Statistics: Analyzing Various Industries

Part 1
Fifth Avenue industries, a nationally known manufacturer of menswear produces four varieties of ties. One is an expensive, all-silk tie, one is an all-polyester tie, and two are blends of polyester and cotton. The following table illustrates the cost and availability (per monthly production planning period) of the three materials used in the production process:

Material Cost per Yard ($) Material available per month (Yards)
Silk 21 800
Polyester 6 3,000
Cotton 9 1,600

The firm has fixed contracts with several major department store chains to supply ties. The contracts require that capital Fifth Avenue industries supply a minimum quantity of each tie, but allow for a larger demand in Fifth Avenue choices to meet that demand. (Most of the ties are not shipped with the name Fifth Avenue on their label, incidentally, but with "Private Stock" labels supplied by the stores.)

The following table summarizes the contract demand for each of the sour style of ties, the selling price per tie and the fabric requirements of each variety. (Please view attachment)

(A) If Fifth Avenue's goal is to maximize its monthly profit, clearly define the decision variables and create a LP to solve this production mix problem.

Use any available application software (e.g., The Management Scientist, Excel Solver, Lingo, Lindo, STORM), print out the Optimal Solution/Results, Sensitivity Analysis outputs, and attach the outputs to this assignment. Based only on the software output you printed out, answer questions (B)-(G). Explain/ support your answers, citing specific parts of the output used to answer the question and, where appropriate, showing how the information was applied.
(B) What is the optimal solution and what is the monthly profit contribution?
(C) Which constraints are binding?
(D) Interpret the shadow prices (dual prices) for the binding constraints.
(E) An additional 50 yards of silk have become available at a cost of $30.00 per yard. Should Fifth Avenue purchase the extra 50 yards? Will this change your profit contribution? If yes, what is the new profit contribution for Fifth Avenue?
(F) If an additional 50 yards of silk can be purchased and an additional 100 all-silk ties are required to meet a new monthly minimum contract, how will this affect the optimal solution? Will the optimal solution change? What is the new profit contribution?
(G) How much of the following materials are being used as part of the optimal solution found in part b? Silk, polyester and cotton?

Part 2
North York Manufacturing has been awarded a contract to produce 1000 radar units. The delivery schedule is exact 400 units on April 30 and exact 600 units on May 30. Deviations from this schedule are not permitted. Radar units produced in April can be held in inventory to meet demand in Man at a cost of $30 per unit.

Each radar unit requires 30 hours of assembly labor. At the start of the contract (April 1) North York Manufacturing has 80 trained workers and can hire new workers on April 1 and or on May 1. Each trained worker produces 150 hours of assembly labor per month. Each new worker requires 50 hours of initial training and will therefore produce only 100 hours during the first month of employment. All workers receives a fixed salary of $2,500 per month. All workers may work a maximum of 20 hours on overtime in a month. Overtime is paid $15 per hour. The cost of hiring new workers is $200 per worker and all hiring is done on the first of a month. North York Manufacturing wishes to restrict their entire work force to a maximum of 100 workers at any time. North York Manufacturing does not lay-off any workers. The company seeks to minimize the cost of production plan.
(a) Clearly define the decision variables.
(b) Write the objective function for the linear programming.
(c) Write complete constraints for the linear programming.

PART 3
Consider a capital budgeting problem where five possible projects are being considered for execution over the next three years. The expected returns for each project, the yearly expenditures, and annual budget (in thousands of dollars) are given in the table below. Assume that each approved project will be executed over the entire three-year period.

Expenditure
Project Year 1 Year 2 Year 3 Expected Return
1 5 1 8 20
2 4 7 10 40
3 3 9 2 20
4 7 4 1 15
5 8 6 10 30
Budget 25 30 35 -

(a) Assume that unused funds cannot be carried forward. Define clearly decision variables.
(b) Formulate an integer linear programming problem to maximizing total expected return. Use the decision variables that you defined in question (a).
(c) Suppose unused budget in a year can be carried forward for use in succeeding years. Define new decision variables and write a new integer linear programming formulation.
(d) What constraint(s) must be added to the formulation if the following is imposed: When project 5 is selected, then either project 1 or project 2 may be selected but not both.
(e) What constraint(s) must be added to the formulation if the following is imposed: If project 4 is selected, then either project 3 or project 2 or both must be projected.