Econ 172A Slides from Lecture 10 Joel Sobel October 21 2010 Econ 172A Sobel Announcements I Important Midterm room seating assignments There are three rooms and assigned seats I Midterm on October 28 2010 Material for Midterm I I I I I Problem Formulation Graphing Duality all topics Interpreting Excel Spreadsheets I can ask you to fill in the blanks in a sensitivity table I will not ask you sensitivity questions like the ones I discuss in today s lecture This means material through 10 19 lecture but today s lecture and Tuesday s lecture practice useful skills for the exam Example Simple formulation exercise I A furniture company that makes tables and chairs I A table requires 40 board feet of wood I A chair requires 30 board feet of wood I Wood costs 1 per board foot and 40 000 board feet of wood are available I It takes 2 hours of labor to make an unfinished table or an unfinished chair I 3more hours of labor will turn an unfinished table into a finished table I 2 more hours of labor will turn an unfinished chair into a finished chair I There are 6000 hours of labor available I No need to pay for this labor Prices The prices of output are given in the table below Product Unfinished Table Finished Table Unfinished Chair Finished Chair Econ 172A Price 70 140 60 110 Sobel Formulation I Objective Describe the production plans that the firm can use to maximize its profits I Variables Number of finished and unfinished tables and chairs I Let TF and TU be the number of finished and unfinished tables I Let CF and CU be the number of finished and unfinished chairs I Revenue 70TU 140TF 60CU 110CF I Cost is 40TU 40TF 30CU 30CF because lumber costs 1 per board foot I Profit Revenue Cost 30TU 100TF 30CU 80CF I The constraints are 1 40TU 40TF 30CU 30CF 40000 2 2TU 5TF 2CU 4CF 6000 Econ 172A I The first constraint says that the amount of lumber used is no more than what is available I The second constraint states that the amount of labor used is no more than what is available Sobel Solution I Excel finds the answer to the problem to be to construct only finished chairs 1333 333 I It is crazy to produce I This means that the LP formulation is not literally correct I Now We pretend that fractional chairs are ok I Later We will introduce methods to deal with the constraint that variables should be integers I The profit is 106 666 67 1 3 chair Sensitivity questions I What would happen if the price of unfinished chairs went up I I I I Currently they sell for 60 Allowable increase in the coefficient is 50 it would not be profitable to produce them even if they sold for the same amount as finished chairs If the price of unfinished chairs went down then certainly you wouldn t change your solution What would happen if the price of unfinished tables went up I I I I I I The allowable increase is greater than 70 Even if you could sell unfinished tables for more than finished tables you would not want to sell them How could this be At current prices you don t want to sell finished tables Hence it is not enough to make unfinished tables more profitable than finished tables you must make them more profitable than finished chairs Doing so requires an even greater increase in the price More Questions I What if the price of finished chairs fell to 100 I I I I Econ 172A This change would alter your production plan it involves a 10 decrease in the price of finished chairs and the allowable decrease is only 5 Complete answer resolve problem The best thing to do is specialize in finished tables producing 1000 and earning 100 000 If you continued with the old production plan your profit would be 70 1333 31 93 333 13 so the change in production plan was worth more than 6 000 Sobel I How would profit change if lumber supplies changed I I I I I The shadow price of the lumber constraint is 2 67 The range of values for which the basis remains unchanged is 0 to 45 000 This means that if the lumber supply went up by 5000 then you would continue to specialize in finished chairs and your profit would go up by 2 67 5000 10 333 If lumber supply increased by more you run out of labor and want to reoptimize If lumber supply decreased then your profit would decrease but you would still specialize in finished chairs I How much would you be willing to pay an additional carpenter I I I I Skilled labor is not worth anything to you You are not using the labor than you have You would pay nothing for additional workers Suppose that industrial regulations complicate the finishing process so that it takes one extra hour per chair or table to turn an unfinished product into a finished one How would this change your plans This problem differs from the original one because the amount of labor to create a finished product increases by one unit I I I I I I You cannot read your answer off the sensitivity table but a bit of common sense tells you something The change cannot make you better off To produce 1 333 33 finished chairs you ll need 1 333 33 extra hours of labor You do not have that available So the change will change your profit Using Excel it turns out that it becomes optimal to specialize in finished tables producing 1000 of them and earning 100 000 New Activity Econ 172A I The owner of the firm comes up with a design for a beautiful hand crafted cabinet I Each cabinet requires 250 hours of labor this is 6 weeks of full time work and uses 50 board feet of lumber I Suppose that the company can sell a cabinet for 200 would it be worthwhile Sobel One Approach Econ 172A I Change problem by adding an additional variable and an additional constraint I The coefficient of cabinets in the objective function is 150 which reflects the sale price minus the cost of lumber I The final value increases to 106 802 7211 I The solution involved reducing the output of unfinished chairs to 1319 727891 and increasing the output of cabinets to 8 163265306 I You could not have guessed these figures in advance but you could figure out that making cabinets was a good idea Sobel Using Sensitivity Analysis I Value the inputs to the production of cabinets I Cabinets require labor but labor has a shadow price of zero I They also require lumber I The shadow price of lumber is 2 67 which means that each unit of lumber adds 2 67 to profit I Hence 50 board feet of lumber would reduce profit by 133 50 I Since this is less than the price at which you can …