You're facing time constraints in linear programming models. How can you optimize solutions effectively?

This article appears to be treating Linear Programs (LP) (where the variables are continuous) the same as Mixed Integer Programs (MIP)/Integer Linear Programs (ILP) (where there are integer valued variables). LPs are generally "easy" to solve. The Simplex Method, perhaps the most well known algorithm for solving LPs, has a polynomial average runtime complexity. There are also interior point methods that can solve LPs in polynomial runtime complexity under the worst case scenario. MIPs/ILPs, on the other hand, can be very difficult to solve to optimality within a practical amount of time.

Prioritize Key Variables: Focus on the variables that have the greatest impact on the objective function. Combine Similar Constraints: Merge constraints with similar effects to simplify the model. Eliminate Redundancies: Remove non-essential constraints that don’t affect the solution. Use Approximation: Apply approximation techniques to speed up computations without losing accuracy.

Apply a "gross oversimplifying assumption"...something that almost trivializes the problems. Not that you want to "assume the problem away", but you want to get to an answer. Once you do that, it becomes more reasonable to improve the quality of your answer by relaxing your assumption.

Simplifying models while under duress is a sure fire recipe for disaster. It is akin to locking the garage door after the car has been stolen. If process improvement has kept pace you should not have non criticals to jettison. Dynamic does Not mean “All over the place “ punching holes in your product.

Pure linear models are easy to solve, therefore linear constraints should not be prioritized. Mixed Integer and Linear models on the other hand may have bennefits on changing the formulation, howewer making a model faster is not as simple as just removing model components. A better strategy is explore alternative formulations to the base problem (e.g: modelling a sequencing problem as an travelling salesman problem). Also, avoid using Big M constraints, since they worsen the performance of the model. As a last resource, explore other solvers and if possible go for a paid one.

Some practical approaches: 1- Remove trivial solutions (if you already know the value of variable X then remove it from model and replace it with a parameter) 2- Simplify the model (remove unnecessary big domains from integer variables, specify the right up and low limits) 3- Provide feasible initial solutions (this helps solvers a lot) 4- If some constraints are independent of each other then separate them and solve them in two different models

In addition to the suggestions already mentioned, an interesting approach that often yields good results is to add a primal solution. To do this, you can use a good heuristic before starting the solving process and incorporate it correctly into the solver.

You need to ensure that the input is clean, accurate, and ready for analysis. It directly affects the quality of the solution as well as the reliability of the model's outcomes. It can help in formulating the problem more clearly. You can see an example from one of my research publications* where preprocessing steps are identified before the LP model. Caliskan F., Reeves K.A., Ozcan O., Zeller D. The impact of collaboration network Position on innovative performance. Technology and Innovation, Vol. 14, pp. 351–363, 2012. * Available at research.net or upon request.

Outlier Detection: Identify and handle outliers to prevent them from skewing results or causing infeasibility. Standardize Units: Ensure all data is in consistent units to avoid scaling issues in the model. Data Imputation: Fill in missing values with appropriate estimates to maintain data completeness. Feature Selection: Focus on the most relevant data features to reduce model complexity and computation time.

Preprocessing data before solving your LP model is crucial. Clean and normalize your data to avoid numerical instability and improve efficiency. Aggregating similar constraints and using shadow prices can help identify which constraints are binding and which can be relaxed.

Parallel Computing: Leverage solvers that support parallel processing to significantly speed up computations. Warm Starts: Use previous solutions as a starting point for similar models to accelerate the solving process. Solver-Specific Preprocessing: Enable built-in preprocessing features in solvers to automatically simplify and optimize the model. Heuristic Methods: Consider heuristic approaches for large-scale problems to obtain near-optimal solutions quickly when exact solutions are not feasible.

On LP problems the most impactful settings with great results on minimal efforts are gaps and tolerances: - Absolute and Relative gaps: choose them according with the level of precision needed (it is not relevant to solve a million dollar objective function with a $0.000001 distance of the actual optimal) - Feasibility tolerances: the same as above, however looking at allowed constraint viollations. Other than these, enable the pre solver and experiment with some cuts. If the time constraint on model solving time is hard, set the time limit parameter which will force the solver to give the best solution up to that point (it may not be the optimal, but its an efficient solution).

I believe that leveraging advanced solvers for linear programming is crucial in today’s fast-paced business environment. By optimizing algorithm selection and fine-tuning settings, organizations can enhance decision-making speed and accuracy, gaining a significant competitive edge. Investing in these tools is essential for navigating complex challenges and driving operational efficiency.

Ensure that you standardize the tolerance limits that you use. Tolerance limits must be at the same level at every step of the problem solving process. You miss a zero and your tolerance level will vary by an order of magnitude skewing your results.

Advanced solvers can get bogged down if parameters aren't precisely tuned. Dive into adjusting pivot rules or pre solve techniques to strip away redundancy. Infeasibility or slow convergence often occurs in large-scale models, so switching between primal-dual and barrier methods can recalibrate your approach. Solver-specific parameters like dual simplex settings or crossover methods may shave off critical minutes. Test on smaller datasets first to detect the nuances in computational performance, and always revisit problem formulation to ensure you’re not over-complicating. Time isn’t always on your side—your solver setup should be.

In many problems, finding a feasible solution quickly can be challenging due to the size and complexity of the model. Additionally, the difference between the first feasible solution and the eventual optimal solution is often small enough that spending extra time to prove optimality is not always worth it. One common heuristic approach in optimization is the feasibility pump, which alternates between finding integer-feasible solutions and improving the objective value until a good feasible solution is found quickly. Relax-and-fix heuristics are also popular in large-scale models, where you iteratively solve relaxed versions of the problem (e.g., fixing some variables and relaxing others) to quickly home in on a high-quality solution.

Flexibility: Heuristics can adapt to various problem types, making them versatile for complex or non-linear models. Faster Convergence: These methods quickly find near-optimal solutions, ideal when speed is more important than precision. Scalability: Heuristics handle large-scale problems efficiently, which may overwhelm traditional methods. Combine with Exact Methods: Use heuristics for initial exploration and refine with exact solvers to balance speed and accuracy.

Use Deductive Modelling for problems with large data sets that require near-instant solutions. The advantage with this approach is that we can achieve faster convergence. As we approach the solution we can refine our assumptions and with addition of a few more steps, if time allows, arrive at a precise solution.

Heuristic methods, like genetic algorithms or simulated annealing, help find good-enough solutions quickly when exact solutions aren’t critical. They explore the solution space efficiently, providing approximate solutions faster than traditional methods, which is useful when you’re short on time.

We’re balancing speed with solution quality, knowing full well that the clock is unforgiving in linear programming models. Techniques like tabu search or greedy algorithms offer you rapid approximations, trading off precision for operational feasibility. You’re not expecting the perfect answer but need one that works, especially under real-world pressures like fluctuating demand or tight production schedules. These methods shine when exact solutions are computationally expensive, offering you an edge in identifying feasible paths while sidestepping the exhaustive searchof optimality.

Em situa??es práticas que eu enfrentei, busquei verificar inicialmente se seria possível reduzir o número de variáveis e ou restri??es do modelo. Funcionou em alguns casos. Como já sugerido aqui, para se obter uma boa solu??o em problemas de grande porte no tempo requerido, o desenvolvimento de um método heurístico pode ser uma abordagem adequada.

Improved Efficiency: Parallel processing splits tasks across multiple cores, drastically reducing computation time for large models. Scalable Solutions: It easily scales with the problem size, making it ideal for handling complex, data-heavy models. Concurrent Scenario Analysis: Run multiple scenarios or simulations simultaneously, speeding up decision-making processes. Cloud Integration: Utilize cloud-based computing resources for flexible and cost-effective parallel processing capabilities.

Parallel computing speeds up LP model solutions by distributing the workload across multiple processors or machines. This method is especially useful for handling big data or analyzing multiple scenarios at once, making it possible to solve complex models much faster.

We can split the workload across multiple processors, optimizing performance by addressing large-scale models more efficiently. The challenge often lies in balancing data distribution, especially when synchronisation delays creep in. Decomposing the model into smaller, independent tasks requires careful consideration of interdependencies, ensuring that core computational tasks aren't bottlenecked. This method is especially effective in multi-scenario analysis, where speed matters. The key is fine-tuning resource allocation across nodes, allowing you to scale performance without compromising accuracy.

By dividing large datasets and complex models into smaller tasks, each processor can handle a portion of the workload simultaneously, significantly reducing processing time. However, challenges arise in coordinating data dependencies and synchronization between processors. To mitigate this, use parallel solvers that optimize task allocation, ensuring minimal communication delays. For mixed-integer programming models, decomposition methods like Dantzig-Wolfe and Benders can further leverage parallelism, enhancing performance without compromising solution accuracy.

Early Insights: Solve simpler versions to get quick, actionable insights before tackling the full problem. Focus on Key Drivers: Identify and refine the most critical variables first to maximize impact with minimal effort. Easier Debugging: Simplifying the model initially helps detect errors or inconsistencies early in the process. Progressive Complexity: Gradually increasing complexity ensures more manageable computation and clearer problem-solving pathways.

Start by focusing on critical variables or constraints that drive the most significant outcomes. By structuring your model in phases, you reduce computational intensity, focusing on key parameters first. Early-stage results help you make informed trade-offs, particularly when facing competing objectives or limited resources. Prioritise high-impact constraints while leaving secondary factors for later iterations. This method lets you rapidly assess the most important aspects, keeping precision while staying within your time limits.

An incremental approach to solving LP models involves starting with a simplified version and gradually adding complexity. This method helps you get quick preliminary results and decide where to focus next, managing time constraints effectively by prioritizing key factors first.