Glossary

6.4 Transportation and assignment problems

5 min read•july 30, 2024

Transportation and assignment problems are key components of network flow optimization. They focus on efficiently moving resources from sources to destinations while minimizing costs. These problems have wide-ranging applications in logistics, supply chain management, and resource allocation.

The chapter explores specialized algorithms for solving these problems, including the and the . Understanding these techniques provides valuable insights into network optimization and lays the foundation for tackling more complex real-world scenarios.

Transportation Problems as Minimum Cost Flows

Network Representation and Problem Structure

Top images from around the web for Network Representation and Problem Structure

  • Cost Minimization Policy for Manufacturer in a Supply Chain Management System with Two Rates of ... View original

    Is this image relevant?

  • Bipartite graph - Wikipedia View original

    Is this image relevant?

  • Cost Minimization Policy for Manufacturer in a Supply Chain Management System with Two Rates of ... View original

    Is this image relevant?

1 of 3

Top images from around the web for Network Representation and Problem Structure

  • Cost Minimization Policy for Manufacturer in a Supply Chain Management System with Two Rates of ... View original

    Is this image relevant?

  • Bipartite graph - Wikipedia View original

    Is this image relevant?

  • Cost Minimization Policy for Manufacturer in a Supply Chain Management System with Two Rates of ... View original

    Is this image relevant?

1 of 3
  • Transportation problems modeled as specialized linear programming problems with network flow structure
  • representation connects sources (supply nodes) to destinations (demand nodes) via edges (transportation routes)
  • Objective function minimizes total transportation cost subject to constraints
  • ensures total supply equals total demand
  • limit flow on each edge (typically upper bounds)

Formulation as Minimum Cost Flow

  • problem generalizes transportation and assignment problems
  • Network G = (N, A) with node set N and arc set A
  • xijx_{ij} represent flow on arc (i, j)
  • Objective function: min⁡∑(i,j)∈Acijxij\min \sum_{(i,j) \in A} c_{ij}x_{ij} where cijc_{ij} is the unit cost on arc (i, j)
  • Flow balance constraints: ∑j:(i,j)∈Axij−∑j:(j,i)∈Axji=bi\sum_{j:(i,j) \in A} x_{ij} - \sum_{j:(j,i) \in A} x_{ji} = b_i for all i∈Ni \in N
    • bi>0b_i > 0 for supply nodes, bi<0b_i < 0 for demand nodes, bi=0b_i = 0 for transshipment nodes
  • Capacity constraints: lij≤xij≤uijl_{ij} \leq x_{ij} \leq u_{ij} for all (i,j)∈A(i,j) \in A
  • Solving as minimum cost flow allows use of efficient network algorithms (Network Simplex, Successive Shortest Path)

Assignment Problems as Special Cases

  • Assignment problems are special transportation problems with equal number of supply and demand nodes
  • Each supply node has exactly one unit of supply, each demand node requires exactly one unit
  • Binary decision variables xij∈{0,1}x_{ij} \in \{0, 1\} indicate whether assignment is made
  • Can be solved as minimum cost flow with unit capacities on all arcs
  • Hungarian algorithm provides efficient combinatorial solution method
  • Applications include job scheduling, resource allocation, and matching problems (job assignments, organ transplant matching)

Solving Transportation Problems

Initial Basic Feasible Solution Methods

  • Northwest Corner Rule starts in upper-left corner, allocates maximum possible to satisfy row/column
  • Least Cost Method allocates to cell with lowest unit cost, repeats until all supply/demand satisfied
  • Vogel's Approximation Method (VAM) uses opportunity costs to make better initial allocations
    • Calculate penalty for each row/column as difference between two lowest costs
    • Allocate to lowest cost cell in row/column with highest penalty
    • Repeat until all allocations made
  • VAM typically produces better initial solutions, reducing number of iterations needed

Transportation Simplex Algorithm

  • Specialized simplex method exploits network structure for efficiency
  • Maintains basic as spanning tree of the transportation network
  • Iterative improvement process:
    1. Compute (u, v) for current basic solution
    2. Calculate for non-basic variables: cˉij=cij−ui−vj\bar{c}_{ij} = c_{ij} - u_i - v_j
    3. If all reduced costs non-negative, current solution is optimal
    4. Otherwise, select entering variable with most negative reduced cost
    5. Determine leaving variable using minimum ratio test along cycle created by entering variable
    6. Update flow along cycle and repeat
  • Degeneracy handled through perturbation or cycle-finding algorithms to prevent stalling

Sensitivity Analysis and Parametric Programming

  • from dual variables indicate cost changes for small supply/demand variations
  • Reduced costs show how much non-basic variables must improve to enter the basis
  • determines how much a cost coefficient can change without affecting
  • Parametric programming analyzes solution behavior as costs or supply/demand change continuously
    • Useful for strategic planning and understanding problem sensitivity
  • helps identify critical constraints and most influential parameters

The Hungarian Algorithm for Assignment

Algorithm Overview and Steps

  • Efficient combinatorial method for solving assignment problems in O(n^3) time
  • Works directly on the , avoiding full simplex tableau
  • Main steps of the algorithm:
    1. Subtract minimum value in each row from all elements in that row
    2. Subtract minimum value in each column from all elements in that column
    3. Cover all zeros with minimum number of lines (rows/columns)
    4. If number of lines equals matrix dimension, optimal assignment found
    5. Otherwise, find smallest uncovered element, subtract from all uncovered elements, add to elements at line intersections
    6. Repeat steps 3-5 until optimal assignment found
  • Primal-dual interpretation relates to complementary slackness conditions

Implementation and Optimizations

  • Matrix operations can be efficiently implemented using NumPy or similar libraries
  • (e.g., Hopcroft-Karp) can improve worst-case time complexity
  • Sparse matrix representations beneficial for large, sparse cost matrices
  • Early termination possible if partial assignment satisfies all constraints
  • Parallel implementations can exploit multi-core processors for larger problems

Extensions and Special Cases

  • Unbalanced assignment problems handled by adding dummy rows/columns with high costs
  • Maximization problems converted to minimization by negating or subtracting costs from large constant
  • Forbidden assignments represented by setting corresponding costs to very large values
  • Multi-dimensional assignment problems (e.g., three-dimensional) require specialized algorithms
    • Often NP-hard, approximation algorithms or heuristics used for larger instances
  • Applications in computer vision (feature matching), natural language processing (word alignment)

Duality in Transportation and Assignment Problems

Primal-Dual Relationships

  • Duality provides alternative perspective and solution methods for transportation/assignment problems
  • Primal (minimization) problem: minimize total transportation/assignment cost
  • Dual (maximization) problem: maximize total of dual variables subject to reduced cost constraints
  • Strong duality holds dual optimal objective value equals primal optimal objective value
  • Complementary slackness conditions link primal and dual solutions:
    • If primal variable xij>0x_{ij} > 0, then corresponding dual constraint must be tight
    • If dual constraint not tight, corresponding primal variable must be zero

Dual Variables and Shadow Prices

  • Dual variables (u, v) represent shadow prices for supply and demand constraints
  • Interpretation supply node dual variable uiu_i maximum price willing to pay for additional unit of supply
  • Demand node dual variable vjv_j minimum price willing to accept for not satisfying unit of demand
  • Reduced costs cˉij=cij−ui−vj\bar{c}_{ij} = c_{ij} - u_i - v_j measure relative cost efficiency of non-basic variables
  • uses dual information to determine ranges for cost coefficients and supply/demand changes

Algorithmic Insights from Duality

  • Hungarian algorithm implicitly works with dual problem by maintaining feasible dual solution
  • Successive shortest path algorithms for min-cost flow alternate between primal and dual updates
  • Duality used in proving optimality of network simplex algorithm
  • Column generation techniques leverage dual information to identify promising new variables
  • Applications in pricing problems, resource allocation, and mechanism design (auction theory)

Key Terms to Review (27)

Assignment problem: An assignment problem is a type of optimization problem where the goal is to assign a set of resources to a set of tasks in the most efficient way possible. This involves minimizing costs or maximizing efficiency while ensuring that each resource is assigned to exactly one task and each task is assigned to exactly one resource. This concept is closely related to transportation problems, as both involve optimization under constraints, often using similar mathematical techniques.
Augmenting path algorithms: Augmenting path algorithms are techniques used to find maximum flows in flow networks by iteratively searching for paths that can increase the flow from a source to a sink. These algorithms play a crucial role in solving transportation and assignment problems by optimizing resource allocation and ensuring efficient flow through a network. They focus on identifying paths where additional flow can be pushed through, thereby improving overall efficiency in various applications.
Bipartite Graph: A bipartite graph is a type of graph where the set of vertices can be divided into two distinct and independent sets such that no two graph vertices within the same set are adjacent. This structure is useful for modeling relationships between two different groups, allowing connections only between members of different sets. The bipartite nature helps in solving various problems involving network flow, transportation, and assignment scenarios effectively.
Capacity constraints: Capacity constraints refer to the limitations on the amount of resources or outputs that can be produced or managed within a given system. These constraints play a critical role in decision-making processes, particularly in logistics and resource allocation scenarios where there is a need to optimize routes, assignments, or transportation methods while staying within the bounds of available resources.
Cost matrix: A cost matrix is a mathematical representation that outlines the costs associated with transporting goods from multiple sources to multiple destinations. It provides a structured way to analyze and optimize resource allocation, particularly in scenarios like transportation and assignment problems, where minimizing costs while meeting supply and demand is essential. By identifying the cost of each route or assignment, decision-makers can make informed choices to enhance efficiency and reduce expenses.
Decision Variables: Decision variables are the unknown values in an optimization problem that decision-makers seek to determine. They represent the choices available in a model and are essential for formulating the problem, as they directly impact the objective function and constraints that govern the system being optimized.
Dual Variables: Dual variables are associated with the constraints of an optimization problem and represent the sensitivity of the objective function to changes in these constraints. These variables help in understanding how the optimal value of the objective function will vary with small changes in the resources or limits imposed by the constraints.
Feasible Solution: A feasible solution is a set of decision variables that satisfies all the constraints of an optimization problem. This concept is critical as it defines the boundaries within which optimal solutions can be found. Feasible solutions can vary in quality, and while some may be optimal, others may simply meet the necessary conditions without achieving the best possible outcome.
Flow Conservation: Flow conservation refers to the principle that the amount of flow entering a node in a network must equal the amount of flow exiting that node, ensuring that there are no losses or gains at that point. This concept is critical for maintaining balance in various applications, such as routing resources efficiently and ensuring the integrity of supply and demand. It provides a foundation for modeling and solving complex problems involving networks, where understanding how to distribute resources without excess or shortage is vital.
Hungarian Algorithm: The Hungarian Algorithm is an efficient method used to solve assignment problems, where the goal is to minimize the total cost of assigning tasks to agents. It is particularly relevant in scenarios involving a one-to-one relationship between agents and tasks, allowing for optimal pairing based on given costs or weights. This algorithm effectively finds the optimal assignment by leveraging matrix manipulation and graph theory principles.
Iterative methods: Iterative methods are mathematical techniques used to find approximate solutions to problems by repeatedly refining an initial guess or solution through a series of iterations. These methods are particularly useful for solving complex optimization problems, such as those found in transportation and assignment scenarios, where direct solutions may be difficult or impossible to obtain. By leveraging feedback from previous iterations, these methods help converge towards the optimal solution more efficiently.
Minimization Theorem: The minimization theorem is a principle that states that under certain conditions, a linear programming problem will have an optimal solution that minimizes the objective function. This theorem is particularly significant in resource allocation scenarios where costs need to be minimized while satisfying specific constraints, making it essential for solving various problems like transportation and assignment issues.
Minimum Cost Flow: Minimum cost flow refers to the optimization process used to determine the most cost-effective way of transporting goods through a network while satisfying supply and demand constraints. It is an essential concept in operations research, particularly related to transportation and assignment problems, where the goal is to minimize transportation costs while ensuring that all supplies are delivered to meet demands at various destinations.
Modified distribution method: The modified distribution method is an optimization technique used to find the optimal solution for transportation problems by adjusting initial feasible solutions. This method improves upon the basic feasible solution by iteratively redistributing shipments to minimize transportation costs while satisfying supply and demand constraints. It ensures that the solution meets the feasibility criteria of the transportation problem while also working towards cost-effectiveness.
Network Diagram: A network diagram is a visual representation of a system of interconnected nodes and edges that depicts relationships and flow within a network. It is commonly used to illustrate the logistics of transportation and assignment problems, allowing for a clear understanding of the routes, resources, and constraints involved in optimizing movement and allocation.
Northwest corner method: The northwest corner method is a technique used to find an initial feasible solution for transportation problems, which involve distributing goods from several suppliers to multiple consumers while minimizing transportation costs. This method begins at the northwest corner of a cost matrix and allocates as much as possible to the first cell, then moves either down or across to the next available cell until all supply and demand constraints are satisfied.
Optimal Solution: An optimal solution refers to the best possible outcome or result for a given optimization problem, maximizing or minimizing an objective function while satisfying all constraints. Finding this solution is central to various mathematical modeling techniques, as it determines the most efficient or effective way to achieve goals under specified conditions.
Optimality Condition: The optimality condition is a set of criteria used to determine whether a solution to an optimization problem is optimal, meaning it cannot be improved upon without worsening another aspect of the solution. These conditions provide necessary or sufficient criteria for evaluating potential solutions, helping to identify whether further refinement or adjustments are needed. In the context of transportation and assignment problems, understanding these conditions is essential for ensuring that resources are allocated efficiently and effectively.
Post-optimality analysis: Post-optimality analysis refers to the process of assessing how changes in the parameters of an optimization problem affect the optimal solution obtained. This analysis helps decision-makers understand the robustness of the optimal solution and identify how sensitive it is to variations in costs, resources, or constraints. By evaluating these aspects, stakeholders can make more informed choices and adjust their strategies based on potential changes in the scenario.
Range of Optimality: The range of optimality refers to the range of values for the coefficients in a linear programming objective function where the current solution remains optimal. Understanding this concept is crucial for evaluating how changes in the objective function affect the optimal solution and its feasibility. It helps in determining stability in decision-making and guides the analysis of different scenarios without needing to re-solve the entire optimization problem.
Reduced Costs: Reduced costs refer to the amount by which the objective function coefficient of a variable needs to be improved before that variable can enter the basis in a linear programming problem. Essentially, it measures how much the cost associated with an additional unit of a non-basic variable would affect the overall objective function. Understanding reduced costs is vital in various optimization techniques and helps in analyzing the potential impact of changing decision variables.
Sensitivity Analysis: Sensitivity analysis is a technique used to determine how the variation in the output of a mathematical model can be attributed to different variations in its inputs. This method helps to understand which variables have the most impact on the outcome, allowing for more informed decision-making in optimization problems.
Shadow Prices: Shadow prices represent the implicit value of an additional unit of a resource in optimization problems, indicating how much the objective function would improve if that resource were increased by one unit. They connect to various methods of optimization, helping to interpret solutions from different mathematical approaches and understand the economic implications of resource constraints.
Stepping Stone Method: The stepping stone method is an iterative technique used to optimize transportation problems by finding the least cost routes for transporting goods from multiple suppliers to multiple consumers. It involves evaluating the current solution and adjusting allocations based on a graphical or numerical representation of costs, allowing for more efficient transportation solutions. This method identifies potential improvements in the current solution, aiming to minimize overall transportation costs while satisfying supply and demand constraints.
Supply and Demand: Supply and demand is a fundamental economic model that describes how the quantity of a good or service available (supply) interacts with the desire of consumers to purchase it (demand). This model illustrates how prices are determined in a competitive market, with equilibrium reached when the quantity supplied equals the quantity demanded. Understanding this concept is essential for analyzing various problems related to resource allocation and optimizing transportation and assignment processes.
Transportation problem: The transportation problem is a type of optimization issue that seeks to determine the most cost-effective way to transport goods from a set of suppliers to a set of consumers while meeting supply and demand constraints. This problem is crucial in logistics and supply chain management, as it helps in minimizing transportation costs while ensuring that all demands are satisfied.
Transportation Simplex Method: The transportation simplex method is an optimization technique used to solve transportation problems, where the goal is to determine the most cost-effective way to transport goods from multiple suppliers to multiple consumers. This method builds on the simplex algorithm by incorporating specific constraints related to supply and demand, allowing for an efficient allocation of resources. By finding an optimal solution that minimizes transportation costs while satisfying supply and demand constraints, this method plays a crucial role in logistics and operations management.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide