Expected value of profit

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. 

Example 12 Expected Value of Net Profit Let iis consider a contractor who stands to make a net profit of 100,000 on a contract. The cost of preparing the bid on the contract is 10,000. There are four competing contractors, each with a probability pi = 0.25 of obtaining the contract. Thus, each contractor has a probability p2 = 0.75 of not obtaining the contract. Therefore, the expected value of the project is... 

We leave it as an exercise for the reader to find the condition under which the expected value of the profit is maximum i.e., maximize ... 

Suppose your marginal cost is 10. Based on the least squares regression, compute a 95% confidence interval for the expected value of the profit maximizing output. 

Technology Product flexibility Levelised cost [ /kg] Expected profit [M ] Expected value of flexibility [M ]... 

At the end of this stage, variance of the mature costs will be rather high. However, the expected value of this cost has been determined. If this cost is too high to offer adequate hope of profit (i.e., to offer hope that the technology will be commercialized) or if the cost necessary to commercialize lies within the variance but at an inadequately low probability, then the technology is not commercial. The project is dropped or perhaps the Invention stage is reentered. 

The coefficient jS > 0 represents the weight given to the conservative part of the decision. If /3 is large, then the above optimization problem tries to find a solution with minimtil profit variance, while if /3 = 0, then problem (8) coincides with problem (4). Note that since the variance Var[G(x , D)] = E[(G(x, D) — E[G(x, D)]) ] is itself an expected vrilue, from a mathematical point of view, problem (8) is similar to the expected value problem (4). Thus, the problem of optimizing the expected value of an objective function G(x, D) is very genertil—it could include the means, variances, qtrantiles, and almost any other aspects of random variables of interest. 

For instance, one may be interested in maximizing the expected value of a profit given (Albritton et al. 1999) ... 

Since the size of the orders that has to be request is chosen between three values, and, in addition, there are four time instants in which an order has to be placed, there are 81 possible combinations for simulating in each scenario. For each one of these configurations, taking into account each scenario, the expected values of the profit (Eiprofit)) has been calculated and the one with the largest expected profit is picked. In the deterministic case the values for the demand have been chosen in 7 units of product A each five simulations steps and 3 units of B with the same frequency. Moreover, the value for the safety inventory level has been set in 50 units for all the entities except in SIP and PI where the selected value has been 100 units. In the first case below described, a variance of 3 and 2 has been added to both deterministic values of the demand size. In the second case, a variance of 30 has been added to the deterministic value of the safety inventory level. 

This is more than an 8 percent increase in profitability relative to the policy of ordering the expected value of 1,000 parkas. 

The expected profit from ordering 350 pairs of skis can be evaluated as 45,718. Thus, ordering 468 pairs results in an expected profit that is almost 8 percent higher than the profit obtained from ordering the expected value of 350 pairs. 

Both methods assume that the money earned can be reinvested at the nominal interest rate. Suppose the rates of return calculated are after tax returns and the company is generally earning a 5% or 6% return on investment. Is it reasonable to expect that all profits can be reinvested at 23% or even 20% No, it isn t Yet this is what is assumed in the Rate of Return method. Sometimes the rate of return may be as high as 50%, while a reasonable interest rate is less than 15%. Therefore if a reasonable value for the interest rate has been chosen (this is discussed later in this chapter) and the two methods differ, the results indicated by the Net Present Value method should be accepted. 

Optimization techniques are procedures to make something better. Some criteria must be established to determine whether something is better. The single criterion that determines the best among a number of alternatives is referred to as the performance index or the objective function. Economically, this is the expected profit for a plant design. It may be expressed as the net present value of the project. 

Two-phase optimization Maximize expected profit across multiple price scenarios subject to the constraint that a given minimum profit value is reached. From a practical point of view, this approach seems to be more appropriate in situations where a high variability of profit can be expected and the risk of low profit outcomes shall be minimized. 

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.

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. 

In general, the coefficients of variation decrease with smaller values of 02. This is definitely desirable since it indicates that for higher expected profits there is diminishing uncertainty in the model, thus signifying model and solution robustness. It is also observed that for values of 02 approximately greater than or equal to 2, the coefficient of variation remain at a static value of 0.5237, thus indicating overall stability and a minimal degree of uncertainty in the model. 

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. 

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