating realistic market conditions. The intertemporal nature of the model necessitates a balance condition, extending the work of previous researchers by incorporating dynamic elements that capture the evolution of inventory and pricing decisions over time. This practical approach enhances the relevance and applicability of the study.

The study takes a dynamic approach, integrating flow conservation of the lot-sizing type to maximize a firm’s utility. This is achieved by carefully considering the price, quantity produced, and stored inventory. Additionally, the analysis of demands between substitute products within a company is established as a focal area, highlighting the complexity and interdependencies in real-world inventory systems.

Given the model’s high nonlinearity and recursive structure, traditional optimization methods were deemed unsuitable. Instead, a solution algorithm integrating two PSO-type metaheuristics was designed. The first metaheuristic focuses on finding the global solution, while the second ensures maintaining balanced intertemporal conditions for each period and product. This dual approach is vital for handling the model’s intricate dynamics. The same algorithm is also used to determine optimal storage through cutoff capacity constraints and to calculate the probability of demand based on the products’ utility.

The proposed model’s performance was validated using a testing scenario with a five-period time horizon and two substitute products. This setup was carefully chosen to balance complexity and clarity, effectively demonstrating the model’s capabilities. The results highlighted how the model adapts to changing conditions through its dynamic nature, providing valuable insights into inventory management decisions over time. The findings showed that the model could dynamically adjust inventory levels in response to potential availability constraints, thus optimizing the overall utility for the firm.

This research introduces a novel and robust tool for optimizing inventory and pricing decisions in environments characterized by substitute products and soft supply constraints. The unique aspect of this research lies in the innovative use of PSO metaheuristics and the incorporation of a dynamic demand model, marking significant advancements in the field. These contributions enhance the theoretical understanding of inventory management under scarcity conditions and offer practical insights for implementing effective inventory and pricing strategies in competitive markets. The study’s findings underscore the importance of integrating advanced optimization techniques to address complex, nonlinear problems in modern supply chain management.