\$title Strategic Petroleum Reserve (MARKOV,SEQ=82) \$onText This is a linear programming formulation for optimal petroleum stockpile policy based on a stochastic dynamic programming approach. Each state of the Markov process is a pair (s,i) where s is the size of the inventory and i is the state of the world (normal or disrupted). However, we assume the probability of entering state (s',j) from state (s,i) is independent of the stockpile levels. Teisberg, T J, A Dynamic Programming Model of the U.S. Strategic Petroleum Reserve. Bell Journal of Economics (1981). Keywords: linear programming, stochastic dynamic programming, Markov process, energy economics \$offText Set s 'level of the reserve' / empty, 3, 6, 9, 12, 15, 18, 21 / i 'state of the oil market' / normal, disrupted /; Alias (s,sp,spp), (i,j); * remember that supply is fixed at q = 110 million barrel per year * and the shape of the demand curve is : d(p) = d + k*p**-e Scalar b 'discount factor' / .95 / beta / .0625 / g 'u.s. demand' / .25 / e / .1 / q 'supply' / 110 / d / -130 / k / 342 / pn 'normal price (us\$ per bbl)' / 34.526 / h 'storage cost' / .32 /; Table pr(i,j) 'transition probability of the word oil market' normal disrupted normal .8 .2 disrupted .5 .5; Parameter lev(s) 'level of reserve' dis(i) 'disruption' / disrupted 11 / p(s,sp,i) 'price affected by action a' c(s,sp,i) 'cost of taking action a' pi(s,i,sp,j,spp) 'probability matrix for problem'; lev(s) = 3*(ord(s)-1); p(s,sp,i) = (k / (q - dis(i) - d - (lev(sp)-lev(s))))**(1/e); c(s,sp,i) = g*(d*(p(s,sp,i) - pn) + k*(p(s,sp,i)**(1 - e) - pn**(1 - e))/(1 - e)) + p(s,sp,i)*(lev(sp) - lev(s)) + h*lev(sp); pi(s,i,sp,j,sp) = pr(i,j); display lev, dis, p, c, pi; Variable z(s,i,sp) 'multiple of joint probability' pvcost 'present value of expected cost'; Positive Variable z; Equation constr(s,i) equil(s,sp) cost 'cost definition'; constr(sp,j).. sum(spp, z(sp,j,spp)) - b*sum((s,i,spp), pi(s,i,sp,j,spp)*z(s,i,spp)) =e= beta; equil(s,spp).. z(s,"disrupted",spp)*(ord(spp) - ord(s)) =l= 0; cost.. pvcost =e= sum((s,i,spp), c(s,spp,i)*z(s,i,spp)); Model strategic / all /; solve strategic using lp minimizing pvcost;