REVIEW OF LITERATURE

Transcription

1 CHAPTER II REVIEW OF LITERATURE In any researcher study the work done in the past is of great importance since it forms the very basis of the work to be carried out. It is rather a continuous process. The different types of problems taken for investigation, the various methods of approach to solve the problem are all quite important. Hence a brief idea of the work done in the past, the continuity etc should all be clearly stated and furnished. It forms the very basis of review of literature. In this chapter a brief review of literature especially the development of latest techniques in inventory control theory is given. Since the development of inventory control has been over the past six decades, it would be almost impossible to cover all the relevant research papers. Hence some selected research papers which are of greater importance with possibility of practical applications are taken up for review. Inventory control is a major discipline of Operations Research. The concept of optimization forms the very basis for inventory control

2 24 theory. Hence in most of the problems which are taken up for study the very aim is to determine the optimal value such as optimal reorder quantity, optimal lot size etc. It is interesting to note that the very famous EOQ formula due to Willson and Harris (1915) has given a basic model for the determination of the optimal reorder quantity. This formula provides the basis for many of the inventory models which have been developed subsequently. Many of the inventory models of classical nature have been under the assumption that the demand, lead time etc., are of deterministic nature. Considerable changes in the models have been introduced only by assuming that the variables like demand, lead time etc., are probabilistic in nature. Such models are known as the probabilistic models. In this context it is interesting to note that the news boy problem, the base stock system inventory models have been developed. An excellent discussion of the usefulness of inventory control methods is found in Barber (1925). The author has discussed the need for inventory management and various models of inventory control. It is interesting to note that the concept of inventory management is not confined to the determination of the optimal reorder size which is

3 25 otherwise known as the economic order quantity and similarly the quantity to be kept in the stock pile. There are many other areas of applications of inventory policies and this includes the so called warehouse problems in which the size warehouse is determined and the capacity of the warehouse creates effective constraints on purchasing and selling decisions. Bellman(1956, 1956) has applied the theory of dynamic programming problem to warehousing. Optimal reorder size is an important parameter of interest and when the demand is price dependent the determination of optimal reorder quantity has been discussed by Hanssmann (1962). Tadikamalla (1978) has derived an EOQ inventory model under the assumption that the rate of deterioration follows Gamma distribution. It is assumed that the demand is constant K. The cost of product in question, inventory holding cost per unit time the reordering cost, rate of deterioration are all defined and the expression for h(t) namely the hazard rate or the rate of deterioration has been obtained. The differential equation as a function of T, which is the cycle time has been obtained and the optimal value of T has been derived by solving the differential equation. By using computer packages the numerical solution has been

4 26 worked out. In addition to this, a comparison is made that the optimal EOQ values by assuming the rate of deterioration follows Weibull and Gamma variates. The series of stations is an interesting area of inventory models. In this case the system comprises of stations in series and the output of the first station is the input for the second and so on. It is likely that if there is a failure of any one station the input to the next station is stopped and therefore the system suffers. In this aspect, reserve inventory of semi finished products between successive stations is necessary. The optimal size of the reserve inventory of semi finished products is to be determined taking into consideration the cost of averages and shortages. An interesting model with two stations (machines) in series has been considered by Hanssmann(1962) and the optimal solution is derived. Ramachandran and Sathiyamoorthi (1981) have extended this model to the case where the consumption of semi finished products by the second machine is a random variable. The optimal solution taking different demand distributions has been attempted. Ramanarayanan (1981) has discussed an inventory model based on the Markov processes. The essential difference between this model and

5 27 the conventional model is that it uses the phases type (PH) distribution for the representation of the lead time distributions. The concept of phase type distribution and its applications have been studied by Neuts (1975). In this model, it is assumed that the demands occur according to a Poisson process with parameter and rate of demand is one unit at a time. The inventory capacity is denoted as S and the reorder level is s. The explicit steady state solution has been derived and it gives the reordering rule at different points of the demand epochs. An inventory model which is very interesting under the assumption that ordering levels has been discussed by Thangaraj and Ramanarayanan (1983). In this work, they have considered the conventional (s, S) inventory system, where s is the reordering level, and S is the reordering quantity. It is assumed that the reordering quantity can be either small or large. The smaller reorder level is denoted as a and the longer reorder level is denoted as b. So the lead time for a- type and b- type reorders also taken to be random variables. The interarrival times between successive demands are taken to be i.i.d random variables. The steady state probabilities P n (t) which denotes the probability that the stock level is n at the time point t have been computed. Assuming appropriate costs the expression for the cumulative profit has also been obtained.

6 28 Another interesting contribution on the base stock system for patient customers is studied by Ramanarayanan, Ramachandrean and Sathiyamoorthi (1988). In this model it is assumed that the lead time demand is a random variable. There are N demand epochs during the lead time and the demands during such epochs are random variables denoted as X 1, X 2,, X N. The interarrival times between successive demand epochs are taken to be random variables which are identically distributed but not independent. They are constantly correlated random variables. Under these assumptions the optimal base stock levels have been derived by using the distribution of sums of correlated random variables which has been discussed by Gurland (1955). Rogers and Tsubakitani(1991) have discussed a Multiechelon (Multilevel) inventory model and using the results of the news boy model for obtaining the optimal solution. It is very common that the inventory may be at different levels of a production system and the centralized decisions for the location and control of inventories is an important aspect. The inventories that are to be maintained at different levels of a production oriented system is very important. The determination of the optimal inventory arises at different locations and at each level the demand may be different. The demand function for the common

7 29 component as following normal distribution N( u, 2 ) has been taken. The optimal values of the decision variables have been obtained by taking a Hessian matrix (H) and using the Lagrangian Multiplier technique. Ramasesh et al (1993) have considered an inventory problem in which the replenishment of inventory takes place from two sources. In this model, the concepts of micro studies and macro studies have been discussed. In macro studies the examination of merits of the procurement policies with cost benefit analysis of dual source competition is considered. In micro studies the modeling and optimization of total cost arising due to reordering inventory holdings and shortages is considered. The solution for sole and duel sourcing models have been developed. The lead times are considered as stochastic and it follows uniform distribution. The expression for expected total cost per unit of time has been obtained. The case of single vendor and also a two vendor problem have been considered. The concept of order statistics is used to represent the inter supply times. John S. Rose (1992) has discussed an entirely different inventory model using the conventional News Boy problem. The news boy problem is one in which the demand is taken to be random variable and

8 30 the optimal size of supply is determined assuming the salvage cost as well as shortage cost. Here in this model a completely reverse situation is considered, where the demand is taken to be known and deterministic. The optimal quantity of supply which is the quantity of replenishment is called the fill. Assuming the material cost (c), shortage cost (p) and inventory holding cost (h) and also is the average of the replenishment quantity; the expected cost is denoted as E cμ p A D F A x F dx h x A F dx F B A Where (-D F, B F ) = Support of F, D F, B F A = the quantity of demand X = the quantity of replenishment. The optimal value of the replenishment has been derived and in doing so the cost function is taken to be normally distributed with mean and standard deviation. This approach of determining the

9 31 replenishment quantity is sometimes new and the practical application of this model is also discussed. Goh (1994) has discussed the concept of EOQ models with general demand and holding cost function. In deriving EOQ formula the conventional model, the assumption is that the holding cost is fixed. In the present model, the holding cost is assumed to be a variable. It is also assumed that the demand rate depends upon the inventory level. Hence the demand rate is taken to be deterministic and is a known function of the level of inventory. It is strictly differentiable with a concave polynomial function of the form β Dq, 0 β 1 R q where D = Constant annual demand rate constant = The inventory level elastic and is a shape parameter. q = On hand inventory level. Under these assumptions the optimal reorder quantity has been obtained for the following two cases.

10 32 i. Instantaneous replenishment with non linear time dependent holding cost. ii. Instantaneous replenishment with non linear stock dependent carrying cost. Another EOQ formula in stochastic inventory control has been developed by Sven Axsater (1996), after considering a single item continuous review inventory model with a stationary stochastic model. The inventory is controlled by (Q, r) policy, where Q stands for the reorder size and r stands for the number of orders of size Q. The inventory position is the inventory on hand + inventory on outstanding back orders. Assuming the ordering cost, holding cost and back order, the optimal values of Q and r have been determined. It may be noted that the expression for the worst case performance ratio has also been obtained and the worst case performance ratio denoted as R. R = C(Q, r(q))/c(q *,r(q * )) where Q * : Optimal order size. The method of obtaining optimal order size has been derived and numerical example also given.

11 33 Brill Percy. H (1995) has discussed about an EOQ model with random variations in demand and also indicated that the objective of the very basic model is to explore the implications of demand disruptions. Three demand rates have been assumed as K 1, K 2 and 0. I (t) is taken to be the difference between the supply and demand. Where I(t) = M(t) demand and M(t) is called the supply size. It is also assumed that the Markov process {M(t), t 0} is a continuous time Markov Chain. M(t) has sojourn times for the three states 1, 2 and 3 which are exponential with parameters 1, 2 and 3. System point level crossing theory has been used for the formulation of system of equations for the model and using the same, the EOQ has been derived. Kumaran and Achray (1996) have discussed an inventory model based on the generalized News Boy problem and assumed that the lead time is a random variable and the lead time demand is determined on the basis of the Generalized type Demand (GLD). The GLD distribution can be used whenever a complete and precise knowledge of the distribution of random variable is not available. The P th quantile denoted as R(p) has been considered and it is based on 1 and 2 called the location and scale parameters. It is assumed the setup cost, purchase cost, salvage cost and penalty cost; the selling price is also taken into account.

12 34 The expressions for the expected profit and loss have been constituted by using the same the objective function. The optimal size to be produced has been determined. Harkan Poletogin and Izzet Sahin (2000) have discussed the method of obtaining the optimal procurement policies under price dependent demand. Here the price of the product is also taken to be a decision variable. In many organizations the management is concerned with simultaneous prizing and procurement quantity discount. They both put together jointly maximize the present value of expected profit over a planning Horzon. The initial work in this direction has been done by Whitin (1955). In the present model, a periodic review inventory system with N periods has been considered. The decision variables are q n, which is the reorder quantity and p n the prize level for the n th period. The expression for n period optimal expected profit, the optimal procurement policy and the optimal prize level have been derived. Rajagopal and Sathiyamoorthi (2003) have considered an extension of the optimal reserve inventory between two machines model and considered the case of three machines M 1, M 2 and M 3 in series. There is a possibility that the machine M 1 may fail so that the input for M 2 is

13 35 stopped. Another possibility is that the both M 1 and M 2 may fail with result that the machine M 3 goes to the idle state. The breakdown durations of M 1 and M 2 are taken to be random variables. The consumption rate of M 2 and M 3 are given by r 1 and r 2 responsively. The inter arrival times between successive breakdowns of M 1 and M 2 are random variables with means 1 and 2 respectively and discussed two different models. In the first model M 1 is in the down state, M 2 and M 3 are in the up state. In the second model both M 1 and M 2 are in down state and M 3 is in the upstate. For both the models, the optimal reserve inventory between M 1 and M 2 and that between M 2 and M 3 have been analyzed.the optimal values are represented as S 1 and S 2. The numerical illustrations are also given taking specified distribution of the random variables involved. Sandeep Jain and Srinivasa Raghavan (2005) have discussed the analysis of the base stock control production inventory system using queueing theory. They have considered a production inventory system which consists of a manufacturing plant and a warehouse. The demands from the customers are supplied from the inventory in the ware house. It is assumed that the demand orders from the customers arrive according to a Poisson process. The finished goods inventory is the base stock and its

14 36 level fixed at K. The finished goods inventory is well defined and each finished goods inventory is attached with production authorization card. The expression for total cost K which is the base stock level at the warehouse has been obtained. Assuming the arrival process to be Poisson the optimal value of K has been determined. Srinivasan, Sulaiman and Sathiyamoorthi (2007) have discussed an inventory model in which the demand over the time interval (0, t) is taken to be a random variable. There is a one time supply denoted as S. The demands at N random epochs in (0, t) are denoted as random variables X 1, X 2 X N. It assumed that the random sample of N observations on demand are taken and arranged in increasing order of magnitude. It is noted that X (1) is the first order statistic and X (n) is the n th order statistic. Using the distribution of X (1) and that of X (n) the expression for the optimal supply size has been obtained Sehik Udhuman, Sulaiman and Sathiyamoorthi (2007, 2008) have discussed the problem of optimal reserve inventory between two machines in series under different assumptions. In this model it has been assumed that the repair time of machine M 1 would be increasing and the inter arrival times between successive break downs of M 1 may become

15 37 shorter. This is due to the fact that ageing of the machine has an impact upon both the repair time and inter arrival times between successive break downs. Hence, it is assumed that the random variables denoting the inter arrival times as well as random variable denoting the repair time of M 1 both undergo a parametric change and satisfy the so called SCBZ property. The optimal size of the reserve inventory has been derived and numerical illustrations are given. The concept of uncertainty was first introduced in EOQ model by Arrow Harris and Marschak model (1951). It is a highly generalized model and many other inventory models can be proved as a special case of this model. In this model, n-periods are taken and the reordering decisions are considered at n-different points called the checking points. The demand is assumed to be a random variable ' r ' and these random variables for the n-periods are taken to be i.i.d random variables. The lead-time is taken to be zero and shortage in one period if it exists is taken to the next period. Under these assumptions the expressions for the expected cost for any period is given in the form of L( x, y) E C( x, y, r) where x =stock on hand before ordering, y = the quantity to be ordered and r = the demand.

16 38 The optimal policy is given as f x x min L x, y y x This model has proved to be very useful and uses the dynamic programming principles for obtaining the optimal solution. In single station models, the concept of dynamic models is very interesting. Several authors have formulated some ordering rules of very general nature. It is interesting to note that a very general model to form the ordering rule has been formulated by Dvoretsky, Kiefer and Wolfowitz (1952). In this paper, the authors have considered dynamic single station model with finite number of decision intervals. The demand in any period '' i is given by a conditional probability distribution f r / B where the vector Bi ( x1, x2,... xi, y1, y2,... yi, r1, r2,... ri 1) is a summary description of the history of demands () r and the stock level before' n ' and after ordering( y ), including the stock levels x i and yi of the present period '' i. Consequently, the expected cost in period '' i is a function of the form i i C ( x, y / B ), where " i i i i 1 B ( x, x,... x, y, y,... y, r, r,... r ) " i i i i 1 An ordering policy is defined by a set function

17 39 y g x B " i i( i / i ), i = 1, 2, 3, n which obey the restrictions i i y x, i = 1, 2, 3, n Another interesting paper developed by Gaver (1959) is on the socalled base stock level inventory. In this model, a given period (0, t ) is taken and subdivided into smaller intervals of equal length like a week etc. The demand during each time sub-interval is taken to be a random variable. As and when a demand occurs, it is satisfied or supply is made from the available stock on hand. The order for supply or the reordering is done at each demand epoch and supply will be received at the next period. If the stock in hand is exhausted, the customers are patient in the sense that they will not cancel the order but rather take delivery of the goods when the supply is received in the next period. Under these conditions, the inventory can be called as base stock, which means the inventory on hand, and inventory on orders. The optimal value of the base stock is derived and in doing so, the author has considered the stationary distribution of available inventory when the customers wait, in other words when the customers are patient. It is also interesting to note that the author has considered the stationary distribution function of available inventory when the customers are impatient.

18 40 It is very interesting to note that the standard EOQ models have been extended to the perishable products by Ghare and Schrader (1963). In this model, the authors have assumed that the decay of the product is according to the exponential distribution. Taking in to consideration the demand to be a function of the time interval and the decay as exponential, the expression for the Economic Order Quantity has been derived. Another interesting inventory model is the so-called newsboy problem in which a somewhat different concept is introduced. In most of the models, the inventory on hand is such that it can be kept as a stock for any period of time. It is called an infinite process. However, a somewhat different type of problem that arises is that the inventory processes are terminated after a finite period and several models have been developed using this concept. A typical example of this kind of a model is the socalled newsboy problem in which the newspaper supplied at the beginning of the day is sold and the demand for the same is a random variable. The unsold papers will be called the wastage and there is an associated cost for the same. If the supply is less than the demand, again there is shortage cost. The determination of the optimal one time supply is to be determined. Based on this concept several authors have discussed inventory model of similar type.

19 41 Another interesting paper is by Covert and Philip (1973). In this paper, the authors have assumed a variable rate of deterioration of the items. A two parameter Weibull distribution has been used to represent the distribution of time to deterioration. Using this model, they have derived the optimum cycle time for reordering with the assumption of associated costs. A generalization of this model has been attempted by Philip (1974). This model is a generalized version of the model by Covert and Philip (1973) and a three parameter Weibull distribution is used to represent the distribution of time to deterioration. Here three different parameters namely = scale parameter, = shape parameter and = location parameter have been included. The demand is taken to be deterministic and three different costs namely (i) cost of the unit (ii) cost of holding of unit per unit time and (iii) reordering cost have been incorporated into the model. The authors have obtained the optimal values of the optimum cycle time, the Economic Order Quantity and also the total deterioration during the cycle time. Chenniappan and Ramanarayanan (1996) have considered a new type of an inventory situation. In this model the inventories are kept as two different stocks; when a demand occurs, one unit from each of the two inventories is sold. The model is such that the order for the first

20 42 product is supplied along with the second product. Sometimes the first product alone is supplied without the second product. For example, computers are sold with or without a printer. Considering the distributions of the inter-arrival times between demands as following (i) exponential distribution and (ii) general distributions, the authors have derived the steady state probabilities for the inventory levels. They have used the matrix geometric method as proposed by Neuts (1975). It may be noted that the scope of this problem here is only to find out probability of different inventory levels; optimal solution has not been attempted. Production scheduling algorithm form a part of inventory control. Warburton (2004) has developed a model in which the use of continuous differential equations to describe a typical ordering policy and inventory management. In production scheduling algorithms, the use of Laplace Transform technique has been very much in practice. In this paper, the di authors have taken an inventory balance equation as R( t) D( t) dt where It () is the inventory at a timet, Dt () is demand and Rt () is supply received. The use of step functions for demand can improve the accuracy of the model. The authors have taken the demand distribution and its contribution to the inventory level at any time '' t. In addition to the

21 43 demand, the nature of inventory replenishment also contributes to the inventory level. The authors have defined WIP, which refers to the inventory position, which is the total of net stock, inventory on order and items in manufacturing line. The WIP has its own contribution to the ordering policy. Combining all these, the authors have derived the solution for It () and Laplace Transform technique has been used in such a derivation. Sachithanantham, Ganesan and Sathiyamoorthi (2006) have considered an inventory model in which the optimal reserve between two machines in series is derived. Since the output of machine M 1 happens to be the input for the machine M 2, the breakdown of M 1 results in the idle time of M 2 which is very costly. So an inventory between M 1 and M 2 is suggested. Assuming the inventory holding cost and also the shortage cost due to the idle time of M 2 the size of the optimal reserve inventory has been obtained. In doing so the authors have assumed that the repair time or breakdown time of the machine M 1 is a random variable with pdf gt (, ). The repair time is assumed to satisfy the so called the Setting Clock Back to Zero (SCBZ) property as discussed by Raja Rao and Talwalkar (1990). According to this property the pdf of the random

22 44 variable denoting the repair time of machine M 1 undergoes a parametric change at a truncation point denoted as ' x 0 ' which is a constant; and so the pdf is gt (, ) if t x0 and it is * gt (, ) if t x0. The optimal size of the reserve inventory under these assumptions has been derived. The authors have derived the expression for optimal reserve inventory under the assumption that truncation point ' x 0 ' is a random variable, which follows an exponential distribution. This result is discussed by Sachithanantham, Ganesan and Sathiyamoorthi (2007). Liang-Yuh Ouyang (2007) have discussed an integrated Vendor Buyer inventory model with quality improvement and lead-time reduction. The authors have discussed the advantages of the just in time (JIT) production. The concept of JIT is directed towards the shortening of the lead-time and improving the quality of the product. In the previous model of classical inventory theory, it has been implicitly assumed that the quality level of the product is fixed at an optimal level and all the items are assumed to have perfect quality. But in real production environment it can be observed that there may be defective items and these items are rejected, repaired and reworked or refunded to the customers. In all such cases substantial costs are incurred. Therefore investing capital on quality improvement will reduce this kind of cost.

23 45 Hence, the authors have formulated a single vendor-single buyer inventory model with quality issue and lead-time reduction. The total expected cost is given as the total of the following elements. Ordering cost per unit time. Lead time crashing cost per unit time. Vendor holding cost per unit time. Buyers holding cost per unit time. Defective item rework per unit time. Opportunity cost of quality improvement investment. It is a non-linear programming model in which minimizing the total cost is attempted. The authors have suggested an algorithm. Numerical example by assuming specific values of the cost component is also provided by the authors. Ozalp Ozer, Onwr Uncu and Wei Wei (2007) have discussed the problem of dual purchase contract systems in which a new contract form with the manufacturer can

24 46 Push inventory to the retailer, known also as channel stuffing. Create a strict Pareto improvement over the whole sale price contract while inheriting the whole sale price contract s simplicity, and Reduce the manufacturer s profit variability. To do so, the authors have proposed a dual purchase contract that induces a retailer to place two consecutive orders, before and after obtaining the final forecast update. This essentially is a problem in supply chain in which the manufacturer and retailer are in series. The authors have formulated a demand model D X where X is the forecast of demand after a market research, is a random error. Assuming that the random variable has a pdf g (.) and cdf G (.) such that it has an Increasing Failure Rate (IFR), the authors have discussed the maximization of profit of the retailer and determined the optimal order quantity. Several variations of this problem have been taken up and theorems have been established. Sehik Uduman, Sulaiman and Sathiyamoorthi (2007) have considered the problem of the determination of the optimal reserve

25 47 inventory between two machines in series using the concept of order statistics. In this model, they have taken up the problem in which two machines are in series so that the output of machine M 1 happens to be the input for machine M 2. Whenever the break down of M 1 occurs the supply of raw material to M 2 is stopped. Then M 2 becomes idle which proves costly. The expression for optimal reserve inventory between the two machines has been derived, taking into consideration the inventory holding cost and also the idle time cost of M 2. In doing so the authors have introduced the distributions of the order statistics to find the mean interarrival time between successive breakdowns of machine M 1. The expression for optimal inventory has been obtained both in the case of the first order statistic as well as the th n order statistic which represent the interarrival times between the breakdowns of machine M 1. Sehik Uduman, Sulaiman and Sathiyamoorthi (2007) have discussed the newsboy inventory model with demand satisfying the socalled SCBZ property. The newsboy problem is an inventory model with a finite process. This implies that the product in question can be sold only for a finite duration, after which the product cannot be used and so it has only a salvage cost. The similar situation exists in the case of

26 48 newspapers. The newspaper of the day should be sold within the same day itself. It cannot be sold the next day as it has only the value of a waste paper. If the supply is inadequate, the shortage cost arises. So the determination of the optimal supply size is important. The authors have taken up this problem under the assumption that the random variable denoting the demand for newspaper is such that it satisfies the SCBZ property. Under these assumptions, the optimal supply size has been determined. Wen-Chuan Lee, Jang Wuu Wn and Chia Ling Lei (2007) have discussed an optimal inventory policy involving back order discount. In this paper, the lead-time is taken to be a stochastic variable and hence it is not under control. But the authors suggest that the lead-time can be reduced and made a control variable. This is done by using the partition of lead-time as the following components. Order preparation, order transit, supplier lead-time, delivery time and set-up time. So at an added cost these components can be crashed or made smaller with result that the lead-time can be made shorter which results in lower safety stock; also the shortage cost will become lesser. The authors have extended the result of the inventory model, by Liao and Shyu (1991). In this model they have taken the lead-time as consisting of m-components and each

27 49 component has a different crashing cost for reducing lead-time. So the crashing cost function is a piece-wise linear function. Using this result, the authors have attempted the reduction of lead-time and so the back order rate is taken as a control variable. Using the lead-time demand as a mixture of normal distributions, the authors have derived the optimal crash value of the lead-time.