Problem:
The Thompson Furniture Company produces inexpensive tables and chairs using a production process common for both products. Each table requires 4 hours in the carpentry department and 2 hours in the painting and varnishing department. Each chair requires 3 hours in carpentry and 1 hour in painting and varnishing. During the current production period, 240 hours are scheduled in carpentry and 100 hours are scheduled in painting and varnishing. Also, each product must undergo an inspection and possible rework process. A table requires an average of 1/2 hour for this process, each chair requires 3/5 hour on average. For the current production period, 36 hours have been scheduled for the inspection/rework process. Each table requires 32 linear feet of lumber in the manufacturing process while 10 linear feet are needed to complete each chair. There are 1,248 linear feet of lumber currently available for production of these products.
Each table contributes $23 towards profit while each chair contributes $13. The current schedule reflects a maximum demand for tables at 40 without a similar constraint for chairs.
a. Formulate a linear programming model that can be used to determine the production schedule for the two products for the current period.
b. Additional lumber is available to Thompson at $0.38 per linear foot in minimum quantity orders of 500 linear feet each. An order for the next production period has already been placed under a continuing vendor contract to arrive at the end of the current production period. That order will require all but 300 linear feet of available lumber storage space. A maximum of 8 hours of additional time is available for the current period in each of the carpentry and painting/varnishing departments using part-time employees at an hourly rate of $10.50. A maximum of 4 hours of additional time in inspection/rework is available in the current period at an hourly rate of $9.50. All production hours previously scheduled are fixed by employment contract. All unused resources, including lumber not stored, is considered wasted. Determine and justify which additional resources, if any, would result in a better solution for Thompson.