How do you use optimization tools and frameworks to solve complex and multidisciplinary problems?

These are all powerful tools for deterministic, usually static, optimization problems. I suspect many of these are solved to make decisions that are implemented in a sequential setting, which means that the *problem* is a sequential decision problem (almost always stochastic), while the optimization model is a *policy* for making a decision. Nothing wrong with deterministic optimization as long as we understand how it is being implemented, and how the uncertainties of the real world may affect how the decision performs in the real problem.

Optimization tools and frameworks are software solutions aiding in problem formulation, solution, and analysis across diverse domains. From mathematical optimization software like CPLEX to heuristic frameworks such as DEAP, they offer algorithms, interfaces, and modeling languages for solving complex optimization problems. Machine learning libraries like Scikit-optimize automate parameter tuning, while simulation platforms like AnyLogic simulate and optimize system operations. With options spanning cloud-based platforms to general-purpose programming languages, these tools empower users to make data-driven decisions, enhance processes, and allocate resources effectively.

I'm often asked, "How do optimization and data science differ?" At a glance, they might seem similar. The key difference is that optimization manages choice, while data science manages generalization. Fundamental tools in each field illustrate this difference. Consider linear programming in optimization and least-squares in statistical inference. Both are matrix problems. Yet, linear programming matrices tend to be wide (many columns) while least-squares matrices are usually tall (many rows). In this context, combinations of columns in linear programs represent the many choices we seek to manage. Rows in least-squares signify errors we aim to minimize for a predictive function.

For those that aren't familiar, it would be better if the article described what the various tools were instead of just naming them. Categories for the tools listed: -Programming Languages: MATLAB, Python -Modeling Languages: GAMS, AMPL -Solvers: CPLEX, MOEA (not as familiar with MOEA, but the website states that it provides many algorithms) -Python Packages: Pyomo (used for modeling problems in a way similar to GAMS/AMPL)

Optimization tools truly excel when matched with a deep understanding of the specific problem domain. For example, in software engineering, Python's extensive libraries like NumPy or TensorFlow enable tailored problem-solving. Similarly, in helpdesk support, merging CRM platforms with AI analytics transforms customer interactions. It's about smartly aligning technical capabilities with the unique demands of each task, ensuring both innovative and practical solutions

To choose the right optimization tool or framework, assess problem complexity, available resources, and specific requirements. Consider factors like problem type (linear, nonlinear), scalability, ease of use, and cost. Evaluate features, compatibility, documentation, and community support. Ensure the tool aligns with the problem domain and offers necessary algorithms and interfaces. Conduct trials or prototyping to gauge suitability before committing.

I agree with all of the above, I would also take into consideration the knowledge and skills of the team (if you are working in a team). For example, in my team at my job, I am the only one who has used and knows GAMS, however, everyone knows Python and the Pulp and pyomo package. Therefore, I would not choose to develop an optimisation algorithm in GAMS as there would be far less colleagues who would be able to develop and maintain the code base.

Match the Tool to the Problem: Ensure it fits the type and scale of the issue. Evaluate Data Compatibility: Check integration with your data and resources. Assess Accuracy and Speed: Choose based on your needs for precision and performance. Consider User Skills: Select a tool that aligns with your team's expertise. Check Compatibility and Flexibility: Ensure it works with your problem formulation and solver. Review Cost and Support: Look at pricing, availability, documentation, and support.

Next, what happens after the decision is implemented? Does new information arrive that affects how the solution performs? Is this a decision that is then repeated with the new information? If so, you have a sequential decision problem, that has been studied under many names (dynamic program, stochastic program, optimal control problem, ....). See https://tinyurl.com/sdafield/ for an introduction to the general area of sequential decision problems.

Formulating an optimization problem involves defining objectives, decision variables, constraints, and feasible solutions. Clearly articulate what to optimize and why, identify variables impacting outcomes, and specify constraints. Formulate mathematical expressions representing objectives and constraints. Ensure compatibility with chosen optimization methods and tools. Regularly refine problem formulation based on insights and feedback.

In a hospital setting, for instance, optimizing nurse scheduling and bed distribution involves formulating a problem where decision variables include nurse shifts, patient admission rates, and bed availability. The objective function aims to maximize patient care quality while minimizing staff fatigue. Constraints involve staffing policies, patient-to-nurse ratios, and bed capacity. This optimization often requires balancing staffing efficiency with patient care needs, considering shift preferences, and, more importantly, predicting patient flow. Leveraging advanced analytics and AI helps in predicting patient flow and nurse availability, ensuring optimal resource distribution and, consequently, better patient outcomes.

"The decision variables are the parameters ..." I don't like the use of the term "parameters" here because the term "parameters" is commonly used when describing constants in a mathematical program. Suppose we are given the distance between nodes i and j. We would denote this as d_ij and this would be a parameter in our model. We are NOT solving for it. "The objective function is the measure of the quality" Again, we should be careful with the use of the word "quality" here. Suppose we setup and solve a model that finds the min time it takes for me to reach New York. The optimal solution has an obj value of 2 hours. Suppose I consider any time beyond 1.5 hours to be bad. If 2 hours is the best possible time, then is it actually bad?

I would start by identifying decision variables, which represent controllable parameters. Next, define the objective function, quantifying the goal to be optimized. Then, establish constraints representing limitations or requirements. Choose an appropriate mathematical model, such as linear programming for linear relationships, nonlinear programming for nonlinear relationships, integer programming for discrete decision variables, or dynamic programming for sequential decision-making. Ensure the formulated problem accurately reflects real-world constraints and objectives for effective optimization.

Complexity analysis primarily focuses on the performance of the algorithm used to solve the problem. While this may be an issue, today there are such powerful solvers (and computers to go with them) that deterministic problems tend to be solvable with one of these tools (and I include linear integer problems). I think we need more attention given to how well the solution performs after it is implemented, and whether we keep making decisions as new information arrives. If this describes your situation, you have a sequential decision problem, which also means you have a stochastic optimization problem.

Complexity analysis is not a big part of everyday work for practitioners. For the most part, we assume the problems we face are theoretically intractable, and that we have the solve them anyway. Many algorithmic techniques can find very good solutions to challenging problems quickly. Exact methods like mixed integer programming have become billions of times faster over the last few decades. As practitioners, we are often much more interested in choosing an appropriate solution technique for a problem than we are in that problem's theoretical complexity.

Analyze an optimization problem by assessing objectives, constraints, and feasible solutions. Evaluate problem complexity, sensitivity to parameters, and scalability. Use sensitivity analysis to understand solution stability. Compare different strategies and algorithms. Consider trade-offs between conflicting objectives. Iterate problem analysis to refine formulations and optimize outcomes.

Another factor to consider is to *design* flexibility into the framework so that the user can even override a violated constraint. In realistic settings, constraints may need to be violated in order to have a *solution*, which is much better than having nothing. Off course the user interface needs to inform the user about such violations and how to resolve them. Such flexibility results in a much higher match between the model and the real world, and adds enormous value to the designed solution.

Sensitivity Analysis: Evaluate how changes in parameters or data affect the solution. Feasibility Analysis: Determine if there is a feasible solution that meets all constraints. Optimality Analysis: Verify whether the solution is indeed the optimal one. Complexity Analysis: Assess the computational difficulty or effort required to solve the problem. These analyses help you understand the problem better, identify potential issues, and enhance solution quality and efficiency.

This is the language we use for solving static, deterministic optimization problems. In most cases I have seen, these are for decisions that are made over time. Using a deterministic approximation model to make decisions may work fine, but it is never going to produce optimal decisions *over time*. In other words, if is not an optimal policy and, in fact, may not be very good.

"There are several solvers you can use to solve an optimization problem, such as exact solvers which guarantee to find the optimal solution" The caveat to this is if you gave the solver infinite time and resources. In reality, there are problems that would take many lifetimes to find the provably optimal solution. When using exact solvers in practice, we can usually set a solver parameter to have the solver cut off once a time limit has been reached or once a MIP gap threshold has been met.

Over my career, I have been stunned by the transition of deterministic optimization problems from hard to "just use a package." In the beginning of my career (1981) we could not even solve interesting linear programming problems. By the early 1990s, LPs were considered easy (primarily due to Bixby's development of Cplex), but IPs were still hard. By year 2000, Bixby had conquered general integer, linear programs. Today, Cplex (and the follow on package that Bixby developed, Gurobi) can handle integer programs with 100,000 integer variables (depending on the structure). Steve Boyd has created a powerful solver (CVX) for nonlinear problems. Other packages have also become available.

Assuming your decision problem is stochastic. Things to think about 1. Are you experimenting/learning online or learning from data ? One requires you to refine a policy over time as you experiment, the other assumes you can infer a causal effect from data and then optimise. 2. Set up a causal graphical model representing chance variables, utilities and decision constraints. You can do this if you’re experimenting or learning from observed data. 3. Is your decision problem single shot or sequential? Sequential problems are harder to optimise, you need to use Bellman’s equations . This requires thinking through what is observed between decisions too. Single shot decisions are typically easier.

It would be better to set solver parameters, such as convergence criteria or search strategies, to tailor the solution approach to the problem. This ensures that the solver effectively balances solution quality and computational efficiency, leading to optimized production schedules that meet operational constraints and objectives.

I would say, apply optimization tools to complex, multidisciplinary problems by: 1. Understanding problem requirements and constraints. 2. Breaking down the problem into manageable subproblems. 3. Utilizing appropriate optimization techniques and algorithms. 4. Integrating domain-specific knowledge and constraints. 5. Iteratively refining solutions based on feedback. 6. Collaborating with experts across disciplines. 7. Leveraging advanced modeling and simulation capabilities.

Understand the question that the customer is asking. Understand the problem if there is one. Just because you have a hammer, don’t imagine everything is a nail.