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.
To find the optimum production strategy for the manufacture of these products, we need to consider the constraints and objectives of the problem. Let's break down the problem into smaller parts and identify the key factors involved.
1. Profit: The objective is to maximize the overall profit from the production of these products. The profit for each unit of product X is £a, and for each unit of product Y is £b. In the case of three products, the profit for each unit of X is p1, Y is p2, and Z is p3.
2. Production Limitations: The number of person-hours available per day for production is L. The time required to manufacture one unit of X increases proportionally with the number of units produced, and it requires c person-hours. For product Y, the time required is proportional to (d * y), where d is a constant of proportionality and y is the number of units produced.
3. Manufacturing Resources: In the case of three products, there is a constraint on the primary reagent available. The maximum number of units of the primary reagent per day is M. The manufacture of X requires c1 units of the reagent, and Y requires c2 units.
To find the optimum production strategy, we can use optimization techniques such as linear programming or mathematical modeling. These techniques involve formulating the problem into mathematical equations and solving them to obtain the optimal solution.
In this case, we would formulate the problem considering the profit objectives, production limitations, and resource constraints. The specific mathematical equations and variables would depend on the exact values and relationships involved. To determine the optimum production strategy, we would solve this optimization problem using suitable techniques like linear programming solvers or optimization algorithms.
It is important to note that the exact solution would depend on the specific values of a, b, c, p, d, p1, p2, p3, c1, c2, L, and M. These values would need to be provided in order to solve the problem and determine the optimal production strategy.