Answer :
a) The ratio of water to beans that will minimize the cost of morning coffee is 17:1, while the quantity of coffee is 17 grams.
b) The following is the calculation of the minimum cost of your daily coffee-making process:
$ / day = (16.95 / 12 * 17) + (5.72 / 100 * 0.17) + (18.33 / 100) + (0.11643 / 60 * (5/60 + 0.5)) = 1.413 dollars.
c) The following is the calculation of the maximum cost of your daily coffee-making process:
$ / day = (16.95 / 12 * 16) + (5.72 / 100 * 0.16) + (18.33 / 100) + (0.11643 / 60 * (5/60 + 0.5)) = 1.413 dollars.
d) Variables: amount of coffee beans (c), amount of water (w)
Constraints: 16 ≤ c ≤ 17; 350 ≤ w ≤ 380;
w = 17c
Objective Function: 16.95/12c + 5.72w/100 + 18.33/100 + (0.11643 / 60 * (5/60 + 0.5))
e) Constraints: 16 ≤ c ≤ 17; 350 ≤ w ≤ 380; w = 17c,
graph shown below:
f) The optimal solution(s) can be found at the vertices of the feasible region. Minimizing or maximizing a function over the feasible region means finding the highest or lowest value that the function can take within that region. The optimal solution(s) can be found by evaluating the objective function at each vertex and choosing the one with the lowest value. The minimum value of the objective function is found at the vertex (16, 272) and is 1.4125 dollars. The maximum value of the objective function is found at the vertex (17, 289) and is 1.4375 dollars.
g) The cost of Northampton's water would have to increase to $0.05 per 100 cubic feet for the cheaper option to become a different ratio of water to beans.
h) The potential optimal solutions are all the vertices of the feasible region. Any point in the feasible region cannot be an optimal solution because the objective function takes on different values at different points.
i) The potential optimal solutions are:(16, 272) for α ≤ 0 and β ≥ 0(17, 289) for α ≥ 16.95/12 and β ≤ 0
All other points in the feasible region are not optimal solutions.
ii) The objective function αc + βw is minimized for a particular optimal solution when α is less than or equal to the slope of the objective function at that point and β is greater than or equal to zero.
This is because the slope of the objective function gives the rate of change of the function with respect to c, while β is a scaling factor for the rate of change of the function with respect to w.
Learn more about Constraints:
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