Econ 172A Slides from Lecture 2 Joel Sobel October 2 2012 Econ 172A Sobel Announcements 1 Sections this evening York 226 8 9 or 9 10 2 Podcasts available when I remember to use microphone 3 Textbook on reserve at SSH Library 4 Look at Supplementary Formulation Problems 5 Updated notes posted Econ 172A Sobel PROBLEM FORMULATION 1 Problem Formulation is the process of translating a natural language problem into math 2 I can t teach this but maybe I can illustrate it Econ 172A Sobel DIET PROBLEM 1 Given I I I I I A list of different foods A list of different nutrients The unit price of each food The minimum daily requirement of each nutrient The nutrient contribution of each food 2 Find the cheapest way to minimize all nutritional requirements BASIC DATA 1 n different kinds of food 2 pj price per unit of jth food 3 m different nutrients 4 nutritional requirement of Nutrient i is ci 5 A is technology aij is the amount of the ith nutrient in one unit of the jth food Econ 172A Sobel INFORMALLY 1 Foods lettuce peanut butter bread apple juice Fj the jth food is one of these 2 Nutrients Vitamin B12 iron calcium Ni the ith nutrient is one of these 3 Everything has units 3 1 prices dollars per unit of food 3 2 nutrient requirements units of nutrient 3 3 aij units of nutrient per unit of food Step 1 Identify Variables What are you looking for I You are looking for amounts of food I Variables are quantities of each of the n foods I These are unknowns and need names I Let xj be the number of units of food j purchased I You want to find x x1 xn IMPORTANT I The problem statement typically identify the variables That is it doesn t say Your job is to find x where xj is the quantity of Food j I You must not only define variables you must specify the units Here it is uninteresting xj is the number of units of Fj Step 2 Write Down the Objective Function What are you trying to do Minimize cost Minimize cost of the food that you buy If you buy x you pay p1 x1 pj xj pn xn n X pj xj p x j 1 1 is the objective function That is you want to find x to min p x 1 NOTICE LINEARITY ASSUMPTION Econ 172A Sobel Step 3 Write Down the Constraints I The constraints are that you satisfy nutritional requirements I You need to buy enough food to supply all nutrients in at least the recommended amounts I How much nutrient i do you need ci I How much of this nutrient is supplied when you have x Next page Writing the Constraints I I I I You buy x1 units of the first food You obtain ai1 x1 units of the ith nutrient coming from the first food Notice product is in units of nutrient How much nutrient i do you get from x ai1 x1 aij xj ain xn n X aij xj 2 aij xj ci 3 j 1 I The constraint ai1 x1 aij xj ain xn n X j 1 I describes the ith nutritional constraint The entire problem imposes such a constraint for each nutrient That is we need an inequality for i 1 m Cleaning Up Constraints Ax c summarizes all m constraints Econ 172A Sobel Reflect on Linearity Assumptions Implicit in Constraints Econ 172A Sobel Nonnegativity Implicit in problem x 0 Econ 172A Sobel Step 4 Write Down the Entire Problem The problem is to find x to solve min p x subject to Ax c and x 0 In practice you will be given values for the parameters of the problem A p and c and then would go ahead and try to find a numerical solution http www zweigmedia com RealWorld dietProblem diet html Econ 172A Sobel