Optimal control problem for an imperfect production process using fuzzy variational principle

Abstract

The aim of this paper is to formulate and solve an optimal control problem under finite time horizon in fuzzy environment using fuzzy variational principle. Here an imperfect/defective item is produced to meet a time-dependent demand for a finite time period having no stock at both ends. The unit production cost is a function of production rate and also dependent on raw material cost, development costs due to durability and wear-tear cost. The cost function which consists of revenue, production and holding costs is formulated as a Fixed-Final Time and Fixed State System optimal control problem with finite time horizon. Here production rate is unknown and considered as a control variable and stock level is taken as a state variable. It is formulated to optimize the production rate so that total cost is minimum. For the fuzzy model, the production rate, stock level, inventory cost and development cost are taken as fuzzy. The models are solved by using conventional Variational Principle for crisp model and Fuzzy Variational Principle (FVP) for fuzzy model. For simulation, Mathematica-9.0 and the non-linear optimization technique Generalised Reduced Gradient Method (LINGO 11.0) have been used. The optimum results are illustrated both numerically and graphically. For the fuzzy model, the membership functions of fuzzy outputs are presented. The results of crisp model are also obtained from the fuzzy model.

Keywords

Fuzzy variational principle finite time horizon imperfect production system optimal control problem

1 Introduction

Based on variation of the optimal curve that is controlled directly by Euler’s necessary conditions and Lagrange’s undetermined multipliers, a new mathematical discipline was born which is known as Euler-Lagrange equation. This is known as “Calculus of Variations”. When the connections and differences of the theories and methods to the classical calculus of variations became lucid, one began to develop a unitary theory of optimal control taking into the account its new characteristics. Optimal control theory is an extension of the calculus of variations.

In real life, only five percentage (approx.) out of total information system are deterministic and the rest is uncertain. In conventional optimal control model (cf. Benjaafar et al. [2]) formulation, all the data/parameters/variables in the form of input or output are deterministic. In case of real problems, some model parameters can be only roughly estimated. While in case of classical models the vague data is replaced by “average data”, fuzzy models (cf. Li et al. [9]; Wang et al. [23 –25], Nayeem [15] and Jinquan et al. [7]) offer the opportunity to model with subjective imaginations of the decision-maker as precisely as a decision maker will be able to describe it.

It is well known that optimality of the cost functions are prerequisites for defining the dynamical system under finite time horizon in imprecise environment. Here, one method is explained for the resolution of query regarding the imprecise nature of integrand for the optimal control problem. As a consequence, Fuzzy Variational Principle (FVP) (cf. Farhadinia [5]) is applied for the work. Generally, in variational problems, the task is to find a suitable curve or optimal path for which a given functional fetches its optimum. The notion of fuzzy sets is widely spread to various optimization problems, solving procedure of fuzzy variational problems is seldom used in literatures. However, optimality conditions for fuzzy variational problems are reported for very few articles. It looks that, it is a new idea to optimality conditions for both fuzzy unconstrained and constrained variational problems due to Buckley and Feuring [4]. In a sense, this problem is one of the applications for generalization of calculus.

Earlier many authors solved such problems in different approaches. But these problems were little different from the present one. In earlier problems, either the limits were fuzzy (Mandal et al. [11]) or or the integrand was fuzzy (Mandal et al. [12], Venkata Ramu et al. [22], Baten and Kamil [1]). In 2013, Najariyan and Farahi (cf. [13]) developed optimal control of fuzzy linear controlled system with fuzzy initial conditions. For the problems with fuzzy integrand Mandal et al. [12] solved with the help of Mean Chance Constrained method. Here they expressed the fuzzy integrand in some crisp form. But till now, none used Fuzzy Variational approach for the production inventory problem. Thus the present investigation is unique in this regard.

One of the weaknesses of current production-inventory models is the unrealistic assumption that all items produced are of good item. But production of defective units is a natural phenomenon due to different difficulties faced in a long-run production process. Recently, Salameh and Jaber [20] formulated a model in such situations where all items are not perfect. In the imperfect production system, the items deteriorate slowly with time, may be at constant rate. However, in the present study, we have considered the production inventory model with defective items. For the defective items, as a result of imperfect production process, were initially considered by Porteus [17] and later by several researchers (such as Zequeira, Prida, and Valds [26], Maity and Maiti [10]; Panda et al. [16]; Buscher, Rudert, and Schwarz [3]; Sarkar, Sana, and Chaudhuri [21]; Roul et al. [18] and others).

Another limitation of the dynamic production model is that the stock and production levels and system costs are taken as deterministic. But, in real life situation, production rate in a firm varies due to the different workers’ efficiencies, quality of the raw material, machine breakdown etc. Hence the realistic production rate is imprecise in nature. Moreover, the stock level also becomes imprecise as the stock level partly depends on production rate. Again, the system costs i.e development costs, holding costs etc. are not deterministic over the time. These always vary and may be considered as fuzzy. Once the above parameters are taken as fuzzy, the integrand for the optimal control problem becomes fuzzy. This consideration promoted as to consider the investigation using Euler-Lagrange method.

Thus with respect of the above consideration, there are several gaps for the investigation of the dynamic imperfect production inventory control problem.

The present investigation is a real life application of FVP. Here the production and stock levels, inventory costs and development costs are taken as fuzzy.

For the first time where FVP is applied for production inventory model.

For simulation, Mathematica-9.0 is taken as an another tool and used to compare the results.

Here the results of fuzzy model are compared with the results of deterministic model. Min cost,optimum production and stock are calculated for the different values of α like α = 0.25, 0.5, 0.75, 1.00 and presented by appropriate membership functions.

Taking the above lacunas into account, in this paper, imperfect production inventory models are formulated over a finite time horizon with known dynamic demands in crisp or fuzzy environment. Here, production rates which are functions of time are the control variables. For this system, total production cost is considered as crisp and fuzzy. A general model is formulated as an optimal control problem over a finite time horizon with defective items for the cost minimisation. By the theory of Fuzzy Variational Principle (cf. Farhadinia [5]), the imprecise integrand is converted into crisp one by using α- measure techniques. Then, the equivalent crisp problem is solved by Variational Principle, fixed-final time and free-final state system (cf. Naidu [14]) and Generalised Reduced Gradient (GRG) Technique (Gabriel and Ragsdell [6]). The behaviour of optimal production rate, stock and demand with respect to ^′α′ over the time are depicted. The total Min Cost with respect to different values of α are graphically presented. For α = 1.00, the Min cost in imprecise environment is equal to the result in crisp environment which is established here. This is numerically established here.

2 Preliminaries: Basic concepts of fuzzy algebra and calculus

2.1 Fuzzy algebra

Fuzzy Number: (cf. Zimmermann [27]). If the sets A and B are two fuzzy numbers, defined in the positive real line then the fuzzy numbers are expressed as follows: $\tilde{A} = [a, b]$ and $\tilde{B} = [c, d]$ where a, b, c, d > 0. Then operations of Addition, Subtraction, Multiplication and Division are written as, $\tilde{A} + \tilde{B} = [a + c, b + d]$ ; $\tilde{A} - \tilde{B} = [a - d, b - c]$ ; $\tilde{A} . \tilde{B} = [ac, bd]$ ; $\tilde{A} / \tilde{B} = [a / d, b / c]$ .

Triangular Fuzzy Number (TFN): A TFN $\tilde{A}$ is specified by the triplet (a₁, a₂, a₃) and is defined by its continuous membership function(MF) $μ_{\tilde{A}} (x) : X \to [0, 1]$ as follows (cf. Fig. 1). $\begin{matrix} μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - a_{1}}{a_{2} - a_{1}} & if a_{1} \leq x \leq a_{2} \\ \frac{a_{3} - x}{a_{3} - a_{2}} & if a_{2} \leq x \leq a_{3} \\ 0 & otherwise \end{matrix} \end{matrix}$

α-cut of Fuzzy Number: α -cut method is a standard method for performing different arithmetic operations like addition, multiplication, division, subtraction. If a fuzzy number represented with three points as follows: $\tilde{A} = (a_{1}, a_{2}, a_{3}) .$ The α-cut of fuzzy number $\tilde{A}$ determines it crisp one. So $A_{α} = [A_{α}^{l}, A_{α}^{r}] = [a_{1} + (a_{2} - a_{1}) α, a_{3} - (a_{3} - a_{2}) α]$ for αɛ [0, 1] .

For example a TFN $\tilde{B}$ is specified by the triplet (1, 3, 5) and is defined by its continuous MF $μ_{\tilde{B}} (x) : X \to [0, 1]$ as follows (cf. Fig. 1.1).

$μ_{\tilde{B}} (x) = {\begin{matrix} \frac{x - 1}{3 - 1} & if 1 \leq x \leq 3 \\ \frac{3 - x}{5 - 3} & if 3 \leq x \leq 5 \\ 0 & otherwise \end{matrix}$

α-cut of Fuzzy Number B: For α-cut, we have $\frac{x - 1}{3 - 1} = α$ and $\frac{3 - x}{5 - 3} = α$ or, x = 1 +2α and x = 5 -2α. The α-cut of fuzzy number $\tilde{B}$ determines it crisp one. So $B_{α} = [B_{α}^{l}, B_{α}^{r}] = [1 + (3 - 1) α, 5 - (5 - 3) α]$ for αɛ [0, 1] . For α = 0.5, $B_{α} = [B_{α}^{l}, B_{α}^{r}] = [2, 4]$ for αɛ [0, 1] .

2.2 Fuzzy calculus

Differentiability of a fuzzy function: Let F the set of all fuzzy numbers on R. Suppose that S ⊆ R and $\tilde{f} :$ S → F is a fuzzy-valued function with $\tilde{f} (\tilde{x}) (α) = [f^{l} (x^{l}, x^{r}, α), f^{r} (x^{l}, x^{r}, α)] .$ If the partial derivatives of f^l (x^l, x^r, α) and f^r (x^l, x^r, α) exist with respect to $\tilde{x}$ and the interval $[\frac{d f^{l} (x^{l}, x^{r}, α)}{{dx}^{l}}, \frac{d f^{r} (x^{l}, x^{r}, α)}{{dx}^{r}}]$ for $\tilde{x} ɛ R; α ɛ [0, 1]$ defines the α-level set of a fuzzy number, then it is called differentiable. Then the differentiable of fuzzy function $\tilde{f} (\tilde{x}, α)$ with respect to $\tilde{x}$ is denoted by $\frac{d \tilde{f} (\tilde{x}, α)}{d \tilde{x}} = [\frac{d f^{l} (x^{l}, x^{r}, α)}{{dx}^{l}}, \frac{d f^{r} (x^{l}, x^{r}, α)}{{dx}^{r}}]$ for $\tilde{x} ɛ R; α ɛ [0, 1]$ (cf. Farhadinia [5]).

Integrability of a fuzzy function: A fuzzy valued function $\tilde{f}$ is said to be integrable with respect to $\tilde{x}$ if f^l (x^l, x^r, α) and f^r (x^l, x^r, α) are Lebesgue integrable functions for $\tilde{x} ɛ R; α ɛ [0, 1]$ and [∫f^l (x^l, x^r, α) dx, ∫f^r (x^l, x^r, α) dx] defines the α-level set of a fuzzy number. The integral value of fuzzy function $\int \tilde{f} (\tilde{x}, α) dx = [\int f^{l} (x^{l}, x^{r}, α) dx, \int f^{r} (x^{l}, x^{r}, α) dx]$ for $\tilde{x} ɛ R; α ɛ [0, 1]$ (cf. Farhadinia [5]).

Fuzzy Variational Principle: Let ${\tilde{x}}^{*} = {\tilde{x}}^{*} (t)$ be an admissible fuzzy curve, i.e., it is twice continuously differentiable fuzzy curve that joins the given endpoints. Then ${\tilde{x}}^{*} (t)$ gives a relative (local) minimum to the fuzzy functional $J (\tilde{x}, \tilde{\dot{x}}, t)$ in (FVP) (cf. Farhadinia [5]). In this section, FVP (Fuzzy Variational Principle) is derived. To derive this, let us consider the functional $J (\tilde{x}, \tilde{\dot{x}}, t)$ . The α-cut of $J (\tilde{x}, \tilde{\dot{x}}, t)$ is $[g^{l} (x^{l}, x^{r}, {\dot{x}}^{l}, {\dot{x}}^{r}, t, α), g^{r} (x^{l}, x^{r}, {\dot{x}}^{l}, {\dot{x}}^{r}, t, α)]$ where $g^{l} (x^{l}, x^{r}, {\dot{x}}^{l}, {\dot{x}}^{r}, t, α)$ stands for $g^{l} (x^{l} (t, α), x^{r} (t, α), {\dot{x}}^{l} (t, α), {\dot{x}}^{r} (t, α), t)$ and $g^{r} (x^{l}, x^{r}, {\dot{x}}^{l}, {\dot{x}}^{r}, t, α)$ stands for $g^{r} (x^{l} (t, α), x^{r} (t, α), {\dot{x}}^{l} (t, α), {\dot{x}}^{r} (t, α), t)$ .

Here ${\tilde{x}}^{*} = {\tilde{x}}^{*} (t)$ be an admissible fuzzy curve. Now we may deform the fuzzy curve ${\tilde{x}}^{*} (t)$ by choosing an arbitrary twice continuously differentiable fuzzy function $\tilde{η} (t)$ such that $\tilde{x} (t) = {\tilde{x}}^{*} (t) + ɛ \tilde{η} (t)$ be admissible for any real number ɛ. Now, left-increment and right increment $g^{l} (x^{l}, x^{r}, {\dot{x}}^{l}, {\dot{x}}^{r}, t, α)$ are follows as D_lg^l (α) and D_rg^l (α) respectively and by the property of fuzzy weak neighbourhood, one can consider D_lg^l (α) ≥0 and D_rg^l (α) ≥0. Considering x^*l + ɛη^l stands for x^*l (t, α) + ɛη^l (t, α) and x^*r + ɛη^r stands for x^*r (t, α) + ɛη^r (t, α) for αɛ [0, 1]. Then, we have $\begin{matrix} D_{l} g^{l} (α) \\ = \int_{t_{0}}^{t_{f}} [g^{l} (x^{* l} + ɛ η^{l}, x^{* r}, {\dot{x}}^{* l} + ɛ {\dot{η}}^{l}, {\dot{x}}^{* r}, t) \\ - g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α)] dt \geq 0 . \end{matrix}$ (1) $\begin{matrix} D_{r} g^{l} (α) \\ = \int_{t_{0}}^{t_{f}} [g^{l} (x^{* l}, x^{* r} + ɛ η^{r}, {\dot{x}}^{* l}, {\dot{x}}^{* r} + ɛ {\dot{η}}^{r}, t) \\ - g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α)] dt \geq 0 . \end{matrix}$ (2)

Now, by expanding the integrands $g^{l} (x^{* l} + ɛ η^{l}, x^{* r}, {\dot{x}}^{* l} + ɛ {\dot{η}}^{l}, {\dot{x}}^{* r}, t)$ of (1) and $g^{l} (x^{* l}, x^{* r} + ɛ η^{r}, {\dot{x}}^{* l}, {\dot{x}}^{* r} + ɛ {\dot{η}}^{r}, t)$ of (2) in Taylor series about the point $(x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r})$ and for the conditions η^l (t₀, α) =0; η^l (t_f, α) =0; we have $\begin{matrix} \int_{t_{0}}^{t_{f}} η^{l} [\frac{\partial}{\partial x^{l}} g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α) \\ - \frac{d}{dt} (\frac{\partial}{\partial {\dot{x}}^{l}} g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α))] dt \geq 0; \end{matrix}$ (3) $\begin{matrix} \int_{t_{0}}^{t_{f}} η^{r} [\frac{\partial}{\partial x^{r}} g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α) \\ - \frac{d}{dt} (\frac{\partial}{\partial {\dot{x}}^{r}} g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α))] dt \geq 0; \end{matrix}$ (4)

We know that for a continuous real valued function h (t) and if

$\int_{t_{0}}^{t_{f}} h (t) g (t) dt = 0$ (5) for every continuous real valued function g (t) in [t₀, t_f], then h (t) must have a zero in [t₀, t_f] . Using (5) in (3) and (4), we have $\begin{matrix} \frac{\partial}{\partial x^{l}} g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α) \\ - \frac{d}{dt} (\frac{\partial}{\partial {\dot{x}}^{l}} g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α)) = 0; \end{matrix}$ (6) $\begin{matrix} \frac{\partial}{\partial x^{r}} g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α) \\ - \frac{d}{dt} (\frac{\partial}{\partial {\dot{x}}^{r}} g^{l} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α)) = 0; \end{matrix}$ (7)

Again, considering the left-increment and right-increment of $g^{r} (x^{l}, x^{r}, {\dot{x}}^{l}, {\dot{x}}^{r}, t, α)$ , they are expressed as D_lg^r (α) (≥0)and D_rg^r (α) (≥0) for αɛ [0, 1] respectively. Proceeding as before, we have

$\begin{matrix} \frac{\partial}{\partial x^{l}} g^{r} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α) \\ - \frac{d}{dt} (\frac{\partial}{\partial {\dot{x}}^{l}} g^{r} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α)) = 0; \end{matrix}$ (8)

$\begin{matrix} \frac{\partial}{\partial x^{r}} g^{r} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α) \\ - \frac{d}{dt} (\frac{\partial}{\partial {\dot{x}}^{r}} g^{r} (x^{* l}, x^{* r}, {\dot{x}}^{* l}, {\dot{x}}^{* r}, t, α)) = 0; \end{matrix}$ (9)

3 Proposed optimal control models

For the optimal control problem under finite time horizon, following assumptions and notations are used.

3.1 Assumptions

It is a single period production inventory model with finite time horizon tɛ [0, T];

Defective rate is known;

Shortages are not allowed;

There is no repair or replacement of defective units over whole time period;

The development cost of the system increases the durability of the system;

Unit production cost depends on produced-quantity, raw material, wear-tear and development costs;

Deterministic and fuzzy variational principle are used crisp and fuzzy models respectively.

3.2 Notations

T : total time for the system;

u (t) : production rate at time t which is taken as control variable;

$\tilde{u} (t) :$ fuzzy production rate which is a fuzzy number and u (t, α) = the α-cut of fuzzy number $\tilde{u} (t)$ i.e [u^l (t, α) , u^r (t, α)] for αɛ [0, 1] .

x (t) : stock level at time t which is a state variable;

$\tilde{x} (t) :$ fuzzy stock level which is a fuzzy number and x (t, α) = the α-cut of fuzzy number $\tilde{x} (t)$ i.e [x^l (t, α) , x^r (t, α)] for αɛ [0, 1] .

h : holding cost per unit; $\tilde{h} :$ fuzzy holding cost and (h₁, h₂, h₃) be a fuzzy number. h (α) = the α-cut of fuzzy number $\tilde{h}$ i.e [h^l (α) , h^r (α)] = [h₁ + (h₂ - h₁) α, h₃ - (h₃ - h₂) α] for αɛ [0, 1] .

L : fixed cost like labour, energy, etc; $\tilde{L}$ is fixed fuzzy cost (L₁, L₂, L₃) be a fuzzy number.

L (α) = the α-cut of fuzzy number $\tilde{L}$ i.e [L^l (α) , L^r (α)] = [L₁ + (L₂ - L₁) α, L₃ - (L₃ - L₂) α] for αɛ [0, 1] .

N : cost of technology, design, complexity, resources, etc; $\tilde{N}$ be the corresponding fuzzy cost, (N₁, N₂, N₃) be a fuzzy number.

N (α) = the α-cut of fuzzy number $\tilde{N}$ i.e [N^l (α) , N^r (α)] = [N₁ + (N₂ - N₁) α, N₃ - (N₃ - N₂) α] for αɛ [0, 1] .

c₁₀ : constant material cost;

d₀ : constant demand at initial stage;

c_d (l) = N + L : development cost to improve the quality of the product; ${\tilde{c}}_{d} (l)$ is fuzzy development cost, (c_d (l) ₁, c_d (l) ₂, c_d (l) ₃) be a fuzzy number.

c_d (l) (α) = the α-cut of fuzzy number ${\tilde{c}}_{d} (l)$ i.e [c_d (l) ^l (α) , c_d (l) ^r (α)] = [c_d (l) ₁ + (c_d (l) ₂ - c_d (l) ₁) α, c_d (l) ₃ - (c_d (l) ₃ - c_d (l) ₂) α] for αɛ [0, 1] .

$[c_{d} (l)_{α}^{l}, c_{d} (l)_{α}^{r}] = [L_{1} + (L_{2} - L_{1}) α + N_{1} + (N_{2} - N_{1}) α, L_{3} - (L_{3} - L_{2}) α + N_{3} - (N_{3} - N_{2}) α] = [L_{1} + N_{1} + (L_{2} + N_{2} - L_{1} - N_{1}) α, L_{3} + N_{3} + (L_{2} + N_{2} - L_{3} - N_{3}) α]$

β₁₀ : wear-tear cost for the system.

$\tilde{(}) :$ represents the fuzzy nature of that quantity;

p : is the selling price of defective units.

3.3 Model-1: Formulation of optimal control model for dynamic demand in crisp environment

Here, we consider a deterministic imperfect dynamic production model. As, in real-life, all production are not perfect, let δ be the imperfect fraction of production. Moreover, the demand of product in the market always changes with time, Hence, it is a function of time. With these assumptions, the differential equation for stock level x (t) regarding above system during a fixed time-horizon, T is

$\frac{d}{dt} (x (t)) = (1 - δ) u (t) - d (t)$ (10) where d (t) = d₀ - ae^-bt with (d₀ - a) , a, b > 0.

The unit production cost is considered as a function of produced-quantity, raw material cost, wear-tear and development costs(cf. Khouja [8]). So the total production cost is

$c_{u} (t) u (t) = (c_{10} u (t) + c_{d} (l) + β_{10} u^{2} (t))$ (11)

The total holding cost over the finite time interval [0, T] for the stock x (t) is $\int_{0}^{T} hx (t) dt$ .

Let p be the selling price of a defective unit. Then the revenue from produced defective units is pδu (t). Thus the problem reduces to minimization of the cost function J subject to the constraint satisfying the dynamic production-demand relation. $\begin{matrix} Min J = \int_{0}^{T} (h x (t) + c_{u} (t) u (t) - p δ u (t)) dt \\ sub to \frac{d}{dt} (x (t)) = (1 - δ) u (t) - d (t) \end{matrix}$ (12) $where u (t) \geq 0, x (t) \geq 0$ (13)

The above Equations (12)–(13) are defined as an optimal control problem with control variable u (t) and state variable x (t). Then the expression (12) takes the form

$\begin{matrix} J_{\min} (x (t), \dot{x} (t), t) = \int_{0}^{T} (h x (t) + c_{d} (l) \\ + (\frac{c_{10}}{(1 - δ)} - p δ) (\dot{x} (t) + d_{0} - {ae}^{- bt}) \\ + \frac{β_{10}}{(1 - δ)^{2}} (\dot{x} (t) + d_{0} - {ae}^{- bt})^{2})) dt \\ where x (t) \geq 0 \end{matrix}$ (14)

The above problem (14) is defined as an optimal control problem with state variable x (t). Here, (14) contains u (t) implicitly.

Using Euler-Lagrange’s equation, Fixed-Final Time and Fixed State System (i.e, here final time T is specified and $x (t) \equiv x, \dot{x} (t) \equiv \dot{x}$ , x (0) =0, x (T) =0); we have

$\frac{\partial J}{\partial x} - \frac{d}{dt} (\frac{\partial J}{\partial \dot{x}}) = 0$ (15)

Using equation (15), we get $\begin{matrix} h - \frac{d}{dt} ((\frac{c_{10}}{(1 - δ)} - p δ) \\ + \frac{2 β_{10}}{(1 - δ)^{2}} (\dot{x} (t) + d_{0} - {ae}^{- bt})) = 0 \end{matrix}$

Considering the end conditions x (0) =0 and x (T) =0, the stock level

$\begin{matrix} x (t) & = & \frac{a}{b} (1 - e^{- bt}) + \frac{at}{bT} (e^{- bT} - 1) \\ + \frac{ht (1 - δ)^{2}}{4 β_{10}} (t - T) \end{matrix}$ (16)

From (10), we have the production rate

$\begin{matrix} u (t) & = & \frac{1}{1 - δ} (\frac{a}{bT} (e^{- bT} - 1) \\ + \frac{h (1 - δ)^{2}}{4 β_{10}} (2 t - T) + d_{0}) \end{matrix}$ (17)

Then the cost functional J takes the form J_min

$\begin{matrix} = & h (\frac{a}{b^{2}} (bT + e^{- bT} - b) + \frac{aT}{2 b} (e^{- bT} - 1) \\ - \frac{h (1 - δ)^{2} T^{3}}{24 β_{10}}) + {Tc}_{d} (l) \\ + (\frac{c_{10}}{(1 - δ)} - p δ) (\frac{a}{b} (e^{- bT} - 1) + d_{0} T) \\ + \frac{β_{10}}{(1 - δ)^{2}} (\frac{a^{2}}{T b^{2}} (e^{- bT} - 1)^{2} + T d_{0}^{2} \\ + \frac{(1 - δ)^{2} h^{2} T^{3}}{12 β_{10}} + \frac{2 a d_{0}}{b} (e^{- bT} - 1)) \end{matrix}$ (18)

3.4 Model-2: Formulation of optimal control model for dynamic demand in fuzzy environment

Here, we consider a dynamic production model with imprecise production u (t) due to the variation in the efficiencies of workers, raw material quality, machine breakdown etc. and imprecise stock level as it partly depends on the production. Taking the stock variable x (t) and control variable u (t) are fuzzy and putting them in (10), the differential equation for stock level $\tilde{x} (t)$ regarding the above system during a fixed time-horizon, T is

$\begin{matrix} \frac{d}{dt} (\tilde{x} (t)) & = & (1 - δ) \tilde{u} (t) - d (t) \\ with d (t) = d_{0} - {ae}^{- bt} \end{matrix}$ (19)

Considering p is the selling price of defective units. Then the revenue from produced defective units is $p δ \tilde{u} (t)$ . The unit production cost is considered as a function of produced-quantity, raw material cost, wear-tear and development costs (cf. Khouja [8]). Here some parameters like h (>0) for holding cost per unit, c_d (l) (>0) for labour and technology cost are taken as fuzzy. Following (11), the total production cost for fuzzy environment is ${\tilde{c}}_{u} (t) \tilde{u} (t) = (c_{10} \tilde{u} (t) + {\tilde{c}}_{d} (l) + β_{10} {\tilde{u}}^{2} (t))$ (20) and the total holding cost over the finite time interval [0, T] for the stock $\tilde{x} (t)$ is $\int_{0}^{T} \tilde{h} \tilde{x} (t) dt$ .

Thus the problem reduces to minimize the cost function $\tilde{J}$ subject to the constraint satisfying the dynamic production-demand relation. $\begin{matrix} Min \tilde{J} = \int_{0}^{T} (\tilde{h} \tilde{x} (t) + {\tilde{c}}_{u} (t) \tilde{u} (t) - p δ \tilde{u} (t)) dt \\ sub to \frac{d}{dt} (\tilde{x} (t)) = (1 - δ) \tilde{u} (t) - d (t) \end{matrix}$ (21) $\begin{matrix} where \tilde{u} (t) i . e [u^{l} (t, α), u^{r} (t, α)] and \\ u^{l} (t, α) \geq 0, u^{r} (t, α) \geq 0; \\ \tilde{x} (t) i . e [x^{l} (t, α), x^{r} (t, α)] and \\ x^{l} (t, α) \geq 0, x^{r} (t, α) \geq 0; \end{matrix}$ (22)

The above Equations (21)–(22) are defined as an optimal control problem with control variable $\tilde{u} (t)$ and state variable $\tilde{x} (t)$ . Then the expression (21) takes the form $\begin{matrix} Min \tilde{J} (\tilde{x} (t), \tilde{\dot{x}} (t), t) \\ = \int_{0}^{T} (\tilde{h} \tilde{x} (t) + {\tilde{c}}_{d} (l) + \frac{c_{10}}{(1 - δ)} \tilde{\dot{x}} (t) \\ + \frac{c_{10}}{(1 - δ)} d (t) - p δ \tilde{\dot{x}} (t) - p δ d (t) \\ + \frac{β_{10}}{(1 - δ)^{2}} {\tilde{\dot{x}}}^{2} (t) + \frac{β_{10}}{(1 - δ)^{2}} d^{2} (t) \\ + \frac{2 β_{10}}{(1 - δ)^{2}} \tilde{\dot{x}} (t) d (t)) dt where \tilde{x} (t) \geq 0 \end{matrix}$ (23)

The above problem (23) is defined as an optimal control problem with state variable $\tilde{x} (t)$ . The above integral can be taken as α-level set since it is Lebesgue Integrable. Then the α-level of $\tilde{J}$ is $\begin{matrix} Min \tilde{J} (\tilde{x} (t), \tilde{\dot{x}} (t), t) [α] \\ = [G^{l} (x^{l} (t, α), x^{r} (t, α), {\dot{x}}^{l} (t, α), {\dot{x}}^{r} (t, α), t), \\ G^{r} (x^{l} (t, α), x^{r} (t, α), {\dot{x}}^{l} (t, α), {\dot{x}}^{r} (t, α), t)] \end{matrix}$ taking as [G^l (α) , G^r (α)].

Hence

$\begin{matrix} G^{l} (α) \\ = \int_{0}^{T} (h^{l} (α) x^{l} (t, α) + \frac{c_{10}}{1 - δ} {\dot{x}}^{l} (t, α) \\ + c_{d}^{l} (l) (α) + \frac{c_{10}}{1 - δ} d (t) - p δ d (t) \\ - p δ {\dot{x}}^{r} (t, α) + \frac{β_{10}}{(1 - δ)^{2}} {\dot{x}}^{l 2} (t, α) \\ + \frac{β_{10} d^{2} (t)}{(1 - δ)^{2}} + \frac{2 β_{10}}{(1 - δ)^{2}} {\dot{x}}^{l} (t, α) d (t)) dt \end{matrix}$ (24) and

$\begin{matrix} G^{r} (α) \\ = \int_{0}^{T} (h^{r} (α) x^{r} (t, α) + \frac{c_{10}}{1 - δ} {\dot{x}}^{r} (t, α) \\ + c_{d}^{r} (l) (α) + \frac{c_{10}}{1 - δ} d (t) - p δ d (t) \\ - p δ {\dot{x}}^{l} (t, α) + \frac{β_{10}}{(1 - δ)^{2}} {\dot{x}}^{r 2} (t, α) \\ + \frac{β_{10} d^{2} (t)}{(1 - δ)^{2}} + \frac{2 β_{10}}{(1 - δ)^{2}} {\dot{x}}^{r} (t, α) d (t)) dt \end{matrix}$ (25)

According to the theory of FVP in subsection-2.2, the left-increment and right-increment of G^l (α) i.e. $G^{l} (x^{l} (t, α), x^{r} (t, α), {\dot{x}}^{l} (t, α), {\dot{x}}^{r} (t, α), t)$ are non-negative because it is a cost function. Using (6) and (7) and taking $A = \frac{2 β_{10}}{(1 - δ)^{2}}$ . we have from (24) $h^{l} (α) - A \frac{d}{dt} ({\dot{x}}^{l} (t, α) + d (t)) = 0$ (26) $and 0 - \frac{d}{dt} (- p δ) = 0$ (27)

Similarly, using (8) and (9) and taking $A = \frac{2 β_{10}}{(1 - δ)^{2}}$ . we have from (25) $h^{r} (α) - A \frac{d}{dt} ({\dot{x}}^{r} (t, α) + d (t)) = 0$ (28) $and 0 - \frac{d}{dt} (- p δ) = 0$ (29)

Here out of (26), (27), (28) and (29), (26) and (28) are two governing equations and (27) and (29) are two redundant form. Using (26) and (28) with the conditions Fixed-Final Time and Fixed State System (i.e, here final time T is specified and x^l [0, α] =0, x^l [T, α] =0 ; x^r [0, α] =0, x^r [T, α] =0); and substituting the value of A, we have $\begin{matrix} x^{l} (t, α) & = & \frac{a}{b} (1 - e^{- bt}) + \frac{at}{bT} (e^{- bT} - 1) \\ + \frac{(1 - δ)^{2}}{2 β_{10}} \frac{h^{l} (α) t}{2} (t - T) \end{matrix}$ (30) $\begin{matrix} x^{r} (t, α) & = & \frac{a}{b} (1 - e^{- bt}) + \frac{at}{bT} (e^{- bT} - 1) \\ + \frac{(1 - δ)^{2}}{2 β_{10}} \frac{h^{r} (α) t}{2} (t - T) \end{matrix}$ (31)

From (19) and using (30) and (31), we have the production rate $\begin{matrix} u^{l} (t, α) & = & \frac{1}{1 - δ} (\frac{a}{bT} (e^{- bT} - 1) + d_{0} \\ + \frac{(1 - δ)^{2}}{2 β_{10}} \frac{h^{l} (α)}{2} (2 t - T)) \end{matrix}$ (32) $\begin{matrix} u^{r} (t, α) & = & \frac{1}{1 - δ} (\frac{a}{bT} (e^{- bT} - 1) + d_{0} \\ + \frac{(1 - δ)^{2}}{2 β_{10}} \frac{h^{r} (α)}{2} (2 t - T)) \end{matrix}$ (33)

Then the cost function J takes the form for left-cut of α i.e. $\begin{matrix} G^{l} (α) \\ = h^{l} (α) (\frac{a}{b^{2}} (bT + e^{- bT} - b) + \frac{aT}{2 b} (e^{- bT} - 1) \\ - \frac{h^{l} (1 - δ)^{2} T^{3}}{24 β_{10}}) + {Tc}_{d}^{l} (l) (α) + (\frac{c_{10}}{(1 - δ)} - p δ) \\ (\frac{a}{b} (e^{- bT} - 1) + d_{0} T) + \frac{β_{10}}{(1 - δ)^{2}} (\frac{a^{2}}{T b^{2}} (e^{- bT} - 1)^{2} \\ + T d_{0}^{2} + \frac{(1 - δ)^{2} h^{2 l} (α) T^{3}}{12 β_{10}} + \frac{2 a d_{0}}{b} (e^{- bT} - 1)) \end{matrix}$ Similarly, the cost function J takes the form for right-cut of α i.e. $\begin{matrix} G^{r} (α) \\ = h^{r} (α) (\frac{a}{b^{2}} (bT + e^{- bT} - b) + \frac{aT}{2 b} (e^{- bT} - 1) \\ - \frac{h^{r} (1 - δ)^{2} T^{3}}{24 β_{10}}) + {Tc}_{d}^{r} (l) (α) + (\frac{c_{10}}{(1 - δ)} - p δ) \\ (\frac{a}{b} (e^{- bT} - 1) + d_{0} T) + \frac{β_{10}}{(1 - δ)^{2}} (\frac{a^{2}}{T b^{2}} (e^{- bT} - 1)^{2} \\ + T d_{0}^{2} + \frac{(1 - δ)^{2} h^{2 r} (α) T^{3}}{12 β_{10}} + \frac{2 a d_{0}}{b} (e^{- bT} - 1)) \end{matrix}$

In case of imprecise environment, the min cost lies in the interval of above two values. So the required min cost is

${\tilde{J}}_{\min} (\tilde{x} (t), \tilde{\dot{x}} (t), t) [α] = [G^{l} (α), G^{r} (α)]$ (34)

4 Numerical experiments

4.1 Input data for Crisp and Fuzzy environment

To illustrate the models, in Crisp and Fuzzy environments i.e for Model-1, 2, we assume the following input data. For fuzzy environment, the level of α = 0.5 is taken.

4.1.1 Input data for the Crisp environment

4.1.2 Input data for the Fuzzy environment

4.2 Output for the Fuzzy and Crisp control problems

Using the above input data, the numerical values of optimal path of the system is derived using a non-linear optimization technique(GRG)-with the help of software LINGO-11.0 and optimal states of the unknowns x (t) , u (t) are evaluated in crisp and fuzzy environments. For the defective units with δ = 0.01, the required min cost = 248319.7$ and the values of production u (t), demand d (t) and stock x (t) for the time period [0, T] are given in Table 4 and these are graphically depicted in Fig. 2.

For fuzzy environment, the min cost is J^l (α = 0.5) =248264.3$ and J^r (α = 0.5) =248369.1$ i.e J [248264.3 $ , 248369.1 $] and the values of production (u^l (t, α), u^r (t, α)); stock (x^l (t, α), x^r (t, α)) and demand d (t) for the time period [0, T] with α-level = 0.5 are given in Table 5.

At the level of α = 0.25, the others values of the model i.e, the values of production u^l (t, α), u^r (t, α); stock x^l (t, α), x^r (t, α), demand d (t) are evaluated and given in the Table 6. Similarly, the level of α = 0.75, values are given in Table 7. At α = 0.5, the values of production, demand and stock over the time are graphically presented in Fig. 3.

The characteristics of production, demand and stock over the time at the level of α = 0.25, 0.75 are same like Fig. 3.

Here the Min cost for different level of α are:

248236.6 $ _{(α_left=0.25)} < 248264.3 $ _{(α_left=0.5)}

<248292.3 $ _{(α_left=0.75)} <248319.7 $ _(α=1.00)

<248344.4 $ _{(α_right=0.75)} <248369.1 $ _{(α_right=0.5)}

<248393.8 $ _{(α_right=0.25)}.

As a consequence, the nature of min cost at different level of α are pointed in Fig. 4. Also, the Min cost for the level of α = 1.00 is equivalent to the min cost in crisp environment; i.e $\begin{matrix} 248319.7 $_{(fuzzy environment) (α = 1.00)} \\ = 248319.7 $_{(crisp environment)} . \end{matrix}$

4.2.1 Outputs for the crisp environment

4.2.2 Outputs for the fuzzy environment

4.3 Fuzzy representation of decision variables

Following Fig. 4, the membership function (MF) of total cost is defined by its continuous membership function $μ_{\tilde{cost}} (x)$ as follows $\begin{matrix} μ_{\tilde{cost}} (x) = {\begin{matrix} \frac{x - J_{1}}{J_{2} - J_{1}} & if J_{1} \leq x \leq J_{2} \\ \frac{J_{3} - x}{J_{3} - J_{2}} & if J_{2} \leq x \leq J_{3} \\ 0 & otherwise \end{matrix} \end{matrix}$ where the cost triplet (J₁, J₂, J₃) = (248236.6 $ , 248319.7 $ , 248393.1 $) Similarly, following Figs. 5 and 6, the MFs of production at t = 12 and stock at t = 3 are respectively presented as $\begin{matrix} μ_{\tilde{prod}} (x) = {\begin{matrix} \frac{x - u_{1}}{u_{2} - u_{1}} & if u_{1} \leq x \leq u_{2} \\ \frac{u_{3} - x}{u_{3} - u_{2}} & if u_{2} \leq x \leq u_{3} \end{matrix} \end{matrix}$ where (u₁, u₂, u₃) = (205.135, 205.580, 206.026) $\begin{matrix} μ_{\tilde{stock}} (x) = {\begin{matrix} \frac{x - x_{1}}{x_{2} - x_{1}} & if x_{1} \leq x \leq x_{2} \\ \frac{x_{3} - x}{x_{3} - x_{2}} & if x_{2} \leq x \leq x_{3} \end{matrix} \end{matrix}$ where (x₁, x₂, x₃) = (74.804, 75.797, 76.789).

5 Output for the optimal control problems in Crisp and Fuzzy Environment using Mathematica-9.0

Results of numerical experiment in Mathematica-9.0 are almost same the results in Lingo-11.0. For this reason, some precise results are given in the following section and Tables 8 and 9. Using the above input data, the numerical values of optimal path of the system is derived and optimal states of the unknowns x (t) , u (t) are evaluated in crisp and fuzzy environments. In crisp environment, for the defective units with δ = 0.01, the required min cost = 248319$ and the values of production u (t), demand d (t) and stock x (t) for the time period [0, T] are given in Table 8.

For fuzzy environment, the min cost is J^l (α = 0.5) =248264$ and J^r (α = 0.5) =248369$ i.e J [248264 $ , 248369 $] and the values of production (u^l (t, α), u^r (t, α)); stock (x^l (t, α), x^r (t, α)) and demand d (t) for the time period [0, T] with α-level = 0.5 are given in Table 9.

The characteristics of production, demand and stock over the time using Mathematica-9.0 at the level of α = 0.25, 0.50, 0.75 are same like the characteristic of the result using Lingo-11.0. For these reason, all results, graphical presentations and discussion are considered for one of the tools (Lingo-11.0). Here the Min cost for different level of α are:

248236 $ _{(α_left=0.25)} < 248264 $ _{(α_left=0.5)}

<248292 $ _{(α_left=0.75)} <248319 $ _(α=1.00)

<248344 $ _{(α_right=0.75)} <248369 $ _{(α_right=0.5)}

<248393 $ _{(α_right=0.25)}.

Also, the Min cost for the level of α = 1.00 is equivalent to the min cost in crisp environment; i.e 248319 $ _{(fuzzyenvironment)(α=1.00)} = 248319 $ _{(crispenvironment)}.

5.1 Outputs for the crisp environment

5.2 Outputs for the fuzzy environment

6 Discussion

Here Lingo-11.0 and Mathematica-9.0 are taken as tools for the simulation to validate the result. Most of the results by using the tool Lingo-11.0 are shown the similarity with the results of Mathematica-9.0. The Min cost in imprecise environment (α = 1.00) =248319 $ = Min cost in crisp environment, which is established from Fig. 4. In the Fig. 2 for the crisp environment, initially production is more than demand and hence stock is built up after satisfying the demand. As demand is here dynamic i.e. increases with time, after sometime, demand surpasses the product and then excess demand is satisfied from the stock. Again, due to the deficit in demand becomes so the stock starts to decrease at this point and gradually decreases up to the end of cycle and finally becomes zero. Similarly, for the fuzzy environment, it is observed from the Fig. 3 (drawn for left cut of α, α = 0.5), that initially stock is built up and after some time, gradually decreases to zero following same process as in crisp model. For Model-2, Figs. 7 and 8 represent the production rate and stock level over the time for different values of α respectively. It can be noted that in Fig. 8, stock level curves are different for different values of α. The Figs. 7 and 8 are realistic in nature from the physical consideration. In Fig. 7, production rate increases with time due to the experience and expertism of workers over the time and standardisation of the machine tools. As it is a finite time horizon model, the stock level reaches maximum at a certain time and then becomes zero. This behaviour is observed for the seasonal products.

7 Conclusion

In this paper, an imperfect fuzzy production model is formulated as an optimal control problem and solved by FVP and GRG(Lingo-11.0). For the first time a fuzzy optimal control production-inventory model solved by FVP. For this model, as the input, production rate is fuzzy, the stock and min cost are evaluated as fuzzy numbers and represented by corresponding MFs. The model has been solved for a particular form of dynamic demand d (t) = d₀ - ae^-bt with (d₀ - a) , a, b > 0. As the formulation and solution are quiet general, the results can be obtained for other forms of dynamic demand such as linearly increasing demand, ramp demand etc. The model can be extended to include multi-item fuzzy inventory problem with fuzzy space and budget constraints. Major limitations of the present investigation is that here the limits of integration are deterministic. Hence in future, optimal control may be developed for the problem with integration limits and integrand both as fuzzy (cf. [19]).

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