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Classical Optimization Methods
Published in A Vasuki, Nature-Inspired Optimization Algorithms, 2020
These functions could be of different types such as continuous or discrete, differentiable or non-differentiable, deterministic or non-deterministic with some random parameters inculcated into them. The traditional algorithms are usually applicable if the function is continuous and differentiable. When the function is constrained, it is a constrained programming problem whereas if it is unconstrained, it is an unconstrained programming problem. If there is only a single objective to be optimized, it is single-objective optimization, whereas if there are multiple objectives, it is multi-objective optimization. The function could be concave or convex as determined by the contours of the surface plot of the function. Concave functions have one maximum whereas convex functions have one minimum. A problem is said to be feasible if there exists at least one set of variables that satisfy the objective function and the constraints. A problem is infeasible if it is not possible to find at least one set of design variables that satisfy the objective(s) and the constraints. If the problem is infeasible the solution set will be empty. For an unbounded problem there is no optimal solution since it is always possible to find a solution that is better than the existing solution to the problem.
Forecasting in Global Supply Chain Engineering
Published in Erick C. Jones, Supply Chain Engineering and Logistics Handbook, 2020
Bowman (1956) suggested a transportation formulation of the aggregate planning problem when back orders are not permitted. Several authors have explored solving aggregate planning problems with specially tailored algorithms. These algorithms could be faster for solving large aggregate planning problems than a general linear programming code such as LINDO. In addition, some of these formulations allow for certain types of nonlinear costs. For example, if there are economics of scale in production, the production cost function is likely to be a concave function of the number of units produced. Concave cost functions are not as amenable to linear programming formulations as convex functions. We believe that a general-purpose linear programming package is sufficient for most reasonable-sized problems that one is likely to encounter in the real world. However, the reader should be aware that there are alternative solution techniques that could be more efficient for large problems and could provide solutions when some of the cost functions are nonlinear. A paper that details a transportation-type procedure more efficient than the simplex method for solving aggregate planning problems is Erenguc and Tufekci (1988).
Variables, functions and mappings
Published in Alan Jeffrey, Mathematics, 2004
A convex function is one which has the property that a chord joining any two points A and B on its graph always lies above the graph of the function contained between those two points. Similarly, a concave function is one which has the property that a chord joining any two points A and B on its graph always lies below the graph of the function contained between those two points. Thus the function y = |x| shown in Fig. 2.9(c) is convex on the interval ( − ∞, ∞) whereas the function shown in Fig. 2.9(d) is only concave on the closed interval [− a, a]. In some books, instead of saying that a function is convex, they say it is concave up and, conversely, instead of saying a function is concave, they say it is concave down. Thus the function y = |x| shown in Fig. 2.9(c) is concave up, while the function shown in Fig. 2.9(d) is concave down in [− a, a].
Quasiconcavity of a separable product of utility functions
Published in Optimization, 2018
Mohammed Berdi, Abdelhak Hassouni
One can also consider the function . The two problems just above are equivalent to the quasiconcavity of the product function Transposing convexity indices to the multiplicative case, Crouzeix-Kebbour [6] have introduced multiplicative indices of concavity for positive functions. If f is quasiconcave, then all are log-concave, except perhaps one which is then log-concave transformable. These results are easily derived of the results on the additive case. Recall that log-concave functions and distributions are of first importance in stochastic optimization. The multiplicative aspect of separability seems to be easier in application, and has a great interest in many models of mathematical programming with probabilistic constraints. See Précopa[7] and Norkin-Roenko [8].
A comparative study of progressive carbon taxation strategies: impact on firms’ economic and environmental performances
Published in International Journal of Production Research, 2021
Amina Chelly, Imen Nouira, Atidel B. Hadj-Alouane, Yannick Frein
To understand the price setting option, we suggest analysing the theoretical expression of the optimal price. Therefore, we compare the optimal price decisions for two cases, when and , under one of the progressive carbon taxes (i.e. the linear progressive carbon tax). Without loss of generality, we omit the inflation discount rate to simplify our calculations. We note that, for both situations, all demand is sold. Based on our numerical results, for , the optimal investment decision is to install the green technology in period 5 and to set the unit selling price to . Meanwhile, for , the optimal investment decision is not to invest in green technology and to sell the final product at per unit. We aim to analytically determine the theoretical optimal selling price based on investment and production decisions for each case. To do so, we write the profit of the company as a function of the price for these two cases. As a concave function, the maxim is found by setting its derivative with respect to P to zero. This yields what we refer to as the optimal theoretical price . Considering our numerical example, we deduce the value of for both studied cases and then compare them to , the optimal prices obtained by solving our cases on Cplex. Table 5 summarises the obtained results.
Competition model of cruise home ports based on the cruise supply chain – based on China cruise market
Published in Maritime Policy & Management, 2019
Heying Sun, Qingcheng Zeng, Hui Xiang, Chao Chen
The second derivative of to is positive; thus, is concave function of . Therefore, there isn’t an optimal unit subsidy aimed at total profit maximization .