Calculus

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As you know, Chemproc, Inc. is a premier manufacturer of chemicals and chemical waste. Further, as has been recently highlighted on the local news, we are in the process of expanding our Lonlinc, Skanebra chemical manufacturing plant to add two or three new products to our elegantly packaged and painstakingly marketed product line, which will result in our hiring at least four local workers. Our engineers have, however, determined that for these products to be successful in the competitive market into which we are making our foray we must be careful to obtain the absolute highest profit possible from our overall production.
We are for reasons of corporate secrecy unable to reveal the exact names of the products that we will be producing, and therefore refer to them herein as products X and Y (for the case in which two products are produced), respectively. We will be manufacturing x and y units of these per day, and expect to realise a profit of £a and £b per unit respectively per unit on each of the products (again, we are unable to divulge the actual values determined by our marketing department).
The actual number of units of these that we are able to produce is, however, limited by the number of person-hours that the newly hired local workers have available for this production process on any given day. This will be L person-hours. To manufacture one unit of X requires c person-hours, and the hours required increases proportional to the number of units of X produced. For the second product, however, economies of scale are much more pronounced, so that the number of person-hours required to produce y units of it is proportional to , with constant of proportionality d, where 0 < p < 1.
It is imperative that we determine the best production strategy for the manufacture of these two products. Find the optimum production strategy.

It is imperative that we determine the best production strategy for the manufacture of these two products. We are additionally interested in the case in which we manufacture three products, X, Y, and Z, and hope that you will find it possible to also investigate this possibility. In this case the economy of scale indicated above is only applicable to the third product, while the time required for the other two is linear in the number of units produced. The profit on these is p1, p2 and p3 dollars per unit. Additionally, both of X and Y require a primary reagent of which we are only able to allocated M units per day, and to manufacture one unit of X requires c1 units for this reagent, while manufacture of Y requires c2 units thereof. We would be very interested in your determination of the optimum production strategy in this case as well.

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