Stochastic demand

Due to stochastic demand in China, stochastic production yields in Europe and some stochastic variations in transport times between the two it was decided to support the decision between these alternatives by means of simulation. The structure of the simulation model is shown in Figure 2.2. 

In particular results on tank capacity are a typical output of simulations on the plant engineering level. It could be argued that these results may be obtained without simulation as well and this is true as long as the stochastic impact on supply and demand is within certain boundaries. As soon as the facility needs to be able to handle stochastic supply and demand with significant variations static calculations reach their limits. These limitations become even more critical if a multi-product process is analyzed as quite often is the case. 

The stochastic tools used here differ considerably from those used in other fields of application, e.g., the investigation of measurements of physical data. For example, in this article normal distributions do not appear. On the other hand random sums, invented in actuary theory, are important. In the first theoretical part we start with random demand and end with conditional random service which is the basic quantity that should be used to decide how much of a product one should produce in a given period of time. 

Especially in the process industries various stochastic methods can be applied to cope with random demand. In many cases, random demands can be described by probability distributions, the parameters of which may be estimated from history. This is not always possible, the car industry is an example. No two cars are exactly the same and after a few years there is always a new model which may change the demand pattern significantly. 

Figure 6.10 shows the data flow of the software tool BayAPS PP for optimal capacity assignment for given stochastic demands. Transaction data about demand and inventories is typically imported from SAP R/3 as indicated, production capacity master data and side conditions are stored in the software tool. Forecasts can be taken from a forecast tool or from SAP R/3. The output ofthe tool is a list ofpriorities of products and their lot sizes, which are optimal based on the presently available information. Only the next production orders are realized before the computation is repeated, and the subsequently scheduled production is only a prediction. 

The moving horizon scheme using the two-stage model is shown in Figure 9.6. In contrast to the deterministic scheduler which uses the expected value of the demands dls and ds i (see Section 9.2.2), the stochastic scheduler updates the demand in form of the distribution given in Table 9.1 d, df, d]+1, and dj+1. 

The sequence of decisions obtained from the stochastic scheduler for all possible evolutions of the demand for the three periods is provided in Figure 9.7. The sequence of decisions obtained by the stochastic scheduler differs from that obtained by the deterministic one, e.g., xi(ti) = lOinstead ofxi(ti) = 6. The average objective for the stochastic scheduler after three periods is P = —17.65. 

I. (2004) Approximation to multi-stage stochastic optimization in multi-period batch plant scheduling under demand uncertainty. Industrial and Engineering Chemistry Research, 43, 3695—3713. 

Deterministic vs. stochastic an optimization problem can be based on deterministic parameters assuming certain input data or reflect uncertainty including random variables in the model in value chain management deterministic parameters are the basic assumptions extended models also model specifically uncertain market parameters such as demand and prices as stochastic parameters based on historic distributions in chemical commodities, this approach has some limitations since prices and demand are not normally distributed but depend on many factors such as crude oil prices (also later fig. 37). 

With respect to demand certainty, demand is forecasted with bid character and is not stochastic following for example a normal distribution pattern, since demand is influenced by the price development. With respect to demand volatility, demand prices and quantities are not stable but monthly volatile. The total demand elasticity is smaller or equal to 1 with respect to average prices. That means that average prices for total demand can change, if more or less sales quantity is sold in the market. 

Normal gas-source mass spectrometers do not allow meaningful abundance measurements of these very rare species. However, if some demands on high abundance sensitivity, high precision, and high mass resolving power are met, John EUer and his group (e.g., Eiler and Schauble 2004 Affek and Eiler 2006 EUer 2007) have reported precise (<0. l%c) measurements of CO2 with mass 47 (A47-values) with an especially modified, but normal gas-source mass spectrometer. A47-values are defined as %o difference between the measured abundance of all molecules with mass 47 relative to the abundance of 47, expected for the stochastic distribution. 

Chemical process systems are subject to uncertainties due to many random events such as raw material variations, demand fluctuations, equipment failures, and so on. In this chapter we will utilize stochastic programming (SP) methods to deal with these uncertainties that are typically employed in computational finance applications. These methods have been very useful in screening alternatives on the basis of the expected value of economic criteria as well as the economic and operational risks involved. Several approaches have been reported in the literature addressing the problem of production planning under uncertainty. Extensive reviews surveying various issues in this area can be found in Applequist et al. (1997), Shah (1998), Cheng, Subrahmanian, and Westerberg (2005) and Mendez et al. (2006). 

To ensure that the original information structure associated with the decision process sequence is honored, for each of the products whose demand is uncertain, the number of new constraints to be added to the stochastic model counterpart, replacing the original deterministic constraint, corresponds to the number of scenarios. Herein lies a demonstration of the fact that the size of a recourse model increases exponentially since the total number of scenarios grows exponentially with the number of random parameters. In general, the new constraints take the form ... 

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... 

We demonstrate the implementation of the proposed stochastic model formulations on the refinery planning linear programming (LP) model explained in Chapter 2. The original single-objective LP model is first solved deterministically and is then reformulated with the addition of the stochastic dimension according to the four proposed formulations. The complete scenario representation of the prices, demands, and yields is provided in Table 6.2. 

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. 

Risk factors e e3 Optimal objective value Expected deviation between profit V(z0)(E + 7) Expected total unmet demand/ production shortfall Expected total excess production/ production surplus Expected recourse penalty costs Es Deviation in a —, /, V +W(Ps) recourse penalty costs W(ps) Expected profit E[z0] E[zo]-Es c Stochastic a n Deterministic... 

This chapter addresses the planning, design and optimization of a network of petrochemical processes under uncertainty and robust considerations. Similar to the previous chapter, robustness is analyzed based on both model robustness and solution robustness. Parameter uncertainty includes process yield, raw material and product prices, and lower product market demand. The expected value of perfect information (EVPI) and the value of the stochastic solution (VSS) are also investigated to illustrate numerically the value of including the randomness of the different model parameters. 

This implies that if it were possible to know the future realization of the demand, prices and yield perfectly, the profit would have been 2 724 040 instead of 2 698 552, yielding savings of 25 488. However, since acquiring perfect information is not viable, we will merely consider the value of the stochastic solution as the best result. These results show that the stochastic model provided an excellent solution as the objective function value was not too far from the result obtained by the WS solution. 

The results of the model considered in this Chapter under uncertainty and with risk consideration, as one can intuitively anticipate, yielded different petrochemical network configurations and plant capacities when compared to the deterministic model results. The concepts of EVPI and VSS were introduced and numerically illustrated. The stochastic model provided good results as the objective function value was not too far from the results obtained using the wait-and-see approach. Furthermore, the results in this Chapter showed that the final petrochemical network was more sensitive to variations in product prices than to variation in market demand and process yields when the values of 0i and 02 were selected to maintain the final petrochemical structure. 

The above formulation is an extension of the deterministic model explained in Chapter 5. We will mainly explain the stochastic part of the above formulation. The above formulation is a two-stage stochastic mixed-integer linear programming (MILP) model. Objective function (9.1) minimizes the first stage variables and the penalized second stage variables. The production over the target demand is penalized as an additional inventory cost per ton of refinery and petrochemical products. Similarly, shortfall in a certain product demand is assumed to be satisfied at the product spot market price. The recourse variables V [ +, , V e)+ and V e[ in... 

---