Formulation and Computer Solution for Larger LP Problems 733Upon execution, the spreadsheet-based optimization program instantly com-
putes all optimal values consistent with satisfying all constraints. From the spread-
sheet solution in Table 17.1, we see that the minimum total number of regular-time
officers is 1,150, as shown in cell I5. Note that the bare minimum number of offi-
cers is present on five of the six shifts; only the second shift has excess numbers.
Moreover, officers begin work on five of the six shifts; no officers begin work on
shift 6 at 4 A.M. (This illustrates the earlier general result: Since there are five bind-
ing constraints, there are exactly five nonzero decision variables.)^9
The spreadsheet also lists the shadow price associated with each constraint.
For instance, the shadow price of requiring an extra officer on the fourth shift
(moving from 400 to 401 officers) is .125. How can this extra officer be obtained
for only a fractionalincrease in the workforce? The answer is by hiring one fewer
officer beginning in shift 2 (where we already have surplus personnel) and hir-
ing one more officer beginning in shift 3. This trade satisfies the new constraints.
The net increase in cost comes from the difference between the hourly costs
on shifts 2 and 3: 1.25 1.125 .125. This confirms the shadow price.(^9) This is not the only optimal plan. A second solution of the LP problem is x 1 75, x 2 250, x 3 0,
x 4 400, x 5 100, and x 6 75. This plan also requires the minimum number of regular-time offi-
cers (1,150).
Clean
Coal
Energy production—whether to provide electric power, heat homes and other
buildings, or fuel all kinds of transportation—is a dirty business. The most
abundant and economical forms of energy, coal and oil, are also the most
harmful to the environment. As energy demand surges in China, India, and
other fast-developing nations, the risks of environmental harm and global cli-
mate change only increase. Consequently, developing “green” technologies
and alternative energy sources is surely one of the paramount economic chal-
lenges and opportunities of the new century. Rapid innovation has raised the
promise of technologically feasible solar and wind power. Though use of these
and other renewable energy sources has more than tripled during the last
decade, they still account for less than 5 percent of electricity production in the
United States. Moreover, they are costly; for instance, in generating electricity,
solar power is more than five times as costly as coal.
To understand the energy challenge, one must come to grips with a funda-
mental constraint. Absent a miraculous technological breakthrough (and the evi-
dence makes such a leap extremely unlikely), the share of alternative green
sources in energy production will be very small, and green energy is and will be
expensive. Instituting a carbon tax would help level the “playing field” where dirty
and clean fuels compete with respect to cost. But clean fuels would still be severely
constrained by the problem of scale. (Spreadsheet problem S3 at the end of the
chapter models the choice among alternative energy sources as a linear program.)
Arguably, the best immediate means for reducing environmental harm (and,
in particular, the rate of global warming caused by greenhouse gas emissions) is
to institute incremental improvements in the combustion of traditional fossil
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