Production shortfall/surplus

A 5% standard deviation from the mean value of market demand for the saleable products in the LP model is assumed to be reasonable based on statistical analyses of the available historical data. To be consistent, the three scenarios assumed for price uncertainty with their corresponding probabilities are similarly applied to describe uncertainty in the product demands, as shown in Table 6.2, alongside the corresponding penalty costs incurred due to the unit production shortfalls or surpluses for these products. To ensure that the original information structure associated with the decision process sequence is respected, three new constraints to model the scenarios generated are added to the stochastic model. Altogether, this adds up to 3 x 5 = 15 new constraints in place of the five constraints in the deterministic model. 

Although increasing 02 with fixed value of 0 corresponds to decreasing expected profit, it generally leads to a reduction in expected production shortfalls and surpluses. Therefore, a suitable operating range of 02 values should be selected to achieve a proper trade-off between expected profit and expected production feasibility. Increasing 02 also reduces the expected deviation in the recourse penalty costs under different scenarios. This, in turn, translates to increased solution robustness. In that sense, the selection of 0j and 02 values depends primarily on the policy adopted by the decision maker. 

Operational risk factor 02 Optimal objective value Expected variation in profit V(z0)(E + 8) Expected total unmet demand/ production shortfall Expected total excess production/ production surplus Expected recourse penalty costs Es Expected variation in recourse penalty costs Vs p = E[z ] - Es c a P... 

First- stage variable Stochastic solution Product (i) Production shortfall zjf or surplus Zy (t/d) Scenario 1 Scenario 2 Scenario 3 ... 

One of the reasons why the pair of decreasing values of 0, with a fixed value of 03 leads to increasing profit is due to the decrease in production shortfalls and, at the same time, increase in production surpluses. Typically, the fixed penalty cost for shortfalls is lower than surpluses. A good start would be to select a lower operating value of 0 j to achieve both high model feasibility as well as increased profit. Moreover, lower values of 03 correspond to decreasing variation in the recourse penalty costs, which implies solution robustness. 

Equations 9.11 and 9.13 represent the refinery production shortfall and surplus as well as the petrochemical production shortfall and surplus, respectively, for each random realization Ic C N. These variables will compensate for the violations in Equations 9.11 and 9.13 and will be penalized in the objective function using appropriate shortfall and surplus costs C r 1 and for the refinery products, and C et + and C 1 for the petrochemical products, respectively. Uncertain parameters are assumed to follow a normal distribution for each outcome of the random realization Although this might sound restrictive, this assumption imposes no limitation on the generality of the proposed approach as other distributions can be easily incorporated instead. Furthermore, in Equation 9.13 an additional term xi 1 was added to the left-hand-side representing the flow of intermediate petrochemical... 

Compensating slack variables accounting for shortfall and/or surplus in production are introduced in the stochastic constraints with the following results (i) inequality constraints are replaced with equality constraints (ii) numerical feasibility of the stochastic constraints can be ensured for all events and (iii) penalties for feasibility violations can be added to the objective function. Since a probability can be assigned to each realization of the stochastic parameter vector (i.e., to each scenario), the probability of feasible operation can be measured. In this... 

It is desirable to demonstrate that the proposed stochastic formulations provide robust results. According to Mulvey, Vanderbei, and Zenios (1995), a robust solution remains close to optimality for all scenarios of the input data while a robust model remains almost feasible for all the data of the scenarios. In refinery planning, model robustness or model feasibility is as essential as solution optimality. For example, in mitigating demand uncertainty, model feasibility is represented by an optimal solution that has almost no shortfalls or surpluses in production. A trade-off exists... 

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