Econ 172A - Slides from Lecture 2Joel SobelOctober 2, 2012Econ 172A SobelAnnouncements1. 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 Problems5. Updated notes posted.Econ 172A SobelPROBLEM FORMULATION1. Problem Formulation is the process of translating anatural-language problem into math.2. I can’t teach this, but maybe I can illustrate itEcon 172A SobelDIET PROBLEM1. Given:IA list of different foods.IA list of different nutrients.IThe unit price of each food.IThe minimum daily requirement of each nutrient.IThe nutrient contribution of each food.2. Find the cheapest way to minimize all nutritionalrequirements.Econ 172A SobelBASIC DATA1. n different kinds of food.2. pjprice per unit of jth food.3. m different nutrients.4. nutritional requirement of Nutrient i is ci.5. A is technology (aijis the amount of the ith nutrient in oneunit of the jth food).Econ 172A SobelINFORMALLY1. Foods: lettuce, peanut butter, bread, apple juice. Fj, the jthfood, is one of these.2. Nutrients: Vitamin B12, iron, calcium, . . . . Ni, the ithnutrient, 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”Econ 172A SobelStep 1: Identify Variables.What are you looking for?IYou are looking for amounts of food.IVariables are quantities of each of the n foods.IThese are unknowns and need names.ILet xjbe the number of units of food j purchased.IYou want to find x = (x1, . . . , xn).Econ 172A SobelIMPORTANTIThe problem statement typically identify the variables. Thatis, it doesn’t say:“Your job is to find x, where xjis the quantity of Food j.”IYou must not only define variables, you must specify the units.(Here, it is uninteresting: xjis the number of units of Fj.)Econ 172A SobelStep 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 payp1x1+ · · · + pjxj+ · · · + pnxn=nXj=1pjxj= p · x. (1)(1) is the objective function. That is, you want to find x tomin p · x .Econ 172A SobelNOTICE LINEARITY ASSUMPTIONEcon 172A SobelStep 3: Write Down the Constraints.IThe constraints are that you satisfy nutritional requirements.IYou need to buy enough food to supply all nutrients in (atleast) the recommended amounts.IHow much nutrient i do you need? ci.IHow much of this nutrient is supplied when you have x? Nextpage.Econ 172A SobelWriting the ConstraintsIYou buy x1units of the first food.IYou obtain ai1x1units of the ith nutrient coming from thefirst food.INotice: product is in units of nutrient.IHow much nutrient i do you get from x?ai1x1+ · · · + aijxj+ · · · + ainxn=nXj=1aijxj. (2)IThe constraint:ai1x1+ · · · + aijxj+ · · · + ainxn=nXj=1aijxj≥ ci(3)describes the ith nutritional constraint.IThe entire problem imposes such a constraint for eachnutrient. That is we need an inequality for i = 1, . . . , m.Econ 172A SobelCleaning Up ConstraintsAx ≥ csummarizes all m constraints.Econ 172A SobelReflect on Linearity Assumptions Implicit in ConstraintsEcon 172A SobelNonnegativityImplicit in problem:x ≥ 0.Econ 172A SobelStep 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 theproblem (A, p, and c) and then would go ahead and try to find anumerical solution.http://www.zweigmedia.com/RealWorld/dietProblem/diet.htmlEcon 172A