Econ 172A Fall 2004 Midterm II A Instructions 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 last three questions Please read and answer these questions carefully 5 The table below indicates how points will be allocated on the exam Score I II III IV Exam Total 1 Possible 10 21 33 36 100 1 Consider the linear programming problem P max c x subject to Ax b x 0 where A is a matrix with two columns and three rows Answer the questions below You do not need to justify your answers No partial credit will be awarded of this problem a How many variables does the dual of P have b Suppose that the solution 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 2 2 Consider the linear programming problem Find x1 and x2 to solve P max subject to x1 x1 2x1 x1 x2 2x2 x2 x2 x 4 4 3 0 You must provide justifications for your answers to the questions below In particular say what 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 a candidate solution to the dual d Is 2 1 a solution to P Explain 3 3 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 sell for 15 20 and 25 per case respectively Each case of red wine costs 10 to produce and requires one hour to process and six hours to bottle Each case of white wine costs 12 to produce 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 in the Winery s factories Pallo stores its wine in a warehouse with a capacity of 3000 cubic feet Each week they ship out their entire inventory White and red wine occupy 1 5 cubic feet per case The blue wine occupies two cubic feet per case Pallo has a contract with a local wine seller to deliver at least 60 cases of red wine each week Here is a formulation of the problem as a linear programming problem I denote by xi the number 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 is max 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 of the dual variables in words including a specification of the units in which they are measured an economic interpretation of the meaning of these variables and interpretations of both the constraints and objective function of the dual c Does the Pallo wine problem have a solution Answer this question as completely as possible 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 5 4 A local bakery makes three different kinds of bread A loaf of whole wheat bread uses one pound of whole wheat flour and an ounce of yeast A loaf of oatmeal rye bread uses three quarters of a pound of white flour one quarter pound of rye flour one quarter pound of oatmeal and an ounce of yeast A loaf of white bread uses three quarters of a pound of white flour and two ounces of yeast The bakery can sell a loaf of whole wheat bread for 2 00 a loaf of oatmeal rye bread for 2 50 and a loaf of white bread for 1 50 Each day the bakery has available 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 most 125 loaves each day The bakery wishes to know how many loaves of each type of bread to produce in order to maximize profits subject to the constraints above In order to formulate the problem I defined the variables WHOLE number 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 subject to WHOLE 2 5OATRYE 1 5WHITE 120 0 75OATRYE 0 75WHITE 100 0 25OATRYE 50 0 25OATRYE 30 WHOLE OATRYE 2WHITE 140 WHOLE OATRYE WHITE 125 WHOLE OATRYE WHITE 0 I solved this problem using Excel The output follows this problem Use the output to answer the questions on the next page Answer the questions independently so that a change described in one part applies only to that part You must justify your answers by providing brief but complete descriptions of how you arrived at them 6 a What is the bakery s profit maximizing output How much does it earn b What is the most that the bakery would be willing to pay for an additional pound of white flour c What is the most that the bakery would pay for another pound 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 loaves each day f The baker notices that loaves of white bread are smaller than the others and decides that he could fit two of these loaves in the oven where he could only place one of the other loaves As a result the last resource constraint above changes to WHOLE OATRYE 5WHITE 125 Is it worthwhile for the baker to produce white bread now g The baker invents a new recipe for all grain …