The management of One Mile Island Power Company (OMIPC) wants to make better short and long run decisions regarding the operation of its Keyport electric generation plant. Management's goal is to be able to run the plant to achieve maximum power generation during peak load hours. Power output is limited both by engineering capacity of the equipment and by environmental considerations. In particular, the plant must comply with the following emission standards:

Maximum SO2 emissions are 3,300 parts per million (PPM) in the stack gas at all times.

Maximum particulate emissions are 12 kg per hour.

Coal is brought to the Keyport plant by railroad and dumped into stockpiles in the coal yard. From there it is carried by conveyor belt to a pulverizer and fed directly into the combustion chamber at the desired rate. The heat produced by the combustion produces steam, which drives the generation turbines.

Two types of coal are used by the plant: hard coal which is expensive but has low sulfur content and softer chunky coal which, while cheaper, is smoky and has higher sulfur content.

Technical specs on the coals are:

When mixing the hard and soft coals, the sulfur content in the output is a weighted average of the two outputs. For example, a mixture consisting of equal parts of the coals would emit 2,800 PPM of sulfur gas. Similarly, a mixture consisting of three times as much hard coal as soft coal would emit 2,300 PPM of SO2.

If the pulverizer were fed with hard coal only, it could handle 16 tons per hour.  Similarly, if it were fed with soft coal only, it could handle 24 tons per hour.  Thus, an input of 8 tons/hour of hard coal would require 50% of the entire pulverizer capacity. An input of 18 tons/hour of soft coal would require 75% of the entire pulverizer capacity.  The pulverizer can be fed with any combination of hard and soft coal that does not exceed its capacity.

The conveyor can handle up to 20 tons per hour of either coal. The conveyor can be fed with any combination of hard and soft coal that does not exceed its capacity.

The management of OMIPC wants to determine the mix of coals that will maximize the amount of power (in BTU) produced per hour. This will be its summer-time peak hour operating plan. Algebraically formulate an LP to find the optimal plan, but you are not required to solve the LP. Define the decision variables and their units of measurement and label each constraint in terms of the requirements of the problem.

(Hint: This is a nontrivial formulation. Define your decision variables in terms of tons of coal used per hour. The constraints on PPM of SO2 and the pulverizer capacity might be the most difficult to formulate.  For the former, it may help to express the SO2 content of a coal mix as the weighted average of the individual SO2 contents. Note that a nonlinear expression of the form x / y ≤ 1 can be converted into the equivalent linear form: x - y ≤ 0.

For the pulverizer capacity constraint, one way of reasoning is the following. Using H tons of hard coal requires H/16 of the entire pulverizer capacity; similarly, S tons of soft coal requires S/24 of the entire pulverizer capacity. No more than 100% of the pulverizer capacity can be used in an hour.)