What are the costs for producing the three products in-house? Develop a linear programming model for this problem. What production schedule minimizes total cost including outsourcing, and what is the corresponding value of the objective function?
About the company: The company C produces and distributes a wide range of cosmetics offerings through a subscription-based e-commerce model. Customers can choose from different pricing plans, and the products are delivered to their homes monthly. The subscription model provided C with a stable income stream while continuing to build brand loyalty, particularly with regards to its 3 flagship products: face cream, body cream, and hand cream. With sales approaching $150 million annually, C had experienced double-digit growth over the past few years. Its marketing department estimated that the demand for the 3 products for the upcoming quarter are 12,000 cartons of face cream, 8,000 cases of body cream, and 18,000 cases of hand cream. The manufacturing process consisted of a two-stage production procedure that used four ingredients: purified water, oil, scents and colours, and emulsifiers. Stage 1 involved materials preparation and initial mixing while Stage 2 focused on final blending and packaging. The cost for raw materials is $1 per pound for purified water, $1.50 per pound for oil, $3 per pound for scents and colours, and $2 per pound for emulsifiers. C's available first-shift capacity for the next quarter was 15,000 labour-hours for stage 1 and 10,000 for stage 2. The first-shift hourly rate was $8.50 for stage 1 and $9.25 for stage 2. A second shift was available with a 10% reduction in capacity and a 10% increase in wage rates. C could also subcontract with a local supplier identified by CEO for face cream and body cream at a cost of $40 per carton and $55 per carton, respectively. This vendor had the capacity to meet the demand requirements that were in excess of C's capabilities. The production department had available 200,000 pounds of purified water, 50,000 pounds of oil, 7,500 pounds of scents and colours, and 15,000 pounds of emulsifiers.