
From:  Kelly, Jeff (ON0F) 
Subject:  RE: [Helpglpk] LP problem with variable coeffcients (parametric LPsimplex) 
Date:  Mon, 4 Apr 2011 15:15:16 0400 
Paolo; That’s ok, I’m a chemical engineer. Unless you want to write your own MISLP you will have to linearize your cost profiles and then have the binaries select what region or segment the costs should be in. It is actually a relatively easy thing to do. Jeff From: Paolo Rossi [mailto:address@hidden Hi Jeff, thanks for that. I had figured it out in the meantime (while cradling my son to sleep!) that the first one could be solved by something like INJi <= MaxInj * BInj Withi <= MaxWith * BWith BInj + BWith = 1 which is I think exactly what you mention. So thank you very much for your reply, as I was wondering if my reasoning was making sense! At the very beginning I thought I'd need to introduce BInj*INJi in the objective function  that is why I stopped due to the introduction of nonlinearity in the objective function. I get the hint from your remark on my second question but I really have to think about how to implement it. Sorry my background is not in operations Research.. Paolo
On 4 April 2011 19:23, Kelly, Jeff (ON0F) <address@hidden> wrote: Paolo; Your first issue of INJi * WITHi = 0 complementarity is easily modeled by adding two binary variables for each quantity i.e., yINJi and yWITHi then you need three constraints: 1. INJi <= uINJi * yINJi 2. WITHi <= uWITHi * yWITHi 3. yINHi + yWITHi = 1 or <= 1 4. yINJi, yWITHi are binary The first two constraints are semicontinuous and third is a SOS1/GUB where the “u” prefix is the upper bound on the quantities. A lower bound is an exercise for you. Your second issue requires piecewise linear approximation of the cost curves due to most likely economizes/diseconomiesofscale. You will need to define regions of linearity and create extra binary variables for these regions with either SOS1 or SOS2 constraints depending on how you implement the “separable programming” aspects. I hope this helps  Jeff From: helpglpkbounces+jeff.kelly=honeywell.com@gnu.org [mailto:helpglpkbounces+jeff.kelly=honeywell.com@gnu.org] On Behalf Of Paolo Rossi Hi everyone, I am trying to replicate the modelling Byers, 2006. Commodity Storage Valuation: A linear optimization based on Traded Instruments, Energy Economics. The author is quite concise on how the model has been specified but it says that he used LpSolve The paper assesses the value of a gas storage facility. The value is a function of:  Injected quantity: INJ  Withdrawn quantity: WITH  Price paid for injections: Pi  Price paid for withdrawals: Pw  cost ofinjecting one unit of gas: ci  cost of withdrawing one unit of gas: cw If one takes two periods, Max INJ1 x pi,1 + WITH1 x pw,1 – ci,1 x INJ1 – cw,1 x WITH1 – INJ2 x pi,2 + WITH2 x pw,2 – ci,2 x INJ2– cw,2 x WITH2 Constraints  For each time period i, if INJi > 0 then WITHi = 0 and if WITHi > 0 then INJi = 0  you can either withdraw or inject. I thought of using a binary variable but then I realised that it would need to multiply INJi and WITHi so I got stuck as it would violate linearity of objective functions   For each time period I, cw,I and ci,I (cost of withdrawing and cost of injecting) are a function of the gas stored in the facility. The right curve here would be something similar to an exponential function through the origin for ci, i.e. the more gas you have in the facility the more it costs to push an extra unit of gas in. If one works with strep functions, the formulation would be something like ci = 1 if sum of (inj – with ) over the periods up to i is <= 3 2 if sum of (inj – with ) over the periods up to i is > 3 and <= 6 3 if sum of (inj – with ) over the periods up to i is > 6 For cw, the curve would be symmetric to the one above. cw = 3 if sum of (inj – with ) over the periods up to i is <= 3 2 if sum of (inj – with ) over the periods up to i is > 3 and <= 6 1 if sum of (inj – with ) over the periods up to i is > 6 I am pretty stuck here so thanks a lot for any help Paolo 
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