The Cosmetic Co is a company producing a variety of cosmetic creams and lotions. The creams and lotions are sold to a variety of retailers at a price of $23.20 for each jar of face cream and $16.80 for each bottle of body lotion. Each of the products has a variety of ingredients, with the key ones being silk powder, silk amino acids and aloe vera. Six months ago, silk worms were attacked by disease causing a huge reduction in the availability of silk powder and silk amino acids. The Cosmetic Co had to dramatically reduce production and make part of its workforce, which it had trained over a number of years, redundant.
The company now wants to increase production again by ensuring that it uses the limited ingredients available to maximise profits by selling the optimum mix of creams and lotions. Due to the redundancies made earlier in the year, supply of skilled labour is now limited in the short term to 160 hours (9,600 minutes) per week, although unskilled labour is unlimited. The purchasing manager is confident that they can obtain 5,000 grams of silk powder and 1,600 grams of silk amino acids per week. All other ingredients are unlimited. The following information is available for the two products:
|silk powder (at $2.20 per gram)||3 grams||2 grams|
|Aloe vera (at $1.40 per gram)||1 gram||0.5 grams|
|Silk amino acids (at $0.80 per gram)||4 grams||2 grams|
|skilled ($12 per hour)||4 minutes||5 minutes|
|Unskilled (at $8 per hour)||3 minutes||1.5 minutes|
Each jar of cream sold generates a contribution of $9 per unit, whilst each bottle of lotion generates a contribution of $8 per unit. The maximum demand for lotions is 2,000 bottles per week, although demand for creams is unlimited. Fixed costs total $1,800 per week. The company does not keep inventory although if a product is partially complete at the end of one week, its production will be completed in the following week.
Write the constraints, which are required to draw on a graph to identify feasible area, in form of inequalities.
Assuming the company wants to maximize its contribution, define the objective functions.
Assuming that the optimal solution lies at the intersection of the constraints for skilled labour and silk powder, find the optimum production plan and total contribution that can be earned.
Explain slack and shadow price.
Calculate the shadow price for silk powder and the slack for silk amino acids. All workings MUST be rounded
to 2 decimal places.