C O U R S E I N S T R U C T O R : D R . M O H A M M A D A L D U R G A M

S T U D E N T: M O H A M M A D A L M A R H O U N

“Optimal tool replacement with product quality deterioration and random tool failure”

Weigang Xua & Le Caoaa State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, ChinaPublished online: 10 Sep 2014.

OUTLINE

INTRODUCTION

QUALITY LOSS ESTIMATION

TOOL REPLACEMENT MODEL

DECISION MAKING MODEL

THANKS

INTRODUCTION

Production system: one of the processes have a machine with deteriorated cutting tool.

Tool wears as the cutting process proceeds

Maintenance actions, such as tool replacements, are taken to avoid operating the machine in an undesirable state.

Excessive tool replacements may increase tool replacement cost and production capacity loss due to increased production interruptions.

What’ve been done before?

Tool replacement policy has been studied extensively in previous studies:

Iakovou, Ip, and Koulamas (1996) developed an analytical model for simultaneously determining the optimal cutting speed and tool replacement time with stochastic tool life by minimising the machining cost per part.

Vagnorius, Rausand, and Sorby (2010) proposed a method to determine the optimal replacement time by balancing tool replacement cost and penalty cost for tool failure.

Recently,Cao et al. (2014) indicated that the uncertainty of time duration

for performing tool replacement has a significant impact on the efficiency of production line. They introduced a tool replacement method for machine tool to decrease the efficiency losses of production line.

In the previous papers, tool replacement decision with stochastic tool life is investigated, but the product quality loss is not well considered. Drezner and Wesolowsky (1989) proposed a tool replacement

policy so as to minimize the expected cost per part. Jeang and Yang (1992) presented a tool replacement model

to minimize the average cost of quality loss and tool replacement in the long run.

Sheikh et al. (1999) developed a model for tool replacement and resetting through minimising the cost of producing parts out of specifications

Pearn and Hsu (2007) investigated the relationship between tool replacement and product quality, and presented a policy that the tool replacement should be performed once the process capability drops blow a critical point.

Moreover, there are several studies which determine the optimal tool replacement time based on balancing quality loss, tool replacement cost as well as penalty cost for tool failure, such as Makis (1995), Jeang (1998) and Hui, Leung, and Linn (2001). Zhang, He, and Li (2009) and Jeang (1999) further extended the costs by adding adjustment cost of the tool.

In these papers, most of them use Taguchi loss function to quantify the product quality loss.

Taguchi loss function cannot: Reflect the inherent deterioration characteristics of

product quality during cutting process. Be used to predict product quality deterioration in a

given environment.

introduction

•Problem of deteroated cutting machine lead to poor product quality and a cause of machine failure.

•When ?

•Maintenance Action taken is: Replacement.•This paper presents an approach for determining the optimal tool replacement time for cutting process.

•A tool replacement model is proposed based on balancing the product quality loss, penalty cost for possible tool failure, production capacity loss and tool replacement cost.

QUALITY LOSS ESTIMATION

PRODUCT QUALITY DETORIATION MODEL

Colledani and Tolio (2006, 2011) stated that tool wear could cause a transition of a machine tool from the in-control state to the out-of-control state.

AlDurgam and Duffuaa (2013) indicated that the percentage of products not meeting specifications increases as the manufacturing system deteriorates. Quality Failure Rate p(t) with respect to t

QUALITY LOSS ESTIMATION

INTRODUCTION OF LOGISTIC FUNCTION

In logistic function, the term rN is defined as an exponential growth rate, where r is a rate constant and N is a population increasing in size or

numbers.

In the beginning, Verhulst (1838) defines rN as the rate of change of N such that dN/dt = rN.

Since unlimited growth is almost impossible, Verhulst adds a mortality term, m, such that dN/dt = rN – mN^2, and the growth ceases when rN is equal to mN^2. Then setting K = r/m, the basic differential equation for bounded population growth is as follows:

r is considered as a factor that fundamentally controls the growth. | K is the maximal production of the population limited by some necessary conditions.

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QUALITY LOSS ESTIMATION

MODELING OF QUALITY FAILURE RATE

Let λ0 denote the quality failure rate of the first stage. Thus, we have p(t) = λ0 when t ≤ t0, for the assumption that p(t) is constant in the first stage. From time point t0, the quality failure rate p(t) begins to increase because of the impact of severe tool wear, and it will approach to one as time goes to infinity if tool failure does not happen.

QUALITY LOSS ESTIMATION

Separating the variables and integrating both sides we get:

Integrating both sides and using the condition that p(t0) = λ0, we have:

Then the quality failure rate p(t) can be formulated as

QUALITY LOSS ESTIMATION

ESTIMATING THE QUALITY LOSS

Let μ denote the production rate, define quality loss as the cost related to product quality failure. Let Cf denote the average loss for a quality failure. Let L be the time period starting from the last tool replacement, and let D(L) be the accumulated quality loss for the time period L. Then D(L) is given by

Combining Equations we get:

TOOL REPLACEMENT MODEL

TOOL LIFE MODEL

Weibull distribution is well known and widely used in many situations due to its high flexibility. It can be used to model different shapes of failure rate behaviours: decreasing, constant and increasing (Rausand and Høyland 2004; Jiang and Murthy 2008).

The probability density function F(t) is represented as:

The reliability function of cutting tools is:

where λ is the scale parameter with λ > 0, α is the shape parameter with α > 0.

TOOL REPLACEMENT MODEL

According to Jiang and Murthy (2008), the failure rate function is given by:

And the shape of r(t) is increasing if α > 1, constant if α = 1, and decreasing if 0 < α < 1. In our case, the cutting tool has an increasing failure rate, i.e. α > 1, since the tool fails more easily as it wears.

TOOL REPLACEMENT MODELDECISION MAKING MODEL

Let Y denote the tool replacement time, i.e. the interval time between successive tool replacements, and let C(Y) denote the total cost during a tool replacement cycle. According to Jiang and Murthy (2008), the expectation of C(Y) is given by:

The expected cost for two replacement situations are given by:

The cost for each situation involves a product quality loss, the production Cp =capacity loss for a tool replacement

Cr = cost of replacing a used tool. And for the first situation there is an additional penalty cost for a tool failure,

Cc=which includes the cost of rework, scrapping or other extra charges.

TOOL REPLACEMENT MODEL

According to Jeang (1998), the expectation of C(Y) can be expressed as:

Let U(Y) denote the cutting time of the tool during a replacement cycle. According to Jiang and Murthy (2008), the expectation of U(Y) is given by:

According to He (2008) and Jiang and Murthy (2008), the tool replacement model can be treated as a reward renewal process and the average cost per time slot in the long run is given by

THANKS