Harrison and Lara have discovered a counterfeit jewelry operation. Two workers, Choot and Wanlah, sell fake "gold" watches and "diamond" rings for a notoriously frugal mob boss. They need to gather intel and file a report on their discovery.
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 honouring 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?
To solve this optimization problem, we need to define the objective function and the constraints.
Objective:
The objective is to minimize the total cost of labor while ensuring that the mob boss's demands are met. The total cost of labor can be calculated by multiplying the hourly wage by the number of hours worked by each worker.
Let's denote:
Cw = Total cost of labor for worker Choot
Cm = Total cost of labor for worker Wanlah
Wc = Hourly wage for worker Choot (in dollars per hour)
Ww = Hourly wage for worker Wanlah (in dollars per hour)
Thus, the objective function can be written as:
Minimize: Cw + Cm
Constraints:
1. The minimum number of watches to be sold per day is 12.
2. The minimum number of rings to be sold per day is 30.
3. The total number of watches sold by both workers should be greater than or equal to 12.
4. The total number of rings sold by both workers should be greater than or equal to 30.
5. The total number of watches sold by Choot plus the total number of rings sold by Choot should be equal to or less than the number of watches or rings sold by Wanlah (since Choot is more efficient).
6. The total number of watches sold by Wanlah plus the total number of rings sold by Wanlah should be equal to or less than the number of watches or rings sold by Choot (since Choot is more efficient).
Let's denote:
Xc = Total number of watches sold by Choot
Xw = Total number of watches sold by Wanlah
Yc = Total number of rings sold by Choot
Yw = Total number of rings sold by Wanlah
Thus, the constraints can be written as:
Xc + Xw ≥ 12 (Minimum number of watches constraint)
Yc + Yw ≥ 30 (Minimum number of rings constraint)
Xc + Yc ≤ Xw + Yw (Efficiency constraint for Choot)
Xw + Yw ≤ Xc + Yc (Efficiency constraint for Wanlah)
The optimization equation can now be written as:
Minimize: Wc * Hc + Ww * Hw
Subject to:
Xc + Xw ≥ 12
Yc + Yw ≥ 30
Xc + Yc ≤ Xw + Yw
Xw + Yw ≤ Xc + Yc
To find the optimization equation, we need to determine the objective function and the constraints.
Objective function: The objective is to minimize the total cost of the workers' wages.
Constraints:
1. The total number of watches sold each day must be at least 12.
2. The total number of rings sold each day must be at least 30.
Based on the provided information, we can assume that the wages depend on the number of items sold by each worker.
Let's define the following variables:
x = number of watches sold by Choot each day
y = number of rings sold by Choot each day
w = number of watches sold by Wanlah each day
z = number of rings sold by Wanlah each day
The total cost of the workers' wages can be calculated as follows:
Total cost = (Choot's wage per watch * x) + (Choot's wage per ring * y) + (Wanlah's wage per watch * w) + (Wanlah's wage per ring * z)
Based on the given information, Choot's wage per watch is $4, Choot's wage per ring is $3.50, Wanlah's wage per watch is $3, and Wanlah's wage per ring is $4.
Now we can write the optimization equation as follows:
Minimize: 4x + 3.5y + 3w + 4z
Subject to the constraints:
x + w ≥ 12
y + z ≥ 30