Last week we took a quiz regarding a prescriptive analytics problem.
The problem:
RMC Inc. is a small firm that produces a variety of chemical products. In a particular production process, three raw materials are blended (mixed together) to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of 2/5 ton of material one and 3/5 ton of material three. A ton of solvent base is a mixture of 1/2 ton of material one, 1/5 ton of material two, and 3/10 ton of material three. After deducting relevant cost, the profit contribution is 40 per every ton of fuel additive produced and 30 for every ton of solvent base produced. RMC’s production is constrained by a limited availability of the three raw materials. For the current production period, RMC has available the following quantities of each raw material. Assuming that RMC is interested in maximizing the total profit contribution, find the optimal solution.
Upon solving the problem by using the graphical solution, I got the values of $x=25$ and $y=20$ for the objective function. Assuming that it is correct, I was now given the task to solve the problem by using the Northwest, Least-Cost, and Vogel's approximation methods.
Is it correct to use the $x=25$ and $y=20$ as the demands and add a dummy column, or is there a correct way to solve this?
To start,
Have $F$ to denote amount of the fuel additive (in tons) used in the production process.
Have $S$ denote the amount of solvent base (in tons) used in the production process.
Then the model will go as follows:
$$\text{max } z = 40F + 30S$$ Subject to, $$\frac{2}{3}F+\frac{1}{2}S\le 20$$ $$\frac{1}{5}S\le 5$$ $$\frac{3}{5}F+\frac{3}{10}S\le 21$$ $$F, S \ge0$$
However, notice for every unit of $F$ or $S$ added, it adds the amount related to the mixture to the amount used of the resource pool. For more information on blended problems in Operations Research, look at this cool document I happened to find
It’s okay to use the graphical method to solve a modeling problem. However, it is important to double-check the values you got by plugging it back into the constraints of the model to ensure it did not invalidate any of them (in other words, an infeasible solution). So for your $x$ and $y$ values you got from the graphical method, you should plug it back in to the original problem and see if the amount of the mixtures did not overtake the amount of resources available. If it didn’t, then you’re in the clear.