2) a) Find the solution set for the following systems graphically and graph the iso profit line on the same diagram. (8 marks)

b) Using the iso-profit line, determine the optimal solution to each system. (2 marks)

c) Which constraints are binding, and which ones are non-binding? (5 marks) i) C = 2x +3y ii) P = 3x + y S.t. 4x + 2y 12 S.t. x + 3y 9 3x + 5y 25 5x + 2y 15 x 0 2x + 3y 10 y 0 x 0 and y 0

3) Explain the meaning of shadow pricing and binding constraints in your own words. (5 marks) Shadow price:In linear programming problems, when the right-hand side (RHS) of the constraint is increased by one unit, the shadow price of the constraint is the difference between the optimal value of the objective function and the value of the objective function, calculated on an optional basis. Binding constraints:In linear programming problems, binding constraints are some restrictions that make the optimal solution and feasible solution greatly change due to some changes, thus making the feasibility criteria more stringent. Stricter constraints adversely affect the optimality of the solution, while looser constraints lead to an increase in the region in which the optimal solution resides.

4) Consider the following linear programing problem: P = 2x + 3y subject to 2x + 5y 20 Resource 1 3x + 2y 17 Resource 2 as well as x 0 and y 0.

a) Use the method of corners (showing your graph and corner points) to solve this problem. (i.e. Find the number of x and y so that P is optimized) (5 marks)

b) Will the optimal solution you found in part a remain optimal if coefficient of x changes to 1 in the objective function? How do you know? (3 marks)

c) Will the solution remain optimal is amount of Resource 2 changes to 20? How do you know? (2 marks)

5) Formulate the following LP problems (Do not attempt to solve):

a) Soundex produces two models of satellite radios. Model A requires 15 min of work on assembly line 1 and 10 min of work on assembly line 2. Model B requires 10 min of work on assembly line 1 and 12 min of work on assembly line 2. At most, 25 labor-hours of assembly time on line 1 and 22 labor-hours of assembly time on line 2 are available each day. It is anticipated that Soundex will realize a profit of $12 on model A and $10 on model B. How many satellite radios of each model should be produced each day to maximize Soundexs profit? (5 marks)

b) Patricia has at most $30,000 to invest in securities in the form of corporate stocks. She has narrowed her choices to two groups of stocks: growth stocks that she assumes will yield a 15% return (dividends and capital appreciation) within a year and speculative stocks that she assumes will yield a 25% return (mainly in capital appreciation) within a year. Determine how much she should invest in each group of stocks to maximize the return on her investments within a year if she has decided to invest at least 3 times as much in growth stocks as in speculative stocks. (5 marks)