How to Maximize Preferred Outcomes with Linear Optimization
Kfir Azoulay is Vice President of Marketing at Lumenis in the Netherlands and a member of the MIT EMBA Class of 2015.
One of the highlights of the MIT Executive MBA program is without doubt course 15.730, Data, Models, and Decision (referred to as “DMD”).
A more apt name for this course, in my opinion, would be Management Science, given its solid roots in rigorous scientific tools, concepts, and techniques that change the way managers reach conclusions and make decisions. Some of the fundamental concepts used in DMD to aid business leaders in dealing with uncertainty and in optimizing managerial decision processes are, to name a few: decision trees, probability and statistics, simulations, regression, linear and nonlinear optimization, and discrete optimization. The fact that the course is (co) taught by the person who co-wrote the book on this very subject, Prof. Dimitris Bertsimas, is undoubtedly one of the many appeals that MIT Sloan has to prospective EMBA students.
Concepts taught during the DMD course can be (and are!) applied in numerous fields, ranging from ordering spare parts, through minimizing production and distribution costs, maximizing sales, and all the way to developing better medical treatments (as I examine here).
The Application of Linear Optimization in Radiation Therapy
Cancer is the scourge of our era. According to the American Cancer Association, approximately 1,660,290 new cancer cases were diagnosed in 2013, and 580,350 Americans were projected to die of cancer (almost 1,600 people a day). Cancer remains the second most common cause of death in the US, accounting for nearly 1 of every 4 deaths.
An effective tool for treating cancer in the radiologist’s armamentarium is radiation therapy (a beam of high-energy photons fired into the patient in order to kill cancerous cells.) However, to reach the tumor, radiation also passes through healthy tissue, and damages both healthy and cancerous tissues.
Using advanced diagnostic imaging modalities such as computed tomography (CT) and magnetic resonance imaging (MRI); our ability to pinpoint the size, shape and location of a tumor has improved dramatically.
Linear optimization is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
In this example, linear optimization provides an efficient and systematic way of designing a radiation treatment beam that maximizes radiation absorption (measured in Gy) and minimizes absorption and radiation damage to healthy tissue.
The following illustration demonstrates the constraints of irradiating tissues (represented as volume pixels, or “Voxels”) from different angles, with the aim of maximizing the radiation absorption in the cancerous areas, and minimizing absorption in the healthy or critical areas.
The constraints given in this example are to deliver at least 7 Gy to the cancerous (tumor) voxels and at most 5 Gy to the spinal cord.
The model, therefore, is the following:
Decision area: X1, X2, X3, X4, X5, X6
Our objective is: min 3X1 + 4.5X2 + 2.5X3 + X4 + 2X5 + 4X6
While our constraints are:
2X1 + X5 => 7 X2 + 2X4 + => 7
1.5X3 + X4=> 7 1.5X3 + X5 => 7
2X2 + 2X5 <= 5
X1, X2, X3, X4, X5, X6 => 0
Fast forward to the solution: using linear optimization we achieve an objective value of 22.75 and the optimal solution to the presented challenge, vis-à-vis the radiation levels delivered from each of the beamlets. As you can see in the following illustration, by taking into consideration the abovementioned constraints, we are able to optimize the radiation to maximize the absorption in the cancerous areas, without exceeding the safety levels to critical areas, such as the spinal cord.
Today, several of the leading radiation machines on the market are equipped with treatment planning software that implements and solves optimization models.
Despite my choice of subject to illustrate its advantages, linear optimization is used widely in practically every field in the global economy, including telecom, transportation, operations, strategic planning and decision-making, etc. Although it would have been fairly simple to give a classic business example of linear optimization in action, I wanted to illustrate its usefulness in a technological area.
It may be Sloan, but it is also MIT, after all…
How have you used analytical tools to optimize outcomes?
Disclosure: This blog entry is based on an EMBA class (and slide-deck) taught by Profs. Bertsimas and O’Hare in the spring of 2014.