# Maximum Profit Configurations of Commercial Engines

## Abstract

**:**

^{m})] in commodity flow processes, in which effects of the price elasticities of supply and demand are introduced, is presented in this paper. Optimal cycle configurations of commercial engines for maximum profit are obtained by applying optimal control theory. In some special cases, the eventual state—market equilibrium—is solely determined by the initial conditions and the inherent characteristics of two subsystems; while the different ways of transfer affect the model in respects of the specific forms of the paths of prices and the instantaneous commodity flow, i.e., the optimal configuration.

## 1. Introduction

^{−1})] for maximum work output. Chen et al. [31] investigated effects of heat leakage on the optimal cycle configuration of a heat engine with a finite thermal capacity reservoir and the linear phenomenological heat transfer law for maximum work output. Some studies on the optimal configuration of variable- temperature heat reservoir heat engine for maximum power output were also performed, with the generalized radiative heat transfer law [q ∝ Δ(T

^{n})] [32], generalized convective heat transfer law [q ∝ (ΔT)

^{m}] [33], mixed heat resistance [34], and generalized heat transfer law [q ∝ (Δ(T

^{n}))

^{m}] [35], respectively. Andresen and Gordon [36] and Badescu [37] further optimized a class of heat transfer processes, with generalized radiative heat transfer law for minimum entropy generation [36] and minimum lost available work [37], respectively. Based on the generalized heat transfer law [q ∝ (Δ(T

^{n}))

^{m}], Chen et al. [38] and Xia et al. [39] derived the optimal temperature configurations of heat transfer processes for minimum entropy generation [38] and minimum lost available work [39]. Xia et al. [40] further investigated the minimum entransy dissipation of heat transfer processes with the generalized radiative heat transfer law.

^{m})], where the exponent m is closely related to the price elasticity of supply and demand, De Vos [49,50] further investigated the optimal performances of endoreversible commercial engines. Martinas [51] investigated the similarities and differences between irreversible thermodynamics and irreversible economics. Tsirlin [52], Tsirlin et al. [53,54,55], and Amelkin et al. [56] established an analogy between the processes in microeconomics and irreversible thermodynamics, and defined a physical quality in economics that could be used to measure the irreversibility of commodity exchange processes, i.e., capital dissipation, which is analogous to the physical quality of entropy generation in thermodynamics. Amelkin [57] investigated limit performances of a class of resource exchange processes in complex open microeconomic systems including sequential structure and parallel structure. Tsirlin and Kazakov [58] investigated the optimal cycle configuration of a commercial engine with a finite capacity economic subsystem and the linear transfer law for maximum profit.

^{m})] [48,49,50], which in economics represents possibility of different preferences. By applying the methods of finite time thermodynamics, this paper will provide the optimal cycle configuration of the commercial engine and give a straightforward and intuitive demonstration of price convergence in the model.

## 2. Model Description

_{1}. The commodity price in the subsystem is P

_{1}, whose initial value is given by P

_{1}(0) = P

_{10}. In addition, the dynamics of P

_{1}satisfies the equation:

_{1}dP

_{1}/ dt = −dN

_{1}/ dt

_{1}described by Equation (1) is compatible with the common assumption in economics of diminishing marginal utility. Similarly, for the finite capacity high-price subsystem, the capacity is constant C

_{2}, commodity price is P

_{2}with initial value P

_{2}(t

_{1}) = P

_{20}(t

_{1}is the initial time for selling) and dynamics:

_{2}dP

_{2}/ dt = −dN

_{2}/ dt

_{1′}and P

_{2′}, respectively, with P

_{1}< P

_{1′}< P

_{2′}< P

_{2}.

^{m}) [48,49,50]:

_{1}(P

_{1}, P

_{1′}) and n

_{2}(P

_{2′}, P

_{2}) are commodity flows corresponding to low-price and high-price sides of the commercial engine, α

_{1}(t) and α

_{2}(t) are the corresponding transfer coefficients, and exponents m

_{1}and m

_{2}are indicators of price elasticity of supply or demand. To elucidate, elasticity is a measure of the responsiveness of supply or demand to price changes, mathematically:

_{1}and m

_{2}don’t necessarily have to be the same, because different m’s may represent different preferences of suppliers and demanders.

_{1}and ΔN

_{2}, respectively. They are given by:

**τ**is the given cycle period. Additionally, market equilibrium condition requires that:

_{1}= Δ N

_{2}= Δ N

_{1}), the commercial engine purchases commodity from the low-price subsystem; and at time t (t

_{1}< t < τ), the commercial engine sells commodity to the high-price subsystem. Therefore, α

_{1}(t) and α

_{2}(t) have the following forms:

_{1}and α

_{2}are positive constants.

## 3. Optimization

_{1'}and P

_{2'}, the optimal values of t

_{1}and ΔN to maximize Equation (10) subject to Equations (1), (2) and (8).

#### 3.1. Problem 1

_{1}and λ

_{1}are Lagrangian multipliers.

_{1′}yields

_{1}is a constant to be determined.

_{1′}are uniquely characterized by Equation (19), which, combined with Equations (18), (13) and the initial value of P

_{1}, determines the paths of both P

_{1′}and P

_{1}.

#### 3.2. Problem 2

_{3}and λ

_{4}are Lagrangian multipliers. First order condition with respect to P

_{2′}yields:

_{2}is a constant to be determined.

_{2′}are uniquely characterized by Equation (28), which, combined with Equations (22), (27) and the initial value of P

_{2}, determines the paths of both P

_{2′}and P

_{2}. In sum, the optimal paths of P

_{1′}and P

_{2′}are described by Equations (19) and (28). However, analytical solutions to these differential equations exist only for a few exponents such as 1 and −1. For other exponents which do not admit analytical solutions, numerical method should be adopted.

_{1}, and I, one merely needs to substitute the paths of P

_{1′}, P

_{1}, P

_{2′}and P

_{2}into Equation (10) and solve the system of first order conditions.

## 4. Special case with m_{1} = 1 and m_{2} = 1

#### 4.1. Analytical Solutions

_{1′}− P

_{1}= k

_{1}and , respectively, in this case. Solving the system gives the paths of P

_{1}and P

_{1′}, respectively:

_{1}= (ΔN / C

_{1}t

_{1})t + P

_{10}(0 ≤ t ≤ t

_{1})

_{1′}= (ΔN / C

_{1}t

_{1})t + P

_{10}+ ΔN / α

_{1}t

_{1}(0 ≤ t ≤ t

_{1})

_{2′}− P

_{2}= k

_{2}and , respectively, in this case. Solving the system gives the paths of P

_{2}and P

_{2′}, respectively:

_{2}= −[ΔN / C

_{2}(τ − t

_{1})](t − t

_{1}) + P

_{20}(t

_{1}≤ t ≤ τ)

_{2′}= −[ΔN / C

_{2}(τ − t

_{1})](t − t

_{1}) + P

_{20}− ΔN / α

_{2}(τ − t

_{1}) (t

_{1}≤ t ≤ τ)

_{20}− P

_{10})ΔN − [(1 / 2C

_{2}+ 1 / 2C

_{1}+ 1 / α

_{2}(τ − t

_{1}) + 1 / α

_{1}t

_{1}]ΔN

_{1}= 0 yields:

#### 4.2. Results and Discussion

_{1}

^{*}and P

_{1′}

^{*}increase linearly in the time, while both P

_{2}

^{*}and P

_{2′}

^{*}decrease linearly in the time; commodity flows n

_{1}(P

_{1}, P

_{1′}) and n

_{2}(P

_{2′}, P

_{2}) are constants over time; the optimal exchange time t

_{1}

^{*}is determined only by ratio of the transfer coefficients α

_{1}and α

_{2}.

_{1}

^{*}, P

_{1′}

^{*}, P

_{2}

^{*}and P

_{2′}

^{*}. Mathematically:

_{1′}* and P

_{2′}* also indicates that the instantaneous profit gained by the commercial engine diminishes to 0 as the cycle period approaches infinity, which further indicates that the total profit earned cannot be infinite. Equation (36) serves as an apt substantiation of this point:

## 5. Special Case with m_{1} = − 1 and m_{2} = − 1

#### 5.1. Analytical Solutions

_{1}and P

_{1′}, respectively:

_{2}and P

_{2′}, respectively:

_{1}

^{*}and ΔN

^{*}are jointly determined by first order conditions ∂I / ∂t

_{1}= 0 and ∂I / ∂(ΔN) = 0. However, the system of polynomials cannot be solved explicitly.

#### 5.2. Results and Discussion

_{1}

^{*}, P

_{1′}

^{*}, P

_{2}

^{*}and P

_{2′}

^{*}as τ →∞. From Equations (44) and (46) one can obtain:

_{1}

^{*}, P

_{1′}

^{*}, P

_{2}

^{*}and P

_{2′}

^{*}converge eventually. Then ΔN* is bounded; and thus as τ →∞, Equation (48) becomes:

^{*}determined by Equation (52) supports such an optimal solution, substitute Equation (52) into Equations (49) and (50):

_{1}< P

_{1′}< P

_{2′}< P

_{2}, there must be:

_{1}= m

_{2}= −1 with those of the previous case with m

_{1}= m

_{2}= 1, one finds that the equilibrium price P

_{e}, the optimal amount of commodity exchange ΔN

^{*}, and the maximum profit I

_{max}are the same. It should be noted that this phenomenon is not a coincidence. Actually, in this model, the eventual state—market equilibrium—is solely determined by the initial conditions and the inherent characteristics of two subsystems; while the different ways of transfer (reflected by different values of m

_{1}and m

_{2}) affect the model in respects of the specific forms of the paths of prices and the instantaneous commodity flow, i.e., the optimal configuration.

## 6. Conclusions

^{m})] during commodity flow processes, in which the effects of the price elasticities of supply and demand are introduced, are investigated in this paper. The optimal cycle configurations of the commercial engines for maximum profit are obtained by applying optimal control theory. The optimal cycle configuration of the commercial engine with the linear transfer law [n ∝ ΔP] is that both the price estimation of finite capacity low-price economic subsystem and the commodity-buying price of the commercial engine change with time linearly and the difference between them is a constant, and the selling price of the commercial engine is a constant when it exchanges commodity with the infinite capacity high-price economic subsystem. The optimal cycle configuration of the commercial engine with the transfer law [n ∝ Δ(P

^{−1})] is that both the price estimation of finite capacity low-price economic subsystem and the commodity-buying price of the commercial engine change with time non-linearly and the ratio between them is a constant, and the selling price of the commercial engine is a constant when it exchanges commodity with the infinite capacity high-price economic subsystem. The research in this paper further extends the research lines and methods of finite time thermodynamics to applications in fields of non-conventional thermodynamics. It is worthwhile to note that several authors [60,61,62,63,64] have criticized finite time thermodynamics (emphasis on the endoreversible model and the corresponding study results) in recent years. The responses to those articles can be seen in [65,66,67,68,69], especially, Chen et al. [67].

## Acknowledgements

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Chen, Y.
Maximum Profit Configurations of Commercial Engines. *Entropy* **2011**, *13*, 1137-1151.
https://doi.org/10.3390/e13061137

**AMA Style**

Chen Y.
Maximum Profit Configurations of Commercial Engines. *Entropy*. 2011; 13(6):1137-1151.
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2011. "Maximum Profit Configurations of Commercial Engines" *Entropy* 13, no. 6: 1137-1151.
https://doi.org/10.3390/e13061137