Planning under correlated and truncated price and demand uncertainties

Abstract

Until now, most research work on uncertainty assumes that the demand and price are independent because of the difficulty in computing the bivariate integral originated from the correlated demand and price. This can cause significant discrepancies in revenue calculation and hence yield sub-optimal planning strategies. This paper presents a novel approach to handle correlated and truncated demand and price uncertainties. To compute the expectation of plant revenue, which is the main difficulty for a planning problem under uncertainty, we use a bivariate normal distribution to describe demand and price. The double integral for revenue calculation is reduced to several single integrals after detailed derivation. The unintegrable standard normal cumulative distribution function in the single integrals is approximated by polynomial function. Case studies show that, for a large enough CV of a product, assuming independent price and demand may underestimate the revenue by up to 20%. Since the real world demands or prices vary in limited ranges, integrating over the whole range of a normal distribution, which some research has done, may give incorrect results. This paper thus approximates a bivariate double-truncated normal distribution for demand and price. The influence of degree of truncation on plant revenue is studied. To handle possible unmet customer demands, the hard-to-specify penalty functions of the two-stage programming are avoided and replaced by two of the decision maker's service objectives, namely the confidence level and fill rate objective. Confidence level or the type I service level, which is the probability of satisfying customer demands, is commonly used in chance-constrained programming. However, fill rate or the type II service level, which is the proportion of demands that are met from a plant, is a greater concern of most managers. In this paper, fill rate is efficiently calculated using the derived formulae and the maximal plant profit that satisfies certain fill rate objectives can thus be obtained. Case studies show that a planning strategy that satisfies certain confidence level objectives might be too generous compared to a strategy that satisfies a fill rate objective. Case studies including refinery planning problems were used to illustrate the proposed approach. The proposed approach can be generally applied for modeling other chemical plants under uncertainty.