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).
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;
```