3 25 points An auto manufacturer produces two versions of it

3. (25 points) An auto manufacturer produces two versions of its flagship sedan, one with an automatic transmission (A) and one with a manual transmission (M). The manufacturer can make a profit of $40,000 for each A version sold and $20,000 for each M version sold. Each version is made with two types of labor, assembly labor and finishing labor. Automatic cars require 20 labor hours of assembly and 30 labor hours of finishing. The manual version requires 20 labor hours of assembly and 10 hours of finishing. The goal of the manufacturer is to maximize profit given the following three daily production constraints: 1. The manufacturer can produce a maximum of 100 automatic transmissions per day. 2. The manufacturer can employ a maximum of 6,000 labor hours for assembly per day. 3. The manufacturer can employ a maximum of 4,200 labor hours for finishing per day. (a) Formulate and write out the auto manufacturer’s linear programming problem. Clearly specify the objective function, decision variables, and constraints. (b) Illustrate the constraints on a graph with manual cars (M) on the y-axis and automatic cars (A) on the x-axis and specify the feasible region. (Hint: Rewrite the constraints in a way that can be easily graphed in M-A space.) (c) Rewrite the objective function in a way that can be graphed with the constraints. (Hint: Expresses M as a function of A.) (d) Find the optimal values of M and A that solve the linear programming problem. Also compute the maximum profit at this point. (Hint: Compare the slope of the objective function you derived in part (c) to the slopes of the constraints you found in part (b).) (e) Suppose the maximum amount of assembly labor is lifted from 6,000 to 6,400 hours per day. Calculate the new optimal values of M, A, and profit. How does profit change when the assembly labor constraint is relaxed by 400 hours? What is the shadow price of assembly labor hours? Explain this shadow price in words

Solution

Problem is

Entering =X1=X1, Departing =S2=S2, Key Element = 3030

R2R2 (new) =R2=R2 (old) ÷30=R2÷30=R2 (old) 130130

R1R1 (new) =R1=R1 (old) 20R2-20R2 (new)

Entering =X2=X2, Departing =S1=S1, Key Element = 403403

R1R1 (new) =R1=R1 (old) ÷403=R1÷403=R1 (old) 340340

R2R2 (new) =R2=R2 (old) 13R1-13R1 (new)

Since all CjZj0Cj-Zj0,

Optimum Solution is arrived with value of variables as :
X1=60

X2=240

Maximise Z=7200000