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Walters_Karl_thumbOne of the most powerful tools in the analytics toolkit for decision making is the development of mathematical models, also known as mathematical programming (MP). Mathematical programming encompasses a wide-range of techniques including linear programming (LP), dynamic programming (DP), simulation and others. In most applications of mathematical programming models, there is an overriding objective that represents a business outcome that managers would like to maximize (profit) or minimize (cost). Depending on the problem at hand, the model may incorporate resource limits or other constraints as well.

Walters_Karl_fullOne of the most powerful tools in the analytics toolkit for decision making is the development of mathematical models, also known as mathematical programming (MP). Mathematical programming encompasses a wide-range of techniques including linear programming (LP), dynamic programming (DP), simulation and others. In most applications of mathematical programming models, there is an overriding objective that represents a business outcome that managers would like to maximize (profit) or minimize (cost). Depending on the problem at hand, the model may incorporate resource limits or other constraints as well.

Now that we understand what it is, there remains the question of what benefits are there to using it? Consider the duties of a state facilities manager who is charged with maintaining for commercial or institutional buildings, such as hospitals, clinics, schools, office complexes, sports arenas or convention centers. It is the role of the facility manager to coordinate and oversee the safe, secure, and environmentally-sound operations and maintenance of these assets in a cost effective manner aimed at long-term preservation of the asset. Duties may include air conditioning, electric power, plumbing and lighting systems; cleaning; decoration; grounds-keeping and security.

The role of facility managers in particular is evolving to the extent that many managers have to operate at two levels: strategic-tactical and operational. In the former case, owners need to be informed about the potential impact of their decisions on the provision of space and services. In the latter, it is the role of a facility manager to ensure proper operation of all aspects of a building to create an optimal environment for the occupants to function. Coordinating all of the disparate maintenance activities comprises a very large decision space, one in which downstream effects are commonplace, and potentially very costly.

Mathematical programming is really just a mechanism for considering many alternatives simultaneously and for systematically choosing among them to provide the best answer. The process of developing a MP model forces managers to evaluate potential decisions critically because of the requirement to put metrics to them. Perhaps one decision is whether to invest in a new HVAC system for an office building, one which is expected to significantly reduce future energy consumption (and carbon emissions). In addition to the HVAC system, the model can also incorporate several smaller energy-saving initiatives. When the model is solved, the new HVAC system may or may not be identified as the best solution. Moreover, an unanticipated combination of initiatives could be identified that results in significant cost savings by doing them together (e.g., replacing the HVAC system 5 years from now along with a new, solar-reflecting roof).

Although these models sound complex, a good one must be understandable to the managers using the results; no good manager should base a decision on something he or she doesn’t understand. That doesn’t mean that the results should not be at times counter-intuitive or surprising. Indeed, if your models don’t occasionally surprise you or cause you to rethink some assumptions, you probably included insufficient alternatives to consider, or overburdened your model with constraints such that the expected answer is the only one possible. For example, suppose the MP model in the previous example only considered replacement of the HVAC system in the current fiscal year, and did not include the other energy-saving options. If a goal is to immediately reduce carbon emissions, it is very likely the decision would be to implement the HVAC system immediately because there would be no other means to achieve that goal; moreover, the lower-cost solution discussed earlier would not have been even considered. Perhaps the facilities manager is correct that the immediate replacement of the HVAC system is a good idea, but not in combination with all the other initiatives at the same time. Diverting some budget from other sources, or investing in a less expensive HVAC system may yield better results. 

It is important to note that the results are not simply implemented exactly as formulated in the model, but form the basis for reassessing some assumptions and generating new ideas for further testing. Good models generally start out quite simple and undergo step-wise refinement (adding/dropping alternatives).  One of the great advantages to MP models is that once all the alternatives are enumerated, the manager can quickly determine how different objectives and constraints affect the outcome, and in turn gain a far better understanding of the enterprise overall.

My last point is that you do not have to have “all your ducks in a row” to use mathematical programming effectively. If data is limited, use the best information you have available, and estimate the others. Use the MP model to test how sensitive solutions are to the estimated numbers. If the solutions change wildly with small changes, a conservative approach may be warranted until data collection efforts can be implemented. If you wait until all the data is “perfect” you may be waiting a very long time, and foregoing significant benefits in the meantime.

For more information about mathematical programming, visit The Science of Better, produced by the Institute for Operations Research and Management Science. For a simple overview of LP, click here.

Karl R. Walters is senior solutions analyst at Remsoft; e-mail: karl.walters at remsoft.com

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