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Show yopt, l " ^ min ~ Vpf t a = t l , t2 i- 1 yoP,. i= v+ 1 V p . j t- vt.+ 1,; i- j3 (") yopt, 4 ~ ^ max j- 1 where y0pt, i is the optimal release from the stretched thread model in the years tj to tj+ i, Ptmin an^ Pjnax are respectively the minimum and the maximum release rates from the stretched thread method, ypf is the firm yield from the yield model, ypSj is the j t n secondary yield from the yield model with reliability p, and a p t : is a ( 0,1) reliability variable corresponding to the jth secondary yield in year t. If the proper number of secondary yields and their failure years are selected by the yield model, its solution duplicates that of the stretched thread method. This is however not an easy task a priori, or through the use of the model advocated by Loucks et al ( 1981). The generalization of the yield model to yields for multiple uses and with different economic values and bounds is however straightforward. The modifications to the yield model as presented in Chapter 5, however achieve the equivalence with the stretched thread method in a more direct manner. Consider, a firm yield ypf and a single secondary yield ypS. Now, if the variable < Xpt associated with the failure of the secondary yield is allowed to take any value between 0 and 1, i. e a variable degree of failure, rather than supply or failure is permitted, the equivalence between the stretched thread and the modified yield models may be directly established, subject to a qualification regarding the values taken by the reliability variable ( Xpt. This is stated as: ( 6.7) where ctpt j is a constant reliability level between years tj and tj+ i, corresponding to the i" 1 optimal yield from the stretched thread model. Noting that all the yield variables for the stretched thread model are represented in terms of a linear function of the firm and secondary yields from the modified yield model, a second modification to further reduce the complexity of the yield model, while maintaining the equivalence between the two models is apparent. A single yield variable may be considered for the yield model, with the equivalence with the stretched thread method achieved through a single set of yield reliability variables. This may be stated as: yopt, l " ^ min ypf tmtVt2 y = ' opt. i v+ Viyps t = t., t+ 1, i= 2,3 yopt, 4 ~ ^ max yPr+ yps t = t4, t5 160 |