Econ 172A, Fall 2004: Midterm II-AInstructions.1. Please check to see that your name is on this page. If it is not, then you are in the wrong seat.2. The examination has four questions. Answer them all.3. If you do not know how to interpret a question, then ask me.4. No justification is required for the first question (each part will be graded on a “right” or“wrong” basis – no partial credit will be awarded). You must justify your answers to the lastthree questions. Please read and answer these questions carefully.5. The table below indicates how points will be allocated on the exam.Score PossibleI 10II 21III 33IV 36Exam Total 10011. Consider the linear programming problem P: max c · x subject to Ax ≤ b, x ≥ 0, where A isa matrix with two columns and three rows. Answer the questions below. You do not need tojustify your answers. No partial credit will b e awarded of this problem.(a) How many variables does the dual of P have?(b) Suppos e that the s olution x∗to P satisfies x∗1> 0 and the first constraint is binding.i. Which constraints must bind in the corresponding solution to the dual?ii. Which dual variables must be positive in the corresponding solution to the dual?22. Consider the linear programming problem:Find x1and x2to solve P:max x1+ x2subject to x1+ 2x2≤ 42x1− x2≤ 4x1+ x2≤ 3x ≥ 0You must provide justifications for your answers to the questions below. In particular, saywhat you need to do to check for feasibility and the basis for your inferences in part (c).(a) Write the dual of the problem P.(b) Verify that (x1, x2) = (2, 1) is feasible for P.(c) Assuming that (2, 1) is a solution to P, use Complementary Slackness to determine acandidate solution to the dual.(d) Is (2, 1) a solution to P? Explain.33. The following is a formulation problem that appeared on an old midterm.The Pallo Winery produces three varieties of wine – red, white, and blue. These products sellfor $15, $20, and $25 per case, respectively. Each case of red wine costs $10 to produce andrequires one hour to process and six hours to bottle. Each case of white wine costs $12 toproduce, needs two hours of processing time and eight hours of bottling time. Blue wine costs$21 per case to produce, uses three hours of processing time, and eight hours in bottling time.Each week there are 220 hours of processing time and 480 hours of bottling time available inthe Winery’s factories.Pallo stores its wine in a warehouse with a capacity of 3000 cubic feet. Each week they shipout their entire inventory. White and red wine occupy 1.5 cubic feet per case. The blue wineoccupies two cubic feet per case. Pallo has a contract with a local wine seller to deliver atleast 60 cases of red wine each week.Here is a formulation of the problem as a linear programming problem. I denote by xithenumber of cases of each variety of wine produced in a week, for i = red, white, and blue.Taking into account the costs of production, the objective function ismax 5xR+ 8xW+ 4xB.In addition to non-negativity constraints, the problem has four other constraints:• xR+ 2xW+ 3xB≤ 220 (processing time)• 6xR+ 8xW+ 8xB≤ 480 (bottling time)• 1.5xR+ 1.5xW+ 2xB≤ 3000 (warehouse capacity)• −xR≤ −60 (contracted delivery of red wine)Use this information to answer the questions on the following page.4(a) Write the dual of this problem.(b) Provide an interpretation of the dual. Your interpretation must include: a definition ofthe dual variables in words, including a specification of the units in which they are mea-sured; an economic interpretation of the meaning of these variables; and interpretationsof both the constraints and objective function of the dual.(c) Does the Pallo wine problem have a solution? (Answer this question as completely aspossible. If you can demonstrate that the problem does (or does not) have a solution,then do so. If you need additional information, describe what information you need.)54. A local bakery makes three different kinds of bread. A loaf of whole wheat bread uses onepound of whole wheat flour and an ounce of yeast. A loaf of oatmeal-rye bread uses threequarters of a pound of white flour, one quarter pound of rye flour, one quarter pound ofoatmeal, and an ounce of yeast. A loaf of white bread uses three quarters of a pound of whiteflour and two ounces of yeast. The bakery can sell a loaf of whole wheat bread for $2.00, a loafof oatmeal-rye bread for $2.50, and a loaf of white bread for $1.50. Each day the bakery hasavailable 120 pounds of whole wheat flour, 100 pounds of white flour, 50 pounds of rye flour,30 pounds of oatmeal, and 140 ounces of yeast. In addition, its ovens are able to bake at most125 loaves each day.The bakery wishes to know how many loaves of each type of bread to produce in order tomaximize profits subject to the constraints above. In order to formulate the problem, I definedthe variables:WHOLE = numb e r of loaves of whole wheat bread produced.OATRYE = number of loaves of oatmeal-rye bread produced.WHITE = number of loaves of white bread produced.The bakery’s problem is then: find values for WHOLE, OATRYE, and WHITE to solve:max 2WHOLE + 2.5OATRYE + 1.5WHITEsubject to WHOLE ≤ 1200.75OATRYE + 0.75WHITE ≤ 1000.25OATRYE ≤ 500.25OATRYE ≤ 30WHOLE + OATRYE + 2WHITE ≤ 140WHOLE + OATRYE + WHITE ≤ 125WHOLE, OATRYE, WHITE ≥ 0.I solved this problem using Excel. The output follows this problem. Use the output toanswer the questions on the next page. Answer the questions independently (so that a changedescribed in one part applies only to that part). You must justify your answers by providingbrief (but complete) descriptions of how you arrived at them.6(a) What is the bakery’s profit maximizing output? How much does it e arn?(b) What is the most that the bakery would be willing to pay for an additional pound ofwhite flour?(c) What is the most that the bakery would pay for another p ound of oatmeal?(d) Would the baker produce more oat-rye bread if the price of oat-rye bread doubled?(e) How much would the profits of the bakery decrease if it were only able to bake 120 loaveseach day?(f) The baker notices that loaves of white bread are sm aller than the others, and decides thathe could fit two of these loaves in the oven where he could only place one of the otherloaves. As a result, the last resource constraint above changes to:WHOLE + OATRYE + .5WHITE ≤ 125Is it worthwhile for the baker to produce white bread now?(g) The baker invents a new