Lighting is often considered “low-hanging fruit” from a sustainability standpoint. It is not difficult to change to a more energy efficient light bulb that can significantly lower electricity usage, which in turn reduces cost and emissions. There can be other trade-offs, a well-known one being that the more energy efficient CFL bulbs often contain mercury, a hazardous material, requiring more careful and costly disposal. These and other factors lead to a nice O.R. problem on lighting.

This past semester we ran a student project on re-designing lighting in a classroom space. The problem was to determine the optimal locations of bulbs as well as the kinds of bulbs to use in order to minimize costs (bulb purchase, bulb disposal, and electricity) while meeting standard illumination thresholds.

There are fairly sophisticated computer programs that will do this for you (see DIALux). But having students develop their own model makes sense because it is something they can realistically complete in one semester, and the process can be eye-opening for them. There is another reason related to the discussion here on life-cycle assessment software from a while back. That post described a demonstration in which two different programs were used to calculate the carbon footprint of a simple construction and each subsequently came up with estimates that differed by three orders of magnitude. The models used by these programs will have underlying assumptions, some made clear in the program or its documentation, others buried more deeply. So a good part of the exercise in developing such a program from scratch is to then compare its results with off-the-shelf software, as well as its assumptions. Lighting is simpler than life-cycle assessment, so the challenge of aligning models should be within reason.

To formulate the problem, the ceiling and desk level (where the illumination is to be measured) were broken up into grid cells (see figure). Each ceiling cell can contain at most one bulb from a list of several types. The contribution to the illumination at a given desk level location from a ceiling bulb is calculated using the formula illuminance at desk (measured in lux) = light intensity of bulb (measured in lumens) / distance^2. The overall illumination at the desk location is found by summing these contributions over all of the ceiling cells.

The students found the current room to be significantly over-illuminated. Their solution came in just above the illumination threshold with a smaller number of bulbs spaced across the ceiling.

A (somewhat paraphrased) portion of the formulation follows.

Definitions $x_{i,j,k} = 1$ if bulb type i is in location j,k; 0 otherwise $l_i =$ light intensity (in lumens) of bulb type i $c_i =$ purchase cost of bulb type i $d_i =$ disposal cost of bulb type i $p_i =$ power usage (in watts) of bulb type i $s_i =$ # of life-spans of bulb type i over the given time frame $b_i = \sum_{j,k} x_{i,j,k} \; \; :$ total # of bulbs of type i in the design $r_{j,k,l,m} =$ distance from ceiling cell (j,k) to desk cell (l,m) $T =$ illumination threshold (in lux)

Objectives

Minimize $\sum_i c_i b_i s_i \; \; :$ bulb purchase cost

Minimize $\sum_i d_i b_i s_i \; \; :$ bulb disposal cost

Minimize $\sum_i p_i b_i \; \; :$ bulb power usage

Constraints $\sum_{i,j,k} \frac{x_{i,j,k} l_i}{r_{j,k,l,m}^2} \ge T$ for each desk cell (l,m) (illumination constraint) $\sum_{i} x_{i,j,k} \le 1$ for each ceiling cell (j,k) (at most one bulb per cell)

Note the objectives are left somewhat vague here. For instance the power objective can be converted to cost using the price of electricity (\$0.16/kwh in this region), or to emissions (1.2 lbs of CO2 per kwh here). The three objectives can be combined in a multiple objective optimization with weights as desired by the decision maker, some can be turned into constraints, etc. Many other considerations could be added such as lighting fixtures used, windows, occupancy timers, room obstructions, light tubes, maintenance issues, light quality, etc.

Another aspect of this project was that it was run in conjunction with similar ones from the Civil Engineering and Management Departments. The civil group focused more on general sustainable design in the room including but not limited to lighting. The interaction ended up not being extensive but exchanges of information (e.g. CAD drawings of the room) and ideas took place throughout the semester. These projects provide a great opportunity for multidisciplinary collaboration. That includes not only the academic disciplines but also the real-world operational side of the institution as well.

The reference is Analyzing Light Optimization at the United States Coast Guard Academy: Minimizing Cost and Environmental Impact by T. Cassel, A. Donato, and R. Mozolic, USCGA Dept. of Mathematics Capstone Report, Spring 2010, Advisors: I. Frommer and M. Case.