A company manufacturing fine glass accessories produces color glass vases with a flower pattern. Each vase is produced from liquid glass by an artist glass-blower and then set in a storage room to cool to the room temperature. The vases are made in two sizes: large and small but since the production processes are nearly identical, the two types of vases share the same resources. Each vase, irrespective of its size, takes 20 minutes of the artist's work. The artist works 40 hours each week. A small and a large vase require 10 oz. and 20 oz. of colored glass, respectively. A total of 1,600 oz. of colored glass is available per week. In addition, a small vase occupies 2 units of storage space, whereas a large vase occupies 3 units of storage space. The total available storage space is 260 units. A small vase realizes an earnings contribution of $10.00 and a large vase realizes an earnings contribution of $12.00.

(i) Formulate the above problem as a linear optimization model with decision variables S and L, where S is the number of small vases produced and L is the number of large vases produced.

(ii) Solve the linear optimization model graphically. Plot the constraints, identify each constraint, and identify the feasible region. What is the optimal solution? Which constraints are binding at the optimal solution? What is the value of the objective function at the optimal solution?

(iii) Consider the storage capacity constraint. Suppose that the number 260 was changed to 270, i.e., the company has obtained an additional 10 units of storage space. Re-solve the equations of the binding constraints to compute the new optimal solution.

(a)How does the new contribution to earnings differ from the contribution to earnings in part (ii)?

(b)What is the marginal value of the storage space?