(582g) Planning under Correlated and Truncated Price and Demand Uncertainties

Due to the volatile raw material prices, fluctuating product demands, and other changing market conditions, many parameters in a planning/scheduling model are uncertain. Even though almost 50 years have passed since the seminal works on uncertainty appeared, Dantzig (Horner, 1999) still considers planning under uncertainty as one of the most important open problems in optimization. Upon realization of uncertainty, a schedule built on a deterministic approach will be non-robust or in some cases even infeasible.

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. By regressing real world demand and price data from EIA (Energy Information Administration of the U.S.), the correlation coefficient between gasoline (New York Harbor Gasoline Regular) price and its demand is 0.44 for the year 2003 to 2004. For world crude oil in 2003 and 2004, the correlation coefficient is 0.30. These data show that the demand and price are far from independent.

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 functions. Case studies show that, as the standard deviation of price increases, the revenue increases slightly. However, as the standard deviation of demand increases, the revenue decreases significantly. The revenue difference between the independent and correlated demand and price depends on the CV (Coefficient of Variation) of a product. If the CV of a product takes value of 0.2, the revenue difference is about 2%. This difference is about 5% for CV of 0.3 (According to the regressed data from EIA, we assume that the correlation coefficient is 0.4 for correlated demand and price). The revenue at different correlation coefficients when CV is 0.5 is listed in Table 1. It can be seen that, the revenue at correlation coefficient of 0.4 is 21.1% higher than the revenue calculated by assuming independent demand and price (correlation coefficient=0.0). That means, if for a large enough CV of a product, assuming independent price and demand may underestimate the revenue by up to 20%.

                                       Table 1 Revenue at different correlation coefficient (CV=0.5).

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 approximates a bivariate double-truncated normal distribution for demand and price. Some results are shown in Table 2 (assume that the mean of demand is 50, standard deviation of demand 25; mean of price 3215, standard deviation of price 600 and a production rate of 39.565). It can be seen that, if the price and demand vary inside two standard deviations of their mean values, integrating over the whole range will underestimate the revenue by more than 2%. If the price and demand vary inside one standard deviation of their mean values, integrating over the whole range will underestimate the revenue by about 10%.

                                       Table 2. Comparison between truncated and non-truncated cases

After deriving the formulae for inventory cost calculation under correlated and truncated price and demand uncertainties, we present a general formulation for revenue and cost calculations of a chemical plant by considering both the raw material and the product demand uncertainty.

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 effectiveness of the approach proposed in this paper. The proposed approach can be generally applied for modeling other chemical plants under uncertainty.

Literature Cited

Horner, P. Planning under uncertainty. Questions & answers with George Dantzig, OR/MS Today, 1999, 26, 26-30