Question

The Province of Quebec is facing power shortfalls. Quebec Hydro plans to address this problem by investing in developing coal fired energy and hydropower facilities in the coming seven years, and have $1.5 billion with which to do so.   Every $ million spent on coal fired plants will result in 15 megawatts of power capacity when the plant construction is completed, and every $ million spent on hydropower dams will result in 20 megawatts of power capacity when the dam construction is completed. Completion of coal fired plants takes 3 years and completion of hydropower dams takes 4 years.    Once these facilities are built they will require additional operating funds that are equivalent to 25% of their initial cost annually. Funds set aside for plant operations cannot be used to invest in new plants or dams, but can be invested in the stock market and net an interest of 12%. Any unspent funds can also be invested in this way. Quebec Hydro currently has 40 megawatts of capacity and it can price energy so as to keep the demand at this level in the next two years. Thereafter, however demand will rise to 56, 60, 68, 75, and 87 megawatts of additional power in the third, fourth, fifth, sixth and seventh years, respectively. Assume that once a plant is operational, it is always operational.

a) FORMULATE the Linear Programming decision model that meets the power demand, and maximizes the funds available to Quebec Hydro at the end of the seventh year. Define all decision variables and label (1-5 word description) each constraint. Define C_t and HP_tto be the amount invested in coal fired plants and hydropower dams, respectively, in year t. Define F_t to be the money invested into the stock market in year t. NOT SOLVE THE MODEL.

Answer 1

Ans To formulate: Linear programming model that meets the power demand and maximizes the funds available to Quebes Hydro at the end of 7th year.

Let, Ct be the amount invested in Coal field plants( in $ million) in year t. Here time i.e. t is 3 years.

HPt be the amount invested in Hydropower dams ( in $ million) in year t. Here time i.e. t is 4 years.

Let Ft be the money invested in stock market in year t.

Let ct and ht be the demand of coal and hydropower respectively in year t.

So,

Maximize Z= 15* Ct + 20* HPt + (Ft)* 12%

= 15*C3 + 20* HP4 + F7* 12% ( in millions)

Subject to constraints,

(C3+ HP4 ) + 25%* (C3' + HP4') <= 1.5/1000 ( in millions)

where C3' = C3/3 I.e. cost of investment in coal fired annually in those 3 years.

and, HP4' = HP4/4 i.e. cost of investment in hydropower plants annually in those 4 years.

(c1+h1)= 40 (demand costraint in year 1)

(c2+h2)= 40 (demand costraint in year 2)

(c3+h3)= 56 (demand costraint in year 3)

(c4+h4)= 60 (demand costraint in year 4)

(c5+h5)= 68 (demand costraint in year 5)

(c6+h6)= 75 (demand costraint in year 6)

(c7+h7)= 87 (demand costraint in year 7)

Ct, HPt are non negative constraints.