Profit-risk model

Cost of Inherent Safety and the Profit-Risk Model... 

The profit-risk modeling approach involves development of a mathematical equation relating to the profit, the probability of an equipment failure, and the probability that a safety device fails as follows ... 

Consideration of the expected value of profit alone as the objective function, which is characteristic of the classical stochastic linear programs introduced by Dantzig (1955) and Beale (1955), is obviously inappropriate for moderate and high-risk decisions under uncertainty since most decision makers are risk averse in facing important decisions. The expected value objective ignores both the risk attribute of the decision maker and the distribution of the objective values. Hence, variance of each of the random price coefficients can be adopted as a viable risk measure of the objective function, which is the second major component of the MV approach adopted in Risk Model I. 

Tables 6.3-6.5 show the computational results for Risk Model II over a range of values of risk parameter 02 with respect to different recourse penalty costs, for three representative cases of 0 = 1E — 10, IE — 7, and 1.55E — 5, respectively. An example of the detailed results is presented in Table 6.6 for 02 = 50 of the first case. Figure 6.2 illustrates the corresponding efficient frontier plot for Risk Model II while Figure 6.3 provides the plot of the expected profit for different levels of risk.

Figure 6.3 Risk Model II plot of expected profit for different levels of risk as represented by the economic risk factor 0n and the operational risk factor 02.

From Table 6.7 and the corresponding efficient frontier plot in Figure 6.4, similar trends to Risk Model II (and also the expected value models) are observed in which decreasing values of 0 correspond to higher expected profit until a certain constant profit value is attained ( 81 770). The converse is also true in which a constant profit of 59330 is reached in the initially declining expected profit for increasing values of 0i. 

Recently, risk started to be defined in terms of another point measure introduced by J.P. Morgan, value at risk or VaR (Jorion, 2000). This is defined as the difference between the expected profit and the profit corresponding to 5% cumulative probability. Many other mean-risk models use measures such as tail value at risk, weighted mean... 

In the considered value chain planning problem, the uncertainty of spot sales prices impacts the profitability of the overall value chain plan, since volume decisions can lead to profit-suboptimal plans, if the average sales price cannot be realized as planned. Therefore, price volatility is considered as an external (stochastic) influence in the considered value chain planning problem. The following model extensions account for this uncertainty and try to derive methods to achieve more robust plans with respect to profit results with contributions from Habla (2006). The objective of the proposed modeling approach is to maximize profit for the entire value chain network. It is assumed that the company behaves risk-averse in face of the price uncertainty. 

The first approach adopts the classical Markowitz s MV model to handle randomness in the objective function coefficients of prices, in which the expected profit is maximized while an appended term representing the magnitude of operational risk due to variability or dispersion in price, as measured by variance, is minimized (Eppen, Martin, and Schrage, 1989). The model can be formulated as minimizing risk (i.e., variance) subject to a lower bound constraint on the target profit (i.e., the mean return). 

The model is subject to the same set of constraints as the deterministic model, with 0i as the risk trade-off parameter (or simply termed the risk factor) associated with risk reduction for the expected profit. 0j is varied over the entire range of (0, oo) to generate a set of feasible decisions that have maximum return for a given level of risk, which is equivalent to the efficient frontier portfolios for investment applications. 

As remarked in Approach 1, a potential complication with Expectation Model I lies in computing a suitable range of values for the operational risk factor 0j. Therefore, an alternative formulation of minimizing variance while adding a target profit constraint is employed for Expectation Model II ... 

---