Project #61414 - stat homework



-   Answer all 9 questions. For full or partial credit, show all work. Just writing the final answer

         will not get you full credit.

-    There are 110 points, 10 p0ints are for any silly mistakes which I hope you do not make,  any

            extra point will be credited to you final.

-    You should not use any computer LP solvers including the LINDO, OR Graphical Internet

 Software, all must be done by hand computations.

-  Please take only two continuous hours to do this exam in one sitting.




1.          Solve the following system of 2 equations with 2 unknowns X, and Y:


X + Y = 5

X + 2Y = 6                                                                                                                   (10%)




2-          Graph the following feasible region:


X1 + X2 £ 5,                                                                                                               (10%)

X1 + 2X2 £ 6,

and   X1 ³ 0, X2 ³ 0.




3. Solve the following linear program problem


Max 2X2 - X1,


Subject to

X1 + X2 £ 5,

X1 + 2X2 £ 6, and   X1 ³ 0, X2 ³ 0,


by the graphical method indicate the optimal solution for both X1 and X2, as well as the

optimal value for the objective function.                                                (15%)



4.          Construct the dual problem of the above linear program                 (10%)






5.          Joe's Garage specializes in oil changes and tune-ups.  Profit per oil change is $7 and $15 per tune-up.  Joe's has a fleet account customer, which guarantees 30 oil changes per week.  Each oil change requires 20 minutes labor and $8 in supplies.  A tune-up takes one hour and costs Joe's $15 in supplies.  Mechanics are paid $10 per hour and Joe's currently employs two mechanics working 40 hours each per week.  Every week Joe's orders $1,750 in supplies, Joe's wishes to maximize profit. Formulate Joe’s decision problem as a linear program (do not solve it).                                                                                                                                                                (15%)




6-         Which of the following are correct and why?

a) Zero divided by any number is zero

b) Any number divided by itself is 1.                                                                                     (5%)



7.         Explain in not more than a short paragraph what are the most significant three topics you have learned in this course up to now?                                                                                        (10%)



8.        Consider the following LP problem:


            Max   7X1+10X2  (Objective Function)


Subject to:


5X1 + 6X2 £ 3600 (Cowhide)

X1 + 2X2 £ 960     (Production time)

X1 £ 500                (Production limit of baseballs)                                                                 (30%)  

X2 £ 500                (Production limit of softballs)

X1, X2 ³ 0             (Non-negativity)






Answer the following questions:


8.1. What is the optimal solution and optimal value for the problem?


8.2. Which of the constraints are binding?


8.3.  What is the impact on the optimal solution and optimal value if we decrease the cost coefficient c(1) = 7 to 6.1? Why?


8.4. What is the shadow price for the RHS # 1.? How do you interpret it?


8.5. What are the solution and optimal value for the dual problem? Why?


8.6. What is the impact on the optimal value if we decrease the right-hand side of constraint # 1 by  500.




9.          Given X = 1 then which one of the following holds?


a) X is £ 1

b) X is ³ 1                                                                                                                    (5%)

c) X is equal to one.



Subject Mathematics
Due By (Pacific Time) 03/08/2015 09:47 pm
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