The transportation problem, a fundamental aspect of operations research, seeks to achieve efficiency. The main objective is to find the best cost-effective method for transporting items from several production units to many warehouses. This challenge requires lowering costs while meeting supply and demand requirements. It will assist the business in making optimum decisions and developing effective advertising strategies. Suppliers, customers, their capabilities, and transportation costs are often arranged in a matrix arrangement in a standard representation. The goal is to distribute product quantities from suppliers to customers efficiently by meeting needs, adhering to supply constraints, and reducing prices. Linear programming methods are often used to produce an optimum solution by determining how resources should be distributed throughout the network for the greatest results. The transportation problem originated in the mid-20th century when Koopmans (Koopmans, 1949) developed its formal formulation in 1947 to solve logistical problems, especially during World War II. This mathematical problem requires the optimal distribution of commodities from different suppliers to various demand locations in order to minimize transportation expenses. The development of the simplex approach (Dantzig, 1963) during the same era significantly boosted the optimization field by enhancing the efficiency of solving linear programming problems, such as the transportation problem. With the rise of computers in the 1960s and 1970s, the range of solutions to problems increased to handle intricate real-world situations. The problem’s applications evolved further, leading to the emergence of current algorithms such interior point approaches in the 1990s for improved efficiency in finding answers. Currently, in a time of complex supply chains, the transportation problem is still relevant, with sophisticated optimization software consistently expanding its uses in many sectors. Picture a large industrial complex looking to enhance the efficiency of its transportation network. Stochastic programming develops mathematical models to achieve stability, punctual delivery, and cost savings while considering operational constraints. Stochastic modelling and flexible optimization are used to manage unexpected elements like supply chain disruptions and demand variations, ensuring that the transportation plan can adapt to unanticipated events. Implementing this strategy enhances a manufacturing facility’s sustainability and competitive edge within its sector by tackling transportation problems and optimizing supply chain operations. This enhances resource distribution, environmental stewardship, and cost efficiency.

The linear fractional transportation problem involves managing resource allocation to reduce costs or enhance producer profits. It combines linear programming techniques, which optimize linear objective functions under linear constraints, with fractions. Resources may be subdivided into fractional units, which is very useful in situations such as delivering components of a product. The linear plus linear fractional transportation problem further complicates this complexity. This advanced version includes fractional elements and adds another level of complexity. This layer establishes linear linkages in both the goal function and restrictions. Picture it as a puzzle with two levels - optimizing fractions and managing linear connections—creating a more complex and engaging challenge for those solving problems.

Stochastic programming (SP) is a powerful tool for decision-making in manufacturing, especially in dealing with uncertainty. It combines mathematical optimization with probability theory to create solutions that navigate smoothly through the constantly changing and mysterious circumstances of reality. Imagine it as a navigator on rough seas, plotting a path that not only endures the storm but flourishes in it. SP’s canvas spans several fields, with transportation and industry being notable areas. In the transportation industry, it coordinates routes, manages fleets, and responds to emergencies, all while navigating through unpredictability. SP excels in the industrial environment by optimizing production processes, maintenance, energy distribution, and risk management. It guarantees consistent efficiency in directing decision-makers with knowledge, even in an uncertain environment. Imagine a corporation that specialized in creating and transporting perishable goods. Here, SP is the focal point, creating an exceptional plan for manufacturing and distribution tactics. It achieves this by carefully considering unpredictable factors such as changing demand and irregular supply. Stochastic programming reveals the best production levels and distribution routes by considering various situations such as increased demand during holidays or supply delays caused by unexpected transportation problems. The firm reduces expenses, increases earnings, and remains robust and prepared for the uncertain market. SP, the master of flexibility and anticipation orchestrates this symphony of achievement.

Utilizing a Weibull distribution optimization method in mathematical modelling helps producers achieve equilibrium among conflicting goals, such as reducing costs, minimizing lead times, and enhancing service quality. Implementing supply chains, using stochastic modelling to manage demand variations, and optimizing transportation routes are some of these strategies. Commence construction work. Construction is under progress. This includes the design. Seek practical uses, particularly in multi-objective optimization. Excellence in environmental sustainability and customer service. Manufacturers may address difficult supply chain and transportation difficulties efficiently by using this advanced technology. This ultimately improves customer satisfaction and operational efficiency by enabling them to be flexible and responsive in a constantly evolving production environment.

In this article, we explore uncertainty using probability theory, focusing on Stochastic Programming (SP). We are exploring stochastic programming challenges, which include the integration of probability and mathematical programming to illuminate decision-making processes influenced by random factors. We begin our voyage by investigating a Stochastic Transportation Problem, which is a complex maze with many linear fractional objective functions. The MOSLFTP model is shaped by the limitations provided by normal random variables, forming a complex mathematical structure. Our goal is to discover a collection of Pareto-optimal solutions by balancing opposing goals inside MOSLFTP, rather than achieving an exact approximation of the true Pareto front. Researchers have explored many approaches (Table 1) to solve complicated problems. Stochastic programming techniques are essential tools that provide assistance in dealing with optimization difficulties that include random parameters. They explore a broad range of situations where model coefficients are influenced by uncertainty and follow certain probability distributions. Stochastic Programming has a wide-reaching impact on several fields such as management science, engineering, and technology. In fields where input data is unpredictable and models are created from imperfect information, SP finds its refuge. In recent years, its potential has greatly expanded due to advancements in computer science and optimization methods. It establishes itself and thrives in the industrial sector, revolutionizing energy resource planning, finance, telecommunications, transportation, production scheduling, and supply chain management into successful environments. SP showcases the limitless adaptability of probabilistic exploration in decision-making.

The journey of the linear fractional transportation problem began with Swarup’s pioneering suggestion (Swarup, 1966), marking the genesis of this mathematical challenge. Since then, the literature (Sadia et al., 2016; Safi and Seyyed, 2017) has meticulously chronicled its systematic evolution, with scholars delving deeper into its intricacies. In the realm of solving transportation problems with fractional objectives, valuable methods were laid out by (Pradhan and Biswal, 2015), offering practical solutions for real-world scenarios where parameters may not always be precise, requiring a touch of educated guesswork. The murkiness of real-world problems, often obscured by a lack of specific data, led Liu (Liu, 2007) to unearth the uncertainty theory, reshaping our approach to these enigmas. Within this ever-expanding landscape, various scholars have turned to Stochastic Programming (SP) as a powerful tool for wrangling uncertainty. The SP model itself had its origins in the visionary work of Dantzig (Dantzig, 2011), and over time, it has blossomed into various incarnations with different scholars proposing their own SP models (Goicoechea and Duckstein, 1987). The intersection of stochasticity and fractional objectives has been a rich field of exploration, as evidenced by numerous investigations into the Stochastic Fractional Transportation Problem and its accompanying solution methodologies (Charles and Dutta, 2005; Jain and Arya, 2013; Jadhav and Doke, 2016). Javaid (Javaid et al., 2017) introduced a Transportation Problem model adorned with multiple fractional objectives and unpredictable parameters, adding layers of complexity to the puzzle. In the more recent annals of research, Saini’s work in 2022 (Saini et al., 2022) employed a fuzzy approach to untangle the knots of an MFL (Multi-Fractional Linear) paradox within a multi-objective transportation problem. Meanwhile, Joshi (Joshi et al., 2022a) embarked on an exploration of fractional transportation problems within the realm of neutrosophic situations in the same year, where all the objective function coefficients, demands, and availabilities existed in the realm of speculation. These contemporary inquiries highlight the ongoing pursuit of novel solutions to complex problems in the ever-evolving world of optimization and uncertainty management.

A technique for resolving MOSSTP in the presence of uncertainty by recasting it as a chance-constrained programming problem and employing fuzzy goal programming and the global criterion method to quickly find effective solutions (Das and Lee, 2021). A weighted goal programming method that, using a numerical example, discovers compromise solutions for multi-objective transportation problems following the decision-maker’s priorities (Joshi et al., 2022b). The Weibull distribution and multiple cost coefficients are used in a solution methodology for the multi-choice stochastic transportation problem (Roy, 2014). Using a Weibull distribution to describe the uncertain parameters, the research presents three models for dealing with stochastic fuzzy transportation problems (SFTPs) with mixed-type constraints (Buvaneshwari and Anuradha, 2022). Using a fuzzy technique, multiple-choice parameters, and random variables, a transportation problem with multiple objectives is solved (Nasseri and Bavandi, 2020).

The MOSLFTP is a complex mathematical conundrum that goes beyond the limits of a basic enigma. It explores the complex process of maximizing resource allocation from suppliers to consumers while managing several competing goals, similar to meeting various preferences at the same time. A mist of uncertainty overlays this intricacy, with specific numbers adhering to the elusive Weibull distribution pattern. Picture a complex labyrinth with several destinations, each with fluctuating circumstances. Your objective is to map out the most optimal route while managing competing objectives and moving through the haze of uncertainty. The MOSLFTP and MOSLPLFTP are intricately connected via the LINGO^® 18.0 Software, which acts as a set of tools that work together to optimize processes. An example will demonstrate the effectiveness of this technique by leading us through ambiguity and helping us make well-informed judgments. This research serves as a crucial tool for companies seeking guidance in navigating complicated issues and uncovering the keys to success. Recent research has shed light on creative paths in manufacturing, urban transportation, and multi-objective optimization, paving the way for a more enlightened future (Kumar et al., 2020; Kumar and Gulati, 2021; Boadh et al., 2022; Kumar et al., 2022; Kumar et al., 2023a; Kumar et al., 2023b; Yadav et al., 2023).

This study is an innovative investigation into stochastic programming, focusing on a complex problem involving multiple linear and linear fractional objective functions. The problem incorporates supply and demand parameters that adhere to the Weibull distribution. The complex mathematical task is named MOSLPLFTP, representing a significant advancement in the subject. An extensive examination of current literature has shown a notable finding—there is a lack of study on these particular topics. This work is unique and innovative, exploring uncharted territory in research. This research also includes supply and demand variables that follow the Weibull distribution (Krishnamoorthy, 2006). By doing this, it creates a more complex and captivating specialization within the field of stochastic programming. This study is pioneering an examination of uncharted mathematical area, expanding the limits of knowledge, and creating opportunities for future investigations in optimization and uncertainty management.

A continuous probability distribution called the Weibull distribution can be used to model a variety of variables, such as the time to failure, the interval between events, and extreme values. It was first thoroughly described in 1939 by Swedish mathematician Waloddi Weibull, after whom it is called. The Weibull distribution has a wide range of applications, including Reliability engineering, Life data analysis, Engineering, Biology, Economics etc. The Weibull distribution’s probability density function with the known parameters η , β and χ are given by f   s , η , β , χ = η β s − χ β η − 1 . e − s − χ β η

and F s = 1 − e − s − χ β η where s ≥ 0 , s ≥ 0   o r χ , η > 0 , β > 0 , − ∞ < χ < ∞ . The Weibull distribution has three parameters: ➢

Shape parameter η : This parameter regulates how the distribution looks. An exponential distribution is produced by a shape parameter of 1, while a shape parameter higher than 1 result in a distribution with a longer tail.

The remainder of this paper is thoughtfully organized into several sections, each playing a crucial role in advancing the understanding and exploration of the research topic. Basic definitions, notation and Weibull distribution-based mathematical models defined for MOSLFTP and MOSLPLFTP are discussed in Section 2. Then, the solution methodology is presented in Section 3. While Section 4 offers additional approaches. Numerical examples are discussed in the Section 5. The presentation of results and insightful discussions take center stage in Sections 6, with the ultimate conclusions elegantly summarized in Section 7. This organized structure guides readers through a comprehensive journey of discovery and insight in the realm of optimization and uncertainty management.

The constraints within the presented mathematical model, as detailed in Section 2, incorporate random values due to the consideration of the Weibull distribution. Consequently, direct solution using traditional mathematical techniques becomes infeasible. To address this challenge, the random constraints are ingeniously transformed into deterministic constraints, a transformation method elaborated upon in the following three scenarios outlined in Section 3.1. This transformation ensures that the inherent unpredictability in the constraints, attributed to the Weibull distribution, is effectively managed and incorporated into the optimization process.

These encompass mathematical optimization, where intricate formulations are analysed to achieve optimal solutions that align with multiple objectives while considering uncertainties. Complementing this, heuristic algorithms offer rapid approximate solutions by intelligently exploring solution spaces, particularly suitable for larger instances. Metaheuristic methods take the helm by employing sophisticated strategies like genetic algorithms and simulated annealing to navigate complex landscapes effectively. Simulation-based approaches step in, simulating real-world scenarios to comprehend the effects of various decisions under uncertain conditions. Lastly, interactive decision support systems offer an intuitive platform for decision-makers to visualize and assess different scenarios, thus untangling the complexity of MOSLFTP and MOSLPLFTP and facilitating well-informed choices.

This section exemplifies the tangible impact of our MOSLFTP and MOSLPLFTP research, unveiling its practicality through numerical demonstrations. In Section 3, we introduced MOSLFTP and MOSLPLFTP models and, employing our method alongside the weighted sum approach, derived optimal compromise solutions. The deterministic equivalents of these mathematical models were implemented using the Lingo 18.0 solver, grounding our research in real-world applicability.

In this illustrative section, we immerse ourselves in a practical scenario within the realm of Third-Party Logistics (TPL). Picture a dynamic transportation network where a TPL company plays a pivotal role. This network features four suppliers, symbolized as source nodes: a 1 ,   a 2 ,   a 3 , and   a 4 . Additionally, there are three wholesale market halls denoted as demand nodes:   b 1 ,   b 2 and   b 3 . Each supplier provides a specific type of produce:   a 1 offers pears,   a 2 supplies lemons,   a 3 offers peaches, and   a 4 provides kiwis. Notably, all the produce is perishable, adding complexity to the scenario. Each supplier has varying quantities of their products destined for the wholesale market halls, with each hall having its own unique demand for different items. The overarching objective of this transportation network is to achieve the optimal balance between minimizing total transportation costs, maximizing profits, and ensuring efficient delivery times. This example showcases the challenges and complexities faced in real-world logistics, where factors like product type, perish ability, varying quantities, and market demand must be considered for effective decision-making.

In this dynamic environment, the supply of fresh goods experiences variability influenced by uncontrollable factors like weather conditions and labour availability at the farms. This unpredictability transforms the availability of such produce into a probabilistic phenomenon, particularly when supply stability is uncertain. Consequently, the likelihood of obtaining the desired quantity of produce from provider a₁ is represented by θ a 1 . Similar probability distributions are assigned to suppliers   a 2   ,   a 3 and   a 4 , denoted as θ a 2 , θ a 3 , and θ a 4 , respectively. These probabilities encapsulate the uncertainty surrounding the availability of fresh goods from each supplier in our scenario.

The demand for fresh fruit is naturally unpredictable. Inaccurate demand forecasts, demand volatility, or unanticipated delivery delays are the main causes. Therefore, the chance that the anticipated demand is necessary for market hall   b 1 is   δ b 1 . The definitions of probability δ b 2 and δ b 3 for market halls   b 2 and   b 3 .