Human beings have been performing economic and business activities since the time civilizations began to mature. The activities included managing agricultural goods and products, trading, taxation, etc. All such activities involve some level of planning and organization also known as “decision making.” For e.g. determining the quantity and season to produce different crops, fixing the selling price of products, etc. With time, the size, nature, and complexity of these activities grew manifold. Most importantly, business activities witnessed tremendous growth with the advent of industrialization and globalization.

Early decision-makers could rely on their knowledge, experience, intuition and simple calculations for better decision making as variables/factors which affect the decision output were limited. However, modern-day business planners face a matrix-like scenario as most often, there are hundreds or thousands of different factors that now come into play to influence the output at the same time. Intuition, knowledge, and experience in these scenarios are still important, but in most cases, they will lead to sub-optimal decisions. However, when coupled with advanced mathematics, miraculous improvement in results is observed.

What is Management Science?

The Applied Mathematical Programming textbook by Bradly, Hax, and Magnanti defines Management Science as follows:

“Management science is characterized by the use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information, or in seeking further information if current knowledge is insufficient to reach a proper decision.“

It began during World War II when the allied forces hired scientists from different fields to assist in military operations. The war ended, the practice didn’t. People started applying those techniques to different fields and business operations. Businesses do better when there is an efficient use of limited resources and technologies to achieve maximum output. This leads to better utilization of resources, reductions in makespan, minimization of costs, etc. Helping businesses achieve this lies at the heart of Management Science. Modern times have witnessed many such real-world applications. The growth in computation power has enabled quicker resolution of problems with millions of decision variables.

Although Management Science is a large umbrella and encompasses a spectrum of techniques, in this blog we are going to focus our attention on Mathematical Programming Optimizations. More specifically, linear, non-linear and integer programming optimization problems. These kinds of problems are encountered in many domains and can be solved optimally using modern-day optimizer software. Some of the common industry problems which are solved using this technique are as follows:

Optimization of Distribution Channels

Businesses having warehouses, distribution centers and supply centers at different locations have to make transportation decisions so that the supply is adequate at each of the supply points while aptly handling constraints such as ensuring the number of goods at each warehouse never exceed the warehouse’s capacity limits, etc. There are multiple ways of delivering goods to a particular supply point by traversing through different warehouses resulting in different travel routes. As the costs associated with different travel routes are different, it becomes a challenging task to come up with a distribution system that has the minimum cost while satisfying different constraints.

Product Mix

Manufacturing companies generally produce different products or different varieties of the same product. The manufacturing costs and selling prices of each of these products are different. The goal here then is to find the optimal amount of each product to produce to have the maximum profit.

Chemical Production Scheduling

For chemical manufacturing plants, the production of some materials is a continuous process, whereas for others it is a batch or a job process. The chemical manufacturing processes are accompanied by a galaxy of decisions such as the allocation of different tasks like heating, mixing, etc. to different units (machines such as heaters, reactors). Moreover, some units can perform multiple tasks, and there can be multiple units available for the same kind of task. The arrangement allows a lot of combinations of possible schedules, but not all of them meet the required product demand and other process constraints.

Some schedules which do meet all such requirements may not have the best makespan, operating cost or other similar objectives. The problem statement here is to find an optimized schedule that satisfies all of the requirements and at the same time minimizes or maximizes the required objective.

Personnel Scheduling

Certain companies such as airlines are often involved with activities whose intensity changes according to the time of day or the day of week/month etc. The number of personnel or employees required by these companies at different times varies according to the intensity of activities. Furthermore, not all employees are available at all times and the wages of these employees depend on their varying shift timing every day. The optimization challenge is to find an optimal schedule for employing the workforce which should incur the minimal cost to the company while meeting the required number of employees at different times.

Mathematical Programming Project Overview/Pipeline

Outlining Problem Statement and Collecting Required Data

Every Management Science project begins with a precise definition of problem scope. It dictates what lies within and out-of-scope of the problem statement and marks key metrics to optimize. It is followed by a collection of relevant data sets. For an inventory management problem, it could be sales data of past periods, for scheduling optimization, it could be task processing times, etc.`

Making Suitable Model Assumptions

A model represents interactions between the decision variables and outcome. It is not practical to model every minute detail of the real process into a mathematical model. Acceptable assumptions are made to formulate a tractable and solvable model.

Formulating the Model and Objective Function

Once the required data and assumptions are ready, the formulation of the mathematical model begins. Based on the use-case, a suitable mathematical formulation is picked. The formulation is then enriched with the specific details of the processes. The business requirements and criteria required to be met by the model output is modeled into mathematical constraints along with an equation for the objective function. The mathematical model is then modeled into a software environment using a modeling language tool.

Running the Optimization

This is the core part of an optimization project where the software model is passed to solvers which run the actual optimization and search for an optimal solution/output in the solution space. The solvers come with a very efficient implementation of optimization algorithms. They output a solution that meets all the modeled constraints with the best value of the objective function.

Sensitivity Analysis and Post-optimality Analysis

Sensitivity analysis and Post-optimality analysis builds our trust in the mathematical model. In these analyses, we change certain input parameter values and observe the response of the model. If the response of the model is in line with the business process, the model is taken to be a good representation of the real process. Also, these analyses help in identifying the parameters which have the most impact on the objective function.

Challenges Involved in Mathematical Programming Projects and Their Remedies

These projects come with their own challenges and adventures. The most common ones are –

1. Data Collection and Data Verification: The output which we get after the optimization depends on the data inputted to the model. Often projects of moderate sizes require a variety of data and information to be collected from various sources that go as input parameters to the model. Inconsistencies in these data values lead to an erroneous model.

Remedy: A proper precaution is to build a data checker which checks the validity of the data values based on known business rules.

2. Acceptable Model Assumptions: As mentioned before, certain assumptions are often necessary while formulating the model. That said, adding assumptions to model properties can sometimes deviate the model from the actual objective. Therefore, necessary care must be taken while making assumptions.

Remedy: It can be avoided by discussing these assumptions with the team operating those processes and performing a thorough variation analysis.

3. Organizational Support: Shifting from a traditional methodology to an unconventional one is often met with resistance and mistrust from the decision-makers of the organization, especially when these decisions are highly associated with revenues and losses. A successful OR (Operations Research) project needs the support and willingness of the management to accept the results of optimizer models.

Remedy: Involve concerned business executives throughout the implementation phase and make them aware of the challenges, assumptions and the rationale behind using those assumptions.

4. The Validity of the Models: Some sensitive parameters are not known in advance, hence, their predicted value is used in the optimization model. As the model output depends on those predicted values, if they aren’t accurate, the model may give suboptimal solutions. It is, therefore, required to validate the models before their final deployment.

Remedy: A rigorous sensitivity analysis helps in identifying sensitive parameters and in understanding the response of the model concerning different values of those parameters. There should be a correlation between model output and the expected output.

Modeling Tools and Optimizers

• Modeling Tools – The software which provide an interface to build and code optimization models. Some of the most popular modeling tools are – GAMS (General Algebraic Modeling System), IBM ILOG Optimization Studio, Python, AMPL.
• Solvers – Core of the optimization process which runs the actual optimization. They come with a very efficient implementation of optimization algorithms. Both commercial and open-source solvers are available. Some of the prominent ones are:
  • Commercial – CPLEX, Gurobi, BARON
  • Open-Source – GLPK, Scipy

A limitation of open-source solvers is that their performance starts to degrade with an increase in model size. The choice of the optimizer depends on the type and scale of the problem being solved.

Conclusion

Management Science offers vast avenues to boost revenues and suppress costs. Along with that, it also highlights strategies to minimize delays and increase product quality which in turn increases customer satisfaction. Nowadays, more and more businesses are rushing to leverage these advanced mathematical tools to enhance their processes. Sticking with antiquated and traditional methods is not an option anymore. For a business to withstand new entrants and beat existing competitors in today’s highly dynamic markets, it has to upgrade its decision-making process with unprecedented approaches that, thanks to technological advancement, are now possible.

Innova Solutions with its deep understanding of various businesses/verticals and skilled personnel with the requisite technical acumen can help accelerate this process for you as a customer. We, at Innova Solutions, bring our experience of developing efficient optimization models of huge scales that you can leverage to optimize your business processes.