# Essay on Benders' Decomposition Approach To Solve Network Design Problem

Length: 1135 words (3.2 double-spaced pages)

Rating: Strong Essays

#### Essay Preview

The approach used to solve the network design problem is based on a Benders' decomposition method where the sub-problem is a mixed integer programming problem. The master problem consists of choosing the best configuration $Q$ given the current set of constraints, where $Q$ is the warehouses capacity vector. Once this configuration is found, a cut is generated by searching for the smallest stable transportation cost for this configuration. The problem of finding the smallest stable transportation cost is itself solve by bender decomposition where the master problem search for the worst demand for the fixed configuration and the cut generating sub-problems are simple flow problems using fixed configuration and demands. This section first present some useful definitions, then successively proposes the formulations for the flow sub-problem, the stable transportation cost sub-problem and finally the global design problem.

The distribution network contains mills, warehouses and customers zones. The problem, for every period, consist of transiting manufactured products through warehouses to the customers. The following definitions will be useful.
The flow of products is defined by the two following set of variables.
The quantity of product transiting from the mill to the warehouse.

Since later we want to find a demand that maximizes the cost of the flow problem, it is useful to consider the dual of the previous problem in order to have a maximization objective function.

where $\lambda$, $\alpha$, $mu$ and $\sigma$ are, respectively, the dual variables of constraints \ref{lambda}, \ref{alpha}, \ref{mu}, and \ref{sigma}. We call $\Phi(Q,d)$ the optimal value of the objective function for the linear program (\ref{FlowDual}).
...

... middle of paper ...

...st $r_j$, the variable inventory cost $i_j$ and the transportation cost $\Phi(Q)$ knowing that we will have to face the worst demand for that network. A binary variable $o_j$ determines whether or not warehouse $j$ is open.
To ensure that there is always enough space in the warehouses to fulfill any demand we require that $\sum_{j \in W} Q_j \geq \sum_{s \in S} d_s$. The design problem is.

Let $\Delta$ be the finite set of possible value of the binary variables $\delta^+, \delta^-, w, f$ and let $h_r(\delta) = (\sigma_{r,\delta},z_{r,\delta},\alpha_{r,\delta},\mu_{r,\delta}), r = 1 \ldots m_\delta$ be the set of extreme points of the problem (\ref{WorstBin}) with the value of the binary variable fixed. Since problem \ref{Design1} is convex with respect to variables $\gamma$ we can use a Benders' decomposition approach to solve the following network design problem.

## Need Writing Help?

Get feedback on grammar, clarity, concision and logic instantly.

## My Design At Tall Buildings Essay

- Honestly, before I took this class I had no clue on the restraint on structure that can possibly affect the architecture. I always looked up at tall buildings and tried to understand how they really stood up. It was no mystery on how the transfer loads worked and the importance of the column design. Most column placings in a building are not randomly places they mainly follow a grid. I Know last year this helped me out with my design in studio and my design while studying abroad. Something I found extremely useful during class and assignments was the awesome free body diagrams....   [tags: Problem solving, Problem, How to Solve It]

Strong Essays
709 words (2 pages)

## Essay on The Secrets Of Less Stress And Greater Problem Solving

- 7 Secrets to Less Stress and Greater Problem Solving Do you consider yourself to be a resourceful person. Are you actually able to own most of your own mistakes. Do you believe you have overall more strength than stress. While most of us would consider ourselves "to blessed to be stressed", still we manage to feel most days have become an ongoing battle to try to tame our stress levels. Just making it through the day becomes our daily mission. Evening Out the Joy and Sorrow Ratio Most of us would agree that there are times when we just seem to have more stress, issues, and problems than overall happiness....   [tags: Problem solving, Problem, How to Solve It]

Strong Essays
1023 words (2.9 pages)

## Essay on Techniques on How to Solve a Problem: The 5 Why Technique

- ... 5. Selection – Select the best solution by using suitable decision trees. 6. Planning – Plan the implementation by coming up with action plans. 7. Sell Idea – Find a reliable source to support the solution presenting the idea. 8. Action – Act on solution to resolve the problem. The advantage of this process is that it is easy to use. For someone who has trouble with problem solving this could be a useful process as it is straight forward. It has many basic steps which allow a clear view of what is needed to be done....   [tags: the simplex process, problem solving]

Strong Essays
2537 words (7.2 pages)

## Relationship Between The Ability And Solve Mathematical Complex Tasks And Problem

- The relationship between the ability to solve mathematical complex tasks and problem representation: A review of literature 26284973 Concordia University Among the children born and raised in Quebec, less than 78% of them enrolled in mainstream education classes are able to pass the compulsory end-of-cycle situational problem examination (Tétrault & Desrosiers, 2013). The ministerial exam, composed of multiple criteria, encompasses skills that the children should have practiced throughout their elementary school years....   [tags: Problem solving, Mathematics, Problem]

Strong Essays
1236 words (3.5 pages)

## Design Problematization: The Role of Individual Preferences and Beliefs in the Design Process

- DESIGN PROBLEMATIZATION: the role of the individual preferences and beliefs in the design process Several factors are to be taken into consideration in designing a structure. Among other things, the problem solving of the given “initial” problem, the usage of the structure and the aesthetic outcome of the design are given the most importance when designing. What are usually placed at the bottom of the priority in designing are those that deal with the qualitative qualities in designing. These left out qualities ranges from the intellectual to the emotional....   [tags: Design]

Strong Essays
974 words (2.8 pages)

## Design And Design Of Interior Design Essay

- Interior design is one of the most important professions. When thinking of design and architecture industry, it is necessary to keep in mind that the professionals emphasis on matters of everyday life when commencing a project. Important sectors of interior design consist of: residential, commercial, hospitality, healthcare, education and corporate designs. Thus, the field deals with numerous factors and affects almost everyone in some way. Design is goal-oriented– it strives to achieve a certain purpose....   [tags: Design, Designer, Interior design]

Strong Essays
748 words (2.1 pages)

## Essay about The Student Will Use Equations Solve Real World Problems

- Learning Objective 1: The student will use equations to solve real world problems. (Behaviorism) Teachers who use the behaviorist theory in the classroom create rewarding experiences for students. Students work for things that make them feel good and proud of themselves. They cherish the feeling of getting approval from a teacher they admire. Most behavior can be learned or unlearned. The learning objective for a sixth grade at the school is: The teacher starts the lesson by reviewing what equations are and how they are used in everyday life....   [tags: Learning, Education, Problem solving, Lesson plan]

Strong Essays
1404 words (4 pages)

## Design Of Design Thinking Research Essay

- DESIGN THINKING RESEARCH ABSTRACTS Researching about Design Thinking and the vital role reinterpretation plays in the design. The different strategies individuals design think before creating the outcome. It isn’t clear, which method is superlative. Through research finding out that the way people design products is through sketching, through inspirations from blogs, group work and other methods. Also, through further researching and by realising by the fact that this depends on the individuals’ strengths....   [tags: Design, Design management, Typography]

Strong Essays
867 words (2.5 pages)

## Design Thinking Essay

- Design thinking is a process for practical, creative resolution of problems or issues that looks for an improved future result. It is the essential ability to combine empathy, creativity and rationality to meet user needs and drive business success. Unlike analytical thinking, design thinking is a creative process based around the building up of ideas. There are no judgments early on the design thinking (Simon, 1969, p. 55). Design thinking includes imagination and reason, a combination of convergent and divergent thought, and creativity....   [tags: creative problem resolution]

Strong Essays
1192 words (3.4 pages)

## Design Patterns Essay

- Design Patterns What is a Design Pattern. A design pattern solves a problem within a given context. The solution that is offered by implementing a pattern results in a system design structure, which balances the concerns of the design problem in a manner most appropriate for the given context. In “Understanding and Using Patterns in Software Development”, Dirk Riehle and Heinz Zullighaven offer a rather nice definition of what a design pattern is: “A pattern is the abstraction from a concrete form which keeps recurring in specific non-arbitrary contexts.” The ability to write good patterns is very difficult....   [tags: Definition Design Pattern]

Strong Essays
1551 words (4.4 pages)