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Design under uncertainty

# Design under uncertainty - DESIGN DESIGN UNDER UNCERTAINTY...

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Unformatted text preview: DESIGN DESIGN UNDER UNCERTAINTY Prof. Miguel Bagajewicz CHE 4273 1 TwoTwo-Stage Stochastic Optimization Models Stochastic Philosophy Philosophy • Maximize the Expected Value of the objective over all possible realizations of Expected uncertain parameters. • Typically, the objective is Profit or Net Present Value. Typically the objective is Profit or Net Present Value • Sometimes the minimization of Cost is considered as objective. Cost Uncertainty • Typically, the uncertain parameters are: market demands, availabilities, market prices, process yields, rate of interest, inflation, etc. • In Two-Stage Programming, uncertainty is modeled through a finite number Programming is modeled thro finite of independent Scenarios. Scenarios • Scenarios are typically formed by random samples taken from the probability random distributions of the uncertain parameters. th 2 Characteristics Characteristics of Two-Stage TwoStochastic Optimization Models Stochastic FirstFirst-Stage Decisions • Taken before the uncertainty is revealed. They usually correspond to structural decisions (not operational) decisions (not operational). • Also called “Here and Now” decisions. • Represented by “Design” Variables. • Examples: −To build a plant or not. How much capacity should be added, etc. −To place an order now. −To sign contracts or buy options. −To pick a reactor volume, to pick a certain number of trays and size the condenser and the reboiler of a column, etc 3 Characteristics Characteristics of Two-Stage TwoStochastic Optimization Models SecondSecond-Stage Decisions • Taken in order to adapt the plan or design to the uncertain parameters realization. • Also called “Recourse” decisions. called Recourse decisions • Represented by “Control” Variables. • Example: the operating level; the production slate of a plant. • Sometimes first stage decisions can be treated as second stage decisions. In such case the problem is called a multiple stage problem. 4 Example: Example: Vinyl Chloride Plant HCl recycle Air or O2 Oxychlorination Ethylene Cl2 Light ends EDC purification Direct chlorination EDC pyrolysis VCM purification VCM EDC recycle Heavy ends 5 Example: Example: Vinyl Chloride Plant Consider the following forecasts: Forecasted prices of raw materials product Forecasted prices of raw materials product Year Forecasted excess demand over current capacity Ethylene Chlorine Oxygen VCM \$/ton \$/ton \$/ft3 \$/ton 2004 492.55 212.21 0.00144 499.19 2004 3602 2005 499.39 214.14 0.00144 506.19 2005 5521 2006 506.22 216.07 0.00143 513.18 2006 7355 2007 513.06 218.00 0.00142 520.18 2007 9551 2008 519.90 219.93 0.00141 527.18 2008 11888 2009 526.73 221.86 0.00140 529.17 2009 14322 2010 533.57 223.79 0.00139 535.17 2010 16535 2011 540.41 225.72 0.00138 543.17 2011 18972 Std. Dev 24.17 10.56 0.00010 26.15 Year VCM lb-mol/hr Consider building (in 2004) for three capacities to satisfy excess demand at 2004, 2006 and 2011 Plants will operate under capacity until 2006 or 2011 in the last 2006 and 2011. Plants will operate under capacity until 2006 or 2011 in the last two two cases. These are 3 different first stage decisions. 6 Example: Example: Vinyl Chloride Plant The different investment costs are: Plant Capacity 4090 MMlb/yr 6440 MMlb/yr 10500 MMlb/yr TCI \$47,110,219 \$68,886,317 \$77,154,892 Consider the following calculation procedure Random Number Generation Total Product Cost Income from selling VCM Random numbers are Raw material Cost Gross Income Cash Flow Net Profit obtained for each year for obtained for each year for raw raw materials and product prices using sampling from a Net Present Worth normal distribution. This can be done in Excel Risk & Probability 7 Example: Example: Vinyl Chloride Plant Probability vs. Net Present Worth Histograms and Risk Curves are 0.35 0.3 Pro b ability 0.25 Notice the asymmetry in the distributions. 0.2 0 .15 0.1 pec(j) pec(j) 0.05 0 - 4500 Risk at Different Capacity -3000 -1500 0 6 1500 NPW (\$10 ) 6.44E9 lb/yr 4.09E9 lb/yr 3000 1.05E10 lb/yr Cummulative Probability 1 0.8 0.6 The risk curves show a 36% chance of losing 0.4 money for the 10.5 billion lbs/year capacity, 0.2 31.7% for the 6.44 billion lbs/yr capacity and 0 -6.00E+09 -4.00E+09 -2.00E+09 0.00E+00 41% chance for the 41% chance for the 4.09 billion lbs/year capacity. billion lbs/year capacity 2.00E+09 4.00E+09 Net Present Worth (\$) 6.44E9 lb/yr 4.09E9 lb/yr 6.00E+09 Expected Profits are: 24%, 25% and 20%. 1.05E10 lb/yr 8 SCENARIO SCENARIO GENERATION SCENARIO GENERATION Consider each parameter’s probability distribution. di Discretize Discretize it. Option 1: pick values of probabilities. probabilities. For example, for 3 values, pick 25%, 50% and values, 25% probability and find the va values. Use the cumulative th curve curve to locate the numbers. P(θ) θ1 θ2 θ3 θ4 θ5 θ SCENARIO GENERATION Consider each parameter’s probability distribution. di Discretize Discretize it. Option 2: pick values pick (equidistantly or randomly) and and find the probability that corresponds to them from the area area they “span”. Use the cumulative cumulative curve for this. P(θ) θ1 θ2 θ3 θ4 θ5 θ SCENARIO GENERATION Consider each parameter’s probability distribution. di Discretize Discretize it. Option 3: pick equal probability va values and find parameter fi values. values. For example, for 3 values, pick 33% and locate the points Use points. Use the cumulative curve to to do this. P(θ) θ1 θ2 θ3 θ4 θ5 θ FOR FOR A LARGE NUMBER OF SAMPLES SAMPLES WE USE THIS OPTION SCENARIO GENERATION Each scenario is constructed by picking one realization for each parameter. EXAMPLE: EXAMPLE: 2 parameters (θ1, θ2). If each parameter parameters is discretized in three instances (θi,low, 25%, θi,avg 50%, θi,hig 25%) SCENARIO GENERATION Scenario Probability Scenario Probability θ1,low, θ2,low 6.25% θ1,hig, θ2,low 6.25% θ1,low, θ2,avg 12.5% 12.5% θ1,hig, θ2,avg 12.5% 12.5% θ1,low, θ2,hig 6.25% θ1,hig, θ2,hig 6.25% θ1,avg, θ2,low 12.5% 12.5% θ1,avg, θ2,avg 25.0% SUM OF ALL 1,avg, 2,avg θ1,avg, θ2,hig 12.5% 12.5% PROBABILITIES=1 SCENARIO GENERATION Effect of Small Number of Samples 1.2 SAMPLING 1 ACTUAL 20 Scenarios 0.8 0.6 0.4 0.2 0 75 85 95 105 115 125 1.2 SAMPLING 1 ACTUAL 0.8 500 Scenarios 0.6 0.4 0.2 0 75 85 95 105 115 125 SCENARIO GENERATION Effect of the Number of Samples on Results (Gas in Asia) 1.0 1s 4.666 0.8 10s 4.357 100s 4.565 0.6 200s 4.678 0.4 50s 4.793 0.2 0.0 0 1 2 3 4 5 6 7 8 9 10 REGRET REGRET ANALYIS 17 MINIMAX MINIMAX REGRET ANALYSIS Motivating Example s1 H igh Maximize Average…select A Optimistic decision maker s3 L ow Average A Traditional way s2 M edium 19 14 -3 10 B 16 7 4 9 C 20 8 -4 8 D 10 6 5 7 M ax 20(C) 14(A) 5(D) 10(A) MaxiMax … select C 20 Pessimistic decision maker C \$ Million MaxiMmin … select D A 15 10 5 B D 0 1 -5 s 1 2 s2 3 s3 18 MINIMAX MINIMAX REGRET ANALYSIS Motivating Example A … regret = 8 @ low market C … regret = 9 @ low market D … regret = 10 @ high market s1 High s2 Medium s3 Low Maximum Regret A 1 0 8 8 B Calculate regret: find maximum regret 4 7 1 7 C 0 6 9 9 D 10 8 0 10 10 8 B In general, gives conservative decision general gives conservative but not pessimistic. \$ Million B … regret = 7 @ medium market MINIMAX D B 6 C 4 2 A 0 s1 1 s2 2 s3 3 19 MINIMAX MINIMAX REGRET ANALYSIS TwoTwo-Stage Stochastic Programming Using Regret Theory Theory s1 d1 19.01 d2 11.15 d3 12.75 d4 5.41 d5 15.09 Max 19.01 s1 d1 d2 d3 d4 d5 0.00 7.86 6.26 13.60 3.92 NPV s4 15.48 20.54 22.25 32.02 12.48 32.02 s2 10.38 14.47 7.81 9.91 7.40 14.47 s3 10.57 8.87 16.02 12.63 8.81 16.02 s5 10.66 10.58 9.16 8.08 15.05 15.05 ENPV 13.22 13.12 13.60 13.61 11.77 13.61 s2 4.09 0.00 6.66 4.56 7.07 Regret s3 s4 s5 5.45 16.54 4.39 7.15 11.48 4.47 0.00 9.77 5.89 3.39 0.00 6.97 7.21 19.54 0.00 Min Max Min 19.01 10.38 20.54 8.87 22.25 7.81 32.02 5.41 15.09 7.40 32.02 10.38 Max 16.54 11.48 9.77 13.60 19.54 9.77 20 SAMPLING SAMPLING ALGORITHM Generate h(s) d=d+1 d=1 Outer Loop yes h=h(d) EXIT Fi Fi Fix First-Stage Variables s=1 h=h(s) Solve Deterministic Model no d<=d_max Solve Deterministic Model no Inner Loop yes s= s+1 s<=s_max Store Store NPV(d,s) This generates several solutions 21 UPPER UPPER AND LOWER BOUNDS Risk 1 D 0.8 C B 0.6 Lower Bound 0.4 Upper Bound Bound A 0.2 0 0.0 2.0 4.0 6.0 8.0 10.0 This is very useful because it allows nice decomposition, That is, there is no need to solve the full stochastic problem 22 UPSIDE UPSIDE POTENTIAL Point measure for the upside Risk 1.0 0.9 0.8 0.7 OV =3.075 0.6 OV =0.75 0.5 E(Pro f it ) = 3.4 E(Pro f it ) = 3.0 0.4 0.3 0.2 0.1 VaR =0.75 VaR =1.75 0.0 0 1 2 3 4 5 Profit 6 7 8 9 10 23 AREA AREA RATIO Comparison measure Risk 1.0 0.9 Risk(x2,NPV) O_Area 0.8 0.7 Risk(x1,NPV) 0.6 0.5 0.4 R_Area 0.3 0.2 0.1 0.0 0 1 ENPV2 ENPV1 2 3 4 5 6 7 8 9 NPV 10 24 CONCLUSIONS CONCLUSIONS • Regret Analysis can help in identifying good solutions solutions (It can also fail) • The sampling Algorithm is an important tool to identify upper bounds and good solutions. • The upper potential is important to be Th considered. considered. 25 ...
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