A BI-OBJECTIVE INTEGRATED TRANSPORTATION AND INVENTORY MANAGEMENT UNDER A SUPPLY CHAIN NETWORK CONSIDERING MULTIPLE DISTRIBUTION NETWORKS

. In order to respond to the customer’s needs effectively and efficiently, logistics is characterized as a part of the supply chain that executes and handles forward and reverse movement and storage of products, services, and related data. An efficient logistic network is needed for the supply chain that executes forward and reverses products’ movement. This study resolves the supply chain network’s logistic problem to determine the appropriate order allocation of products from multiple plants, warehouses, and distributors to minimize total transportation and inventory costs by simultaneously determining optimal locations, flows, shipment composition


Introduction
Flexibility and integration in the supply chain (SC) are essential in reducing overall costs and improving overall performance.SC strategy involves a supply chain network (SCN) comprising economic, logistical, and financial decision-making concerns.Economic decisions contribute to long-term growth and supply chain management (SCM), whereas logistical choices reconnect to optimal use of different resources.With increasing competitiveness, many businesses have been engaged in purchasing, processing, and transportation activities in multiple states and cities in recent decades.Dealing with increasing competition and creating a worldwide SCN with lower costs, increased flexibility, and a higher standard of customer support are now critical business problems.The designed optimization models can help the managers place the warehouse center and transport channels to use goods from one place to another.Traditionally, transportation cost is the primary purpose of optimization in SCN.Still, companies are increasingly concerned with inventory cost issues, and there is a growing need to consider goals, such as holding costs.
SCM defines the connection between logistics and other business functions such as acquisition, development, management of the system distribution channel, and customer service [35].Since the early 1990s, business conditions have changed, and competition required that buyers buy the right products at the right time, at the right moment, and in a suitable situation, at the lowest cost.Operations outsourcing positions allow a company to concentrate on its core skills.It will enable companies to leverage their money better to efficiently handle operations while enabling world-class service suppliers and use their equipment and human resources infrastructure.Logistics have become part and parcel of every business today.Without logistics support, a company's distribution, manufacturing, or project management cannot survive [21].Charles et al. [14] formulated a multi-objective optimization model (MOOM) of a supply chain network with order allocation problems to minimize total transportation cost and delivery time.They solved the developed model by three different goal programming approaches to obtain the optimal solution.Ali et al. [3] developed a new mathematical programming model of inventory management by considering conflicting linear fractional objective functions with different factors such as holding cost, purchasing price, selling price, demand, and ordering cost.The authors developed two separate inventory management models with linear and non-linear membership functions and solved them using the intuitionistic fuzzy goal programming technique.
Tuzkaya and Önüt [53] designed a new model to assess the best supply of products between multiple vendors, warehouses, and producers.The goal was to reduce net inventory, warehouse, producer, and penalty costs for vendors, distributors, and warehouses.Bashiri et al. [9] developed a modern multi-item mathematical model of strategic and operational production with varying times to control decisions for a multi-stage network.The primary purpose was to optimize overall net profits over periods measured by subtracting total expenses from total sales.The net fee covers capital costs for construction, the addition of facilities, running facilities, and costs related to raw materials, production, procurement, and transportation.Sadigh et al. [42] used the updated genetic algorithm to construct a deterministic multi-objective SCN problem of supply selection that intertwines with the output decision and distributor position.Three goals have been considered: the first one is related to minimization of the overall expense, including order cost, delivery cost, location of fulfillment centers, and cost of shipping; the second one is associated with the depreciation of the delivery time of the transport of imported goods to customers, and third on related with maximization of component efficiency.Sarrafha et al. [47] developed a multi-objective model for the SCN consisting of manufacturers, factories, delivery centers, and consumers.A novel multi-objective biogeography-based optimization technique is designed to solve the two goals' problem; minimize overall SCN costs and reduce average commodity tardiness to distribution centers.A simulation-based multi-objective model was developed by Tsai and Chen [52].They specified the required configurations of the reorder point and order quantities to reduce at the same time three objective features, which are the estimated values of the overall product cost, the average production volume, and the frequency of stock-outs.Sarkar et al. [46] proposed economic and environmental sustainability by introducing innovative products.The production system consisted with remanufacturing unit.Mohammed and Duffuaa [36] developed an efficient tabu search algorithm to solve the multi-stage SCN problem.The model developed in this paper has three aims: to increase revenue to generate as much capital for investors in the supply chain; the second aim is to reduce risk.The third and last aim is to mitigate overall emissions from the supply chain.
The main contributions of this paper include: -An integrated transportation and inventory model is formulated to optimize total supply chain cost.Transportation cost is the combination of costs incurred from plants to warehouses and warehouses to distributors.-Total inventory cost includes holding costs, ordering costs, and cost of working capital blocked up in the inventory due to delays in transportation.Develop a solution approach based on the value function approach.-A real-world case study of TV sets validates the applicability of the proposed model.The value function approach has been used to determine the proper shipment allocation.
-The robustness of the model has been assessed with other scalarization techniques, i.e., goal programming and fuzzy programming.The decision is established for the model's relevance by carrying out the sensitivity analysis.
The paper's remaining part is structured as follows: Section 2 provides a detailed overview of supply chain issues in the literature, focusing on multi-optimization.In Section 3, we discuss the supply chain of ABC Company.Section 4 presents the supply chain's mathematical model formulation, using the proposed optimizing technique to solve the formulated problem.Section 5 provides a case illustration to demonstrate the application of the proposed model.Finally, concluding remarks are made in Section 11.

Literature review
A detailed review of recent literature on newly developed SCN has introduced different features and general patterns in various models.

Supply chain network
Xu et al. [56] formulated the SCN model to identify the network structure that can concurrently meet the goals of minimization of total expense.It consists of the fixed costs of plants and distribution centers, inbound and outbound transport costs, and maximization of consumer satisfaction, which can provide adequate delivery time.Guiffrida and Jaber [22] examined the consequences of enhancing distribution efficiency when the distribution period is perceived to determine delivery output in sequential supply chains.The variability of a distribution to final customers is modeled as an investment function.Bilgen [11] discussed production and distribution planning problems in an SCN that included allocating quantities for output across different production lines in factories and supplying goods to distribution centers.Moon et al. [37] concentrated on reliability of a smart production system that was emissions-controlled.A two-tier supply-location attribution model with accidental demand was proposed by Wang et al. [55] to minimize the setup, interconnections, and profit-maximizing between production and distribution.Peidro et al. [39] suggested a model that tackles strategic planning problems for constructing, producing, and distributing SCs in the ceramic field.They maximized the total gross profit, eliminated backorder volumes, and decreased idle time in multi-supplier multi-level distribution centers.Seidscher and Minner [48] have evaluated constructive and reactive trans-shipment approaches in a multi-locational distribution network.They believe each sector's demands can meet from a default collection of warehouses, considered constructive transfers to a warehouse outside the range.They concluded that productive management of transitions would contribute to considerable cost reductions.Bandyopadhyay and Bhattacharya [8] developed a multi-objective SCN problem concerning buyers and suppliers.The objectives were to optimize the two-tier supply chain's average cost, decrease order quantity volatility, and reduce the overall inventory of the product.Alvarez et al. [4] introduced side exchanges for spare parts from two warehouses to distinguish preferred consumers.If the preferred customer's requirement in one warehouse cannot be complied with by the stock available, a transshipment flow from the other warehouse would consider.Otherwise, the central depot uses emergency restoration, or the order is backed up.Díaz-Madroñero et al. [17] considered an SCN where the first-tier manufacturer represents a manufacturing client and decides the supplier's production strategy using details from the vehicle assembler, truck capacity, and stock levels.For suppliers, producers, retailers and variable demand driven by selling and advertisement prices, Mahapatra et al. [32] model has established co-op advertising, in which all the actual expenses are treated as fuzzy.The author optimized the revenue by considering the SC decision support system, variable periods, deliveries, pricing, and marketing costs.Taki et al. [15] designed the three-echelon SCN design with the model's goals: to reduce the SC's overall expense and improve the distribution efficiency, which may be equal to reducing the risk of not supplying the goods concerning an increase in demand.
In recent years, various researchers have identified multiple approaches to SCM.Among them, Gupta et al. [23] formulated a bi-objective SCM model to optimize the conventional facility location models to incorporate a range of logistic system elements, such as transportation costs and different inventory costs, in a multi-product, multi-site network.Sarkar et al. [44] developed an artificial neural network (ANN) with multithreading to solve a model having multiple items under uncertainty.Akbarpour et al. [2] proposed a bi-objective model for building a pharmaceutical relief network under unexpected demand conditions for perishable goods.Arasteh [6] and Bera et al. [10] proposed a fuzzy multi-objective linear model to maximize the product's quality in uncertain situations by considering the three-layer supply chain.They considered some limitations: lack of order, market demand, and manufacturing capacity for minimizing total cost.Zandkarimkhani et al. [58] used fuzzy multigoal programming to reduce total costs in the pharmaceutical SC under uncertain situations.Formulating the model included multi-products, multi-period, multi-locations, and vehicle routing.Sometime, Delfani et al. [16] developed a robust fuzzy model to minimize total cost in the multi-layer pharmaceutical supply chain in uncertainty.They considered the product's reliability and the delivery time as significant constraints.

Effectiveness of multi-capacitated logistics under SCN
To support e-tailers optimally and fulfill customer orders while reducing their logistics costs in e-tail environments, Torabi et al. [51] built a mixed-integer programming model.The goal was to create an optimum distribution schedule for customer orders by reducing overall freight and transportation costs after satisfying the demand.In a study of merchandise distribution networks, Ahmadi et al. [1] incorporated constructive transshipments for a company selling seasonal and non-seasonal goods to different consumers.The fundamental goal of this paper is to optimize the overall cost and consumer satisfaction.The overall cost function includes the production output and inventory price, shipping to all destinations, and the chosen facilities' annual fixed fee.The second objective is related to the quality of service rendered by the network and evaluates consumers' satisfaction over the planning period.In light of uncertain demand for an SCN, Amin and Baki [5] introduced a MOOM with mixed-integer linear programming (MILP) that considers global factors, including several manufacturers, suppliers, factories, distribution centers, market channels, storage centers, and produced goods.This study optimizes the delivery time and total profit of the SCN.A closed-loop network for reprocessing waste products has been developed by Nurjanni et al. [38] as a MOOM to reduce overall transportation costs and emissions in the SCN.The developed model has been solved using an optimization process consisting of three scalarization approaches: weighted sum method, weighted Tchebycheff, and augmented weighted Tchebycheff.Bilir et al. [12] have proposed a new mathematical model integrating the idea of SCN modeling into competitive location considerations (e.g., evolving market demand and customer care-related functions).Sabri and Beamon [40] developed a stochastic multi-objective responsive SCN model to analyze performance parameters like flexibility (volume), lead time, total cost, and customer service levels in the considered SC.This model integrates production distribution, supply, and demand complexities and establishes a multi-objective efficiency vector for the SC network.
Farrokh et al. [29] dealt with an SCN model design that includes recycling and disposal processes and formulated a mixed-integer programming model that optimized SCN configuration, fluctuations, and operational risks.Fattahi et al. [19] tackled a multi-period network modernization challenge, where consumer zones have stochastic price-dependent demand for multiple products.To build a network of the minimum cost against disruptions and provide decision-makers with the flexibility to prioritize network resilience over prices, Margolis et al. [34] developed a novel deterministic MOOM.Singh and Goh [49] incorporated supply chain strategy techniques into delivery plans to the hospital level, from sourcing raw materials to developing standardized medicines.The model consists of two goals with unknown parameters, including medicine demand, cost and time coefficients, and limitations at various levels related to the pharmaceutical supply chain.Gupta et al. [25] designed a MOOM for SCN's production-distribution process to discover how many units of the item could be delivered from the beginning to the end such that all the volume manufactured is fully utilized.All the demand levels are fully met such that no inventory will leave in the stock.Kugele et al. [28] developed a novel solution approach using goal programming for finding the unique solution with degree of difficulty.The application was in the smart production system.Han et al. [26] considered an automated production output, procurement, and outbound delivery scheduling that originated in a three-stage SC composed of a producer, a retailer, and many customers.The goal was to set down plans for processing jobs and define distribution plans from the producer to all consumers such that the total number of late jobs, manufacturing costs, and shipping costs can reduce.

Multi-capacitated SCN with multi-objective studies
Liang [30] formulated a fuzzy MOOM with a linear membership function to solve the integrated multi-time production and distribution scheduling problem.This work aimed to reduce production costs and overall processing time regarding inventory volumes, available machine flexibility, workforce requirement at each source, and estimated demand and usable warehouse space at each start and total expenditure.Sarkar et al. [45] considered a complex multi-phase manufacturing system that can control defective production rate automatically.Tapia-Ubeda et al. [50] researched incorporating SCN design and control for conventional spare parts operations.They introduced a generic network management method for simultaneously simulating warehouse sites and stock control decisions, thereby reducing the overall expense of the supply chain network of spare parts.Zandkarimkhani et al. [58] suggested considering the inventory location and transportation concerns and demand ambiguity for logistics operations and decomposable pharmaceuticals delivery.A bi-objective integer non-linear programming model was developed based on a vendor-managed inventory strategy in a three-echelon SC, including one manufacturer, one vendor, and multiple retailers, the maximum efficiency of SCN [41].Avci and Selim [7] considered a three-echelon heterogeneous SC structure consisting of a consumer, a retailer, and many suppliers and formulated a MOOM with premium freight in convergent SC.This paper aimed to evaluate the product's demand adjustment factor, the optimum stock level, and the distributor's versatility levels for products that provide the best storage cost and premium freight performance.
A multi-objective MILP model formulated for three-level global SC, which includes manufacturer, distributor, and customer, with the primary aim to minimize total cost, lead time, and lost sales in the process industry [18,31].Kadziński et al. [27] formulated MOOM for the SC, minimizing transportation cost, transportation time, and dust emission.The SC addressed in this study focuses on the distribution of white goods in the South Eastern European market, and they used interactive algorithms to solve the formulated problem SC.Validi et al. [54] proposed a systematic approach that focuses on a capacitated logistics distribution model for Ireland's two-layer dairy SC market demand.This study aimed to provide customized delivery routes based on the dairy supply chain's carbon emissions and production costs for processing milk products.
Zhang et al. [59], in contrast to the traditional SCN, a new conceptual paradigm for modeling SCN with several delivery networks was implemented in this study.The paradigm built in this study helps the consumers by delivering direct goods and services from the facilities available rather than the traditional flow of products and services.Mahmoodi et al. [33] developed a MOOM focused on multi-product transportation at the five stages of SCN that consists of vendors, producers, dealers, distributors, and consumers.This study aimed to assess the output and supply in each SCN node under the competitive market's uncertain demand.The model has three key goals: reducing costs, mitigating risk, and optimizing efficiency.Table 1 gives an overview of the method and applications used in the above studies.
From the above review, one can see that most of the studies have been carried out concurrently to understand the three relevant parameters, costs, resilience, and customer service level.We have identified that none of the studies measure the supply chain cost by optimizing the transportation and inventory cost for different shipments.This paper provides a realistic multi-objective mathematical programming model for the SCN design problem to address the shortcomings.In this work, we consider the optimization of transportation and inventory costs for different shipments of a supply chain process.The parameters of the model formulation include: items, set of cycles-times, set of shipping costs, active manufacturing locations, active warehouses/stores, ordering or setup costs, shipping costs from the manufacturer to the warehouse, shipping costs from the warehouse to the dealer, storage capability of product flow, manufacturer capability of producing goods.This paper introduces a multi-nonlinear shipping model focused on decision-making, including the configuration of the shipping network, choosing transport means, and transferring individual customer shipments through a particular transport system.A decision-maker aims to minimize transportation and inventory costs of complete orders while attaining the promised responsiveness.The sector of electronics is one where logistic movement is carried out worldwide.Electronics components are produced and assembled as finished products in various locations.An electronic product case study has been chosen to investigate how the company handled its nationwide

Case study for ABC company
ABC Group is a leader in Consumer Electronics and Telecommunications in India.It manufactures color TVs, washing machines, refrigerators, microwaves, vacuum cleaners, lanterns, audio systems, video systems, telephones, and monitors.Besides, it provides services like telecom.It has a significant market share in the Color TVs comprise this business group's principal product, and it had a 20.3% market share in the country in the financial year 2000-2001.Despite a slowdown in the consumer durables business and increased taxation by local governments, ABC achieves over 1 438 212 color television sets.ABC company is the first company to sell over a million television sets in three successive financial years.While overall volumes of color television sets declined in 2000-2001 compared to the previous year, ABC retains its premier brand position across all screen sizes (as validated by ORG).ABC has its premier position in frost-free refrigerators, which comprise about 15-20% of the refrigerator market, with a market share of 22%.Many new products and concepts are launched, such as non-CFC refrigerators, four new direct cool refrigerators, and a new washing machine technology called the perfect wash system, preventing fabric damage-two new upper-end models of microwave ovens and auto-ignition for gas tables.ABC company has a 15% share in washing machines.
The ABC company's supplier base comprises 215 suppliers, including overseas suppliers, local suppliers, and group company suppliers.The logistic network includes various branches and warehouses, which operate through distributors.The SC of ABC company depicts in Figure 1.The present study's scope is limited to the CTV distribution system considering only one tier, i.e., the first-level distributor.The ABC company has about 13 warehouses and 17 distributors across different locations in the country.The company has three plants for the assembly of Televisions situated at Noida, Palakkad, and Bangalore.Combined, these plants can produce 140 000 television units per month in the peak season of October and November.Planning completes Bangalore's corporate office, which sets the three units' production; it completes at the company's Noida unit.All three manufacturing plants assemble 20-inch and 21-inch general models of televisions.The turnover of the Noida plant is around Rs. 320 crores.The ABC company's supply chain includes procurement of raw materials, manufacturing operations, and transportation.The company currently has 35 international vendors and around 180 local vendors.
The planning process for CTV manufacturing units conducts at Noida, Bangalore, and Palakkad.The corporate office of ABC company in Bangalore prepares a rolling plan (called sales and operations plan) every month.This plan gives projected sales for the current month and the following three months.A production plan for each manufacturing unit is made based on the projected sales for the next three months and current  stock levels.This production plan is the basis for the entire materials management system for the plant.It holds special significance for imported items because they are essential items.

Distribution process at ABC company
The distribution of color TVs from ABC manufacturing units to distributors/dealers is represented in Figures 2 and 3.
Periodically, orders are electronically sent to the factory.The respective warehouses and distributors send their electronic/manual format requirements to the head office in Bangalore, where the data is recorded/entered into the central server, which contains the database of requirements compiled for the entire country.Then, the factory's planning process starts compiling this information with the monthly plan, and the entire production schedule is generated at the factory.Completed orders are dispatched in the form of a First-In-First-Out (FIFO) basis for a particular warehouse.Care is taken in designing the transportation route (either to cover full truckload or, in the event of the non-existence of full truckload, to club neighboring warehouse requirements accordingly truck size is decided).Distributors place an order on warehouses depending upon requirements received from retailers.Distributors dispatch the material in small trucks or tempos and ensure that the material is delivered on time with minimum cost.The existing distribution network is represented in Figure 3.  -

Loss of orders
The existing distribution channel is shown in Figure 3.Each plant supplies to specific warehouses, and in turn, each warehouse supplies to three or four distributors, depending on the distributor's location.The following problems are faced if any shortage occurs at any warehouse: (a) Owing to no contact between warehouses, a request is sent to the factory every time a shortage occurs at any warehouse, even though the stock may be available at another warehouse.(b) There is no coordination between dealers/distributors; similarly, requisitions are always sent to the plant.Both of these result in the loss of customers to the organization.
-Lack of customer confidence If the retailer does not have an item in the distribution down-line, the customer will not wait.Instead, s/he will buy an item from another company.This results in a decrease in customer confidence.-Lack of reputation If any customer returns without a product, s/he is likely to tell other persons, resulting in reputation loss.These situations reduce customer confidence and, in turn, market share.-Decrease stakeholders' confidence If the company starts losing orders and, in turn, customer confidence, market share will reduce slowly.If company performance is not good, stakeholders' confidence decreases.-Unnecessary pressure on plant If there is a shortage at any warehouse, the warehouse only contacts the plant.If the plant cannot supply, the warehouse does not contact another nearest warehouse.This increases unnecessary pressure on the plant, though it is not necessary.-Procedural delay All this ends in lengthy procedures to transfer data and compilation.
The supply chain manager of an ABC company faces multiple issues related to transshipment, loss of orders during shipment, lack of customer confidence due to delay in delivery time, lack of reputation, decreased stakeholder confidence, unnecessary pressure on plants, and procedural delay.We have designed a multi-objective supply chain model that optimizes transportation and inventory costs with the optimal distribution policy to overcome this issue.The developed framework is illustrated below in the next section.

Complex supply chain networking
A MILP model can be developed for the problem of identifying an ideal two-tier delivery network (Fig. 5).This has been attained as follows: -The goal is to optimize the network's overall expense by minimizing the transportation and inventory cost.
These costs include shipping, material processing, work capital blocked in the warehouse due to transport

Assumptions of the model
We considered the following basic assumptions of SC before designing the proposed mathematical model: (1) Each distributor purchases goods either from warehouses only.They cannot give the order to the plant directly.The plant will send the product to the warehouse first.(2) The overall monthly average demand at separate nodes has been considered independent of one another.
The demand for all markets is independent.There are no relations between markets.It is assumed that the production still has the item ready as the order comes.That means the company is using a push strategy for its market.There is no shortage at the plant level.The large stock is held at the factories for direct selling, and retailers replenish it by the manufacturer at frequent intervals.The model is not considering operating costs at the warehouse.The warehouse is used as a cross-docking strategy.(3) The company manufactures large types of TV.Nevertheless, the model is on two types of TV: 20 ′′ CTV and 21 ′′ CTV.Therefore, we assume various 20 ′′ CTV models as a single model and 21 ′′ CTV models as a single model.

Notation
A deterministic logistics model describes using the following parameters and variables.Here  symbolize the number of items,  symbolize cycle-time,  symbolize shipping category,  symbolize the number of available manufacturing plants,  symbolize the number of available warehouses, and  symbolize the number of shipments from the distribution center,  symbolize the setup cost of an item.The notation is grouped into three categories, as follows:

SC model structure
In the SC, transportation and inventory costs are considered the most critical factors for determining the network's efficiency.Transportation is vital in business logistics; approximately one-third to two-thirds of companies' logistics costs affect shipping.Logistics could not put its benefits into maximum play without welldeveloped distribution networks.Efficient transportation in logistics operations will provide more outstanding performance, optimize operational costs, and improve the standard of service.On the other hand, the cost of inventory can do a business or break it.However, in many companies, the possibilities of minimizing inventory costs are still not discussed at all or not implemented entirely.The primary function of the resulting mathematical model contains the following components:

Transportation costs
Transportation is a significant factor in the industry's SC.It links the company directly with its SCs associates, such as vendors and consumers, and significantly impacts its satisfaction.Transportation costs can be an essential part of the total logistics expenditure of a business.The shipping proportion can be more than 50% with higher fuel prices.Complete transportation costs shall be the amount of the shipping costs incurred within the SC network, which mathematically can be formulated as follows:

Average inventory holding cost
The carrying costs are related to the total expenditures associated with storing or transporting product items from a logistical viewpoint.Essentially, this paragliding concept covers costs incurred for storing and maintaining products over different periods (warehouse rents, facilities, machinery, employment, utilities), insurance rates, inflation, price volatility, damage, stealing, shrinkage, and deterioration.For 99.87% customer service, the standard total inventory can be calculated as 3 +   /2 (decided by the company).The total cost of holding inventory mathematically can be formulated as:

Ordering costs
Ordering costs are the expense reflecting the manufacturer's costs to produce and implement the order.The ordering costs can be assessed by calculating the sum of an economic order for a logistics item.The overall shipping costs would therefore increase along with the sum of orders placed.Full setup costs are the amount of the cost of placing the order within the SC network, mathematically can be formulated as:

Cost of working capital blocked up in the inventory due to delay in transportation
In other terms, this cost is the interest on loans.This is taken to be 1.2% of the monthly unit sales price.Inventory costs for products shipped to the distributor due to the delay in transport time are measured in two parts, the transportation of products from the production plant () via warehouses () to the distributor (), (,  = 1 to ), and inventory cost due to delay in transport time between plant () and warehouses (), mathematically can be formulated as: In the above-formulated model, we have considered 11 constraints that have the following function: Constraint (1) ensures that each item's demand at each destination is met; constraint (2) is a capacity limitation at each facility (3) is the warehouse balance constraint.Constraints ( 4)-( 7) are freight rates on shipment according to weight; constraints (8) and ( 9) ensure that only one cycle time applies to an item.Constraints (10)

Solution approach
In the above-formulated model,  1 deals with optimizing total transportation cost, and  2 deals with optimizing total inventory cost.The solution obtained from the above-formulated model will be the optimum quantity to be transported from different sources to different destinations to minimize an ABC company's transporta-tion and inventory costs.Since many efficient algorithms are available to solve a MOOM, firstly, the problem must be converted into a single objective problem by using some compromise criterion.The above-formulated model can be expressed as a value function approach model, which is given below: a function that reflects the decision-maker's preferences among the objective vectors is called a value function.The value function approach is one of the most common methods for solving multi-objective optimization problems.The value function of an optimization problem gives the value attained by the objective function at a solution while only depending on the problem's parameters.The main advantage of using the value function approach is that it provides a complete ordering of the objective functions according to the decision-maker's preferences [24].  = 1, ∀  = 1, 2, . . .,  are weights according to the relative importance of the objective functions.For each objective function, the weights are frequently seen as broad measurements of relative importance.However, choosing a set of weights indicates a preference for one goal or another is challenging, and a reference is often ambiguous.It is undoubtedly conceivable to arrange or classify, by relative significance, a set of discrete choices since it generally includes some degree of uncertainty in measuring preference.In addition, the solution may not necessarily represent the desired preferences that are purportedly integrated into the weights, even with a complete understanding of the objectives and a good choice of weights.Transportation costs may be one of the most significant logistics expenses for companies with high inventories.However, the efficient management of logistics minimizes transportation costs, increasing companies' profit margins.With decreased transportation costs, companies may cut consumer retail prices and invest resources in more demanding transactions, for instance, manufacturing and inventory management.In this study, the higher weight is given to optimizing transportation cost with the weightage of 0.60, and 0.40 weightage for the optimization of inventory cost, which is decided by the company.

Real data from multi-capacitated ABC company
To explain the proposed work, we emphasize an actual case study of a distribution network of ABC company in India.In this section, we first present the data set of the ABC supply chain, then use the above-defined proposed approach to measure the overall cost of transportation and inventory cost simultaneously.The supply chain manager of a company decides to implement a new inventory management program.The supply chain manager tries to identify shipping policies that simultaneously optimize inventory and transportation costs.The supply chain manager understands that the management planning tool is a gateway to cost savings to implement this task successfully.He developes a mathematical model to optimize inventory and transportation costs together.To show the efficacy of the developed model of ABC-SCN, a numerical instance is with multiple products, warehouses, distributors, plants, shipment size, number of trucks, and the number of or-

Results and discussions
The model runs with different combinations to get insights into various costs on the optimal distribution network.The LINGO software solves the formulated problem, a comprehensive tool designed to build and solve linear and non-linear programming problems.The model is forced on the transportation and various inventory-related costs during lead time with a 99.97% service level decided by the company to achieve customer satisfaction.The current transportation cost of the company is very high (around 30% of product) in both products, which has to min.by this model.Sametime, the model, forces inventory to relate to cost under uncertainty.The model finds bottlenecks of its supply chain in its outbound logistics, i.e., the parts from the finished goods inventory to the customer.The obtained results are interpreted below:

(a) Considering transportation cost
The following optimal distribution network is in Figure 6.The optimal transportation cost for different numbers of shipments is in Tables A. 10 and A.11. Variation of total Transportation cost with the number of shipments is shown in Figure 7.At first, the total transportation cost does not vary much because as the shipment size reduces, it transports the CTV by a small truck.Nevertheless, as the number of shipments increases, shipment size reduces below small truck capacity; here, the total transportation cost is used instead of unit transportation cost.

(b) Considering total inventory cost
Working capital expenses are blocked up during the shipment of CTV because of transport time, the overall cost of maintaining inventories, and the cost of storing supplies.The optimal inventory cost for a different number of shipments/orders is in Table A.12.The variation of total inventory cost with the number of shipments/orders is in Figure 8. From the graph, one can see that the minimum optimal inventory cost comes at eight shipments, and when the number of shipments is six and ten, the optimal total logistics cost is minimum.As shown in

Comparative analysis with closely related models and techniques
In this section, the comparison considers the result with the previous studies of SCN models.The considered model for comparison is with certain assumptions.We employed the dataset used in this paper by making some adjustments in the considered models, and the dataset values have been changed or modified accordingly.A comparison of the result shows below.

Comparison with Charles et al. [14]
Charles et al. [14] proposed a fuzzy goal programming approach to solve the formulated multi-objective SCN problem.Three different goal programming approaches have been used to obtain the proposed approach's optimal solution.The proposed model leads to the development of a new SCN mathematical model generally aimed at reducing shipping costs and distribution times.The formulated model includes the quantity supplied from the vendor to the manufacturing plant and the quantity in the manufacturing plant that cannot surpass the capacity and quantity delivered via the warehouse.Its model covers the customer's demand, amounts of raw material that cannot exceed the volume had to customers from the warehouse, and the amount of the raw material purchased from the supplier.Following the approach suggested, the transportation and delivery time costs are more than by employing the same dataset on the model developed by Charles et al. [14].The transportation and inventory cost of the considered SCN problem is Rs. 4 132 966 and Rs. 26 261 126.The reason for getting the different results is that the new details on non-preference relations have been used in the model.It shows that the present manuscript's approach is advantageous compared to Charles et al. [14].

Comparison with Gupta et al. [25]
Gupta et al. [25] developed an efficient model based on the fuzzy goal programming technique for solving the multi-objective SCN by simultaneously minimizing the total shipping costs and time, including inventory volumes, available initial stock at each source and consumer demand, and availability of storage capacity at each destination, including the overall shipping budget.In the proposed work, after determining the problem's fuzzy goals, a satisfactory solution has been efficiently derived by updating the minimal satisfactory levels with considerations of the overall satisfactory solution.In the existing situation defined by Gupta et al. [25], the Company has only two products for transportation and inventory management.Here, we are discussing transportation and inventory management of 20 ′′ and 21 ′′ CTV.In the current study, the author tried to minimize the sum of positive deviations of the SCN with a limited number of constraints.By making only minor changes in the datasets, we have fit them to the model proposed by Gupta et al. [25].The preferred compromise solution with a deviation of 0.0963 and 0.0557 from the set goal is Rs.4022.351 and Rs. 26 432 610.The obtained solution satisfies all the termination conditions, and it is a satisfactory solution.

Comparison with techniques
Furthermore, we have compared our proposed technique with the other scalarization techniques, i.e., goal programming and fuzzy programming.After solving the formulated model with the goal programming technique, the total transportation and inventory cost are found to be Rs. 4 161 281 and Rs.28 328 860, respectively, with a positive deviation of 0.33 and 0.67 from the set goal.The fuzzy programming technique is a straightforward method.This technique gives us the set of non-dominated (efficient) solutions and an optimal compromise solution.After applying it to the formulated model, the total transportation and inventory costs are found to be Rs. 3 861 783 and Rs.27 254 255, respectively, with the maximum degree of overall satisfaction of 0.72 for the solution.

Sensitivity analysis
In this section, we have conducted a sensitivity analysis (Tab.A.14), through which managers will be able to identify areas for improvement that can boost the system's profitability or customer experience.To demonstrate the sensitivity of the numerical solution on the quantity shipped from one place to another destination, we have performed ten further sensitivity tests with increasing or decreasing demand and capacity limitation at each facility for the various 20 ′′ CTV and 21 ′′ CTV models for multi-distributors available in Bhopal, Calcutta, Delhi, Jaipur, Jamshedpur, Raipur, Ahmedabad, Bangalore, Bhiwandi, Calicut, Chandigarh, Chennai, Cochin, Coimbatore, Bhubaneswar, Ghaziabad, Guwahati, Haryana, Hyderabad, Mumbai, Lucknow, Madurai, Nagpur, Pune, Punjab, Vijayawada, Vishakhapatnam, Aurangabad, Patna, and Varanasi.In all ten cases, all the parameters of the formulated model except demand and capacity limitation are kept constant except for change in demand and capacity units.
Following the same pattern of the original problem, ten new compromise solutions using the value function approach have been generated and presented in Table A. 13.They show that the increase or decrease in demand and capacity units of the firms affects both the transportation cost and inventory cost because of the change in the allocation of units from one source to another destination.With this sensitivity analysis, it is evident that uncertainty in demand and supply directly impacts transportation and inventory costs because it directly impacts the total logistics expenses and costs in other functional areas of the organization.Since transportation and inventory costs are currently a key problem for the logistics sector as well as other businesses, accounting for around 40-50% of overall logistics expenses and 4-10% of final product selling prices.

Limitations and scope for future work
This research seeks to identify the best logistics network by taking into account varying logistical costs.This study is limited to the supply chain network design of Fast-Moving Durable Goods (FMDG), Fast Moving Consumer Goods (FMCG), and Medicine.The model is semi-static because it involves inventory cost, ordering cost, the plant's capacity & warehouse, lead time, and buffer stock parameters in multi-periods and multiproducts.Various other costs (shortage cost, cost of setting up the warehouse, and manufacturing cost of goods) and risk parameters have been proposed in various studies [15,28,29,36,43] can be incorporated in the present model.Some exciting research areas for the future are: -One of the critical problems of the organization is the minimization of the obsolescence of finished items in stock.The model built aims to simulate shipping costs and enforce a strategy to minimize this obsolescence.-This model can be incorporated into production facilities' output preparation and planning ().
-The company's primary concern is to minimize the overall costs of logistics.Just two tiers of destinations are in the present model.The research could be expanded to the retailers' stage.-Analysis can consider that we will transport a full truckload and find the optimal network and logistics cost in a shipment.-If freight costs are charged per unit rather than shipping, total logistics costs can be charged by growing demand.The model can then be generalized by realizing that by choosing the correct route, we can transfer the products to a variety of distributors.-The developed model of logistics can be combined with the geographical information system.Based on some famous case studies, we will try to modify our model that including [13] analyzing efficiency and customer perceptions of corporate social responsibility towards luxurious fashion products using multimethodological optimization techniques in the fashion supply chain industries.Garai and Sarkar [20] designed a multi-objective environmentally conscious closed-loop SCM that was customer-centric.Yadav et al. [57] introduced a flexible production system that involves strategic and operational decision-making by controlling by-products.

Managerial insight
Following are the managerial insight of the study: -The proposed model customizes (in terms of product & lead time) to solve the supply chain network problem in an uncertain environment, identify bottlenecks in the distribution network and develop a suitable model that eliminates the bottlenecks.-Our model includes two objectives.The first one is based on minimizing the total cost of a supply chain, including inventory, transportation & warehousing costs under risk, which leads to profitability; the second objective is to see the impact of lead time on product cost & delivery.The proposed model research is helpful to chain managers, purchasing managers, logistic managers, and operations managers for decisionmaking in SC network design, multi-product & multi-period product distribution problems, and warehouse management under a risk environment.-The traditional SC model for multi-product, multi-period problems considered the cost uncertain, but various unpredicted factors influence the cost.Therefore, this study considers the risk and the forecasted change by non-learning programming at an uncertain cost and time.This planning makes this model more robust, which managers may use in different sectors.-Our model gives the manager a clear vision to decide on the opening of plant, distribution centers, and warehouses to be more receptive to supply chain interruptions.

Conclusions
The present study's objective was to investigate a company's supply chain, identify bottlenecks in the distribution network, and develop a suitable model that eliminates the bottlenecks.The following activities were carried out: A literature review conducted on relevant topics such as supply chain, distribution network design model, and transportation model, which gave us an overview of the supply chain management system.A case company of ABC identified that the company has one of the country's largest distribution networks.Problems related to the ABC company were analyzed.It found that the main bottlenecks of its supply chain were in its outbound logistics, i.e., parts from the finished goods inventory to the customer.The distribution setup

Figure 1 .
Figure 1.Existing multi-directed supply chain of ABC.

Figure 2 .
Figure 2. Distribution information process at ABC company.

Figure 4
Figure 4 depicts the reasons for shortcomings in the existing supply chain of ABC company.

Figure 4 .
Figure 4. Causes and effect for ABC company.

Figure 6 .
Figure 6.Total transportation cost variations with number of shipments.

Figure 8 .
Figure 8.Total inventory cost variation with number of orders.

Figure 9 ,
Figure9, the optimal distribution network and shipment size, freight category, number of orders, and number of frights are required to transport the CTV.

Table 1 .
Shared process information of the reviewed works. parameters Represents the number of trucks required for the shipment of each product from multiple plant to multiple warehouses with cycle time-period  and shipment category    Represents the number of trucks required to transport the product from multiple warehouses to multiple distribution centers with cycle time-period  and shipment category (11)11)restrict the number of open plants and warehouses.The equations described in Section 4.2 have established different linear and non-linear cost and constraint functions.The logistics model has been compiled with all the linear and non-linear functions defined above.

Table A .
4. Transportation cost from multi-capacitated warehouse to multi-distributor by container type truck.

Table A .
4. Continued.Table A.8.Total forecasted demand of various 21 ′′ CTV model for multi-distributor.Table A.9. Shipment size, number of trucks and number of orders from multi-capacitated plant to capacitated warehouse.

Table A .
10. Shipment size, number of trucks and number of orders from multi-capacitated warehouse to multi-capacitated distributor.Table A.11. Shipment size, number of trucks, and number of orders from multi-capacitated plant to multi-capacitated distributor.

Table A .
12. Cost of shipments.

Table A .
13.Comparison with other techniques for 8 shipments.Table A.14. Sensitivity analysis of demand.