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Abstract

The production control of failure-prone manufacturing systems is notoriously difficult because such systems are uncertain and non-linear. Since the introduction of hedging-point policies, many researches have been done in this field. However, there are few literatures that consider the production control problem of tree-structured manufacturing systems. In this article, a hedging-point production control policy is proposed for a multi-machine, tree-structured failure-prone manufacturing system. To obtain the optimal hedging points, an iterative learning algorithm is developed by considering the system’s characteristics. A simulation method is embedded in the iterative learning algorithm to calculate the system cost. To estimate the performance of the proposed algorithm, comparisons are made between our algorithm, genetic algorithm and particle swarm optimization algorithm. The experimental results show that our algorithm works better than others in reducing the computation time and minimizing the production cost.

Keywords

Tree-structured production control simulation method iterative learning algorithm

Introduction

Manufacturing systems tend to be complex and subject to various uncertain disruptions, such as machine breakdown and variable customer demands,^1–5 making it hard for managers to plan and control production processes. As for failure-prone manufacturing systems, they usually have the following characteristics: first, due to failures of machines and other equipments, production instability is inevitable, which may make production deviate from the expected goal; second, production is more often than not dominated by customer requirements, which may change manufacturing processes; and finally, the existence of defective products and other factors may also bring a negative impact on the continuity of manufacturing processes.

The production control problem of failure-prone manufacturing systems has been extensively studied in the last three decades, and many outstanding results have been obtained. Kimemia and Gershwin⁶ demonstrated that the optimal control policy for such systems has a special structure called the hedging-point policy (HPP). In this policy, a nonnegative production surplus of part types is maintained during times when excess capacity is available to hedge against future capacity shortages. Hedging-point control is proved to be the optimal policy if unmet demands are completely backlogged.^7,8 These results were extended to systems without backlog and bounded backlog.^9,10 Proth et al.¹¹ focused on the problem of supply management and inventory/backlogging control in assembly systems with random component yield times. El-Ferik et al.¹² introduced a simple maximal hedging (SMH) policy by which the production performance comes close to that of the optimal control. Feng and Yan¹³ studied the production control of a failure-prone manufacturing system with stochastic demands. Perkins and Srikant¹⁴ investigated a failure-prone manufacturing system with bursty demand arrivals. They proved the HPP to be still an optimal control policy and provided analytical expressions to calculate the hedging point. Gharbi and Kenne¹⁵ proposed a parameterized near-optimal production policy for a multiple-product, multiple-machine manufacturing system and combined the analytical formalism with simulation-based statistical tools to obtain an approximation of the optimal control policy. Kinafar¹⁶ came up with a numerical method to approximate optimal production and maintenance plan in a flexible manufacturing system. Mok and Porter¹⁷ developed an evolutionary stochastic optimization procedure to estimate the short-run optimal hedging points for failure-prone manufacturing systems under crisp-logic control. Yan et al.¹⁸ proposed the concept of state jump system to analyze a knowledgeable manufacturing cell with an unreliable agent. Abou-Kandkil et al.¹⁹ discussed the problem of computing optimal production rates for a failure-prone manufacturing system with multi-machine and multi-product. Mourani et al.²⁰ explored the optimization of failure-prone transfer lines with important delays for material transfer, constant demand and echelon base stock policy for production control. Kenne and Gharbi²¹ presented the optimal flow control method for a one-machine, two-product manufacturing system subject to random failures and repairs based on the combination of the analytical model, simulation experiments, experimental design and response surface methodology. Song and Sun²² considered a manufacturing system with multiple operational modes producing one part type. They proved that the optimal control policy is still of a hedging-point structure.

Based on these pioneering works, many researchers have gone deeper into the production control problem in recent years. Hajji et al.²³ considered joint production control and product specification decision-making in a failure-prone manufacturing system. Mok²⁴ constructed gain-scheduled adaptive controllers for unreliable manufacturing systems with variable demands based on genetically optimized short-run hedging points. Njike et al.²⁵ analyzed the interaction between defective products and optimal control of production rate, lead time and inventory. Berthaut et al.²⁶ proposed a joint preventive maintenance and production/inventory control policy based on a modified block replacement policy (MBRP) and HPPs for a mono-product, single-machine manufacturing cell. Ouaret et al.²⁷ studied the production control problem of a hybrid manufacturing/remanufacturing system where demand is modeled as a diffusion-type stochastic process. Gharbi et al.²⁸ dealt with the production control problem of an adjustable capacity unreliable manufacturing cell responding to a single product type demand.

In most of the existing literatures, a lot of remarkable work has been done on the production control problem of simple manufacturing systems with single machine, two machines or production line. To the authors’ knowledge, little work has been reported on complex manufacturing systems, especially on tree-structured ones. This article’s main contribution lies in the development of a production control policy for such manufacturing systems. Here, the HPP is adopted, and an iterative learning algorithm (ITA) is developed to optimize the hedging points by taking the system’s characteristics into account. Comparisons are performed between our algorithm, genetic algorithm (GA) and particle swarm optimization (PSO). Simulation results show that the proposed algorithm works apparently better than GA and PSO.

Problem description

It is assumed that there are $φ$ failure-prone machines in a tree-structured production system. For the machine $M_{i} (i = 1, \dots, φ)$ , the time between failures is an exponentially distributed random variable with mean $1 / q_{i}^{u}$ , while the repair time is also exponentially distributed with mean $1 / q_{i}^{d}$ . When functioning, $M_{i}$ can produce at any rate up to its maximum $r_{max}^{i} (r_{max}^{i} > 0)$ ; when broken down, $M_{i}$ cannot produce at all. At all times, the demand for final products is constant at a rate $d$ . The binary variable $s_{i} (t) \in {0, 1}$ is used to express the machine state, and $s_{i} (t) = 1$ when $M_{i}$ is up at time $t$ ; otherwise, $s_{i} (t) = 0$ .

Definition 1.

If there are parts flowing from $M_{m}$ to $M_{n}$ , $M_{m}$ is called the former machine of $M_{n}$ and $M_{n}$ the succeeding machine of $M_{m}$ .

Definition 2

If $M_{i}$ has no former machines, it is called the starting machine; if $M_{i}$ has no succeeding machine, it is called the ending machine.

A tree-structured manufacturing system is shown in Figure 1. For a machine $M_{i}$ , it may have more than one former machine/buffer, but with only a single succeeding machine/buffer. Furthermore, $B_{01}, B_{02}, B_{03}, B_{04}$ are the input buffers, and $B_{6}$ is the output buffer.

Figure 1.

A tree-structured manufacturing system.

The assumptions are given as below:

Each machine has a sufficient capacity to ensure that the system keeps a stable state. $M_{φ}$ represents the last machine; then the maximum production rate $r_{max}^{i}$ of $M_{i}$ should meet $((r_{max}^{i} q_{i}^{d}) / (q_{i}^{d} + q_{i}^{u})) > α (i, φ) d$ , where $α (i_{1}, i_{2})$ is the number of parts in buffer $B_{i_{1}}$ required to produce one part in buffer $B_{i_{2}}$ .

If $M_{m}$ is the former machine of $M_{n}$ , the demand of $M_{n}$ can always be met by $M_{m}$ at their maximum production rates $r_{max}^{n}$ and $r_{max}^{m}$ , that is

\frac{r_{max}^{m} q_{m}^{d}}{q_{m}^{d} + q_{m}^{u}} > α (m, n) \frac{r_{max}^{n} q_{n}^{d}}{q_{n}^{d} + q_{n}^{u}}

There are enough materials or parts in the input buffers.

Control strategy and objective

HPP is adopted as the production control strategy for all the machines. The objective function of the system is modeled in an integral form, where two types of state variables are considered: continuous variables (the number of parts) and discrete variables (machine working states). The processing time of each part is given. For a machine $M_{i}$ , state(s) of its former buffer(s) can be expressed by $w_{i} (t) \in {0, 1}$ , that is, for $\forall t \geq 0$ , if no shortage exists in the former buffers of $M_{i}$ , then $w_{i} (t) = 1$ ; otherwise, $w_{i} (t) = 0$ . Suppose that $M_{i}$ is the former machine of $M_{ϕ}$ , $r_{v}^{i} (t)$ is the production rate of $M_{i}$ required to maintain the production of $M_{ϕ}$ , $x_{i} (t)$ is the inventory level of $B_{i}$ and $z_{i}$ is the value of the hedging point of $B_{i}$ . Then the control policy of $M_{i}$ is

u_{i} (t) = {\begin{matrix} r_{max}^{i} & x_{i} < z_{i}, s_{i} (t) = 1, w_{i} (t) = 1 \\ r_{v}^{i} (t) & x_{i} = z_{i}, s_{i} (t) = 1, w_{i} (t) = 1 \\ 0 & others \end{matrix}

(1)

If $M_{i}$ is the last machine, the control policy is

u_{i} (t) = {\begin{matrix} r_{max}^{i} & x_{i} < z_{i}, s_{i} (t) = 1, w_{i} (t) = 1 \\ d & x_{i} = z_{i}, s_{i} (t) = 1, w_{i} (t) = 1 \\ 0 & others \end{matrix}

where $r_{v}^{i} (t) = α (i, ϕ) u_{ϕ} (t)$ should be met to maintain the continuous production of $M_{ϕ}$ . This control policy aims to make the production meet customer requirements, reduce inventory levels and avoid shortages, regardless of the initial inventory states of buffers and working states of machines. As the input buffers are not controlled by machines, their optimization problems are not necessary to be considered here. The optimization objective is to find the optimal production rate for minimizing the expected long-run average production cost

\begin{matrix} J^{z} (x_{0}, s_{0}) = \\ lim_{T \to \infty} \frac{1}{T} E (\int_{0}^{T} g (x (t), s (t)) dt | x (0) = x_{0}, s (0) = s_{0}) \end{matrix}

(2)

where $J^{z} (x_{0}, s_{0})$ is the system cost function. It is the cost incurred using the above control policy and starting from the initial condition $(x_{0}, s_{0})$ . Moreover, at steady state, $J^{z}$ should be independent of $(x_{0}, s_{0})$ . $g (x (t), s (t)) = \sum_{i = 1}^{φ} [c_{i}^{+} x_{i} (t)] + c_{φ}^{-} x_{φ}^{-} (t)$ is the penalty function of inventory and backlog, $x_{i} (t)$ is the inventory level of $B_{i}$ , $x_{φ}^{-}$ is the backlog level of the last buffer $B_{φ}$ , $c_{i}^{+}$ is the holding cost of $B_{i}$ per unit part at per unit time and $c_{φ}^{-}$ is the backlog cost of $B_{φ}$ per unit final product at per unit time.

Suppose $Ω$ is the set of all feasible hedging-point sets $z = (z_{1}, z_{2}, \dots, z_{φ})$ , where $z_{i} \geq 0, i = 1, 2, \dots, φ$ , and $z^{*} = (z_{1}^{*}, z_{2}^{*}, \dots, z_{φ}^{*})$ is the optimal one in $Ω$ . The optimization objective to find the optimal set of hedging points $z^{*}$ can then be transformed into equation (3)

J^{z^{*}} (x_{0}, s_{0}) = inf_{z \in Ω} J^{z} (x_{0}, s_{0})

(3)

Optimization of hedging points

Simulation method

According to equation (2), the system cost consists of inventory cost and backlog cost. For a multi-machine, tree-structured manufacturing system with random disruptions, it is very difficult to get an analytical expression of the cost function $J^{z} (x_{0}, s_{0})$ . Simulation method is thought to be an important and effective solution to this complex problem.^17,29 As the running process of the system can be expressed as a Markov process in a finite state space,³⁰ $J^{z} (x_{0}, s_{0})$ will converge to a certain value under a given hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ , which is why the simulation method is used here to calculate the value of $J^{z} (x_{0}, s_{0})$ .

The hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ is taken as the input variable of the simulation system. For the running time $T (T = Δ T, 2 Δ T, \dots, l Δ T)$ , all values of the cost function $J^{z} (x_{0}, s_{0}) (T)$ should be calculated. If $| (J^{z} (x_{0}, s_{0}) (l Δ T) - J^{z} (x_{0}, s_{0}) ((l - a) Δ T)) / (J^{z} (x_{0}, s_{0}) (l Δ T)) | \leq ε, \forall a = 1, 2, \dots, τ$ , is true, where $ε$ is a threshold value and $τ$ is the number of steps, then $J^{z} (x_{0}, s_{0}) (l Δ T)$ is treated as an approximation of $J^{z} (x_{0}, s_{0})$ , that is, $J^{z} (x_{0}, s_{0}) (l Δ T) \approx J^{z} (x_{0}, s_{0})$ . For the given hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ , if the cost function can converge to a fixed value, the system is thought to be in a cost function steady state of the hedging points called “State 1” for short. The ratio between the number of parts produced by $M_{i}$ and the running time $T$ is called the average production rate of $M_{i}$ .

ITA

The block diagram of the hedging-point optimization process is demonstrated in Figure 2, where the simulation module is employed to realize the above-mentioned simulation method to calculate the production cost $J^{z} (x_{0}, s_{0})$ . The optimization algorithm module is designed to adjust the hedging points of the buffers according to the production information.

Figure 2.

Block diagram of the hedging-point optimization process.

Definition 3

Suppose $M_{i + 1}$ is the succeeding machine of $M_{i}$ . For the given hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ in State 1, $T_{i}^{b}$ denotes the times during which the number of parts in buffer $B_{i}$ is larger than $α (i, i + 1)$ . If the system’s total running time is $T$ , $g_{i} = T_{i}^{b} / T$ is taken as the characteristic value of $B_{i}$ , $g_{i} \in [0, 1]$ .

To facilitate the analysis of the hedging points, a clustering rule is adopted to acquire a new space $V' = {{\tilde{g}}_{i} | {\tilde{g}}_{i} \in {0, 1}}$ from $V = {g_{i} | g_{i} \in [0, 1]}$ . The rule goes as follows: if $g_{i} \geq ξ_{i}$ ( $ξ_{i} \in [0, 1]$ ), then ${\tilde{g}}_{i} = 1$ ; otherwise, ${\tilde{g}}_{i} = 0$ . The value of $ξ_{i}$ approximately equal to 1 can be selected according to experiences of experts for different manufacturing systems.

If ${\tilde{g}}_{i} = 1$ , the number of parts in buffer $B_{i}$ can be considered to be almost always greater than $α (i, i + 1)$ , that is to say, the impact of the shortage in buffer $B_{i}$ on $M_{i + 1}$ is negligible. If $v_{i}$ and $v_{i + 1}$ are the average production rates of $M_{i}$ and $M_{i + 1}$ , respectively, then $v_{i} \geq α (i, i + 1) v_{i + 1}$ holds when ${\tilde{g}}_{i} = 1$ .

If buffer $B_{j}$ locates on any path from buffer $B_{i}$ to the input buffers, $B_{j}$ locates before $B_{i}$ . If $B_{j}$ locates on any path from buffer $B_{i}$ to the output buffer, then $B_{j}$ locates behind $B_{i}$ .

Theorem 1

$B_{i}$ and $B_{j}$ represent any two buffers in the manufacturing system. Suppose the average production rates of $M_{j}$ are $v_{j}$ and $v'_{j}$ when the hedging points of $B_{i}$ are $z_{i}$ and $z'_{i}$ . If $z'_{i} \geq z_{i}$ , $v'_{j} \geq v_{j}$ holds.

Proof

Suppose the average production rate of $M_{i}$ is $v_{i}$ or $v'_{i}$ when the hedging point of buffer $B_{i}$ is $z_{i}$ or $z'_{i}$ . If $z'_{i} \geq z_{i}$ , the succeeding machine of $M_{i}$ requires $M_{i}$ to provide more parts to maintain its uninterrupted production in unit time so that $v'_{i} \geq v_{i}$ can be met. Suppose $M_{i + 1}$ is the succeeding machine of $M_{i}$ and the average production rates of $M_{i + 1}$ are $v_{i + 1}$ and $v'_{i + 1}$ when the hedging points of buffer $B_{i}$ are $z_{i}$ and $z'_{i}$ . As the supplement of $M_{i + 1}$ will increase when $z'_{i} \geq z_{i}$ , so $v'_{i + 1} \geq v_{i + 1}$ holds. According to the theory of constraints (TOC), the production rate of a stable manufacturing system is determined by the bottleneck $M_{b}$ . Suppose when the hedging points of buffer $B_{i}$ are $z_{i}$ or $z'_{i}$ , the average production rates of $M_{b}$ are $v_{b}$ and $v'_{b}$ , and the average production rates of the last machine $M_{φ}$ are $v_{φ}$ and $v'_{φ}$ . The average production rates of $M_{j}$ in State 1 are $v'_{j} = α (j, φ) v'_{φ} = (α (j, φ) v'_{b}) / (α (b, φ))$ and $v_{j} = α (j, φ) v_{φ} = (α (j, φ) v_{b}) / (α (b, φ))$ . Therefore, we have

v'_{j} - v_{j} = \frac{α (j, φ) v'_{b}}{α (b, φ)} - \frac{α (j, φ) v_{b}}{α (b, φ)} = \frac{α (j, φ)}{α (b, φ)} (v'_{b} - v_{b})

(4)

If $M_{b}$ represents the same machine with $M_{i}$ or $M_{i + 1}$ , then $v'_{b} - v_{b} = v'_{i} - v_{i} \geq 0$ or $v'_{b} - v_{b} = v'_{i + 1} - v_{i + 1} \geq 0$ .

If $M_{b}$ represents the different machine from $M_{i}$ or $M_{i + 1}$ , then $v'_{b} - v_{b} = 0$ .

In summary, when $z'_{i} \geq z_{i}$ , the result $v'_{j} \geq v_{j}$ is true, which completes the proof of Theorem 1.

Theorem 2

If the values of the hedging points of $B_{i}$ and the buffers before $B_{i}$ are large enough, the characteristic value ${\tilde{g}}_{i} = 1$ .

Proof

Assume that $B_{j}$ can represent $B_{i}$ or any one buffer before $B_{i}$ . If the value of the hedging point of buffer $B_{j}$ tends to infinity, that is, $z_{j} \to \infty$ , the production control policy of $M_{j}$ can be written as

u_{j} (t) = {\begin{matrix} r_{max}^{j} & x_{j} < z_{j}, s_{j} (t) = 1, w_{j} (t) = 1 \\ 0 & others \end{matrix}

(5)

According to Assumption (2), if $M_{m}$ is the former machine of $M_{n}$ , then we have

\frac{r_{max}^{m} q_{m}^{d}}{q_{m}^{d} + q_{m}^{u}} > α (m, n) \frac{r_{max}^{n} q_{n}^{d}}{q_{n}^{d} + q_{n}^{u}}

(6)

Suppose that $M_{m}$ is a machine before $B_{i}$ ; the average production rates of $M_{m}$ and $M_{n}$ are $v_{m}$ and $v_{n}$ , respectively. According to equations (5) and (6), $v_{m} > α (m, n) v_{n}$ holds. If the number of parts in buffer $B_{i}$ is $Q_{m}$ , then $Q_{m} = \int (v_{m} - α (m, n) v_{n}) dt$ is true. Thus, when $t \to \infty$ , $Q_{m} \to \infty$ holds, so $Q_{i} \to \infty$ is true if $z_{i} \to \infty$ . The number of parts in buffer $B_{i}$ is almost always greater than $α (i, i + 1)$ , that is, $g_{i} = (T_{i}^{b} / T) \to 1$ ; thus, we obtained ${\tilde{g}}_{i} = 1$ .

That completes the proof of Theorem 2.

Suppose $U_{i}$ represents a buffer set which consists of $B_{i}$ and the buffers before $B_{i}$ . In terms of the assumption that there is no shortage in the input buffers, there must be some buffers of $U_{i}$ in shortage of parts if ${\tilde{g}}_{i} = 0$ . That is to say, the hedging points of these buffers called F-buffers tend to be small. $U_{i}^{F}$ denotes the set of F-buffers.

Algorithm 1 is designed to ensure the characteristic value of $B_{i}$ equals 1, that is, ${\tilde{g}}_{i} = 1$ . Its detailed steps are listed below:

Step 1. Initialize the hedging-point set $\tilde{z} = ({\tilde{z}}_{1}, {\tilde{z}}_{2}, \dots, {\tilde{z}}_{φ})$ .

Step 2. Input $\tilde{z}$ into the simulation module in Figure 2 and calculate the characteristic value ${\tilde{g}}_{i}$ of each buffer.

Step 3. For the buffers in $U_{i}^{F} = {B_{1}^{F}, \dots, B_{h}^{F}, \dots, B_{ρ}^{F}}, ρ = | U_{i}^{F} |$ , if the characteristic value ${\tilde{g}}_{i}$ of $B_{i}$ is 0 (i.e., ${\tilde{g}}_{i} = 0$ ), the hedging point of $B_{h}^{F} (h = 1, 2, \dots, ρ)$ is updated by ${\tilde{z}}_{h} = ceil (β {\tilde{z}}_{h} + η')$ , where $β (β > 1)$ and $η' (η' > 0, η' is a very small value)$ are the constants. Once we obtain a new hedging-point set $\tilde{z} = ({\tilde{z}}_{1}, {\tilde{z}}_{2}, \dots, {\tilde{z}}_{φ})$ , go to Step 2; if ${\tilde{g}}_{i} = 1$ , output the hedging point $\tilde{z}$ and terminate the algorithm.

Proposition 1

For a given hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ , when the system is in State 1, if ${\tilde{g}}_{i} = 0$ , we get ${\tilde{g}}_{i} = 1$ from Steps 2 and 3 of Algorithm 1.

Proof

According to Theorem 2, there must be the hedging points of the buffers in $U_{i}^{F}$ which are large enough to ensure ${\tilde{g}}_{i} = 1$ . Therefore, with the ongoing iteration in Steps 2 and 3 of Algorithm 1, the values of the hedging points of the buffers in $U_{i}^{F}$ will increase continuously until the set $U_{i}^{F}$ is empty. Once $U_{i}^{F}$ is empty, ${\tilde{g}}_{i} = 1$ .

Proposition 2

For a given hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ , when the system is in State 1, if ${\tilde{g}}_{i} = 0$ , the characteristic value ${\tilde{g}}_{i}$ of buffer $B_{i}$ will remain unchanged even if the hedging point of $B_{j}$ behind $B_{i}$ increases.

Proof

Suppose $B_{i + 1}$ is the succeeding buffer of $B_{i}$ , $v_{i}$ and $v_{i + 1}$ are the average production rates of $M_{i}$ and $M_{i + 1}$ , respectively, and $v'_{i}$ and $v'_{i + 1}$ are the average production rates of $M_{i}$ and $M_{i + 1}$ , respectively, when the hedging point of $B_{j}$ is increased.

In terms of Theorem 1, the system’s production rate will increase when the upper limit of a buffer is increased. As ${\tilde{g}}_{i} = 0$ , the average production rate of the sub-system, which consists of the machines and the buffers before $B_{i}$ , has reached the maximum value on the condition that the hedging points of the buffers before $B_{i}$ are fixed. Therefore, the production rate of $M_{i}$ remains unchanged, that is, $v_{i} = v'_{i}$ , and by Theorem 1, $v'_{i + 1} > v_{i + 1}$ holds, which indicates that the characteristic value ${\tilde{g}}_{i}$ of buffer $B_{i}$ is invariable after increasing the hedging point of $B_{j}$ , that is, ${\tilde{g}}_{i} = 0$ .

Proposition 3

For a given hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ , when the system is in State 1, if ${\tilde{g}}_{i} = 0$ , the characteristic value of $B_{j}$ behind $B_{i}$ is always 0, that is, ${\tilde{g}}_{j} = 0$ .

Proof

Suppose the succeeding machine of $M_{i}$ is $M_{i + 1}$ , the succeeding machine of $M_{i + 1}$ is $M_{i + 2}$ , and the average production rates are $v_{i}$ , $v_{i + 1}$ and $v_{i + 2}$ . When ${\tilde{g}}_{i} = 0$ , three cases are explored for production rates $v_{i + 1}$ and $v_{i + 2}$ as below:

If $v_{i + 1} \leq α (i + 1, i + 2) v_{i + 2}$ , ${\tilde{g}}_{i + 1} = 0$ .

If $v_{i + 1} > α (i + 1, i + 2) v_{i + 2}$ and $v_{i} > α (i, i + 2) v_{i + 2}$ , the output of $M_{i}$ exceeds the requirement of $M_{i + 2}$ . The hedging point $z_{i + 1}$ of $B_{i + 1}$ is bounded, so $v_{i} > α (i, i + 1) v_{i + 1}$ , which contradicts the known condition ${\tilde{g}}_{i} = 0$ .

If $v_{i + 1} > α (i + 1, i + 2) v_{i + 2}$ and $v_{i} \leq α (i, i + 2) v_{i + 2}$ , the requirement of $M_{i + 2}$ goes over the output of $M_{i}$ , that is, ${\tilde{g}}_{i + 1} = 0$ .

In conclusion, if ${\tilde{g}}_{i} = 0$ , ${\tilde{g}}_{i + 1} = 0$ . By this analogy, the characteristic value of any buffer $B_{j}$ behind $B_{i}$ is always 0.

Inference 1

For a given hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ , when the system is in State 1, if ${\tilde{g}}_{i} = 1$ , the characteristic value of $B_{j}$ before $B_{i}$ equals 1, that is, ${\tilde{g}}_{j} = 1$ .

Proof

Reductio ad absurdum is used here. By Proposition 3, ${\tilde{g}}_{i} = 0$ if ${\tilde{g}}_{j} = 0$ . This result contradicts the known condition ${\tilde{g}}_{i} = 1$ . Thus, Inference 1 is proved.

For a given hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ , suppose the system is in State 1, the characteristic value of $B_{i}$ is 1 (i.e. ${\tilde{g}}_{i} = 1$ ), the duration when the number of parts in $B_{i}$ is greater than $η_{i}$ is ${\tilde{T}}_{i}^{b} (η_{i})$ and the system’s total running time is $T$ . When $(({\tilde{T}}_{i}^{b} (η_{i})) / (T)) \geq ξ_{i} > (({\tilde{T}}_{i}^{b} (η_{i} + 1)) / (T))$ , $η_{i}$ is called the redundancy of $B_{i}$ . If $ψ$ input buffers are equipped by the system, then there are $ψ$ paths from the input buffers to the output buffer. According to the property of the characteristic value and features of tree structures, Algorithm 2 (ITA) is proposed here to optimize the system’s hedging points. Its detailed steps are listed as follows:

Step 1. Initialize the hedging-point set $z^{(n)} = (z_{1}^{(n)}, z_{2}^{(n)}, \dots, z_{φ}^{(n)})$ and set $n = 0$ .

Step 2. Input $z^{(n)}$ into the simulation system and then calculate the system cost $J^{z^{(n)}}$ and the characteristic value ${\tilde{g}}_{i}$ , $i = 1, \dots, φ$ .

Step 3. For each path from the input buffers to the output buffer, set $\tilde{z} = z^{(n)}$ if ${\tilde{g}}_{i} = 0$ , $i = 1, \dots, φ$ . After performing Algorithm 1, set $z^{(n)} = \tilde{z}$ .

Step 4. Repeat Steps 2 and 3 of Algorithm 2 until all the characteristic values are equal to 1, that is, ${\tilde{g}}_{i} = 1$ , $i = 1, \dots, φ$ .

Step 5. Calculate $z^{(n + 1)} = (z_{1}^{(n)} - η_{1}, z_{2}^{(n)} - η_{2}, \dots, z_{φ}^{(n)} - η_{φ})$ and set $n = n + 1$ . (The value of $η_{i}$ is obtained via simulation, being the largest one that ensures ${\tilde{g}}_{i}$ still equals 1 after subtraction.). Input $z^{(n)}$ into the simulation system and then calculate the system cost $J^{z^{(n)}}$ and the characteristic value ${\tilde{g}}_{i} (i = 1, \dots, φ)$ .

Step 6. Set $k = 1$ and $θ_{i} = 0 (i = 1, \dots, φ)$ .

Step 7. Set $μ = φ$ .

Step 8. If $θ_{μ} = 0$ , go to Step 9; if $θ_{μ} = 1$ , $k < ψ$ and $B_{μ}$ is not an input buffer, then find the buffer $B_{ρ}$ which is adjacent to and locates before $B_{μ}$ on the kth path from the input buffers to the output buffer, set $μ = ρ$ and go to Step 8; if $θ_{μ} = 1$ , $k < ψ$ and $B_{μ}$ is an input buffer, set $k = k + 1$ and go to Step 7; if $θ_{μ} = 1$ , $k = ψ$ and $B_{μ}$ is an input buffer, then output $z^{(n)}$ and terminate the algorithm.

Step 9. Set the temporary variable $z_{μ}^{d} = 0$ .

Step 10. Set $z^{(n + 1)} = z^{(n)}$ , $z_{μ}^{(n + 1)} = round [(z_{μ}^{(n)} + z_{μ}^{d}) / 2]$ ( $z_{μ}$ is the µth component of $z$ ) and $n = n + 1$ . Input $z^{(n)}$ into the simulation system and calculate the system cost $J^{z^{(n)}}$ . If $J^{z^{(n)}} > J^{z^{(n - 1)}}$ , set $z_{μ}^{d} = z_{μ}^{(n)}$ and $z_{μ}^{(n)} = z_{μ}^{(n - 1)}$ , and then go to Step 10; if $J^{z^{(n)}} \leq J^{z^{(n - 1)}}$ and $J^{z^{(n - 1)}} - J^{z^{(n)}} < δ$ (The threshold value $δ$ is set to avoid the premature termination of the optimization process of $z_{μ}$ .), then set $θ_{μ} = 1$ and go to Step 7; otherwise, go to Step 10.

Algorithm 2 consists of two parts: the first part (Steps 1–5) aims to obtain the minimum inventory level of each buffer when no part shortages occur in the system. The result is the fundament for further optimization. The second part (Steps 6–10) aims to decrease the inventory levels, from the output buffer to the input buffers, to reduce the system cost and find the optimal hedging levels by relying on a convexity property of the cost function.

In the first part (Steps 1–5), Algorithm 1 is implemented in Steps 2–4. In terms of the definition of buffer redundancy $η_{i}$ , when $η_{i}$ is subtracted, the average production rate remains unchanged, that is, ${\tilde{g}}_{i}$ is still equal to 1. Meanwhile, $η_{i}$ is turned to be 0, which makes $(({\tilde{T}}_{i}^{b} (η_{i})) / T) = (({\tilde{T}}_{i}^{b} (0)) / T) \geq ξ_{i}$ hold. Then, an initial optimization for hedging points is performed in Step 5 on the premise of not changing the value of ${\tilde{g}}_{i}, \forall i = 1, 2, \dots, φ$ .

For a failure-prone manufacturing system, the nonnegative inventory level is used for hedging against future capacity shortage. But too small hedging points may still incur shortage, and thus, the production may not be maintained. In the first five steps, the minimum hedging-point set $z^{(1)}$ , which can ensure no shortage occurs in the whole production process, is obtained for further optimization.

Seen from equation (2), no backlog cost needs to be paid with $z^{(1)}$ in the system, but some inventory cost may be incurred. Therefore, we reduce the values of the hedging points in $z^{(1)}$ for lower inventory cost and even try to reduce the system cost at the expense of higher backlog cost. According to equation (2), only the backlog cost of the last buffer is considered. Following Steps 6–10, the hedging points are decreased from the output buffer to the input buffers. In each iteration process, the value of a hedging point is determined based on the result of the last iteration. Until the system cost cannot be lowered by decreasing the hedging point of the current buffer, the one of the next buffer will be determined. It is assumed that $B_{j}$ is a buffer immediately before $B_{i}$ , $z_{j}$ and $z_{i}$ are the hedging points of $B_{j}$ and $B_{i}$ , respectively, obtained in Step 10, and $z'_{j}$ and $z'_{i}$ are the different values from $z_{j}$ and $z_{i}$ . In Step 10, the system cost will rise if $z'_{j}$ and $z'_{i}$ are chosen to be the hedging points. Therefore, the optimization of the hedging points of any two adjacent buffers can be implemented through the approach whereby the inventory levels are decreased one after another from the output buffer to the input buffer. When all the hedging points are determined, the algorithm is terminated.

Example and analysis

Table 1 lists the production parameters of the machines of the system in Figure 1. The demand rate for the final product is $d = 0.025$ . The values of relationship functions between the buffers are as follows: $α (B_{1}, B_{2}) = 5$ , $α (B_{3}, B_{4}) = 3$ , $α (B_{5}, B_{4}) = 4$ , $α (B_{2}, B_{6}) = 1$ and $α (B_{4}, B_{6}) = 1$ . The holding costs $c_{i}^{+} (i = 1, 2, \dots, 6)$ are 2, 8, 10, 13, 4 and 15 (yuan/h) per unit part/product, and the shortage cost $c^{-}$ of $B_{6}$ is 10,000 yuan per unit product. In the simulation method, $ε = 0.01$ and $τ = 5000$ . In the clustering rule, $ξ_{i} = 0.9$ . The simulation module and the optimization algorithm module are integrated and developed using C++, compiled on a PC with Intel (R) Core (TM) i5-2400 CPU at 3.10 GHz and 4 GB main memory under WinXP. Simulation model is designed to describe the dynamic behavior of the system, covering several subroutines to realize its specific tasks, such as system initialization, failure and repair time generation and control policy implement. For the model, the hedging-point set $z = (z_{1}, z_{2}, \dots, z_{φ})$ is considered as the input, and the corresponding incurred cost $J^{z}$ is viewed as its output.

Table 1.

Production parameters of machines.

Machines	Maximum production rates (piece/h)	Parameters of repair rate	Parameters of failure rate
M ₁	$r_{max}^{1} = 1 / 1$	$q_{1}^{d} = 0.025$	$q_{1}^{u} = 0.0120$
M ₂	$r_{max}^{2} = 1 / 6$	$q_{2}^{d} = 0.020$	$q_{2}^{u} = 0.0125$
M ₃	$r_{max}^{3} = 1 / 3$	$q_{3}^{d} = 0.025$	$q_{3}^{u} = 0.0125$
M ₄	$r_{max}^{4} = 1 / 8$	$q_{4}^{d} = 0.020$	$q_{4}^{u} = 0.0120$
M ₅	$r_{max}^{5} = 1 / 1$	$q_{5}^{d} = 0.020$	$q_{5}^{u} = 0.0120$
M ₆	$r_{max}^{6} = 1 / 10$	$q_{6}^{d} = 0.020$	$q_{6}^{u} = 0.0120$

Performance comparison

GA and PSO have been widely applied to optimization problems.^31–36 And the former is also frequently used to obtain the hedging points of the manufacturing systems.^{17,24,37–39} To evaluate the performance of the proposed hedging-point optimization algorithm, 10 experiments are conducted, respectively, by ITA, GA and PSO. The initial hedging-point set used in ITA is (20, 20, 20, 20, 20, 20). The parameters of GA and PSO are set as follows—the population sizes: both 50; the generation numbers: both 100 (the simulation results show that the optimizing effect sees no conspicuous improvement when the generation number updates); the crossover probability: 0.5; mutation probability: 5%; inertial weight: 0.5 and acceleration constants: both 2.0. The experimental data in Table 2 show that the proposed algorithm exhibits a better performance than GA and PSO in both computation time $t$ (ms) and the average production cost $J^{z}$ (yuan/h).

Table 2.

Comparison of algorithms.

Performance		1	2	3	4	5	6	7	8	9	10	Average
ITA	$J^{z}$	318.5	320.3	319.9	326.6	317.8	330.9	297.0	329.4	324.7	316.2	320.1
ITA	$t$	1438	1453	1235	1359	1390	1343	1422	1266	1266	1094	1327
GA	$J^{z}$	393.6	369.8	388.3	382.1	374.7	381.3	373.6	424.9	409.9	380.1	388.0
GA	$t$	39,968	39,156	38,843	39,687	38,907	38,797	39,531	39,031	39,594	39,453	39,297
PSO	$J^{z}$	345.2	351.0	342.4	334.7	360.2	349.1	355.2	345.5	360.1	347.4	349.1
PSO	$t$	43,000	40,828	39,964	40,015	39,719	38,694	39,451	40,112	40,008	38,845	40,064

ITA: iterative learning algorithm; GA: genetic algorithm; PSO: particle swarm optimization.

Sensitivity analysis

To illustrate the effect of system parameters, a sensitivity analysis is made as presented in Table 3, where the first line is the basic case (No. 1). For each case, 20 experiments were performed, with the best result selected in each instance.

Variation in the inventory cost $c_{i}^{+}$ . As shown in Table 3, the effect of inventory cost is remarkable on the hedging points and production costs. With increasing $c_{i}^{+}$ , the hedging points of the buffers vary and that of buffer $i$ decrease in order to avoid further inventory costs. In addition, the more the inventory cost increases, the more the average production cost $J^{z}$ rises as well. An opposite effect is observed with decreasing $c_{i}^{+}$ .

Variation in the backlog cost $c^{-}$ . Table 3 reveals that when the backlog cost $c^{-}$ rises, the hedging thresholds vary and that of the last buffer increase in order to avoid further backlog cost. In addition, the average production cost $J^{z}$ increases accordingly. Decreasing $c^{-}$ yields the opposite result.

Table 3.

Sensitivity analysis.

No.	$c_{1}^{+}$	$c_{2}^{+}$	$c_{3}^{+}$	$c_{4}^{+}$	$c_{5}^{+}$	$c_{6}^{+}$	$c^{-}$	$J^{z}$	z
1	2	8	10	13	4	15	10,000	302.1	24, 6, 8, 5, 15, 8
2	0.5	8	10	13	4	15	10,000	268.1	41, 4, 5, 6, 11, 7
3	5	8	10	13	4	15	10,000	325.5	14, 6, 5, 6, 15, 9
4	2	2	10	13	4	15	10,000	285.3	24, 8, 5, 5, 15, 7
5	2	12	10	13	4	15	10,000	334.2	29, 4, 8, 5, 15, 8
6	2	8	5	13	4	15	10,000	291.8	24, 6, 11, 4, 15, 8
7	2	8	15	13	4	15	10,000	318.4	24, 4, 5, 5, 15, 7
8	2	8	10	5	4	15	10,000	276.4	29, 4, 5, 10, 11, 7
9	2	8	10	20	4	15	10,000	322.2	19, 5, 8, 2, 15, 7
10	2	8	10	13	2	15	10,000	271.1	24, 4, 5, 4, 23, 7
11	2	8	10	13	8	15	10,000	335.0	24, 5, 5, 5, 11, 7
12	2	8	10	13	4	5	10,000	294.8	24, 7, 5, 14, 15, 9
13	2	8	10	13	4	20	10,000	314.4	29, 4, 5, 4, 15, 6
14	2	8	10	13	4	15	5000	232.4	29, 4, 5, 4, 7, 6
15	2	8	10	13	4	15	15,000	367.2	26, 5, 8, 5, 19, 9

Bold values indicate modification for better visualization.

Conclusion

HPP is widely used in various production problems, by which we study the production control problem of a multi-machine, tree-structured failure-prone manufacturing system. A simulation method was adopted to calculate the system cost. An ITA was developed based on the characteristics of the manufacturing system. We compared our algorithm with GA and PSO, and the experimental data show that the proposed algorithm can obtain a lower production cost in a shorter time.

For further study, it is advisable to verify the optimality of the HPP or to find a better policy for the system in point. What is more, taking the real-life production into account, a more complex manufacturing system with defective products and general distributed repair/failure time (e.g. gamma, lognormal and Weibull) should be studied.

Footnotes

Acknowledgements

We thank the Associate Editors Katrina Newitt and Mr. Martin McDonald and reviewers as well as Professor Li Lu for their valuable comments and suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported, in part, by a key program of the National Natural Science Foundation of China under Grant 60934008, the Fundamental Research Funds for the Central Universities of China under Grant 2242014K10031 and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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Hedging-point control policy for a failure-prone manufacturing system

Abstract

Keywords

Introduction

Problem description

Definition 1.

Definition 2

Control strategy and objective

Optimization of hedging points

Simulation method

ITA

Definition 3

Theorem 1

Proof

Theorem 2

Proof

Proposition 1

Proof

Proposition 2

Proof

Proposition 3

Proof

Inference 1

Proof

Example and analysis

Performance comparison

Sensitivity analysis

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

References