Wnter 2008 CS567 Stochastc Lnear/Integer Programmng Guest Lecturer: Xu, Huan Class 2: More Modelng Examples 1 Capacty Expanson Capacty expanson models optmal choces of the tmng and levels of nvestments to meet future demands of gven product. Here we consder the case of power expanson for electrcty generaton: to fnd optmal levels of nvestments n varous types of power plants to meet future electrcty demand. We dscuss three modelng here: statc determnstc model, dynamc model and stochastc model. 1.1 Statc Determnstc Model Three propertes of a gven power plant can be sngled out: the nvestment cost r, the operatng cost q and the avalablty factor a whch ndcates the percent of tme the power plant can effectvely be operated. Demands for electrcty can be consdered a sngle product, but the level of demand vares over tme. Analysts usually use load duraton curve to descrbe the demand over tme n decreasng order of demand level. The load duraton curve can be approxmated by a pecewse constant curve wth m segments. Let d 1 = D 1, d j = D j D j 1, j = 2,, m represent the addtonal power demand n mode j wth a tme τ j. In the statc stuaton, the problem conssts of fndng the optmal nvestment foe each mode,.e.., to fnd a partcular plant such that the total cost of effectvely producng electrcty durng the tme τ j : { } r + q τ j (j) = arg mn.,,n The soluton of the statc model captures one essental feature of the problem: base load demand should be covered by equpment wth low operatng costs and peak load demand should be covered by equpment wth low nvestment costs. a 1.2 Dynamc Model At least four elements justfy usng a dynamc or multstage model: 1. the long-term evoluton of equpment costs; 2. the long-term evoluton of the load curve; 3. the appearance of new technologes; 4. the obsolescence of currently avalable equpment. Of sgnfcant mportance s the evoluton of demand n both the total energy demanded, and the peak level whch determnes the total capacty that must be avalable. Hence, the followng multstage model s proposed to handle ths: 1
t = 1,, H ndex the perods or stages; = 1,, n ndex the avalable technologes; j = 1,, m ndex the operatng modes n the load duraton curve; a = avalablty factor of ; L = lfe tme of ; g t = exstng capacty of at tme t, decded before t = 1; r t = unt nvestment cost for at tme t; q t = unt producton cost for at tme t; d t j = maxmal power demanded n mode j at tme t; τ t j = duraton of mode j at tme t; x t = new capacty made avalable for technology at tme t; w t = total capacty of avalable at tme t; yj t = capacty of effectvely used at tme t n mode j. The electrcty generaton H-stage problem can be defned as H rw t t + qτ t jy t t j s.t.: mn x,y,w w t = w t 1 + x t x t L, = 1,, n, t = 1,, H, yj t = dt j, j = 1,, m, t = 1,, H, yj t a (g t + w), t = 1,, m, t = 1,, H, x, y, w 0. The objectve functon s the sum of nvestment plus mantenance costs and operatng costs. Compared to the statc model, the factor a goes to the constrant. 1.3 Stochastc Model In contrast to the dynamc model whch assumes that the evolutons are all determnstc, we can also consder them as truly random. Ths leads to the stochastc model. The major dfference here s x t now represent the new capacty of decded at tme t, whch becomes avalable at x t+ ( s the constructon delay). We use boldface to represent random varables: x t = new capacty decded at tme t for equpment ; w t = total capacty of avalable at tme t; 2
ξ = the vector of random parameters at tme t; The stochastc model s then: s.t.: mn x,y,w E ξ H r t w t + q t τ t jy t j w t = w t 1 + x t x t L, = 1,, n, t = 1,, H, yj t = dt j, j = 1,, m, t = 1,, H, y t j a (g t + w t ), = 1,, m, t = 1,, H, x, y, w 0. Notce here, the decson w t, y t only depend on the realzaton of the random vector up to tme t, but can not depend on future realzatons of the random vector. That s, t s non-antcpatve. If the decson varable (w t, y t ) were not dependent on ξ t, the objectve functon can be wrtten as E ξ r t w t + E ξ q t τ t yj t = r w t t + (q τ j )y t j. t j t j That s, t becomes a determnstc formulaton. Problem 1 s a multstage stochastc lner program wth a specal property: block separable recourse. Ths property stems from a separaton that can be made between the aggregate level decson (x t, w t ) and the detaled-level decsons y t. Suppose future demands are always ndependent of the past. In ths case, the decson on capacty to nstall n the future at some t only depends on avalable capacty and does not depend on the outcomes up to tme t. The same x t must then be optmal for any realzaton of ξ. The only stochastc decson s n the operaton-level: y t, whch now depends separately on each perod s capacty. Thus, ths mult-perod problem becoms a less complex two-perod problem. Consder the followng example: we have a two-perod three-operatng-mode problem, wth n = 4, = 1, a = 1; g = 0. The only random varable s d 1 = ξ. The other demands are d 2 = 3 and d 3 = 2. The nvestment costs are r 1 = (10, 7, 16, 6) wth producton costs q 2 = (4, 4.5, 3.2, 5.5) and load duraton τ 2 = (10, 6, 1). We also add a budget constrant to keep all nvestment below 120. The resultng stochastc program s: 3 mn : 10x 1 1 + 7x 1 2 + 16x 1 3 + E ξ [ τj 2 (4y1j 2 + 4.5y2j 2 + 3.2y3j 2 + 5.5y4j)] 2 s.t.: 10x 1 1 + 7x 1 2 + 16x 1 3 120, 3 x 1 + yj 2 0, = 1, 4, y y1 2 = ξ, y yj 2 = d 2 j, j = 2, 3 x, y 0. 3 (1)
Assumng that ξ takes on the values 3, 5 and 7 wth probabltes 0.3, 0.4 and 0.3 respectvely, an optmal soluton ncludes x 1 = (2.67, 4.00, 3.33, 2.00) wth optmal value of 381.85. If we consder the expected parameter formulaton, the optmal soluton s (0.00, 3.00 5.00 2.00 ) wth an objectve value 365. However, f we use ths soluton n the stochastc problem, then wth probablty 0.3 the demands are not satsfed. Formulaton of 1 requres that the demand s always satsfed, n practce t s often relaxed to a probablty constrant: n 1 P [ a (g t + w) t D t j] α, t, for some α (0, 1).Ths s often called a chance constrant or probablty constrant n stochastc programmng. Notce ths constrant s equvalent to a determnstc constrant: n 1 a (g t + w) t (F t ) 1 (α), t, where F t s the dstrbuted functon of m dt j. 2 Desgn for Manufacturng Qualty Consder a desgner decdng varous product specfcaton to acheve some measure of product cost and performance. However, the specfcaton may not completely determne the characterstcs of each manufactured product. Key characterstcs of the product are often random. For example, every tem ncludes varatons due to machnng or other processng, and each customer also does not use the product n the same way. Thus, cost and performance become random varables, and stochastc programmng can help n ths case, because determnstc methods can yeld costly results that are only dscovered after producton has begun. In ths secton, we consder the desgn of a smple axle assembly for a bcycle cart. The desgner must determne the specfed length w and the dameter ξ of the axle. Together, these quanttes determne the performance characterstc of the product. The goal s to determne a combnaton to acheve the greatest expected proft. In the producton process, the actual dmensons are not exactly those that are specfed. For ths example, we suppose that the length w can be produced exactly but the dameter ξ s a random varable ξ(x) that depends on a specfed value x that represents, for example the settng of a machne. We assume the followng trangular dstrbuton of ξ(x): (100/x 2 )(ξ 0.9x) f 0.9x ξ x, f x (ξ) = (100/x 2 )(1.1x ξ) f x ξ 1.1x, 0 otherwse. The decson s to determne w and x subject to w w max and x x max, n order to maxmze the expected utlty. Further assume that we sell as many as we can produce under the maxmum sellng prce, and the maxmum sellng prce depends on the length and s expressed as r(1 e 0.1w ), 4
where r s the maxmum possble for any such products. Our producton costs for labor and equpment are assumed to be fxed, so only materal cost s varable. Ths cost s proportonal to the mean value of the specfed dmenson because materal s acqured before the producton process. That s, c(wπx 2 )/4, for some unt cost c. We have certan qualty constrant: w ξ 3 39.27, w 3 ξ 4 63169. When ether of these constrants s volated, the axle deforms, whch ncurs a qualty cost proportonal to the square of the volaton. That s: Q(w, x, ξ) = mn y {qy 2 w/ξ 3 y 39.27; w 3 /ξ 4 300y 63169}, and the expected qualty cost gven w and x s Q(w, x) = Q(w, x, ξ)f x (ξ)dξ =q 1.1x 0.9x The overall problem s to fnd ξ (100/x 2 ) mn{ξ 0.9x, 1.1x ξ}[max{0, w/ξ 3 39.27, w 3 /300ξ 4 210.56}] 2 dξ. max (total revenue er tem - manufacturng cost per tem -expected qualty cost). Mathematcally, we wrte ths as max z(w, x) = r(1 e 0.1w ) c(wπx 2 )/4 Q(w, x) s.t. 0 w w max, 0 x x max. Let w max = 36, x max = 1.25, r = 10, c = 0.025, q = 1. The optmal soluton s w = 33.6 and x = 1.038 where the objectve functon s 8.94. If we consder the expected parameter formulaton, we get a soluton w = 35.0719 and x = 0.963, wth the optmal value 9.07. It seems that the expected proft s even better than the stochastc soluton. That s because, the optmal value usng the expected parameter overestmate the true expected proft. Substtute ths soluton nto the stochastc formulaton, the expected proft s 26.79. Hence we see that the VSS s 35.73. 5