Q1

Suppose that there are three suppliers S1, S2, and S3 in a distribution system that can supply 5, 5, and 6 units, respectively, of the company's good. The distribution system contains five demand centers, that require 2, 2, 4, 4, and 3 units each of the good. The transportation costs, in dollars per unit, are as follows:

D1 D2 D3 D4 D5

S1 2 1 2 3 3

S2 2 2 2 1 -1

S3 3 3 2 1 2

(a) Compute an optimum shipping schedule. Is the optimal solution unique?

(b) Find the range over which the cost of transportation from S1 to D3 can vary while maintaining the optimal basis solution in part (a).
(c) To investigate the sensitivity of the solution to this problem, consider what happens if the amount supplied from any one supplier and the amount demanded by any one demand center were both increased. Is it possible for the total shipping costs to decrease by increasing the supply and the demand for any particular choice of supply and demand centers?

(d) A landslide has occurred on the route from S2 to D5. If you bribe the state highway crew with $10, they will clear the slide. If not, the route will remain closed. Should you pay the bribe?

Q2

The following is based on a real case.

At an International Cosmetics company (referred to as C)'s board meeting, the firm's chief executive officer (CEO) reported that C was planning its production schedule for the upcoming quarter. She stated that the firm did not have the internal capacity to meet the projected demand and that the only short-term possibility was to outsource some of the demand to a third-party supplier. She indicated that C had been reluctant in the past to use vendors in this way because of the proprietary nature of the company's product line. However, she had been in negotiations with a local supplier that was prepared to sign a secrecy agreement. She indicated that she would need board approval before proceeding down this path. The board's chairman asked how much of the projected product demand might have to be subcontracted out and cautioned against exposing C's complete product line to an outside vendor. She stated that the analysis could be completed along with a recommendation within a week using C's analytics-based linear programming model and a further review of the candidate vendor.

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.

	Face Cream	Body Cream	Hand Cream
Labour (Hours/carton)
Stage 1	1.5	1.8	1.0
Stage 2	0.8	1.0	0.5
Materials (pounds/carton)
Water	8.0	6.0	7.0
Oil	1.0	3.0	2.0
Scents and Colors	0.5	0.3	0.4
Emulsifiers	0.5	0.7	0.6

(1) Use the given labor stages and material requirements provided above, what are the costs for producing the three products in-house (i.e., the internal production costs per carton)?

(2) Develop a Linear Optimization model to determine optimal production strategies, including the use of outsourcing. Specify the decision variables, constraints, objective function, and then solve using Solver.

(3) Using your answer in part (2) what production schedule minimizes total cost including outsourcing, and what is the corresponding value of the objective function?

(4) What labour and material resources are completely used up, and what are the corresponding shadow prices?

(5) What is the production schedule if hand cream can be subcontracted out at a cost of $50 per carton?

(6) Suppose that the board only allows outsourcing one product, i.e., only Face Cream or only Body Cream but not both. Under this additional requirement imposed by the company's board, what production schedule minimizes total cost including outsourcing?