markov.gms : Strategic Petroleum Reserve

**Description**

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.

**Reference**

- Teisberg, T J, A Dynamic Programming Model of the U.S. Strategic Petroleum Reserve. The Bell Journal of Economics 12, 2 (1981), 526-546.

**Small Model of Type :** LP

**Category :** GAMS Model library

**Main file :** markov.gms

$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). $Offtext Sets s level of the reserve / empty, 3, 6, 9, 12, 15, 18, 21 / i state of the oil market / normal, disrupted / Alias(s,sp,spp), (i,j) * remember that supply is fixed at q = 110 million barrel per year * and the shape of the demand curve is : d(p) = d + k*p**-e Scalars b discount factor / .95 / beta / .0625 / g u.s. demand / .25 / e / .1 / q supply / 110 / d /-130 / k / 342 / pn normal price (us$ 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 Parameters 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; Variables z(s,i,sp) multiple of joint probability pvcost present value of expected cost Positive variable z Equations 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;