solved examples of transportation problem in operational research

Transportation Problem in Operational Research

The transportation problem in operational research is concerned with finding the minimum cost of transporting a single commodity from a given number of sources (e.g. factories) to a given number of destinations (e.g. warehouses). These types of problems can be solved by general network methods, but here we use a specific transportation algorithm.

The data of the model include

1. The level of supply at each source and the amount of demand at each destination.

2. The unit transportation cost of the commodity from each source to each destination.

Since there is only one commodity, a destination can receive its demand from more than one source. The objective is to determine how much should be shipped from each source to each destination so as to minimise the total transportation cost.

Types of Transportation Problem in Operational Research

• Balanced Transportation Problem
• Unbalanced Transportation Problem

Details about balanced and unbalanced transportation problem you find in attached pdf notes at end of this article.

Solution of the transportation problem

Stage I: Finding an initial basic feasible solution.

Stage II: Checking for optimality

Existence of Feasible Solution :  A necessary and sufficient condition for the existence of a feasible solution to the general transportation problem is that

Total supply = Total demand

Existence of Basic Feasible Solution:  The number of basic variables of the general transportation problem at any stage of feasible solution must be ( m + n  – 1). Now degenerate basic feasible solution (a feasible solution) involving exactly ( m + n –  1) positive variables is known  as  non-degenerate basic feasible solution  otherwise it is said to be  degenerate basic feasible . These allocations should be independent positions  in case of non-degenerate basic feasible solutions.

• Optimum Solution:  A feasible solution is said to be optimal, if it minimizes the total transportation cost.
• Unbalance TP  If total supply is not equal to total demand, then it balance with dummy source or destination.

Finding an Initial Basic Feasible Solutions

There are three methods as given below

• Northwest corner method
• Least cost method
• Vogel’s approximation method (or Penalty method)

Steps for North-West Corner Method

• Allocate the maximum amount allowable by the supply and demand constraints to the variable x11 (i.e. the cell in the top left corner of the transportation tableau).
• If a column (or row) is satisfied, cross it out. The remaining decision variables in that column (or row) are non-basic and are set equal to zero. If a row and column are satisfied simultaneously, cross only one out (it does not matter which).
• Adjust supply and demand for the non-crossed out rows and columns.
• Allocate the maximum feasible amount to the first available non-crossed out element in the next column (or row).
• When exactly one row or column is left, all the remaining variables are basic and are assigned the only feasible allocation.

Note: Solved example you find in video or in PDF

Steps for Least Cost Method

• Assign as much as possible to the cell with the smallest unit cost in the entire tableau. If there is a tie then choose arbitrarily.
• Cross out the row or column which has satisfied supply or demand. If a row and column are both satisfied then cross out only one of them.
• Adjust the supply and demand for those rows and columns which are not crossed out.

Steps for Vogel’s Approximation Method

• Determine a  penalty cost  for each row (column) by subtracting the lowest unit cell cost in the row (column) from the  next lowest unit cell cost  in the same row (column).
• Identify the row or column with the greatest penalty cost. Break the ties arbitrarily (if there are any). Allocate as much as possible to the variable with the lowest unit cost in the selected row or column. Adjust the supply and demand and cross out the row or column that is already satisfied. If a row and column are satisfied simultaneously, only cross out one of the two and allocate a supply or demand of zero to the one that remains.
• If there is exactly one row or column left with a supply or demand of zero, stop.
• If there is one row (column) left with a  positive  supply (demand), determine the basic variables in the row (column) using the Minimum Cell Cost Method. Stop.
• If all of the rows and columns that were not crossed out have zero supply and demand (remaining), determine the basic  zero  variables using the Minimum Cell Cost Method. Stop.
• In any other case, continue with Step 1.

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• Open access
• Published: 20 July 2023

Solving a multi-objective solid transportation problem: a comparative study of alternative methods for decision-making

• Mohamed H. Abdelati   ORCID: orcid.org/0000-0002-5034-7323 1 ,
• Ali M. Abd-El-Tawwab 1 ,
• Elsayed Elsayed M. Ellimony 2 &
• M Rabie 1  

Journal of Engineering and Applied Science volume  70 , Article number:  82 ( 2023 ) Cite this article

3471 Accesses

3 Citations

Metrics details

The transportation problem in operations research aims to minimize costs by optimizing the allocation of goods from multiple sources to destinations, considering supply, demand, and transportation constraints. This paper applies the multi-dimensional solid transportation problem approach to a private sector company in Egypt, aiming to determine the ideal allocation of their truck fleet.

In order to provide decision-makers with a comprehensive set of options to reduce fuel consumption costs during transportation or minimize total transportation time, a multi-objective approach is employed. The study explores the best compromise solution by leveraging three multi-objective approaches: the Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. Optimal solutions are derived for time and fuel consumption objectives, offering decision-makers a broad range to make informed decisions for the company and the flexibility to adapt them as needed.

Lingo codes are developed to facilitate the identification of the best compromise solution using different methods. Furthermore, non-dominated extreme points are established based on the weights assigned to the different objectives. This approach expands the potential ranges for enhancing the transfer problem, yielding more comprehensive solutions.

This research contributes to the field by addressing the transportation problem practically and applying a multi-objective approach to support decision-making. The findings provide valuable insights for optimizing the distribution of the truck fleet, reducing fuel consumption costs, and improving overall transportation efficiency.

Introduction

The field of operations research has identified the transportation problem as an optimization issue of significant interest [ 1 , 2 ]. This problem concerns determining the optimal approach to allocate a given set of goods that come from particular sources to the designated destinations to minimize the overall transportation costs [ 3 ]. The transportation problem finds applications in various areas, including logistics planning, distribution network design, and supply chain management. Solving this problem relies on the assumption that the supply and demand of goods are known, as well as the transportation cost for each source–destination pairing [ 4 , 5 ].

Solving the transportation problem means finding the right quantities of goods to be transported from the sources to the destinations, given the supply and demand restrictions. The ultimate goal is to minimize the total transportation cost, which is the sum of the cost for each shipment [ 6 ]. Various optimization algorithms have been developed for this problem, such as the North-West Corner Method, the Least Cost Method, and Vogel’s Approximation Method [ 7 ].

A solid transportation problem (STP) is a related transportation problem that centers around a single commodity, which can be stored at interim points [ 8 ]. These interim points, known as transshipment points, act as origins and destinations. The STP involves determining the most efficient means of transporting the commodity from the sources to the destinations, while minimizing transportation costs by going through the transshipment points. The STP has real-world applications in container shipping, air cargo transportation, and oil and gas pipeline transportation [ 9 , 10 ].

Multi-dimensional solid transportation problem (MDSTP) represents a variation on the STP, incorporating multiple commodities that vary in properties such as volume, weight, and hazard level [ 11 ]. The MDSTP aims to identify the best way to transport each commodity from the sources to the destinations, taking into account the capacity restrictions of transshipment points and hazardous commodity regulations [ 12 ]. The MDSTP is more complex than the STP and requires specific algorithms and models for its resolution.

Solving the STP and MDSTP requires identifying the most effective routing of commodities and considering the storage capacity of transshipment points. The goal is to minimize total transportation costs while satisfying supply and demand constraints and hazardous material regulations. Solutions to these problems include the Network Simplex Method, Branch and Bound Method, and Genetic Algorithm [ 13 ]. Solving the STP and MDSTP contributes valuable insights into the design and operation of transportation systems and supports improved sustainability and efficiency.

In the field of operations research, two critical research areas are the multi-objective transportation problem (MOTP) and the multi-objective solid transportation problem (MOSTP) [ 14 ]. The MOTP aims to optimize the transportation of goods from multiple sources to various destinations by considering multiple objectives, including minimizing cost, transportation time, and environmental impacts. The MOSTP, on the other hand, focuses on the transportation of solid materials, such as minerals or ores, and involves dealing with multiple competing objectives, such as cost, time, and quality of service. These problems are essential in logistics and supply chain management, where decision-makers must make optimal transportation plans by considering multiple objectives. Researchers and practitioners often employ optimization techniques, such as mathematical programming, heuristics, and meta-heuristics, to address these challenges efficiently [ 15 ].

Efficient transportation planning is essential for moving goods from their source to the destination. This process involves booking different types of vehicles and minimizing the total transportation time and cost is a crucial factor to consider. Various challenges can affect the optimal transportation policy, such as the weight and volume of products, the availability of specific vehicles, and other uncertain parameters. In this regard, several studies have proposed different approaches to solve the problem of multi-objective solid transportation under uncertainty. One such study by Kar et al. [ 16 ] used fuzzy parameters to account for uncertain transportation costs and time, and two methods were employed to solve the problem, namely the Zimmermann Method and the Global Criteria Method.

Similarly, Mirmohseni et al. [ 17 ] proposed a fuzzy interactive probabilistic programming approach, while Kakran et al. [ 18 ] addressed a multi-objective capacitated solid transportation problem with uncertain zigzag variables. Additionally, Chen et al. [ 19 ] investigated an uncertain bicriteria solid transportation problem by using uncertainty theory properties to transform the models into deterministic equivalents, proposing two models, namely the expected value goal programming and chance-constrained goal programming models [ 20 ]. These studies have contributed to developing different approaches using fuzzy programming, uncertainty theory, and related concepts to solve multi-objective solid transportation problems with uncertain parameters.

This paper presents a case study carried out on a private sector company in Egypt intending to ascertain the minimum number of trucks required to fulfill the decision-makers’ objectives of transporting the company’s fleet of trucks from multiple sources to various destinations. This objective is complicated by the diversity of truck types and transported products, as well as the decision-makers’ multiple priorities, specifically the cost of fuel consumption and the timeliness of truck arrival.

In contrast to previous research on the transportation problem, this paper introduces a novel approach that combines the multi-dimensional solid transportation problem framework with a multi-objective optimization technique. Building upon previous studies, which often focused on single-objective solutions and overlooked specific constraints, our research critically analyzes the limitations of these approaches. We identify the need for comprehensive solutions that account for the complexities of diverse truck fleets and transported products, as well as the decision-makers’ multiple priorities. By explicitly addressing these shortcomings, our primary goal is to determine the minimum number of trucks required to fulfill the decision-makers’ objectives, while simultaneously optimizing fuel consumption and transportation timeliness. Through this novel approach, we contribute significantly to the field by advancing the understanding of the transportation problem and providing potential applications in various domains. Our research not only offers practical solutions for real-world scenarios but also demonstrates the potential for improving transportation efficiency and cost-effectiveness in other industries or contexts. The following sections will present a comparative analysis of the proposed work, highlighting the advancements and novelty introduced by our approach.

Methods/experimental

This study uses a case study from Egypt to find the optimal distribution of a private sector company’s truck fleet under various optimization and multi-objective conditions. Specifically, the study aims to optimize the distribution of a private sector company’s truck fleet by solving a multi-objective solid transportation problem (MOSTP) and comparing three different methods for decision-making.

Design and setting

This study uses a case study design in a private sector company in Egypt. The study focuses on distributing the company’s truck fleet to transport products from factories to distribution centers.

Participants or materials

The participants in this study are the transportation planners and managers of the private sector company in Egypt. The materials used in this study include data on the truck fleet, sources, destinations, and products.

Processes and methodologies

The study employs the multi-objective multi-dimensional solid transportation problem (MOMDSTP) to determine the optimal solution for the company’s truck fleet distribution, considering two competing objectives: fuel consumption cost and total shipping time. The MOMDSTP considers the number and types of trucks, sources, destinations, and products and considers the supply and demand constraints.

To solve the MOMDSTP, three decision-making methods are employed: Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. The first two methods directly yield the best compromise solution (BCS), whereas the last method generates non-dominated extreme points by assigning different weights to each objective. Lingo software is used to obtain the optimal solutions for fuel consumption cost and time and the BCS and solutions with different weights for both objectives.

Ethics approval and consent

This study does not involve human participants, data, or tissue, nor does it involve animals. Therefore, ethics approval and consent are not applicable.

Statistical analysis

Statistical analysis is not conducted in this study. However, the MOMDSTP model and three well-established decision-making methods are employed to derive the optimal distribution of the company’s truck fleet under various optimization and multi-objective conditions.

In summary, this study uses a case study design to find the optimal distribution of a private sector company’s truck fleet under various optimization and multi-objective conditions. The study employs the MOMDSTP and three methods for decision-making, and data on the truck fleet, sources, destinations, and products are used as materials. Ethics approval and consent are not applicable, and statistical analysis is not performed.

Multi-objective transportation problem

The multi-objective optimization problem is a complex issue that demands diverse approaches to determine the most satisfactory solution. Prevalent techniques employed in this domain include the Weighted Sum Method, Minimum Distance Method, Zimmermann Programming Technique, and Global Criteria Method. Each method offers its own benefits and limitations, and the selection of a specific method depends on the nature of the problem and the preferences of the decision-makers [ 21 ].

This section discusses various methodologies employed to identify the most optimal solution(s) for the multi-objective multi-dimensional solid transportation problem (MOMDSTP), which is utilized as the basis for the case study. These methodologies encompass the Minimum Distance Method (MDM), the Zimmermann Programming Technique, and the Global Criteria Method [ 22 ].

Zimmermann Programming Technique

The Zimmermann Programming Technique (ZPT) is a multi-objective optimization approach that was developed by Professor Hans-Joachim Zimmermann in the late 1970s. This technique addresses complex problems with multiple competing objectives that cannot be optimized simultaneously. Additionally, it incorporates the concept of an “aspiration level,” representing the minimum acceptable level for each objective. The aspiration level ensures that the solution obtained is satisfactory for each objective. If the solution does not meet the aspiration level for any objective, the weights are adjusted, and the optimization process is iterated until a satisfactory solution is obtained.

A key advantage of ZPT is its ability to incorporate decision-makers’ preferences and judgments into the decision-making process. The weights assigned to each objective are based on the decision-maker’s preferences, and the aspiration levels reflect their judgments about what constitutes an acceptable level for each objective [ 23 ].

The Zimmermann Programming Technique empowers decision-makers to incorporate multiple objectives and achieve a balanced solution. By assigning weights to objectives, a trade-off can be made to find a compromise that meets various criteria. For example, this technique can optimize cost, delivery time, and customer satisfaction in supply chain management [ 24 ]. However, the interpretation of results may require careful consideration, and computational intensity can increase with larger-scale and complex problems.

 In order to obtain the solution, each objective is considered at a time to get the lower and upper bounds for that objective. Let for objective, and are the lower (min) and upper (max) bounds. The membership functions of the first and second objective functions can be generated based on the following formula [ 25 ]:

Next, the fuzzy linear programming problem is formulated using the max–min operator as follows:

Maximize min \({\mu }_{k}\left({F}_{k}\left(x\right)\right)\)   

Subject to \({g}_{i}\left(x\right) \left\{ \le ,= , \ge \right\}{b}_{i}\mathrm{ where }\;i = 1, 2, 3, ..., m.\)   

Moreover, x ≥ 0.

Global Criteria Method

The Global Criteria Method is a multi-objective optimization method that aims to identify the set of ideal solutions based on predetermined criteria. This method involves defining a set of decision rules that assess the feasibility and optimality of the solutions based on the objectives and constraints [ 26 ]. By applying decision rules, solutions that fail to meet the predetermined criteria are eliminated, and the remaining solutions are ranked [ 27 ].

The Global Criteria Method assesses overall system performance, aiding decision-makers in selecting solutions that excel in all objectives. However, it may face challenges when dealing with conflicting objectives [ 28 ]. Furthermore, it has the potential to overlook specific details, and the choice of aggregation function or criteria can impact the results by favoring specific solutions or objectives.

Let us consider the following ideal solutions:

f 1* represents the ideal solution for the first objective function,

f 2* represents the ideal solution for the second objective function, and

n 1* represents the ideal solution for the nth objective function.

Objective function formula:

Minimize the objective function F  =  \(\sum_{k=1}^{n}{(\frac{{f}_{k}\left({x}^{*}\right)-{f}_{k}(x)}{{f}_{k}({x}^{*})})}^{p}\)

Subject to the constraints: g i ( x ) \(\le\) 0, i  = 1, 2,.., m

The function fk( x ) can depend on variables x 1 , x 2 , …, x n .

Minimum Distance Method

The Minimum Distance Method (MDM) is a novel distance-based model that utilizes the goal programming weighted method. The model aims to minimize the distances between the ideal objectives and the feasible objective space, leading to an optimal compromise solution for the multi-objective linear programming problem (MOLPP) [ 29 ]. To solve MOLPP, the proposed model breaks it down into a series of single objective subproblems, with the objectives transformed into constraints. To further enhance the compromise solution, priorities can be defined using weights, and a criterion is provided to determine the best compromise solution. A significant advantage of this approach is its ability to obtain a compromise solution without any specific preference or for various preferences.

The Minimum Distance Method prioritizes solutions that closely resemble the ideal or utopian solution, assisting decision-makers in ranking and identifying high-performing solutions. It relies on a known and achievable ideal solution, and its sensitivity to the chosen reference point can influence results. However, it does not provide a comprehensive trade-off solution, focusing solely on proximity to the ideal point [ 30 ].

The mathematical formulation for MDM for MOLP is as follows:

The formulation for multi-objective linear programming (MOLP) based on the minimum distance method is referred to[ 31 ]. It is possible to derive the multi-objective transportation problem with two objective functions using this method and its corresponding formula.

Subject to the following constraints:

f * 1 , f * 2 : the obtained ideal objective values by solving single objective STP.

w 1 , w 2 : weights for objective1 and objective2 respectively.

f 1, f 2: the objective values for another efficient solution.

d : general deviational variable for all objectives.

\({{c}_{ij}^{1}, c}_{ij}^{2}\) : the unit cost for objectives 1 and 2 from source i to destination j .

\({{x}_{ij}^{1}, x}_{ij}^{2}\) : the amount to be shipped when optimizing for objectives 1 and 2 from source i to destination j .

Mathematical model for STP

The transportation problem (TP) involves finding the best method to ship a specific product from a defined set of sources to a designated set of destinations, while adhering to specific constraints. In this case, the objective function and constraint sets take into account three-dimensional characteristics instead of solely focusing on the source and destination [ 32 ]. Specifically, the TP considers various modes of transportation, such as ships, freight trains, cargo aircraft, and trucks, which can be used to represent the problem in three dimensions When considering a single mode of transportation, the TP transforms into a solid transportation problem (STP), which can be mathematically formulated as follows:

The mathematical form of the solid transportation problem is given by [ 33 ]:

Subject to:

Z = the objective function to be minimized

m = the number of sources in the STP

n = the number of destinations in the STP

p = the number of different modes of transportation in the STP

x ijk represents the quantity of product transported from source i to destination j using conveyance k

c ijk = the unit transportation cost for each mode of transportation in the STP

a i = the amount of products available at source i

b j = the demand for the product at destination j

e k = the maximum amount of product that can be transported using conveyance k

The determination of the size of the fleet for each type of truck that is dispatched daily from each factory to all destinations for the transportation of various products is expressed formally as follows:

z ik denotes the number of trucks of type k that are dispatched daily from factory i .

C k represents the capacity of truck k in terms of the number of pallets it can transport.

x ijk denotes a binary decision variable that is set to one if truck k is dispatched from factory i to destination j to transport product p , and zero otherwise. The summation is performed over all destinations j and all products p .

This case study focuses on an Egyptian manufacturing company that produces over 70,000 pallets of various water and carbonated products daily. The company has 25 main distribution centers and eight factories located in different industrial cities in Egypt. The company’s transportation fleet consists of hundreds of trucks with varying capacities that are used to transport products from factories to distribution centers. The trucks have been classified into three types (type A, type B, and type C) based on their capacities. The company produces three different types of products that are packaged in pallets. It was observed that the sizes and weights of the pallets are consistent across all product types The main objective of this case study is to determine the minimum number of each truck type required in the manufacturer’s garage to minimize fuel consumption costs and reduce product delivery time.

The problem was addressed by analyzing the benefits of diversifying trucks and implementing the solid transport method. Subsequently, the problem was resolved while considering the capacities of the sources and the requirements of the destinations. The scenario involved shipping products using a single type of truck, and the fuel consumption costs were calculated accordingly. The first objective was to reduce the cost of fuel consumption on the one-way journey from the factories to the distribution centers. The second objective was to reduce the time of arrival of the products to the destinations. The time was calculated based on the average speed of the trucks in the company’s records, which varies depending on the weight and size of the transported goods.

To address the multiple objectives and the uncertainty in supply and demand, an approach was adopted to determine the minimum number of trucks required at each factory. This approach involved determining the maximum number of trucks of each type that should be present in each factory under all previous conditions. The study emphasizes the significance of achieving a balance between reducing transportation costs and time while ensuring trucks are capable of accommodating quantities of any size, thus avoiding underutilization.

Figure  1 presents the mean daily output, measured in pallets, for each factory across three distinct product types. Additionally, Fig.  2 displays the average daily demand, measured in pallets, for the distribution centers of the same three product types.

No. of pallets in each source

No. of pallets in each destination

Results and discussion

As a result of the case study, the single objective problems of time and fuel consumption cost have been solved. The next step is to prepare a model for the multi-objective multi-dimensional solid transportation problem. Prior to commencing, it is necessary to determine the upper and lower bounds for each objective.

Assuming the first objective is fuel consumption cost and the second objective is time, we calculate the upper and lower bounds as follows:

The lower bound for the first objective, “cost,” is generated from the optimal solution for its single-objective model, denoted as Z 1 ( x 1 ), and equals 70,165.50 L.E.

The lower bound for the second objective, “time,” is generated from the optimal solution for its single-objective model, denoted as Z 2 ( x 2 ), and equals 87,280 min.

The upper bound for the first objective is obtained by multiplying c ijkp for the second objective by x ijkp for the first objective. The resulting value is denoted as Z 1 ( x 2 ) and equals 73,027.50 L.E.

The upper bound for the second objective is obtained by multiplying t ijkp for the first objective by x ijkp for the second objective. The resulting value is denoted as Z 2 ( x 1 ) and equals 88,286.50 min.

As such, the aspiration levels for each objective are defined from the above values by evaluating the maximum and minimum value of each objective.

The aspiration level for the first objective, denoted as F 1, ranges between 70,165.50 and 73,027.50, i.e., 70,165.50 <  =  F 1 <  = 730,27.50.

The aspiration level for the second objective, denoted as F 2, ranges between 87,280 and 88,286.50, i.e., 87,280 <  =  F 2 <  = 88,286.50.

The objective function for the multi-objective multidimensional solid transport problem was determined using the Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. The first two methods directly provided the best compromise solution (BCS), while the last method generated non-dominated extreme points by assigning different weights to each objective and finding the BCS from them. The best compromise solution was obtained using the Lingo software [ 34 ]. Table 1 and Fig.  3 present the objective values for the optimal solutions of fuel consumption cost and time, the best compromise solution, and solutions with different weights for both objectives. Figure  4 illustrates the minimum required number of each type of truck for daily transportation of various products from sources to destinations.

Objective value in different cases

Ideal distribution of the company’s truck fleet

The primary objective of the case study is to determine the minimum number of trucks of each type required daily at each garage for transporting products from factories to distribution centers. The minimum number of trucks needs to be flexible, allowing decision-makers to make various choices, such as minimizing fuel consumption cost, delivery time, or achieving the best compromise between different objectives. To determine the minimum number of required trucks, we compare all the previously studied cases and select the largest number that satisfies the condition: min Zik (should be set) = max Zik (from different cases). Due to the discrepancy between the truck capacity and the quantity of products to be transported, the required number of trucks may have decimal places. In such cases, the fraction is rounded to the nearest whole number. For example, if the quantity of items from a location requires one and a half trucks, two trucks of the specified type are transported on the first day, one and a half trucks are distributed, and half a truck remains in stock at the distribution center. On the next day, only one truck is transferred to the same distribution center, along with the semi-truck left over from the previous day, and so on. This solution may be preferable to transporting trucks that are not at full capacity. Table 2 and Fig.  5 depict the ideal distribution of the company’s truck fleet under various optimization and multi-objective conditions.

Min. No. of trucks should be set for different cases

Conclusions

In conclusion, this research paper addresses the critical issue of optimizing transportation within the context of logistics and supply chain management, specifically focusing on the methods known as the solid transportation problem (STP) and the multi-dimensional solid transportation problem (MDSTP). The study presents a case study conducted on a private sector company in Egypt to determine the optimal distribution of its truck fleet under different optimization and multi-objective conditions.

The research utilizes the multi-objective multi-dimensional solid transportation problem (MOMDSTP) to identify the best compromise solution, taking into account fuel consumption costs and total shipping time. Three decision-making methods, namely the Zimmermann Programming Technique, the Global Criteria Method, and the Minimum Distance Method, are employed to derive optimal solutions for the objectives.

The findings of this study make a significant contribution to the development of approaches for solving multi-objective solid transportation problems with uncertain parameters. The research addresses the complexities of diverse truck fleets and transported products by incorporating fuzzy programming, uncertainty theory, and related concepts. It critically examines the limitations of previous approaches that often focused solely on single-objective solutions and overlooked specific constraints.

The primary objective of this research is to determine the minimum number of trucks required to fulfill decision-makers objectives while optimizing fuel consumption and transportation timeliness. The proposed approach combines the framework of the multi-dimensional solid transportation problem with a multi-objective optimization technique, offering comprehensive solutions for decision-makers with multiple priorities.

This study provides practical solutions for real-world transportation scenarios and demonstrates the potential for enhancing transportation efficiency and cost-effectiveness in various industries or contexts. The comparative analysis of the proposed work highlights the advancements and novelty introduced by the approach, emphasizing its significant contributions to the field of transportation problem research.

Future research should explore additional dimensions of the multi-objective solid transportation problem and incorporate other decision-making methods or optimization techniques. Additionally, incorporating uncertainty analysis and sensitivity analysis can enhance the robustness and reliability of the proposed solutions. Investigating the applicability of the approach in diverse geographical contexts or industries would yield further insights and broaden the potential applications of the research findings.

Availability of data and materials

The data that support the findings of this study are available from the company but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request. Please note that some data has been mentioned in the form of charts as agreed with the company.

Abbreviations

Solid transportation problem

Multi-objective solid transportation problems

Multi-dimensional solid transportation problem

Multi-objective multi-dimensional solid transportation problem

Best compromise solution

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Abdelati, M.H., Abd-El-Tawwab, A.M., Ellimony, E.E.M. et al. Solving a multi-objective solid transportation problem: a comparative study of alternative methods for decision-making. J. Eng. Appl. Sci. 70 , 82 (2023). https://doi.org/10.1186/s44147-023-00247-z

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• Decision-making
• Multi-objective
• Solid transpiration

Transportation Problems ¶

Basic set up ¶.

In the Introduction to MIP, we described the wide range of problems that can be modeled and solved with MIP, and described the setup of some widely used problem models. In this notebook, we introduce the transportation problem, as a special type of MIP problem particularly important for logistic operations. In the literature, the general transportation problem deals with the distribution of any commodity (generally goods) from a group of origins of the supply chain, or source facilities, called sources to the endpoints of the model of the supply chain, the receiving centers, called destinations.

There are different types of goods or services that can be modeled with this model. Sources can be production centers, or warehouses. Destinations can be warehouse, demand regions or end customers, depending on the model.

However, in the general problem formulation, we have \(i \in [1, ..., n]\) source nodes and \(j \in [1, ..., m]\) destination nodes. The objective is to minimize the overall costs of the distribution of goods at a minimum costs, subject to a set of constraints.

Regarding the constraints, let us consider two types of constraints:

supply constraints: These constraints represent the limited capacity of the sources. For instance, production centers may have a limited production capacity, or source warehouses may have a limited storage capacity. Let us note \(s_i\) the source capacity of source node \(i\) .

demand constraints: The demand constraints represent the distribution requirements at the destinations: what are the quantities of the goods required at each destination. Let us note as \(d_j\) the demand of destination node \(j\)

Now, based on this, let us define our set of decision variables as:

\(x_{ij}\) : (Integer) units to transport from source \(i\) to destination \(j\)

Assuming a distribution cost of \(c_{ij}\) for each transported unit yields:

\(\min z = \sum_{i=1}^n{\sum_{j=1}^m{c_{ij}*x_{ij}}}\)

\(\sum_{j=1}^m{x_{ij}} \leq s_i \quad \forall i\)

\(\sum_{i=1}^n{x_{ij}} \geq d_j \quad \forall j\)

That is, the total number of units that leave a source cannot exceed the capacity of the source, and the total number of units that reach a destination must be at least equal to the demand.

Note that the model has a feasible solution only if:

\(\sum_{i=1}^n{s_i} \geq \sum_{j=1}^m{d_j}\)

That, is, the problem is only feasible if the overall capacity is sufficient to meet the overall demand.

It is important to note that we may allow ourselves less slack by making any of the three inequalities above equalities.

Sourcing design ¶

In the basic set up above, it is assumed that it is possible to transport units from any source node to any destination. This implies that the design of the distribution network is known. But this may not be always the case. For instance, we may have different possible locations for source warehouse, each with different associated costs. In this case, our decision variables are not only the number of units to transport, but also determine from which subset of source nodes they must be transported. Thus, we can introduce a new set of decision variables:

\(Y_i\) : (Binary) determines whether units to any destination may be delivered from source node \(i\)

We need to introduce new logical constraints to ensure that we do not deliver any units from source nodes that are not selected:

\(\sum_{j=1}^m{x_{ij}} \leq M*Y_i \quad \forall i\)

Where M is a large number. Since \(Y_i\) is binary, this constraint forces that, in case that it is equal to zero, no units are sourced from source node \(i\) .

This logical constraint can be merged with the source capacity constraint as:

\(\sum_{j=1}^m{x_{ij}} \leq s_i*Y_i \quad \forall i\)

This constraint takes into account the logical constraint and the capacity constraint. Since both constraint are equivalent, but the former is more restrictive and renders the first one irrelevant.

We can now take into account in the objective function fixed costs that do not depend on the number of units. For instance, let us consider that the source warehouses have a fixed operation costs \(f_i\) . The problem in the basic set up becomes:

\(\min z = \sum_{i=1}^n{\sum_{j=1}^m{c_{ij}*x_{ij}}} + \sum_{i=1}^n{f_i*Y_i}\)

Single source ¶

The single source constraint imposes that all the demand of a destination node is transported from the same origin. With this constraint, now the decision variables \(x_{ij}\) become binary:

\(X_{ij}\) : (Binary) determines if source \(i\) delivers all demand to destination \(j\) {1 if yes, 0 otherwise}

The demand constraint now becomes:

\(\sum_{i=1}^n{d_j*X_{ij}} = d_j \quad \forall j\)

Note that we now multiply each binary decision variable times the demand of each destination. We have changed the type of the constraint from \(\geq\) to \(=\) to avoid any slack. Although this change is optional, if not applied, the slack will be a multiple of the demand. Note that this is exactly the same requirement:

\(\sum_{i=1}^n{X_{ij}} = 1 \quad \forall j\)

We just divided by \(d_j\) the right hand side and the left hand side.

Also, we need to factor this into the objective function as:

\(\min z = \sum_{i=1}^n{\sum_{j=1}^m{c_{ij}*d_j*X_{ij}}}\)

Since \(X_{ij}\) is binary, and only equal to 1 for a destination node.

Network design ¶

In the set up above, we assume that the cost per unit from sources to destinations is fixed. In many cases however, this cost depends to a great extent on the design considerations taken in the distribution network between sources and destination. Similar to what we did with the sourcing design, we may be interested on evaluating the impact in the cost of the design of the distribution network, for instance, the location of intermediate nodes or junctions between the source nodes and destination nodes. In literature, this is known as the transshipment problem in the literature. Let us come back to the original problem, and assume that the source nodes are production plants, the destination nodes are retailer regions, and that we have intermediate warehouses that we can use to optimise costs. Let us assume that the production plants are fixed (i.e. the sourcing design is known), but that we can select a subset of warehouses from a set of candidate locations.

First, let us modify the indices so that the distribution stages (from production plants through warehouses to regions) follow an alphabetical order:

\(i\) : Production plants \(i \in [1, ..., n]\)

\(j\) : Possible warehouse locations \(j \in [1, ..., m]\)

\(k\) : Retailer regions \(k \in [1, ..., l]\)

Let us use the following notation for the coefficients of our problem:

\(d_k\) : yearly demand in retail region k

\(a_{ij}\) : cost of transporting 1 unit from plant i to warehouse j

\(b_{jk}\) : cost of transporting 1 unit from warehouse j to retailer region k

\(F_{j}\) : yearly operation costs of warehouse j

\(c_i\) : yearly production capacity of plant i

Since we can decide the location of the warehouses, and the units to transport at each stage, the decision variables now become:

\(Y_{j}\) : (Binary) { 1 if a warehouse is placed in location j } {0 otherwise}

\(s_{ij}\) : integer units transported from plant i to warehouse j

\(t_{jk}\) : integer units transported from warehouse j to region k

The objective function is to minimise costs:

\(\min z = \sum_{i=1}^{n}{\sum_{j=1}^{m}{a_{ij}*s_{ij}}} + \sum_{j=1}^{m}{\sum_{k=1}^{l}{b_{jk}*t_{jk}}} + \sum_{j=1}^{m}{F_j*Y_j}\)

Now, for the constraints, clearly, we need to satisfy all the constraints. The units sourced from each production plant cannot exceed the production capacity:

\(\sum_{j=1}^{m}{s_{ij}} \leq c_i \quad \forall i\)

We also need to ensure that the demand at each region is satisfied:

\(\sum_{j=1}^{m}{t_{jk}} \geq d_k \quad \forall k\)

Now, we need to ensure the material flow, that is, that the unit that leave a warehouse are not less than the units that enter the same warehouse:

\(\sum_{i=1}^{n}{s_{ij}} \geq \sum_{k=1}^{l}{t_{jk}} \quad \forall j\)

And that a warehouse is built if it supplies to any region:

\(\sum_{k=1}^{l}{t_{jk}} \leq M*Y_j \quad \forall j\)

Known variations ¶

Combination of single-source in design constraints ¶.

Any combinations of the variations above can be found in a problem instance. For instance, we might be interested in evaluating our network design with a single source constraint on intermediate nodes. Note that when we introduce a single source constraint in our problem, the decision variable that models the transport of units becomes binary. For instance, in the network design problem above, the decision variables Any combinations of the variations above can be found in a problem instance. For instance, we might be interested in evaluating our network design with a single source constraint on intermediate nodes. Note that when we introduce a single source constraint in our problem, the decision variable that models the transport of units becomes binary. For instance, in the network design problem above, the decision variables \(t_{jk}\) become:

\(T_{jk}\) : (Binary) {1 units from warehouse j transported to region k, 0 otherwise}

Since the variable are now binary, the single source constraint modifies the demand constraint, which now becomes:

\(\sum_{j=1}^n{T_{jk}} = 1 \quad \forall k\)

Now, since all \(T_{jk}\) are binary, the logical constraint that rules the construction of warehouses can be written as:

\(T_{jk} \leq Y_{j} \quad \forall j, \forall k\)

Other indices ¶

We may also introduce other indices in the problem, most common indices are:

Type of product : We can have different costs or demands depending on the type of product

Periods : The costs, or the demands can be dynamic and depend on a discrete time period, like the year, the month, or the day, so we might be interested in optimising the distribution network for a set of periods.

When we introduce a new index into the model, the size of the problem (number of coefficients and decision variables) will grow accordingly.

The transportation problem with conflicts

• S.I.: CoDIT2017-Combinatorial Optimization
• Published: 21 August 2018
• Volume 298 , pages 207–227, ( 2021 )

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• Annette M. C. Ficker 1 ,
• Frits C. R. Spieksma 2 &
• Gerhard J. Woeginger 3  

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The transportation problem is a fundamental problem in operations research, where items need to be transported from supply nodes (each with a given supply) to demand nodes (each with a given demand) in the cheapest possible way. Here, we are interested in a generalization of the transportation problem where, each supply node has a (possibly empty) set of conflicting pairs of demand nodes, and each demand node a (possibly empty) set of conflicting pairs of supply nodes. Each supply node may only send supply to at most one demand node of each conflicting pair. Likewise, each demand node may only receive supply from at most one supply node of each conflicting pair. We call the resulting problem the transportation problem with conflicts (TPC). We show that the complexity of TPC depends upon the structure of the so-called conflict graph that follows from the conflicting pairs. More concrete, we show that for many graph-classes the corresponding TPC remains NP-hard, and for some special cases we derive constant factor approximation algorithms.

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Acknowledgements

This research has been partially funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.

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Annette M. C. Ficker

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Frits C. R. Spieksma

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Gerhard J. Woeginger

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Ficker, A.M.C., Spieksma, F.C.R. & Woeginger, G.J. The transportation problem with conflicts. Ann Oper Res 298 , 207–227 (2021). https://doi.org/10.1007/s10479-018-3004-y

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Operations Research/Transportation and Assignment Problem

The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first.

Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money which depends on several factors and varies for each choice of factory and outlet. The total amount of the product a particular factory makes is fixed and so is the total amount a particular outlet can store. The problem is to decide how much of the product should be supplied from each factory to each outlet so that the total cost is minimum.

Let us consider an example.

Suppose an auto company has three plants in cities A, B and C and two major distribution centers in D and E. The capacities of the three plants during the next quarter are 1000, 1500 and 1200 cars. The quarterly demands of the two distribution centers are 2300 and 1400 cars. The transportation costs (which depend on the mileage, transport company etc) between the plants and the distribution centers is as follows:

Which plant should supply how many cars to which outlet so that the total cost is minimum?

The problem can be formulated as a LP model:

The whole model is:

subject to,

The problem can now be solved using the simplex method. A convenient procedure is discussed in the next section.

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Transportation Problem

4.1  introduction.

The term “transportation” is somewhat deceptive it appears to be restricted only to transportation systems, but the case is different. In fact many of the resource allocation problems arising in production systems can be treated as transportation problems. Typical examples are production scheduling, transportation scheduling, etc.

This problem was first introduced by HITCHCOCK (1941) and KOOPMANS (1947), and was solved by DANTZIG (1947) using revised simplex method.

4.1.1  The Transportation Problem can be Described as Follows

Consider m origins (production centres/warehouses) and n destinations (market places) in a transportation system.

The Origin i (i = 1, 2, … m) has s i (s i > 0) units of single product ...

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Transportation Theory ¶

Introduction ¶.

The main motivation is to optimize the transportation of products. Transportation is something that is used all over the world all the time. Transport is used by every society and business. In fact, people are still trying to figure out the optimal way to transport. This is amazing considering how transport has been around for a long time. The theoretical foundation was laid by French mathematician Gaspard Monge in 1781! With how important transportation is and how it is continuing to evolve, we believe that is important for everyone to have a solid grasp on this idea

Monge-Kantorovich optimal transport, or Monge-Kantorovich optimization, has a rich history. As we stated in the introduction, it dates back to the classic 1781 paper of Monge, “Memoire sur la theorie des deblais et des remblais”, related to the most economical way of moving soil from one area to the other. In its most concrete form its application for military and economic purposes is obvious. Monge specified the theory in terms of minimizing the L1 norm of the distance transported. The L2 theory came later, leading to the Monge-Ampere equation, an important nonlinear partial differential equation. The theory received a boost in the 1940’s when Kantorovich generalized it to what is now known as the Kantorovich dual problem, and showed how to deal with it using his newly developed method of linear programming.

The main idea of the optimal transportation problem is that we are trying to minimize cost of transporting products while satisfying the supply and demand restrictions. We set up the problem with m sources and n destinations, called nodes . These nodes are connected by arcs . These arcs represent the transportation cost between each node. Transportation is composed of suppliers of transport services and users of these services. Well-functioning transport markets should allow the transport supply to meet transport demand to satisfy transport needs for mobility. Mobility could not occur or would not occur in a cost-efficient manner. This interdependency can be considered according to two concepts, which are transport supply and demand. Supply is expressed in terms of infrastructures (capacity), services (frequency), and networks (coverage). Capacity is often assessed in static and dynamic terms where static capacity represents the amount of space available for transport. Transport demand similar to transport supply, it is expressed in terms of the number of people, volume, or tons per unit of time and distance.

Example 1 ¶

Let us look at it in the simplest way possible. We have one factory and two warehouses. We have a supply of 16 at the factory. There is also a demand of 11 at warehouse one and a demand of 5. It costs 7 dollars a unit to ship to warehouse one and 10 dollars a unit to ship to warehouse 2. Since there is only one factory, we will ship all our units from it. It costs 77 dollars to satisfy warehouse one and 50 dollars to satisfy warehouse 2. This makes the optimal transport 127 dollars. This is a super simple problem, but it introduces us to the nodes and arcs that we will us for every problem here out.

Example 2 ¶

Problem description ¶.

A boutique brewery has two warehouses from which it distributes beer to five carefully chosen bars. At the start of every week, each bar sends an order to the brewery’s head office for so many crates of beer, which is then dispatched from the appropriate warehouse to the bar. The brewery would like to have an interactive computer program which they can run week by week to tell them which warehouse should supply which bar so as to minimize the costs of the whole operation. For example, suppose that at the start of a given week the brewery has 1000 cases at warehouse A, and 4000 cases at warehouse B, and that the bars require 500, 900, 1800, 200, and 700 cases respectively. Which warehouse should supply which bar?

Formulate ¶

For transportation problems, using a graphical representation of the problem is often helpful during formulation. Here is a graphical representation of The Beer Distribution Problem.

####Identify the Decision Variables In transportation problems we are deciding how to transport goods from their supply nodes to their demand nodes. The decision variables are the Arcs connecting these nodes, as shown in the diagram below. We are deciding how many crates of beer to transport from each warehouse to each pub.

A1 = number of crates of beer to ship from Warehouse A to Bar 1 A5 = number of crates of beer to ship from Warehouse A to Bar 5 B1 = number of crates of beer to ship from Warehouse B to Bar 1 B5 = number of crates of beer to ship from Warehouse B to Bar 5 Let: $ \( \begin{align} W &= \{A,B\} \\ B &= \{1, 2, 3, 4, 5 \} \\ x_{(w,b)} &\ge 0 \ldots \forall w \in W, b \in B \\ x_{(w,b)} & \in \mathbb{Z}^+ \ldots \forall w \in W, b \in B \\ \end{align} \) $

The lower bound on the variables is Zero, and the values must all be Integers (since the number of crates cannot be negative or fractional). There is no upper bound.

####Formulate the Objective Function The objective function has been loosely defined as cost. The problem can only be formulated as a linear program if the cost of transportation from warehouse to pub is a linear function of the amounts of crates transported. Note that this is sometimes not the case. This may be due to factors such as economies of scale or fixed costs. For example, transporting 10 crates may not cost 10 times as much as transporting one crate, since it may be the case that one truck can accommodate 10 crates as easily as one. Usually in this situation there are fixed costs in operating a truck which implies that the costs go up in jumps (when an extra truck is required). In these situations, it is possible to model such a cost by using zero-one integer variables: we will look at this later in the course.

We shall assume then that there is a fixed transportation cost per crate. (If the capacity of a truck is small compared with the number of crates that must be delivered then this is a valid assumption). Lets assume we talk with the financial manager, and she gives us the following transportation costs (dollars per crate):

From Warehouse to Bar A B

Minimise the Transporting Costs = Cost per crate for RouteA1 * A1 (number of crates on RouteA1)

… + Cost per crate for RouteB5 * B5 (number of crates on RouteB5) \(\min \sum_{w \in W, b \in B} c_{(w,b)} x_{(w,b)}\)

Formulate the Constraints ¶

The constraints come from considerations of supply and demand. The supply of beer at warehouse A is 1000 cases. The total amount of beer shipped from warehouse A cannot exceed this amount. Similarly, the amount of beer shipped from warehouse B cannot exceed the supply of beer at warehouse B. The sum of the values on all the arcs leading out of a warehouse, must be less than or equal to the supply value at that warehouse:

Such that: $ \( A_1 + A_2 + A_3 + A_4 + A_5 \leq 1000\\ B_1 + B_2 + B_3 + B_4 + B_5 \leq 4000\\ \sum_{ b \in B} x_{(w,b)} \leq s_w \ldots \forall w \in W \) \( The demand for beer at bar 1 is 500 cases, so the amount of beer delivered there must be at least 500 to avoid lost sales. Similarly, considering the amounts delivered to the other bars must be at least equal to the demand at those bars. Note, we are assuming there are no penalties for oversupplying bars (other than the extra transportation cost we incur). We can _balance_ the transportation problem to make sure that demand is met exactly - there will be more on this later. For now, the sum of the values on all the arcs leading into a bar, must be greater than or equal to the demand value at that bar: \) \( A_1 + B_1 \geq 500\\ A_2 + B_2 \geq 900\\ A_3 + B_3 \geq 1800\\ A_4 + B_4 \geq 200\\ A_5 + B_5 \geq 700\\ \sum_{ w \in W} x_{(w,b)} \geq d_b \ldots \forall b \in B \) $ Finally, we have already specified the amount of beer shipped must be non-negative.

After all this we run it through our code and solve it

References ¶

“The Transportation Problem and the Assignment Problem.” Edited by Ana Zelaia Jauregi, OpenCourseWare, 2012, ocw.ehu.eus/pluginfile.php/40949/mod_resource/content/1/5_Transportation_Exercises.pdf.

Geismar, Niel, et al. “The Integrated Production and Transportation Scheduling Problem for a Product with a Short Lifespan.” INFORMS, vol. 20, no. 1, 2008, pp. 21–33. Winter, doi:10.1287/ijoc.1060.0208.

Coin-or.org . 2021. A Transportation Problem — PuLP v1.4.6 documentation. [online] Available at: https://www.coin-or.org/PuLP/CaseStudies/a_transportation_problem.html [Accessed 27 October 2021].

Taha, H. A. (2017). Chapter 5: Transportation Model and its Varients. In Operations Research An Introduction (Tenth, pp. 179–179). Book, Pearson.

https://cnls.lanl.gov/MK/index.html

https://transportgeography.org/contents/chapter3/provision-and-demand-of-transportation/

Describe what a node and arc is

Define the main goal of the Optimal Transportation problem

Explain the connection between a node and arc

We have one factory and two warehouses. We have a supply of 25 at the factory. There is also a demand of 15 at warehouse one and a demand of 10. It costs 8 dollars a unit to ship to warehouse one and 13 dollars a unit to ship to warehouse two. Find the optimal solution

We have one factory and two warehouses. We have a supply of 40 at the factory. There is also a demand of 20 at warehouse one and a demand of 16. It costs 9 dollars a unit to ship to warehouse one and 11 dollars a unit to ship to warehouse two. Find the optimal solution

We have one factory and two warehouses. We have a supply of 31 at the factory. There is also a demand of 18 at warehouse one and a demand of 13. It costs 3 dollars a unit to ship to warehouse one and 17 dollars a unit to ship to warehouse two. Find the optimal solution

There are two factories and two warehouses. Factory A and Factory B have 50 and 40 cases each. The demand at Warehouse one is 6 and the demand at warehouse two is 2. The cost to transport from A to one is 6 dollars and to transport from A to two is 5 dollars. The cost to transport from B to One is 4 dollars and the cost to transport from B to two is 7 dollars. Find the optimal solution.

There are two factories and two warehouses. Factory A and Factory B have 90 and 60 cases each. The demand at Warehouse one is 7 and the demand at warehouse two is 9. The cost to transport from A to one is 10 dollars and to transport from A to two is 2 dollars. The cost to transport from B to One is 4 dollars and the cost to transport from B to two is 6 dollars. Find the optimal solution.

There are two factories and three warehouses. With factory A and factory B we have 20 and 40 cases respectively. The demand at warehouse one is 7, warehouse two is 3, and warehouse three is 4. It costs 13 dollars to ship from A to one, 4 dollars to ship from A to two, and 8 dollars to ship from A to three. It costs 8 dollars to ship from B to 1, 4 dollars to ship from B to two, and 13 dollars to ship from B to 3. Find the optimal solution

There are two factories and two warehouses. Factory A and Factory B have 30 and 20 cases each. The demand at Warehouse one is 1 and the demand at warehouse two is 9. The cost to transport from A to one is 11 dollars and to transport from A to two is 6 dollars. The cost to transport from B to One is 7 dollars and the cost to transport from B to two is 13 dollars. Find the optimal solution.

Project Idea ¶

Let us look at another example. This time lets take three factories and three warehouses. It costs a certain amount to ship from a factory to a warehouse. It costs 4 dollars to ship from factory 1 to Warehouse 1. It costs 3 dollars to ship from factory 1 to Warehouse 2. It costs 8 dollars to ship from factory 1 to warehouse 3. The supply of factory 1 is 300 units. From factory 2 it costs 7 dollars to ship to warehouse 1, 5 dollars to ship to warehouse 2, and 9 dollars to ship to warehouse 3. The supply for factory 2 is 300 units. From factory 3 it costs 4 dollars to ship to warehouse 1, 5 dollars to ship to warehouse 2, and 5 dollars to ship to warehouse 3. The supply from factory 3 is 100 units. The demand for warehouse 1 is 200 units, the demand for warehouse 2 is 200 units, and the demand for warehouse 3 is 300.

Take two bags of candy and get 3 destinations to take said candy. Each destination will demand a certain amount of candy. It will cost a certain amount to deliver the candy. The idea is to visualize what is happening in the supply and demand chain. Optimizing how each candy is delivered.

Principal authors of this chapter were: Kyle Owen, Raj Sah, and Stanford Obu

Contributors: N.C.Jacob

Network Model

Queuing Theory