The adventurers learn that the mob boss demands a minimum of 12 watches and 30 rings to be sold every day, and has figured out how to get what he wants while paying his two workers the least possible amount (while still honoring their agreed-upon hourly wages). While Lara and Harrison can't tell how many hours each worker spends selling the jewelry, or how many items they actually sell each day, they did learn something about the workers' wages and how efficiently they both work. They wrote down this information in the note that you can read in the image.
what is the optimization equation?
The optimization equation for this scenario can be formulated as follows:
Minimize the total cost of wages for both workers, subject to meeting the minimum requirement of selling 12 watches and 30 rings every day.
Let's assume the two workers are W1 and W2, with hourly wages of x and y respectively.
Define:
w1 = number of hours W1 spends selling jewelry per day
w2 = number of hours W2 spends selling jewelry per day
We are given the following information from the note:
1) Worker W1 can sell 3 watches and 2 rings in 1 hour.
2) Worker W2 can sell 1 watch and 5 rings in 1 hour.
3) The total wages paid to both workers cannot exceed $60.
To minimize the total cost of wages, we need to minimize the expression:
Total cost = (w1 * x) + (w2 * y)
Subject to the following conditions:
1) Minimum number of watches sold per day: (3w1 + w2) >= 12
2) Minimum number of rings sold per day: (2w1 + 5w2) >= 30
3) Total wages cannot exceed $60: (w1 * x) + (w2 * y) <= 60
Therefore, the optimization equation is to minimize (w1 * x) + (w2 * y) subject to (3w1 + w2) >= 12, (2w1 + 5w2) >= 30, and (w1 * x) + (w2 * y) <= 60.
To find the optimization equation in this scenario, we need to determine the objective function and the constraints.
Objective Function:
The objective is to minimize the total cost of wages paid to the two workers, Lara and Harrison. We can represent the total cost of wages as a variable, let's say C.
Constraints:
1. The mob boss demands a minimum of 12 watches and 30 rings to be sold every day. Let's represent the number of watches sold by Lara as Wl and the number of rings sold by Lara as Rl. Similarly, the numbers sold by Harrison can be represented as Wh (watches) and Rh (rings).
2. From the information given, it is known that Lara makes 5 rings per hour, while Harrison makes 8 rings per hour. Moreover, Lara makes 3 watches per hour, while Harrison makes 2 watches per hour. We can represent the number of hours worked by Lara as Hl and by Harrison as Hh.
3. Lara and Harrison are paid $10 per hour and $13 per hour, respectively. We can represent Lara's hourly wage as L and Harrison's hourly wage as H.
With this information, we can set up the following optimization equation:
Minimize C = (L * Hl) + (H * Hh)
Subject to the following constraints:
Wl + Wh >= 12 (number of watches sold)
Rl + Rh >= 30 (number of rings sold)
Wl <= 3 * Hl (Lara's watch production)
Wh <= 2 * Hh (Harrison's watch production)
Rl <= 5 * Hl (Lara's ring production)
Rh <= 8 * Hh (Harrison's ring production)