A Soft Computing based Inventory Model with ...

International Journal of Computer Applications (0975 – 8887) Volume 36– No.4, December 2011

A Soft Computing based Inventory Model with Deterioration and Price Dependent Demand S.R. Singh Department of Mathematics D.N. College, Meerut (U.P.) India E-mail: [email protected]

Tarun Kumar Department of Computer Science Banasthali University, Banasthali (Rajasthan), India

ABSTRACT In most of the classical economic production/order quantity models, the items produced/ received are implicitly assumed to have perfect quality. In this paper we developed an Economic Manufacturing Quantity inventory model for time dependent decaying items and selling price demand using soft computing approach.. Our objective in this paper we are using the application of soft computing techniques like fuzzy logic and genetic algorithms for improve the effectiveness and efficient optimal inventory models with selling price dependent demand. To solve the inventory model we propose to use a fuzzy decision embedded genetic algorithm.

Keywords:

EMQ, Soft Computing, Deterioration, Inventory model, genetic algorithm, fuzzy logic

1. INTRODUCTION Inventory is very important since it helps the company respond quickly to customer demand, which is also an important element of competitive strategy. In real life, the demand rates of a particular type of commodity depend on many factors, such as time, stock level at the showroom, quality of the item and so on. It is a common practice that the higher selling price of an item negates the demand rates of the item whereas less price has the reverse effect.Three important factors of the Economic Manufacturing Quantity (EMQ) model have been dealt with prior significance. First it is assumed that the production facility is not perfect reliable. Second, the production rate (greater than the demand rate) is not predetermined and fixed in advance. Third, the modern facilities are not free from deterioration due to epoch. As a result, random machine shifts from „in-control‟ state to „outof-control‟ state frequently occur during production runs and some percentage of non-conforming items is produced. Further, the process deterioration after a machine shift may result in a machine breakdown in which case the interrupted lot is usually aborted and then the basic EMQ model loses its usefulness. So, from theoretical as well as practical view points, the study of EMQ problem for unreliable manufacturing systems is quite significant and meaningful. There a major vacant space in the area of inventory modeling with machine breakdown. So, we have also taken a stride at the forefront to solve out the machine breakdown problem with flexible manufacturing system. In deteriorating inventory systems, Raafat, (1991) has presented survey on literature on continuously deteriorating inventory models. Goyal and Giri, (2001) have surveyed trends in modeling of deteriorating inventory. Bensoussan et al. (1975) have applied optimal control theory to obtain optimum replenishment policy under the assumption of quadratic cost functions for inventory models with constant and variable rate of deterioration. Choi and Hwang, (1986)

C. B. Gupta Department of Mathematics Birla Institute of Technology and Science, Pilani (Rajasthan), India

used the same approach developed a production inventory model with quadratic production and inventory costs and constant decay rate. Giri, and Dohi, (2005) presented an exact formulation of EMQ model under a general framework in which the time to machine failure, corrective and preventive repair times all are assumed to follow arbitrary probability distributions. But they ignored the study of fuzzy. Chakraborty et al. (2008) presents a generalized economic manufacturing quantity model for an unreliable production system in which the production facility may shift from an „incontrol‟ state to an „out-of-control‟ state at any random time and may ultimately break down afterwards. Widyadana and Wee (2010) develops deteriorating items production inventory models with random machine breakdown and stochastic repair time. The model assumes the machine repair time is independent of the machine breakdown rate. Certainty eventually indicates that we assume the structures and parameters of the model to be definitely known, and that there are no doubts about their values or occurrence. If the model under consideration is a formal model, that is, if it does not pretend to model reality adequately, then the model assumptions are in a sense arbitrary. This means that the model builder can freely decide which model characteristics he chooses. If, however, the model or theory asserts factuality, that is, if the conclusions drawn from these models have a bearing on reality and are supposed to model reality adequately, then the modeling language has to be suited to model the characteristics of the situation under study appropriately. Vojosevic et al. (1996) fuzzified the ordering cost into trapezoidal fuzzy number in the total cost of an inventory without backorder model and obtained the fuzzy total cost. They later used the centroid method and gained the total cost in the fuzzy sense. Chen and Wang (1996) fuzzified the ordering cost, inventory cost and backorder cost into trapezoidal fuzzy numbers and used the functional principle to obtain the estimate of the total cost in the fuzzy sense. Roy and Maiti (1997) rewrote the problem of classical economic order quantity by introducing fuzziness in both the objective function as well as the storage area. Lee and Yao (1999) discussed an inventory model without shortages by fuzzifying the order quantity into a triangular fuzzy number. Yao et al. (2000) explored an inventory model without any backlogging for fuzzy order quantity and fuzzy total demand quantity. Chang et al (2004) considered the fuzzy problems for the mixture inventory model with backorders and lost sales. The total expected annual cost is obtained in the fuzzy sense. Dey et al. (2005) developed a realistic inventory model with imprecise demand and inventory costs and proposed an inventory policy to minimize the cost using man–machine interaction. Chang et al. (2006) considered the mixture inventory model involving a fuzzy random variable and obtained the total cost in the fuzzy sense. Rong et al. (2008) presented an optimization inventory policy for a deteriorating

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International Journal of Computer Applications (0975 – 8887) Volume 36– No.4, December 2011 item with partially/fully backlogged shortages and price dependent demand. Tang et al (2000) presented a multiproduct planning and scheduling using genetic algorithm approach. Y.W. Zhou(2003) proposed a Multi-warehouse Inventory Model for Items with Time-varying Demand and Shortages. Papadrakakis and Lagaros (2003) discussed about soft computing methodologies for structural optimization. Sundarraj and Talluri (2003) developed a multi-period optimization model for the procurement of component-based enterprise information technologies. Yung et. al (2007) discussed on procurement planning of timevariable demand in manufacturing system based on soft computing techniques. Many of the authors have considered single item inventory models with crisp parameter only. In the past, researchers pay less attention towards the coordination of the factor of machine breakdown, volume flexibility and fuzzy environment with multi items which proves a major hindrance to a researcher in this field. It is very much realistic condition for business environment. Produced units deteriorate over time. But most of researchers consider certainty in deterioration. In reality items deteriorate with uncertainty that follows different distributions. Therefore, we have developed this entire concept simultaneously in our model with uniform distribution of deterioration. This paper investigated an Economic Manufacturing Quantity model for time dependent decaying items and selling price demand with volume flexible environment. Controlling market demand through the manipulation of selling price is an important strategy for increasing profit. We assumed that different machines „Ai‟ (1,2….n) are dedicated to the production of different items „i‟ with different production rates „P i‟. The management of production in machine Ai is vested with the management unit „Bi’. It is assumed that a machine may become out of order during its working time. As a result, there is a mean time for every machine between its failure/breakdowns. During a breakdown of a machine, there is demand although there is no production. In such a situation, the demand is met until the inventory level falls below the quantity demanded. When inventory level becomes less than the demand, the concerned management unit Bi is rendered fully idle. This situation occurs when the customer is a wholesaler having the demand of a big lot size and the concerned management unit can‟t meet this demand because the stock size is less than the quantity demanded. Therefore, we considered the idle time of each management unit; this idle time leads to an additional cost for the last man hours. We have considered the capital available for manufacturing the items is limited. In the present study, demand of multi items involved in the study is represented by fuzzy numbers. As a result the total cost function is ultimately obtained as fuzzy. Later on this cost function is defuzzified to obtain a crisp cost function with allowed variations. In this paper we developed an Economic Manufacturing Quantity model with constant deterioration and selling price dependent demand and this is solved by soft computing method.

2. SOFT COMPUTING TECHNIQUES Soft computing, an innovative approach to construct computationally intelligent systems, has recently gained popularity and wide spread use. It is being realized that complex real world problems require intelligent systems that combine knowledge, techniques and methodologies from various sources. These intelligent systems are supposed to

possess human-like expertise within a specific domain and the ability to adapt and learn in changing environments. To achieve this complex goal, a single computing paradigm or solution is not sufficient. Soft computing is a wide ranging term encompassing such varied techniques as fuzzy systems, rough sets, neural networks, genetic algorithms, simulated annealing etc. In this paper we are using only Fuzzy Logic (FL) and Genetic Algorithm (GA) Techniques. FL and GAs have been successfully used for supply chain modelling (2008) and are particularly appropriate for this problem due to their capacity to tackle the inherent vagueness, uncertainty and incompleteness of the data used. A GA (1975) is a heuristics search technique inspired by evolutionary biology. Selection, crossover and mutation are applied to a population of individuals representing solutions in order to find a near optimal solution. FL is based on fuzzy set theory and provides methods for modelling and reasoning under uncertainty, a characteristic present in many problems, which makes FL a valuable approach. It allows data to be represented in intuitive linguistic categories instead of using precise (crisp) numbers which might not be known, necessary or in general may be too restrictive. For example, statements such as „the cost is about n‟, „the speed is high‟ and „the book is very old‟ can be described. These categories are represented by means of a membership function which defines the degree to which a crisp number belongs to the category.

3. ASSUMPTIONS AND NOTATIONS The proposed inventory model is developed under the following assumptions and notations:

3.1 Assumptions: The following assumptions are made for development of mathematical model: 1. Model is developed for multiple items. 2. Demand rate is selling price dependent for each item for first model and time dependent for the second one model. 3. Machine breakdown is considered during the production period. 4. Crisp and fuzzy both the cases are considered. 5. Production rate is considered as a decision variable. 6. Idle time is considered for management of units.

3.2 Notations The following notations are made for development of mathematical model: Qi(t) : On- hand inventory of i-th item at time t Pi : Production rate per unit time for the i-th item (t) : Deterioration rate µi : Mean time between successive breakdown of the machine mi : Mean time of repair of i-th machine i : Mean duration of a breakdown of machine i(ti) : Probability density function of t i i(i) : Probability density function of i Chi : Cost of carrying one unit of i-th item in inventory per unit time Csi : Shortage Cost per unit time of the i-th item Spi : Selling price per unit of i-th item i(Pi): Cost for production of a unit of i-th item Di : Demand rate for the i-th item per unit time Wi : Cost of idle time of management unit Bi CAP : Total capital available for production of all the items

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International Journal of Computer Applications (0975 – 8887) Volume 36– No.4, December 2011

3.3 Proposed inventory model with selling price dependent demand The production cycle begins with zero stock. Production starts at time t=0 and stocks reaches at the highest level Q i(ti) at time ti. After time ti machine becomes out of order, then repairing

Qi ' (t )   Qi (t )  Pi  Di (s) Qi ' (t )   Qi (t )   Di (s)

with Qi (0)=0 with Qi (x)=0

of machines starts and takes time to come back into working state. During the repairing period two cases may arise: one is scenario 1 (a) which is very simple and unrealistic case second is scenario 1 (b) which is very common in manufacturing firms. Hence, our main object is to analyze scenario 1 (b). Differential Equations of the inventory system are

…. (1) ….(2)

Solutions of the above equations are

Qi (t ) 

( Pi  Di ( s))

Qi (t )  



Di ( s)



(1  e t )

(1  e ( x t ) )

…. (.3)

…. (4)

Scenario 1(a):

Figure 1 : Graphical representation of inventory system without shortages

Scenario 1 (b):

Figure 2: Graphical representation of inventory system with shortages One can conclude that the idle times of the management units {Bi, i= 1, 2, 3…..}due to breakdown of the machines {Ai, i=1, 2,3…..} are

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International Journal of Computer Applications (0975 – 8887) Volume 36– No.4, December 2011

3.3.1 Expected cost per breakdown The expected cost per breakdown of the machine {Ai, i= 1, 2, 3……} during idle time, is 



0

Qi ( ti ) Di ( s )

Eic i  Wi  { 3.3.2



( i 

Qi (ti ) )i ( i )d i } i (ti )dti Di ( s)

.... (5)

Expected shortage cost during ideal time

The expected shortage cost for i-th item, during idle item, is 



0

Qi ( ti ) Di ( s )

Esc i  CS i Di ( s)  {



( i 

Qi (ti ) )i ( i )d i } i (ti )dti Di ( s)

.… (6)

Now the total inventory of i-th item is Invi(ti)

= Inventory during[0, ti] + Inventory during [0, x] =

=

( Pi  Di ( s))

ti

 t  (1  e )dt 



( Pi  Di ( s))



Di ( s)



0

[ti 

x

 (1  e

D ( s) e  ] i [ 1







)dt

0

(

e ti

 ( x t )

( Pi  Di ( s )) ti ) Di ( s )





1





( Pi  Di ( s))ti



]

…. (7)

3.3.3 Expected inventory cost for i-th item 

Eic  C i

i h

 Inv (t ) (t )dt i

i

i

i

…. (8)

i

0

3.3.4 Production cost per unit of the item

i ( Pi )  ( Ri 

Gi  ki Pi  )  Pi

…. (9)

Here one can consider the density functions are

 i (ti ) 

1

i

e



ti

i

,

i ( i ) 

1  i mi e mi

3.3.5 Expected total cost Expected total cost breakdown, including the inventory and shortages cost is,

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International Journal of Computer Applications (0975 – 8887) Volume 36– No.4, December 2011 ETC (P1, P2, P3, …..…) = Expected holding cost + Expected cost for idle item + Expected shortage cost P D ( s )  ( iD (is ) ) t i



n

D ( s) e  i Chi  (



i 1



0

n



0

Qi ( ti ) Di ( s )

 Di ( s)Cs  { i 1

1

 )(

 i



i

1



( i 

e



ti

i

n





i 1

0

Qi ( ti ) Di ( s )

)dti  Wi  {



( i 

=

Qi (ti ) )i ( i )d i } i (ti )dti Di ( s)

Qi (ti ) )i ( i )d i } i (ti )dti Di ( s)

….. (10)

ETC (P1, P2, P3……) =  f ( Pi ) Di  g ( Pi )

…. (11)

3.3.6 Expected Production cost of the item …. (12)

Eprc = As the capital for manufacturing the item is limited, the constraints must be satisfied.

3.3.7 Mathematical formulation of the fuzzy model: When the demand rate becomes fuzzy, the objective function can be redefined as ~

ETC (P1, P2, P3……) =

 f ( Pi ) Di (s)  g ( Pi )

Wavy bar denotes the fuzzification of the parameters. We express the fuzzy demand rate (

as the triangular fuzzy number

). Suppose, the membership function of the fuzzy demand rate

 D  Di  1  1   D  2  D  ~ ( Di )   i Di 2  0  

is as follows:

Di  1  D  Di Di  D  Di   2

Here, 0