Question 6 of 10: Integer Programming Formulation — Power-Plant Site Selection and Timing

Question 6: Integer Programming Formulation — Power-Plant Site Selection and Timing (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Sites $i=1,\dots,5$; years $t=1,\dots,20$. Per site $i$: capital cost $K_i$ ($ to build), annual operating cost $O_i$ ($/yr while running), energy capacity $E_i$ (MWh/yr once running). Existing capacity 500,000 MWh/yr. Required system energy demand $R_t$ for each year $t$ — the specific cost/energy table is not reproduced in the exam text, so the model below is built with named symbolic parameters rather than invented numbers.

Find. A binary integer program whose optimal solution is the cost-minimizing set of (site, start-year) commitments that meets demand every year — formulated only, not solved.

Approach. Use one binary variable per (site, year) pair meaning "site $i$'s plant begins operating in year $t$"; this single variable set encodes both the site-selection and timing decisions and lets construction and accumulated operating cost be written as linear functions of it.

1. Decision variables. $y_{it}\in\{0,1\}$ for $i=1,\dots,5$, $t=1,\dots,20$: $y_{it}=1$ if the plant at site $i$ is placed into service in year $t$ (and the site is never built if all $y_{it}=0$).
2. At-most-one-plant-per-site. $\displaystyle\sum_{t=1}^{20}y_{it}\le1$ for each $i=1,\dots,5$ (each site is built at most once, in at most one start year).
3. At-most-one-plant-enters-service-per-year. $\displaystyle\sum_{i=1}^{5}y_{it}\le1$ for each $t=1,\dots,20$.
4. Demand-satisfaction constraint. A plant placed in service in year $\tau$ contributes $E_i$ starting that year and every year after, so cumulative capacity by year $t$ is $500{,}000+\sum_i\sum_{\tau\le t}E_iy_{i\tau}$. This must meet the year's requirement: $$500{,}000+\sum_{i=1}^{5}\sum_{\tau=1}^{t}E_i y_{i\tau}\ \ge\ R_t,\qquad t=1,\dots,20.$$
5. Objective and complete model. Total cost is the sum of each built plant's capital cost plus its operating cost for every year it runs (from its start year through year 20): $$\boxed{\text{Minimize } Z=\sum_{i=1}^{5}\sum_{t=1}^{20}\Big[K_i+O_i(20-t+1)\Big]y_{it}\ \ \text{ s.t. constraints above},\ y_{it}\in\{0,1\}.}$$ This is a complete, defensible binary IP even though the numeric $K_i,O_i,E_i,R_t$ table is not present in the printed exam text.