The L-Shaped BONUS Algorithm

Abstract

This chapter is based on a paper by Shastri and Diwekar . A variant of BONUS is presented here to solve multistage stochastic programming problems with recourse . In stochastic programming problems with recourse , there is action (x), followed by observation, and then recourse r. In these problems, the objective function has the action term, and the recourse function is dependent on the uncertainties and recourse decisions. The recourse function can be a discontinuous nonlinear function in x and r space.

Notations

$cdf_{out}$ :

cumulative probability density function of output

C _i :

per unit cost of chemical i

C _T :

blend product treatment cost per unit reduction in the impurity content

d :

demand

E :

expected value function

f :

a function

g :

inequality constraint function

h :

equality constraint function

$I_{il}(u)$ :

fraction of impurity l in chemical i

${I}_i^*$ :

impurity parameter of chemical i

$\bar{I}_{k}$ :

final impurity parameter of a blend which

depends on the weight contribution of each chemical in a particular blend

${I}_k^{spec}$ :

maximum permitted impurity content in the blend product

$N_{samp}$ :

number of samples

p :

probability values

$P_i()$ :

probabilistic function

$pdf_{in}()$ :

probability density function of input

$Q()$ :

recourse function in farmer’s problem

r :

recourse variable

$R()$ :

recourse function

s _p :

selling price

w _i :

amount of plant i sold

u :

uncertain variable

$W_{ik}$ :

weight of chemical i in blend product k

$\bar{W}_k$ :

total production requirement of blend product k

x :

decision variables

x _j :

planting cost of crop j

$x_{ij}$ :

fraction of component j in chemical i

$\bar{x}_{jk}$ :

specification of component j in blend product k

w _j :

sales cost of crop j

y _j :

purchase cost of crop j

$YY_{i, actual}$ :

actual yield of the crop i

$YY_{i, max}$ :

maximum possible yield if all the conditions are perfect

Y _j :

fractions of the maximum yield due to corresponding effects j

$Z, z$ :

objective function

Greek letters

$\alpha_d$ :

Attack probability of a crop disease

(uniform distribution between 0 and 0.2)

$\alpha_l$ :

importance of a particular impurity in the final product

$\alpha_r$ :

fractional rainfall of the yearly average

$\alpha_s$ :

fractional sunlight of the yearly average

$\theta_k$ :

purification required for blend k to satisfy the impurity constraint

λ:

Lagrange multipliers/dual variables

σ:

standard deviation