Sphero produces toy robots for various companies. Due to a large new order, Sphero predicts that demand for its R2-D2 robot will be 12,500 units in the next quarter. This is a substantial increase relative to previous quarters and will challenge Sphero's manufacturing capabilities. The R2-D2 robot consist of three key components: a housing unit, a motor unit, and a circuitry unit. Sphero is convinced that it will have to subcontract some of these units and purchase a number of these components from outside vendors because its internal capacity is limited. The table below shows both the production cost per unit for each component when the robot is produced internally and the cost per unit for each component when purchased from an outside vendor: Component Vendor Cost Internal Cost Housing $12.10 $8.10 Motor $25 $23 Circuitry $19.50 $14.50 Sphero's manufacturing facility has three departments: Assembly. Quality Control (QC), and Distribution. If produced internally, each component requires a certain amount of time (in hours/unit) in each of these three departments. The following table shows these time requirements; the final row of the table shows the total number of hours available for R2- D2 production in the upcoming quarter. Component Housing Motor Circuitry Capacity Assembly 0.07 0.08 0.04 1650 QC 0.09 0.02 0.06 1650 Distribution 0.06 0.05 0.04 1650 a. Formulate a Linear Program that can help Sphero meet its predicted demand most efficiently. Clearly define the decision variables, the objective function, and the constraints. b. To avoid overreliance on outside vendors for an individual component, Sphero also wants to ensure that the fractions of each component purchased from outside vendors are within 15% of each other. Adjust your formulation to incorporate this. Sphero produces toy robots for various companies. Due to a large new order, Sphero predicts that demand for its R2-D2 robot will be 12,500 units in the next quarter. This is a substantial increase relative to previous quarters and will challenge Sphero's manufacturing capabilities. The R2-D2 robot consist of three key components: a housing unit, a motor unit, and a circuitry unit. Sphero is convinced that it will have to subcontract some of these units and purchase a number of these components from outside vendors because its internal capacity is limited. The table below shows both the production cost per unit for each component when the robot is produced internally and the cost per unit for each component when purchased from an outside vendor: Component Vendor Cost Internal Cost Housing $12.10 $8.10 Motor $25 $23 Circuitry $19.50 $14.50 Sphero's manufacturing facility has three departments: Assembly. Quality Control (QC), and Distribution. If produced internally, each component requires a certain amount of time (in hours/unit) in each of these three departments. The following table shows these time requirements; the final row of the table shows the total number of hours available for R2- D2 production in the upcoming quarter. Component Housing Motor Circuitry Capacity Assembly 0.07 0.08 0.04 1650 QC 0.09 0.02 0.06 1650 Distribution 0.06 0.05 0.04 1650 a. Formulate a Linear Program that can help Sphero meet its predicted demand most efficiently. Clearly define the decision variables, the objective function, and the constraints. b. To avoid overreliance on outside vendors for an individual component, Sphero also wants to ensure that the fractions of each component purchased from outside vendors are within 15% of each other. Adjust your formulation to incorporate this
