An Uncertain Control Model with Application to. Production-Inventory System

Transcription

1 An Uncertain Control Model with Application to Production-Inventory System Kai Yao 1, Zhongfeng Qin 2 1 Department of Mathematical Sciences, Tsinghua University, Beijing , China 2 School of Economics and Management, Beihang University, Beijing , China yaok09@mails.tsinghua.edu.cn, corresponding author: qin@buaa.edu.cn Abstract Uncertain optimal control theory is an important tool to optimize dynamic system in uncertain environment. This paper further develops uncertain optimal control theory, and proposes an uncertain linear quadratic control model. Properties of the model are studied and an optimal solution is obtained. After that, the linear quadratic control model is applied to production-inventory system, and provides an optimal production rate. Keywords: uncertainty theory, optimal control, linear quadratic model, production planning 1 Introduction Control theory finds many applications in science, economics, and engineering. A classical control system is usually described by a deterministic differential equation. Due to the stochastic disturbance, the control system sometimes behaves randomness, and stochastic differential equation is employed to model it. In a stochastic control system, an optimal control is first concerned that maximizes or minimizes some given value functions associated with the system. The most popular result in this area is the Hamilton-Jacobi-Bellman equation. Inspired by stochastic optimal control, an alternative control system characterized by fuzzy differential equation was also studied. It aims at finding an optimal fuzzy process to optimize an objective function under some constraints described by fuzzy differential equation. Zhu [22] gave an optimality equation as a counterpart of Hamilton-Jacobi-Bellman equation. Based on the fuzzy optimality equation, Qin etc [18] proposed linear quadratic control model, and obtained an explicit control for the system. This paper was honored with the best student paper award by Asia Pacific Industrial Engineering and Management Society in

2 Human uncertainty is another kind of undetermined phenomenon. In order to model human uncertainty, an uncertainty theory was founded by Liu [7] in 2007 and refined by Liu [11] in Uncertain process was proposed by Liu [8] to model the evolution of uncertain phenomenon in As a counterpart of Wiener process, Liu [10] designed a canonical process. Based on canonical process, Liu [10] proposed uncertain calculus to deal with the integral and differential of an uncertain process with respect to canonical process. After that, Yao [20] proposed uncertain calculus with respect to uncertain renewal process. Later, Chen [3] generalized uncertain calculus theory, and proposed uncertain calculus with respect to finite variation process. In 2008, Liu [8] proposed uncertain differential equation driven by canonical process. Based on uncertain differential equation, Zhu [23] proposed an uncertain optimal control in 2010, and gave an optimality equation as a counterpart of Hamilton-Jacobi-Bellman equation. In 2011, Xu and Zhu [19] proposed uncertain bang-bang control. In this paper, we will further develop uncertain optimal control theory, and propose a linear quadratic control model in uncertain environment. The remainder of the paper is organized as follows. In Section 2, we will recall some basic concepts in uncertain control system. Section3 will introduce uncertain optimal control model and some important results. In Section 4, we will propose uncertain linear quadratic control model, and give a method to obtain the optimal control. An application to production-inventory system will be given in Section 5. Finally, some remarks are made in Section 6. 2 Preliminary Uncertainty theory was founded by Liu [7] in 2007 and refined by Liu [11] in 2010 to model human uncertainty. Many researchers have contributed a lot in this area such as Gao [5], Liu and Ha [14] and Peng and Iwamura [17]. Nowadays, uncertain theory has become an axiomatic mathematics, and has found many applications in uncertain programming Liu [9], Bhattacharyya etc [1]), risk analysis Liu [13]), finance Liu [8], Chen [4], Peng and Yao [16]), inference Liu [12], Gao etc [6]). The evolution of uncertain phenomenon is described by an uncertain process Liu [8]). As a counterpart of Wiener process in stochastic environment, a canonical process was designed by Liu [10] in uncertain environment. Definition 1. Liu [10]) An uncertain process C t is said to be a canonical process if i) C 0 = 0 and almost all sample paths are Lipschitz continuous, ii) C t has stationary and independent increments, iii) every increment C s+t C s is a normal uncertain variable with expected value 0 and variance t 2, whose uncertainty distribution is Φ t x) = 1 + exp πx )) 1, x R. 3t 2

3 Based on canonical process, a concept of uncertain integral was proposed by Liu [10], which is regarded as an uncertain counterpart of the Ito integral. Definition 2. Liu [10]) Let X t be an uncertain process and C t be a canonical process. For any partition of closed interval [a, b] with a = t 1 < t 2 < < t k+1 = b, the mesh is written as Then the uncertain integral of X t is defined by b a = max 1 i k t i+1 t i. X t dc t = lim 0 i=1 k X ti C ti+1 C ti ) provided that the limit exists almost surely and is finite. For this case, the uncertain process X t is said to be integrable. Remark 1. A stochastic process is integrable in the sense of L 2 -norm, and an uncertain process is integrable in the sense of existence almost surely. Definition 3. Liu [10]) Let C t be a canonical process and Z t be an uncertain process. If there exist uncertain processes µ s and σ s such that Z t = Z 0 + t µ s ds + t 0 0 for any t 0, then Z t is said to have an uncertain differential σ s dc s dz t = µ t + σ t dc t. Liu [11] verified the fundamental theorem of uncertain calculus, i.e., for a canonical process C t and a continuous differentiable function ht, c), the uncertain process Z t = ht, C t ) is differentiable and has an uncertain differential dz t = h t t, C t) + h c t, C t)dc t. Based on the fundamental theorem, Liu proved the chain rule, i.e., for two continuously differentiable functions f and g, the uncertain process fgc t )) has an uncertain differential dfgc t )) = f gc t ))g C t )dc t, and the integration by parts theorem, i.e., for two differentiable uncertain process X t and Y t, the uncertain process X t Y t has an uncertain differential dx t Y t ) = Y t dx t + X t dy t. An uncertain differential equation is essentially a type of differential equation driven by canonical process instead of Wiener process. 3

4 Definition 4. Liu [8]) Suppose C t is a canonical process, and f and g are two given functions. Then dx t = ft, X t ) + gt, X t )dc t 1) is called an uncertain differential equation. Remark 2. If the canonical process is replaced by a Wiener process, then 1) turns into a stochastic differential equation. Chen and Liu [2] gave an existence and uniqueness theorem for uncertain differential equation. Meanwhile, Chen and Liu [2] gave an analytic solution for linear uncertain differential equation dx t = u 1t X t + u 2t ) + v 1t X t + v 2t )dc t. After that, Liu [15] gave an analytic solution for nonlinear uncertain differential equation dx t = ft, X t ) + σ t X t dc t and dx t = µ t X t + gt, X t )dc t. In 2011, Yao and Chen [21] presented a numerical method to solve uncertain differential equation. 3 Uncertain Optimal Control A determined control system is governed by a differential equation, where the state variable is definite. Due to the occasional perturbation of white noise, randomness is introduced to the differential equation, and stochastic control system is proposed, which is governed by a stochastic differential equation. However, human uncertainty also brings vagueness to the observation. Alternately, the control system may be assumed to follow an uncertain differential equation. An uncertain control system is thus introduced as follows, Definition 5. An uncertain control system is a system represented an uncertain differential equation dx t = ft, X t, Z t ) + gt, X t, Z t )dc t 2) where X t is the state variable, C t is a canonical process, and Z t is a control. Remark 3. If the canonical process is replaced by a Wiener process, then 2) models a stochastic control system. Example 1. equation A linear uncertain control system is a system modeled by a linear uncertain differential dx t = αx t + βz t + µ) + γx t + δz t + ν)dc t where α, β, γ, δ, µ and ν are determined real numbers. Optimal control deals with the problem of finding a control for a given system such that a certain optimality criterion is achieved. That is, an objective functional about the state variable and control is optimized under some constraint which is usually described by differential equation. In stochastic 4

5 optimal control, the constraint usually contains randomness and is described by a stochastic differential equation. In uncertain optimal control, the system usually contains human uncertainty and is described by an uncertain differential equation. Zhu [23] first proposed an uncertain optimal control model as follows, [ ] T Js, x) = inf E Z s Rt, X t, Z t ) + W T, X T ) t subject to: dx t = ft, X t, Z t ) + gt, X t, Z t )dc t, X s = x 3) where X t is the state, Z t is a control, C t is an uncertain canonical process, R is the return function and W is the terminal reward function. Note that JT, x) = W T, x) is a boundary condition. Zhu [23] gave a necessary condition of the optimal control Z t for the uncertain control model 3), that is, Z t solves the equation Jt, x) sup + z t ) Jt, x) ft, x, z) Rt, x, z) x = 0. 4) 4 Uncertain Linear Quadratic Control An uncertain linear quadratic control model is an uncertain control model with a quadratic objective functional and a linear uncertain system, i.e., [ T Js, x) = inf E Z s AX 2 t + BZt 2 + CX t Z t + DX t + EZ t + F ) ] + GX T t subject to: dx t = αx t + βz t + µ) + γx t + δz t + ν)dc t, X s = x 5) where A, B, C, D, E, F and G are real numbers, and x is the initial state. The next theorem gives a necessary condition that the value function Js, x) satisfies. Theorem 1. Suppose that Z t Jt, x) is the corresponding value function. If B 0, then is an optimal control of the uncertain linear quadratic model 5), and Z t = 1 β J x Cx E ), 6) and Jt, x) satisfies ) 2 J β αBx βcx + 2µB βe) J J +4B x x t = 4AB C2 )x 2 +2D EC)x+4BF E 2. 7) Proof: By Zhu s optimal control equation, we have sup z αx + βz + µ) J x + J t Ax2 + Bz 2 + Cxz + Dx + Ez + F ) Let Lz) denote the term in the braces. The optimal control Z t satisfies L z = 2βz Cx E + β J x = 0 5 ) = 0. 8)

6 whose solution is just Zt = 1 β J ) x Cx E. Substituting the last equation into Equation 8) yields ) 2 J β αBx βcx + 2µB βe) J J + 4B x x t = 4AB C2 )x 2 + 2D EC)x + 4BF E 2. The theorem is thus verified. Theorem 2. Assume that the value function Jt, x) of the uncertain linear quadratic control model 5) satisfies Jt, x) = Ut)x 2 + V t)x + W t) where Ut), V t), W t) are some determined functions of t. Then ) ) dut) + β2 B Ut)2 + 2α βc C B Ut) + 2 4B A = 0 ) ) dv t) + α + β2 Ut) B βc V t) + 2µ βe B Ut) + CE ) D = 0 dw t) + β2 4B V t)2 + µ βe V t) + E2 4B F = 0. 9) Proof: Since the value function Jt, x) = Ut)x 2 + V t)x + W t), we have and J = 2Ut)x + V t) x J t = Ut) x 2 + t V t) x + t W t). t Substituting these two equations into Equation 7), we obtain that dut) dv t) + + dw t) + + β2 B Ut)2 + α + β2 Ut) B + β2 4B V t)2 + 2α βc B βc µ βe ) C 2 Ut) + )) 4B A ) V t) + 2µ βe B ) ) V t) + E2 4B F = 0 x 2 ) Ut) + CE ) D The equation holds for any x, so we get the ordinary differential equations in Equation 9). Remark 4. Solving the ordinary differential equation 9), we can obtain the expression for Ut), V t) and W t), respectively. Substituting them into the quadratic value function Jt, x), we get the expression for Jt, x). Furthermore, substituting Jt, x) into the Equation 6), we find the expression for the optimal control Z t. x 6

7 5 Production Planning Problem In this section, the uncertain linear quadratic control model is applied to a production planning model with uncertain factors. Consider a factory producing homogeneous goods and storing the goods which are produced but not immediately sold. Once the goods are put into inventory, it results in two kinds of costs, one is the storing cost, and the other is opportunity cost of the capital. Assume that the factory employs the production-inventory system, and adjusts the production rate to meet with the consumer demand and cut down the inventory cost. In order to model the problem mathematically, the following symbols are used, X t : Z t : e : σ : T : x : z : x 0 : p 1 : p 2 : the inventory at time t state variable), the production rate at time t control variable), the demand rate, the diffusion coefficient, the production planning period, the proper inventory level, the proper proper production rate, initial inventory level, the inventory cost coefficient, the production cost coefficient, G : depreciation charge per unit of goods at time T. The factory produces the goods at a rate of Z t and sells the goods at a rate of e. In consideration of the uncertainty in inventory, the inventory level X t is modeled by an uncertain differential equation dx t = Z t e) + σdc t where C t is a canonical process. The factory maintains the inventory X t and production Z t as close as possible to the proper inventory level x and z, respectively. Meanwhile, the factory should minimize the depreciation charge at time T. Then the objective functional is model by [ ] T min E p1 X t x) 2 + p 2 Z t z) 2) + GX T. Z t 0 Thus we obtain a linear quadratic uncertain control model: [ T Jt, x) = min E p1 X Z 0 t x) 2 + p 2 Z t z) 2) ] + GX T t subject to: dx t = Z t e) + σdc t. 10) Theorem 3. The linear quadratic uncertain optimal control model 10) has an optimal control ) Zt p1 1 λt T ) G λt T ) = e + x x) p λt T ) + 2e z) p λt T ). 7

8 Proof: Setting A = p 1, B = p 2, C = 0, D = 2p 1 x, E = 2p 2 z, F = p 1 x 2 + p 2 z 2, α = 0, β = 1 and µ = e in the model 9), we get dut) + 1 p 2 Ut) 2 p 1 = 0 dv t) dw t) + Ut) p 2 V t) + 2z e)ut) + 2p 1 x = 0 + V t)2 4p 2 + z e) V t) p 1 x 2 = 0. Since JT, x) = Gx, we obtain the boundary conditions UT ) = 0, V T ) = G and W T ) = 0. The first ordinary differential equation is equivalent to the following differential equation Thus dut) p1 p 2 + Ut) + dut) p1 p 2 Ut) = 2 p1. p 2 Ut) = c exp2 p 1 /p 2 ) 1 c exp2 p1 p 2 p 1 /p 2 ) + 1 where c is a parameter to be determined. Substituting UT ) = 0 into the equation, we get c = exp 2T p 1 /p 2 ), and Ut) = λt T ) 1 p1 p 2 λt T ) + 1 where λt) = exp2t p 1 /p 2 ). By integration factor method, the second ordinary differential equation has a solution V t) = exp Ut) p 2 ) ) ) Ut) c + 2e z)ut) 2p 1 x) exp p 2 =2p 2 e z) + c λt T ) λt T ) + 2x p 1 p 2 1 λt T ) 1 + λt T ) where c is a parameter to be determined. Substituting V T ) = G into the equation, we get c = 2G 4p 2 e z) and V t) = 2p 2 e z) + 2G 4p 2 e z)) λt T ) λt T ) + 2x p 1 p 2 1 λt T ) 1 + λt T ). The third ordinary differential equation has a solution W t) = p 1 x 2 + e z) V t) V t)2 4p 2 =p 1 x 2 T + p 2 e z) 2 t + T ) + G 2p 2e z)) 2 4p 1 p 2 x 2 2 p 1 p λt T )) λt T ) + 2G 2p 2 e z))x 1 + λt T ) + c where c is a parameter to be determined. Substituting W T ) = 0 into the equation, we obtain c = p 1 x 2 T 2p 2 e z) 2 T G 2p 2e z)) 2 4p 1 p 2 x 2 4 p 1 p 2 G 2p 2 e z))x and W t) = p 2 e z) 2 t T )+ G 2p 2e z)) 2 4p 1 p 2 x λt T ) p 1 p λt T ) G 2p 2e z))x 1 λt T )) λt T ) 8

9 Hence we get the optimal value Jt, x) = Ut) + V t)x + W t)x 2. The optimal control is Zt = 1 ) J 2p 2 x + 2p 2z 2Ut)x + V t) = z + 2p 2 p1 1 λt T ) = e + x x) p λt T ) + ) G λt T ) 2e z) p λt T ). Remark 5. When the current inventory level x is greater than the proper inventory level x, i.e., x > x, the second part of Z t is positive, so the optimal production rate Z t is higher. When x < x, the second part of Z t is negative, so the optimal production rate is lower. Hence, the optimal production rate Z t adjusts according to the current inventory level x, and keeps the inventory level X t close to the proper inventory level x. Remark 6. Note that when T, we have λt T ) 0, so Z t e + x x) p 1 /p 2. Meanwhile, we have Ut) 0, V t) 0 and W t), so Jt, x), which disagrees with the fact. The contradiction happens because the model does not take the discounter rate into consideration. Thus an object value functional should contains the discounter rate when the time span is infinite. 6 Conclusions As a counterpart of stochastic control system driven by Wiener process, an uncertain control system driven by canonical process was introduced, and an uncertain optimal control model was employed to optimize a dynamic system in uncertain environment. Then this paper proposed an uncertain linear quadratic optimal control model, and found an optimal control for the model. After that, the linear quadratic control was applied to a production-inventory system, and an optimal production rate was derived. Acknowledgements This work was supported by National Natural Science Foundation of China Grant Nos and , and Specialized Research Fund for the Doctoral Program of Higher Education of China Grant No

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