Roomba produces two types of robot vacuum cleaners: Basic (just sweeps the floor) and Advanced (sweeps + mops). A local warehouse has received an order for 1000 basic robots and 1200 advanced robots for next month (month 1), and for 750 basic robots and 1000 advanced robots the following month (month 2). The company must fill all the orders. The cost of regular time production for basic robots is $25 and for advanced robots, $40. Overtime production costs 50% more. For each of the next two months there are 1000 hours of regular time production and 500 hours of overtime production available. A basic robot requires .5 hour of production time and an advanced robot, .6 hour. The cost of carrying a robot from one month to the next is $2.

The variables have been defined for you here:

B1R = number of basic robots produced in month 1 on regular time production

B1O = number of basic robots produced in month 1 on overtime production

B2R = number of basic robots produced in month 2 on regular time production

B2O = number of basic robots produced in month 2 on overtime production

A1R = number of advanced robots produced in month 1 on regular time production

A1O = number of advanced robots produced in month 1 on overtime production

A2R = number of advanced robots produced in month 2 on regular time production

A2O = number of advanced robots produced in month 2 on overtime production

Y1 = number of basic robots stored from month 1 to month 2.

Y2 = number of advanced robots stored from month 1 to month 2.

Find the correct objective function for this production scheduling problem.

Question

Roomba produces two types of robot vacuum cleaners: Basic (just sweeps the floor) and Advanced (sweeps + mops). A local warehouse has received an order for 1000 basic robots and 1200 advanced robots for next month (month 1), and for 750 basic robots and 1000 advanced robots the following month (month 2). The company must fill all the orders. The cost of regular time production for basic robots is $25 and for advanced robots, $40. Overtime production costs 50% more. For each of the next two months there are 1000 hours of regular time production and 500 hours of overtime production available. A basic robot requires .5 hour of production time and an advanced robot, .6 hour. The cost of carrying a robot from one month to the next is $2.

The variables have been defined for you here:

B1R = number of basic robots produced in month 1 on regular time production

B1O = number of basic robots produced in month 1 on overtime production

B2R = number of basic robots produced in month 2 on regular time production

B2O = number of basic robots produced in month 2 on overtime production

A1R = number of advanced robots produced in month 1 on regular time production

A1O = number of advanced robots produced in month 1 on overtime production

A2R = number of advanced robots produced in month 2 on regular time production

A2O = number of advanced robots produced in month 2 on overtime production

Y1 = number of basic robots stored from month 1 to month 2.

Y2 = number of advanced robots stored from month 1 to month 2.

Find the correct objective function for this production scheduling problem.

BlackTom AI · Accepted Answer

We need to restate the problem and all answer choices clearly, then evaluate each option.

Question restatement: Roomba produces two types of robots: Basic (B) and Advanced (A). Orders are: month 1: 1000 basic and 1200 advanced; month 2: 750 basic and 1000 advanced. Regular-time costs: basic = $25, advanced = $40. Overtime costs are 50% higher than regular time. Available per month: 1000 regular hours and 500 overtime hours. Time per unit: basic = 0.5 hour, advanced = 0.6 hour. Carrying (holding) cost to move a unit from one month to the next = $2. Decision variables denote production in each month and type of production, plus inventory carried from month 1 to month 2: B1R, B1O, B2R, B2O, A1R, A1O, A2R, A2O, Y1 (basic stored from m1 to m2), Y2 (advanced stored from m1 to m2).

Answer options:
1) MAX 25 B1R + 37.5 B1O + 40 A1R + 60 A1O + 25 B2R + 37.5 B2O + 40 A2R + 60 A2O + 2 Y1 + 2 Y2
2) MAX B1R + B1O + A1R + A1O + B2R + B2O + A2R + A2O – Y1 – Y2
3) MIN 25 B1R + 37.5 B1O + 40 A1R + 60 A1O + 25 B2R + 37.5 B2O + 40 A2R + 60 A2O + 2 Y1 + 2 Y2
4) MIN 25 B1R + 50 B1O + 40 A1R + 120 A1O + 25 B2R + 50 B2O + 40 A2R + 120 A2O + 2 Y1 + 2 Y2

Option-by-option analysis:

Option 1: MAX 25 B1R + 37.5 B1O + 40 A1R + 60 A1O + 25 B2R + 37.5 B2O + 40 A2R + 60 A2O + 2 Y1 + 2 Y2
- First, note the objective is to minimize total production and holding costs to satisfy all demands, given the problem provides costs and capacities but no revenues. A maximization objective would encourage producing more than needed and storing indefinitely, which contradicts the practical goal and the constraints. Therefore, this option is conceptually inconsistent with a typical production-planning problem that uses costs to be minimized.
- The per-unit costs for regular and overtime production are correctly reflected (25 and 37.5 for basic, 40 and 60 for advanced), and the holding costs (2 per unit stored) are included as Y1 and Y2. However, because the objective is stated as MAX, the sign convention and overall aim are wrong, making this option invalid.

Option 2: MAX B1R + B1O + A1R + A1O + B2R + B2O + A2R + A2O – Y1 – Y2
- This option attempts to sum production across both months and subtract inventories. It ignores several essential components:
  - It uses a maximization objective, which is inappropriate for a cost-minimization scheduling problem.
  - It does not incorporate the actual costs of producing by type (regular vs overtime) beyond a flat count, nor does it include the carrying cost for Y1 and Y2 with the correct coefficients.
  - It also omits the time-based constraints (hours available) and the demand fulfillment constraints, which are central to the scheduling problem.
- In short, this becomes a nonsensical objective for the stated problem because it neither matches the cost structure nor the goal of satisfying demand with minimal cost.

Option 3: MIN 25 B1R + 37.5 B1O + 40 A1R + 60 A1O + 25 B2R + 37.5 B2O + 40 A2R + 60 A2O + 2 Y1 + 2 Y2
- This option aligns with the standard structure of a production-cost minimization model:
  - Regular-time costs: basic 25, advanced 40, are correctly captured for B1R, A1R, B2R, A2R.
  - Overtime costs: basic 37.5, advanced 60, are correctly captured for B1O, A1O, B2O, A2O (since overtime is 50% higher than regular: 1.5 × regular).
  - Holding costs: carrying units from month 1 to month 2 is correctly included as 2 per unit for Y1 (basic carried) and Y2 (advanced carried).
  - The coefficients reflect per-unit costs rather than total costs per month, which is appropriate for a linear programming objective that will be combined with constraints (demands, hours, inventories) to minimize total cost.
- Therefore, option 3 correctly represents a cost-minimization objective with the proper unit costs, holding costs, and sign conventions.

Option 4: MIN 25 B1R + 50 B1O + 40 A1R + 120 A1O + 25 B2R + 50 B2O + 40 A2R + 120 A2O + 2 Y1 + 2 Y2
- Here, the regular-time costs for basic and advanced are correct for regular hours (25 and 40), but overtime costs are inflated to 50 for basic and 120 for advanced, which do not reflect the stated 50% overtime premium (which should be 37.5 and 60 respectively).
- Consequently, this option misstates overtime costs, leading to an incorrect objective function structure for the problem as described. The holding costs are correctly included, but the erroneous overtime pricing invalidates this option.

Summary of reasoning: The problem describes a cost-minimization scheduling model with different production modes (regular vs overtime), per-unit costs accordingly, and a carrying cost for month-to-month inventory. Among the four choices, Option 3 is the only one that accurately uses the correct per-unit costs for regular and overtime production and includes the proper carrying costs for Y1 and Y2 while adopting a MIN objective, which matches the intent of minimizing total cost to meet the two-month demand under given constraints. The other options either maximize the objective, misstate overtime costs, or omit/misrepresent essential cost components.

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