Question

A businessman needs 5 cabinets, 12 desks and 18 shelves cleaned out. He has two part time employees, Rashid and Ruby. Ruby can clean 1 cabinet, 3 desks and 3 shelves in a day while Rashid can clean 1 cabinet, 2 desks and 6 shelves in one day. Rashid is paid Rs.22 per day and Ruby is paid Rs.25 per day. In order to minimize the cleaning costs, for how many days should Rashid and Ruby be employed? Formulate the problem as a linear programming problem and find its solution by simplex method.

14 Feb 2018
Answer :
Word Count : 726
### Step 1: Formulating the Linear Programming Problem Let's define the following variables: - \( x \): Number of days Rashid is employed. - \( y \): Number of days Ruby is employed. #### Objective Function We need to minimize the total cost of employing Rashid and Ruby. Rashid is paid Rs. 22 per day, and Ruby is paid Rs. 25 per day. Therefore, the objective function will be: \[ \text{Minimize } Z = 22x + 25y \] #### Constraints The businessman needs the following cleaning tasks completed: - 5 cabinets cleaned. - 12 desks cleaned. - 18 shelves cleaned. Rashid can clean: - 1 cabinet per day. - 2 desks per day. - 6 shelves per day. Ruby can clean: - 1 cabinet per day. - 3 desks per day. - 3 shelves per day. This gives the following system of inequalities: 1. Cabinet constraint: \( x + y \geq 5 \) (since Rashid and Ruby together must clean at least 5 cabinets). 2. Desk constraint: \( 2x + 3y \geq 12 \) (since Rashid and Ruby together must clean at least 12 desks). 3. Shelf constraint: \( 6x + 3y \geq 18 \) (since Rashid and Ruby together must clean at least 18 shelves). 4. Non-negativity constraints: \( x \geq 0 \) and \( y \geq 0 \) (since days employed cannot be
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