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IE426 Problem Set #4 Answers Prof Jeff Linderoth IE 426 – Problem Set #4 Answers 1 My Brown Eyed Girl My long-haired friend Jim Sawyer is down on his luck. He has, however, concocted a new get-rich-quick scheme. Every morning, he will visit the local Pabst Blue Ribbon distributor and purchase a number of beers at a cost of \$0.50 per beer. Jim can purchase at most 500 beers per day, since he can’t fit any more than that into his VW Beetle. After purchasing the beer, Jim will wander the streets of Atlanta selling as many beers as he can at a cost of \$1.50 per beer. At the end of the day, Jim drinks all the beer he doesn’t sell, and for each beer Jim drinks, he counts \$0.10 towards his profit. 1.1 Problem Suppose there are five “beer-demand” scenarios that occur with the probabilities shown in Table 1. Formulate a (linear) stochastic programming instance that will tell Jim the number of beers to purchase to maximize his expected profit. You can, for purposes of this problem, assume that Jim can purchase and return fractional numbers of beers. Table 1: Demand for Beer in Different Scenarios Scenario Demand Probability 1 20 0.05 2 40 0.1 3 300 0.60 4 500 0.15 5 800 0.1 Answer: Make the following definitions: S Set of scenarios ( { 1,2,3,4,5 } ). x Number of beers Jim buys y s Number of beers that Jim sells in scenario s, s S w s Number of beers that Jim drinks in scenario s, s S ρ s The probability of scenario s d s The demand in scenario s Then a (linear) stochastic programming instance that will tell Jim the number of beers to purchase to maximize his expected profit is the following: Problem 1 Page 1

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IE426 Problem Set #4 Answers Prof Jeff Linderoth maximize - 0 . 5 x + s S ρ s (1 . 5 y s + 0 . 1 w s ) x 500 y s + w s = x s S ( Sell + Drink = Purchase ) y s d s s S ( Sell Demand ) y s , w s 0 s S x 0 1.2 Problem Create your model from Problem 1.1 in Mosel, and solve the stochastic program to help Jim determine his optimal policy. Answer: A Mosel model is given here: !\$Id: beerboy.mos,v 1.1 2006-12-04 10:06:32 jeff Exp \$ model "beerboy" uses "mmxprs" declarations S = 1..5 p, d: array(S) of real x: mpvar ! Number to buy y, w: array(S) of mpvar ! sell, and drink (resp.) end-declarations ! Just put data here... d := [ 20, 40, 300, 500, 800 ] p := [ 0.05, 0.1, 0.6, 0.15, 0.1 ] ExpProfit := -0.5 * x + sum(s in S) p(s) * (1.5 * y(s) + 0.1 * w(s)) forall(s in S) y(s) <= d(s) forall(s in S) x = y(s) + w(s) x <= 500 maximize(ExpProfit) writeln("Max Profit: ", getobjval) writeln("Buy: ", getsol(x), " papers") end-model Problem 1 Page 2
IE426 Problem Set #4 Answers Prof Jeff Linderoth And running that Mosel code produces the following output: Max Profit: 244 Buy: 300 papers 1.3 Problem Determine the Expected Value of Perfect Information (EVPI) for Problem 1.1. Answer: Since you know that the optimal policy is to exactly buy the demand, (subject to capacity constraint) you can fill in the following table, where z s is the profit in scenario s .

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