Deterministic demand

Production decisions may change based on the structure of the demand (deterministic vs. stochastic, stationary). Inventory review policies (periodic review vs. continuous review) may affect the production decisions as well. 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

Unpredictable movements in the price level are uncommon in energy markets. The magnitudes of these price shocks can be substantial 1973 and 1979 oil price shocks, electricity price swings in June 1998, and 2000 oil price increases exemplify the potential magnitudes of these price fluctuations. Energy price dynamics usually consist of three components deterministic part, seasonal and cyclical influences, and noise. In a market situation with no demand or sup-... 

Uncertainties in amounts of products to be manufactured Qi, processing times %, and size factors Sij will influence the production time tp, whose uncertainty reflects the individual uncertainties that can be presented as probability distributions. The distributions for shortterm uncertainties (processing times and size factors) can be evaluated based on knowledge of probability distributions for the uncertain parameters, i.e. kinetic parameters and other variables used for the design of equipment units. The probability of not being able to meet the total demand is the probability that the production time is larger than the available production time H. Hence, the objective function used for deterministic design takes the form ... 

Random service levels result when random demands meet available inventories. Throughout this chapter we assume that the available inventory is not random but has a known value. This is justified because in many cases the production process is almost deterministic when compared with the varying demand. 

In order to investigate the performance of a deterministic online scheduler, we apply it to the example problem under demand uncertainty for three periods. The model of the scheduling problem used in the scheduler considers a prediction horizon of H = 2 periods. Only the current production decision Xi(ti) is applied... 

In this fashion, we extend our deterministic model with a prediction horizon of H = 2 to a multi-stage model. The multi-stage tree of the possible outcomes of the demand within this horizon (starting from period i = 1) with four scenarios is shown in Figure 9.5. Each scenario represents the combination fc out of the set of all combinations of the demand outcomes within the horizon. The production decision x has to be taken under uncertainty in all future demands. The decision xj can react to each of the two outcomes of d i, but has to be taken under uncertainty in the demand di. The corrective decisions are explicitly modeled by replacing xj by two variables 2,1 and 2.2 ... 

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. 

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). 

Up to this point, it is assumed that prices are deterministic, which is true for contract demand and procurement but is not necessarily true for spot demand and procurement prices. Therefore, an important value chain planning requirement is the consideration of uncertain prices and price scenarios. Now, uncertain spot demand prices are under consideration and it is illustrated how price uncertainty can be integrated into the model in order to reach robust planning solutions. 

In this chapter, all parameters were assumed to be deterministic. However, the current situation of fluctuating petroleum crude oil prices and demands is an indication that markets and industries everywhere are impacted by uncertainties. For example, source and availability of crude oils as the raw material prices of feedstock, chemicals, and commodities production costs and future market demand for finished products will have a direct impact on final decisions. Thus, acknowledging the shortcomings of deterministic models, the next Chapters will consider uncertainties in the design problem. 

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 ... 

In a data set of normally distributed demands, if the Cv of demand is given as a case problem parameter, the standard deviation is computed by multiplication of Cv by the deterministic demand. Hence, increasing values of Q result in increasing fluctuations in the demand and this is again undesirable. 

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... 

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... 

In a deterministic planning environment the most likely scenario, here scenario 2, would be considered the base case and the optimization model would be solved based on this scenario. The optimal decision would be to open facility 1 in period 1 and facility 3 in period 2 leading to a total profit of 2,590. To assess the robustness of this network to alternative demand scenarios the profit achievable with this configuration in case of the alternative demand scenarios can be assessed. In the example, for scenario 1 the overall profit would be 1,640 and for scenario 3, 2,765 respectively. Considered individually, the optimal decision for scenario 1 would be to open only facility 1 with a total profit of 1,880 and for scenario 3 to open both facilities 1 and 2 in period 1 with a total profit of 2,931. In order to explicitly incorporate the uncertainties caused by the different realization probabilities of the three demand scenarios, the optimization model can be extended into a two-stage decision with recourse ... 

Regression analysis in time series analysis is a very useful technique if an explanatory variable is available. Explanatory variables may be any variables with a deterministic relationship to the time series. VAN STRATEN and KOUWENHOVEN [1991] describe the dependence of dissolved oxygen on solar radiation, photosynthesis, and the respiration rate of a lake and make predictions about the oxygen concentration. STOCK [1981] uses the temperature, biological oxygen demand, and the ammonia concentration to describe the oxygen content in the river Rhine. A trend analysis of ozone data was demonstrated by TIAO et al. [1986]. 

Many systems exhibit nonlinear behavior. This is another systems-level task that is computationally very demanding. Application of bifurcation analysis to simple and complex chemistry hybrid stochastic (KMC)-deterministic (ODE) models has been presented by our group (Raimondeau and Vlachos, 2002b, 2003 Vlachos et al., 1990) for various catalytic surface reactions. Prototype hybrid continuum-stochastic models that exhibit bifurcations were recently explored by Katsoulakis et al. (2004). It was found that mesoscopic... 

C. J. Corbett, U. Karmarkar, Competition and Structure in Serial Supply Chains with Deterministic Demand, Management Science, 47(7), 966-978 (2001). 

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