CHAPTER 1 INTRODUCTION. management. Black [1] described inventory as a buffer between supply and

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1 1 CHAPTER 1 INTRODUCTION 1.1 Inventory Management Various definitions have been given by different authors to inventory management. Black [1] described inventory as a buffer between supply and demand. Jonah et al [2] described it as stock of goods or material awaiting delivery or dispatch. While Monks [3] described it as stock of goods or item held for future use. From the foregoing one can see that good inventory management in a firm would lead to greater profits, minimized losses, greater customer satisfaction, stabilized employment, enhanced product quality and other latent benefits of inventory. Failure to meet demand in any company usually compromises customer satisfaction and attracts high cost that characterizes emergency production. Efficient management of inventory system is therefore very critical in the operations of any firm. Black [1] outlined the basic benefits of inventory management to the customer as off-the-shelf availability of products while to the management as reduced tied-up investment capital on inventory, reduced operating cost and carrying cost associated with warehousing and reduction in the accruing obsolescence of product. From observations it can be said that a lot of failed investments did so as a result of inefficient inventory management.

2 2 In this work our emphasis is on backordering. As Fisher [4] observes, there may be some economic reasons for a company to decide not to satisfy all demand, but rather lose some sales in the interest of the company. We consider a situation where rather than accumulating a lot of buffer stocks and attracting spoilage, some stocks can be back-ordered, some lost and sufficient costumer and company satisfaction achieved. Introduction Partial Back logging On every research on inventory, it is always customary to establish optimal parameters which would ensure effective management of inventory. Two major parameters of interest are optimal order quantity and optimal reorder point. This would ensure a comfortable trade-off between the cost of inventory holding and the cost of shortage. The phenomenon of shortage has been a recurring issue in modern inventory management. Another concept that is akin to it is the concept of yield uncertainty, or yield randomness. Yield uncertainty or yield randomness is simply a situation where the quantity of goods received does not equal the quantity requisitioned, due to factors like defective production, miscounting, breakage and pilferage. Researchers in inventory theory and management have tried, to capture this situation through modeling, so as to enable inventory managers make well informed management decisions. One way this has been done is to consider the

3 3 possibility of keeping all the demands occurring within the time when there are shortages until a new consignment is received to fill the outstanding demands. This approach is known as complete Back-ordering or complete Backlogging. Another way to deal with this problem is to assume that all demands occurring within this period are lost which is known as Total lost sales. However a more dynamic situation is to realize that while some units of the demands occurring within the shortage period can be backordered others are permanently lost. This is an intermediate situation to the two mentioned above and is called Partial Backordering. Partial Backordering posses one difficulty of complicated models which are not easy to handle. A major drawback of this system also is that the reorder point is not systematically determined. That is the reorder point because it is not included in the model is not determined by the conditions within the model. Jonah et al (2) modeled the system using the length of stock-out period and the length of the inventory review period. They also tried to deal with the problem of reorder point by developing a closed form model parameters. In this research a modification of the model by Jonah and Chukwu would be attempted, and an application of same to the Brewing industry would be explored.

4 4 1.3 Classification of Inventories Types of inventories: Basically inventory can be classified into three different forms Raw material inventory: This is the inventory of all the raw materials used in the production process Work in progress inventory: This is the inventory of semicompleted goods (work in progress) calculated at various points in the production process Finished goods inventory: These are the inventory of the products of a firm. They are frequently held throughout a firm s distribution channels even unto retail state. 1.4 Functional Classification of Inventories Apart from classifying inventory according to its form, we can as well classify inventory according to its function, some of which are: Anticipatory Inventory: This inventory is accumulated when a firm produces or purchases more than its immediate requirements in low demand periods to anticipate the needs of high demand periods. By building up its anticipatory inventory the firm smoothes its production requirements. This type of inventory is very helpful when demand is seasonal Cycle Inventory: This is when in order to reduce unit purchase cost (for increased production efficiency) the number of units purchased is greater than the firms immediate needs. It may be more economical for purchasing to order a

5 5 large quantity of units and store some for future use than to make a series of small orders. With some items the firm may be forced to produce a minimum quantity. Production at sizes that exceed immediate requirements are normally chosen to offset the cost of lengthy process set-ups Pipeline Inventory: These are items that have been ordered but not yet received. They are said to be created by materials moving forward through the value chain. It may be inventory moving from supplies to plants and from subcontractors, from one operation to the next within the plant and from the plant to the firm s distribution channel Decoupling Inventory: This inventory enables the synchronization between adjacent processes or operations whose production rates are not synchronized Safety or Buffer Inventory: This is inventory held to offset the risk of unplanned production stoppages or unexpected increases in customer demand. 1.5 Dependent and Independent Demand Inventory: Demand can be said to be dependent when it can be derived from the demand for other items produced by the firm whereas independent demand is when it is unrelated to the demand for other items produced by the firm. 1.6 Classification by Quantities: This brings to bear the idea of ABC classification, where inventory are categorized according to quantities and values.

6 A-class: These are items with higher values but with lower demand or usage in the farm. Production machines and spares can be classified here B-class: These are the items with intermediary demand and values C-class: These are items with lower values but of regular usage. The consumables are usually classified here. 1.7 Deterministic/Probabilistic Demand Inventory: Inventory can further be divided into two areas: the deterministic inventory is a situation where the demand is known, such that for any given period of time the quantity to be ordered is known where as the probabilistic inventory is such where the demand is seasonal and stochastic and is described by probability distribution. 1.8 Background of the Work Researchers for many years have studied the relations to various areas of inventory control, such as raw materials, work in process, finished goods and supplies inventory. Instruments for measurement of inventory like the ABC classification, Bar-coding, inventory counting, inventory turnover and quantity discounting have been established. Theories of precession like the just-in-time system, kanban system and the backlogging systems have also been established. Some of these theories have been made use of by researchers todevelop models which when applied to some specific areas are functional.

7 7 Harris et al [5] is one of the first to appear in print. He developed the basic and widely used Economic Order Quantity (EQQ) which is the reference point of most useful inventory theories today. Oluyele et al [6] presented a system dynamic modeling of Nigerian automotive battery production organization, where policy runs of the system simulations were done to evaluate the impact of the size and pattern of demand. Aderoba, et al [7] developed a model for progressive inventory management for job shops where the developed model incorporates salient characteristics like uncertainty of demand, limitations of space and funds and multiple materials. Giri et al [8] presented a paper which considers an economic lot production quantity problem for an unreliable manufacturing system which the machine is subject to random failure (at most two failures in a production process). They established a model whereby shortages can be managed by accepting the existence of an on-hand inventory. Mc Lachlin [9] established an EQQ model for deteriorating items where the supplier offers a permissible delay in payment. Their model allows not only the partial backlogging rate to be related to the waiting time but also the unit selling price to be larger than the unit purchase cost. On the environs of this research work there are other models like that of Sheng [10] who defined a time dependent partial backlogging rate and introduced an opportunity cost due to lost sales. He established that the larger

8 8 the waiting time for the next replenishment the smaller the backlogging rate would be. Moreover the opportunity cost due to the lost sales should be considered since some customers would not like to wait for backlogging during the stockout period. Faaland et al [11] addressed the economic lot scheduling problem where a manufacturer makes a variety of products types on a single facility or assembly line. The model accounts for the time to set up the facility and charges a penalty on each unit short regardless of whether shortages actually result in lost sales or not. The model considers a situation in which a cycle is complete when one batch of each product type has been set up, and produced. They assumed that production is at constant known rate and a profit maximization firm and also that some lost sales may be attractive in compromise with inventory carrying cost. Netessine et al [12] presented the willingness of customers to backorder as a function of customer incentive that accompany the backorder. They analysed the impact of offering a monetary incentive on the optimal inventory policy by introducing an appropriate relationship between the proportion of backlogging customers and the incentive to backorder. They concluded that under some technical assumption other competitor s optimal inventory policy are monotone in the amount of incentive offered.

9 9 With this background, a further study on partial backordering stock and accepting some lost sale to arrive at a good inventory policy for a company will be made. 1.9 Problem Statement Consider a firm that operates a random placement of orders on raw materials, upholds an infinite production process and delivers to customers when the customers is available. If supplies are made to a clone system of customers the firm may run into a problem of over-production and its attendant consequences. According to Ezema [13] the problem of most companies is that of inadequate planning and control of production activities. Many companies lack the technical know-how while others ignore the practice of inventory control entirely. Another version of problem associated with inventory is the nonplacement of order except there is requisition from customers. Akin to this are delays in supplies to customers and the likely losses of customers and sometimes permanently. This research seeks to establish a good inventory management policy that would bridge the gap between overstocking and understocking and also enhance quick delivery of orders.

10 Objectives of the Work If it is possible it would rather be preferable for a company not to hold inventories since it may mean tying up cash in goods that would have rather improved the company s financial base. However, considering the fact that not holding inventories leads to incessant failures in production, the study of inventory management becomes inevitable. It is to enable a company to arrive at a point in its stock holding capacity where the holding cost will not be at the detriment of the company. In consideration of this fact we derive the objectives as: i) To capture prompt delivery to customers ii) To reduce to the least possible the holding cost by choosing to rather backorder instead of accumulating stocks. iii) To reduce buffer stocks to a known quantity such that even during uncertainties losses are minimized. iv) To smoothen demand even when it is erratic Need and Importance of Work A lot of companies operate without recognition o f the importance of inventory whereas good inventory management determines the wealth of any firm. This work would be important and applicable to companies that operate on either of two classes of inventory, the raw material inventory and finished product inventory. As usual it would be applied to a specific type of firm and if required in others firms, modification should be made to enable it fit into the

11 11 requirement of the given firm. It is therefore expected that good inventory control and infact the application of the model would go a long way to harness the wealth of the company of application Scope of the Work In this research work a model which can be used to determine the total cost of inventory with backordering the quantity and the re-order point will be developed. The model will be tested with a given company and recommendations would be made following the results obtained Methodology The work would be carried out through the following approach: i) The model by Jonah et al [2] will be slightly modified. The research of Jonah and Chukwu is basically a theoretical analysis of a general situation, bringing both total back-ordering, total lost sales and partial backordering cases. In their analysis they made use of figures not obtained from real situation and also guessed some bias factors and variance which they used in their theoretical analysis. However this situation will endeavour to break down the case of partial back ordering by analyzing the development of the model and then employing same in the estimation of the Quantity and Recorder point making use of a real situation. ii) Data will be collected from a given company in the brewing industry covering the following:

12 12 a. Demand rate b. Set-up cost c. Variable costs d. Carrying or holding costs e. Shortage costs f. Backordering costs g. Profits iii) iv) The lower bound for the partial backordering rate will be determined. The length of the inventory period (T), the fill rate (F) and consequently the order quantity (Q) and shortages (S) will be determined v) The re-order point will be established by applying the result above and the re-order point equation Definition of Fundamental Terms Demand: This is the sum total of customers requirements for a given time, usually per year. Lead Time: This is the time limit between the placement of an order and the subsequent arrival of same to fill the inventory. Base Stock Level: This is the maximum level that the replenishment should bring the stock level to. Inventory Level: Is the instant level of on hand inventory.

13 13 Backorder: This is a given quantity of customer demand during stock-out of which he is prepared to wait and receive after replenishment. Continuous Review: Is an inventory replenishment policy whereby replenishment is done at any point. Economic Order Quantity (EOQ): The optimal replenishment level that would best minimize the holding cost and order of on hand inventory. Fill Rate: This is the fraction of the demand that is filled from on hand inventory. Holding Cost: This is the cost per unit of holding stock in inventory and comprises of rents, insurance and opportunity cost of tied up capital. Inventory Level: This is the on-hand inventory less of backorders. Lost Sales: These are sales that are lost due to non-availability of stock. If the waiting time for delivery of an order is too long. Obsolescence: Where stock is no more usable for its intended purpose, by way of expiration, damage, contamination or shift of market. On-hand Inventory: The instant available stock in inventory. Lower Bound: This is the least value that partial backordering rate assumes for the partial backordering model to be applicable. Yield Variance: Is the variance of the yield distribution. The quantity that makes the quantity ordered to differ from the quantity received. Re-order Quantity: The number of items to be ordered during replenishment.

14 14 Safety Stock: Inventory which serves to promote continuous supply when unpredictable demand exceeds forcast, or the delivery of materials from the supplier is delayed. Service Level: This demonstrates the fraction or percentage of order cycle during the year in which there are no stock-outs. Set-up Costs: These are the costs of labour materials and marginal costs of machines or work station set-up. Shortage Cost: The cost incurred when a sub-optimal inventory item must be used to produce an order, due to a stock out of the optimal inventory item. Stationary Demand: Demand having a single probability distribution that does not change order time. Stock: The items in the warehouse to support operations.

15 15 CHAPTER 2 LITERATURE REVIEW 2.1 Preamble Several inventory models addressing different scenarios in inventory theory and management have been developed and applied by researchers Black, [1] Jonah et al [2] Monks, [3]. However in every research on inventory it is customary to establish optimal parameters which will ensure effective management of inventory. This chapter is targeted towards briefly tracing the history of inventory theory in order to place the topic of this work in context. This would then lead into exploring some specific concepts in inventory modeling. Basically the goal of these being; to gradually lead into these optimal parameters such as re-order points and re-order quantities which minimize the total inventory costs. The total inventory costs comprise costs like; ordering costs, set-up costs, purchase costs, holding costs and shortage costs. Before reviewing literature directly related to our focal point, we will relate some fundamental literature so as to reveal some other works by other researchers, such as the theory of EOQ (economic order quantity), ELS (economic lot size), the re-order point and EOQ with price break. 2.2 Fundamental theory in Inventory Management The foremost theory in inventory management is what is known as Economic Order Quantity EOQ where independent demand was used to

16 16 establish cost minimization. This theory was developed in 1913, Green et al [14]. Other early works include those of Harris [15] and Wilson [16] on the economic lot size (ELS) model, which recommends an optimal production batch size by trade-off of the inventory holding cost against production change-over costs. This formed the basis of the EOQ model generally used today. The essence of the model is to assume a continuous review system, a known constant demand and a known lead time and with this minimize total cost. As shown in fig 2.1, the dynamic minimum cost is at the point where the cost curve is minimum. The EOQ is given by Q = 2KD h (2.1) Where K = ordering cost or D = average demand h = holding cost (percent) Black [1] Q = quantity ordered EQQ = 2DS H (2.2) Where D = Annual Demand S = Ordering Cost H = Holding Cost EOQ = Economic order quantity Noori and Radford [17].

17 17 The average order interval per year which also gives the optimal re-order point is given by Average order interval: R = EOQ D, Norri and Radford [17] (2.3) R = Recorder point EOQ = Economic order quantity D = Quantity demand Or simply the re-order point is given by the demand per period multiplied by the lead time in number of periods. Arrow, et al [18] derived an optimal inventory policy for problems in which demand is known and constant and then for a single period problem in which demand is random. They also analyzed the general dynamic problem under the assumption of fixed setup costs and unit order cost, proportional to order size. Taha [18] extended the EOQ model to what he called EOQ with price breaks which is like the EOQ model except that the inventory item may be purchased at a discount if the size of the order y exceeds a given limit, q that is the unit purchasing price C is given by c 1, if y q C = c 2, if y > q, c 1 > c 2 (2.4) He also presented other models as, multi-item EOQ with storage limitations, a no-setup model and setup model

18 18 Order quantity Fig. 2.1: Graphical representation of EOQ, showing where the EOQ is taken 2.3 Other Related Literature: Taha [18] divided inventory into two parts, the deterministic and the probabilistic inventories, showing that the deterministic is a case where the demand is known and certain, while the probabilistic is a case of unknown previous demand. He went further to write on the probabilistic inventory. He studied the case of a mail order retailer selling style goods and receiving large numbers of commercial return. Returned goods arriving before the end of the selling season can be resold if there is sufficient demand. A single order is placed before the season starts. Excess inventory at the end of the season is salvaged and all demand not met directly are lost. Amasaka [19] illustrated a proposition which goes beyond production based on his experience at Toyota Company. The main concept deals with linking quality cost and delivery research activities of all departments concerned with development production and sales.

19 19 Yang [20] presented 4 different inventory shortage models which are developed with deteriorating items and partial backlogging. He assumed that the demand function is positive and fluctuating with time and the backlogging rate is diffentiable and decreasing function of time. He also assumed maximizing profit as the objective to find the optimal replenishment policy. He finally identifies the most profitable alternative. Laan et al [21] worked on the implementation of the probabilistic inventory model. A case study of a photocopier manufacturing company was used to demonstrate the Stochastic inventory model with production, remanufacturing and disposal operations. Customers demand was either fulfilled from the production of new products or by the re-manufacturing of used products. To co-ordinate production remanufacturing and disposal operations they employed what they called the PUSH and PULL strategies to enable the reuse or withdrawal of returned stock. On deterministic inventory model the just in time (JIT) and the material requirement planning (MRP) have been employed by researchers to demonstrate the pre-knowledge of the flow of stock in the planning of replenishment. Gelinas [22] described JIT as an inventory loss elimination process, at all its levels as a means of maximizing high-added valued activities pay-off or to minimize low added value impact. He defined it as a management tool developed for planning, controlling and monitoring and intricate set of nonrepetitive activities.

20 20 Matsuura et al [23] described JIT as a manufacturing philosophy towards improving efficiency through the absolute elimination of waste by continuous improvement and workers involvement. A system developed by Toyota production system in which orders are placed when items are required. They also described MRP as a closed loop system in which functions such as master production scheduling, capacity requirement planning and shop floor control are attached. More researches have also been done on the area of Economic Lot Scheduling (ELS). Faaland et al [11] addressed this with lost sales and set up time considered, where a manufacturer makes a variety of products on a single facility or assembly line. Their model accounts for time to set up facility and charges penalty for each unit short, regardless of whether the shortage results in lost sales or is satisfied by a subcontractor. Hsu [24] also dealt with ELS for perishable products with age dependent inventory and backorder costs. 2.4 Directly related literature on shortages, backordering, partial backordering and lost sales The one general term used to describe shortages and its consequences is deterioration which we earlier described as yield uncertainty or yield randomness. Deterioration in general may be considered as the result of various effects on the stock, some of which are: damage, spoilage, obsolescence, decay, decreasing usefulness, miscounting, breakage or pilferage. Hark and Sohn [25] described this as gradual loss of utility or potential associated with passage of

21 21 time. We may consider a typical uncertainty situation as that of stocking and distribution of day-old chicks where the rate of deterioration is very fast if supplies are not made. Hark and Sohn [25] established the optimal quantity that should be ordered in a situation of combined effect of deterioration and inflation. Ghare et al [26] studied a model having a constant rate of deterioration and a constant rate of demand over a finite planning state. More work on this area was done by Covert et al [27] who added a rate of deterioration and Shah [28] who considered the model allowing complete backlogging of the unsatisfied demand. Wee [29] studied models allowing partial backlogging of unsatisfied demand and deterioration of inventory. Additions to this area are the works of Eilon et al [30] who considered where unit selling price is affected by demand, as low selling prices generated demand, while high selling prices decline demands. They considered perishable items with maximum shelf life and no deterioration before the expiration date. Papachristos [31] made further research in the work of Wee [33] where they established that the demand rate is described by any convex decreasing function of the selling price and instead of a constant rate of partial backlogging considered a variable backlogging rate which is found in the work of Abad [32]. They found the optimal solution and compared it using examples to the approximated results of Wee [33].

22 22 In summary they concluded that the total profit per unit time, NP (T, S,T) is the total average revenue minus total average cost Np (T 1, S, T) K (T 1, S, T) The total average revenue is given as R( T1, S, T) = 1 sd (s) + T Bd (s) T T1 1 + y (T U) du (2.5) While the total average cost is K (T 1 S, T) = φiq + c 1 + c 2 CI + c 3 I b + c 4 I 1 T T T T T (2.6) Therefore deducting the cost from the revenue gives the Net profit (R K = NP) Where T = Cycle length T 1 = Inventory cycle interval with positive stock s = Unit selling price d (s) = Demand rate for product φ 1 = Material cost per unit q = Order quantity c 1 = Fixed cost per order c 2 = Holding cost per unit per time c 3 = Shortage cost c 4 = Sales cost 1 b = Amount of shortages backlogged per cycle I 1 = Lost scales per cycle CI = Inventory carried Y and B = the backlogging parameters

23 23 As earlier mentioned Wee [33] treated a similar case where his model showed a deterministic inventory model with quantity discounts pricing and partial backlogging when the product deteriorates with time. Against the general situation of minimizing cost he like Papachristos worked on profit maximization. In his case, revenue was simply defined as the product of the unit selling price s and the demand rate for the product d(s), i.e. R = s d(s) While the total variable cost per unit time K is the summation of the material cost, the replenishment cost, the carrying cost, backlogging cost and the penalty cost for lost sales. K (T1, S, T) = V 1 (q)q + c 1 + c 2 (q) I T (T 1 ) + c 3 c 4 T T T T I b + T I 1 (2.7) Where V i (q) is the material cost per unit and other notations are same as earlier given. More researchers have also been done not only on the use of Net Profit (NP) but also on the use of total cost (TC) to find the optimal lot sizing Abad [32] developed an optimal lot size for perishable goods under conditions of finite and partial backordering and lost sales. In this case he considered the problem of determining the production lot size of a perishable product that decays at an exponential rate assuming that one

24 24 backlogging of demand is allowed. As in Wee [33] he used same individual costs and solved for the total cost (F) F (J, λ) = C 1 + c + c 2 [p β(j) dj] + C 3 Bdmλ λ + C 4 dmλ 1 B (2.8) θ 2 Where J = T, (in notation) = duration of inventory cycle when there is positive inventory, β = interim time span, λ = duration of inventory cycle when stock out exist and all other notations remain same as in previous notation. Another researcher who worked on optimal lot sizing using cost is Wang [10]. In his work an inventory replenishment policy for deteriorating items with shortages and partial backlogging, he defined as an appropriate time-dependent partial backlogging rate and introduced an opportunity cost due to lost sales. In this article he gave two examples of inventory problems with linear and exponential demand patterns which were taken from Giri et al [34]. He proposed that if the demand rate D(t) is strictly increasing, then, (i) the shortage period and the inventory periods are getting smaller with respect to the number of replenishments and (ii) the inventory level right after a given number of replenishments and shortage level right before the same number of replenishments are increasing, he gave the total cost as Tc= nc1+ C 2 + Cθ Σ I t j-1, t j + C 3 Σ S t j-1, t j α + C 5 H Σ S t j-1, t j J = 1 = nc 1 + C 4 θ -1 Σ tj e θ t j - s j -1 D (t)dt n n n sj J = 1 n + C6 Σ sj tj s f - t D (t) dt J = α sj t / H (2.9) n

25 25 Chern et al [35] extended the inventory lot size model to allow not only for general partial backlogging rate but also for inflation. They established that for seasonal commodities with short life span the willingness for a customer to wait for backlogging during the shortage period is diminishing with the length of the waiting time. Hence the longer the waiting time the smaller the backlogging rate. This is in agreement with Papachristos et al [31] who established a partial backlogging rate inventory model in which the backlogging rate decreases exponentially as the waiting time increase. Chern et al [35] developed an inventory lot size model for deteriorating items with partial backlogging. They also took the time value of money into consideration. They assumed that not only the demand function is fluctuating with time but also the backlogging rate of unsatisfied demand is a decreasing function of the waiting time. They showed that the total relevant costs (i.e. the sum of the holding cost, backlogging, lost sales and purchase costs) is a function of the number of replenishment. Consequently, the search for the number of replenishment was reduced to finding the local minimum. They defined the objective of the inventory problem as to determine the number of replenishments n, the timing of the re-order point (ti) and the shortage points (si) in order to minimize the total relevant cost (TC). TC was derived to be n Tc n, s t, t 1 = Σ (p i + I i + S i ) J = 1 (2.10)

26 26 P i is the purchase cost during the 6th replenishment cycle, I i, the inventory holding cost and S i the shortage cost. The derivations of most of these models are characterized by highly complex and multi-component models. However, Jonah and Chukwu [2] modeled a simpler system using the length of the stock-out period and the length of the inventory review period. They also tried to deal with the problem of re-order point, by developing a closed form model for establishing the reorder point with optimal values of other parameters. Their model integrated both pure backordering, total lost sales and partial backordering. For pure backordering they found the optimal values of Q and S i the optimal quantity with backorders and the maximum stock out as: Q * B = 1 CB + hv 2KD + P 2 D m C B hv C B +hv hv (2.11) S = 1 2 KDhv + (hvs) 2 h + hv - hv (PD) 2 - PD C B + hv C B C B (2.12) They also demonstrated the length of the inventory review period using the lost sales approach and gave the length of the review period as T b = 2KD +hvσ 2 D 2 hv (2.13) This then lead us into the partial backordering model.

27 27 CHAPTER MODEL DEVELOPMENT 3.1 The Partial Back-Ordering Model This model is developed on the assumption that in a company that intends to operate on a zero buffer stock, the expectation should be that not all the customers that arrive during the stock-out period would be willing to wait for the arrival of new stock. However if researches have shown that this is better than holding stock it might then be necessary for the company to do so. As earlier mentioned some companies may device incentive method, another price slash method, all to motivate the customers to wait for the arrival of orders. Despite the motivational approach some customers still would not wait but go to other suppliers. This divided nature of demand during the shortage period creates the need for evaluation of cost and quantities that should be ordered. This is under the assumption that this effect has taken place over a range of time for which studies can be made. 3.2 Other Assumptions Made Include (i) (ii) (iii) (iv) (v) The arrival of orders should be able to meet all backorders and bring the on hand inventory above the re-order points. The carrying cost of the inventory is applicable to only the units of acceptable quantity The study is made on a single item inventory Demand is treated as deterministic Lead time is constant

28 28 (vi) Customer demand is considered linear 3.3 The Partial Model Development Fig. 2: Graph of partial back-ordering model showing the random yield curve From the graph it can be seen that there are two periods when there is inventory and when there are shortages, the shortage period is further divided into two parts, BS and (1 B)S. (µq 1 BS) is the function for the quantity while stock lasts which when divided by the function for total quantity (µq 1 + (1-B) S) we obtain the fill rate. µq 1 BS µq 1 + (1 B)S) A known fraction B of the demand during the lead time is backlogged while the remaining (1-B) is lost. Since S is the maximum stock-out BS is denoting the total amount backlogged while (1 B)S is totally lost. Considering first the length of the inventory review period can be presented mathematically as T = Q 2 BS + S = [Q 2 + (1 B)S] D D D ----(3.1)

29 29 Considering that due to shortages which generate the random yield problem, not all Q 1 will result to Q 2 therefore consider the expectation of the review period The expected length of the inventory review period is given as E T Q 1 = T dq 2 o = o Q2 + (1 B)s = µq1 + (1 B)s D D This is the expected length of the inventory review period. Looking at the total cost TC as a function of the individual cost; V, C 1 C 2 C 3 C B, the expected cost per cycle would be E T Q 1 = C 2 + C 1 v [(Q 2 BS) 2 ] + C 3 S + P (1 B)S + C B BS 2 o 2D 2D i.e., by bringing in all the individual costs to make the total cost. Resolving this (integrating the total cost function) gives TC = C 2 + C 1 V {(δ 2 + µ 2 Q 1 2 ) 2BS (µq 1 ) + (BS) 2 } 2D + C 3 S + P(1 B)S + C B BS 2 2D -----(3.4) dq (3.3) Having found the total cost, the review period and the fill rate (as given by the quantity ordered on the graph) we will proceed to introduce them into the equation of total cost by multiplying the function of total cost by the inverse of T which is called the expected number of cycles per review period. This will

30 30 enable us to identify the review period function and the fill rate function in the equation of cost 1 = D T µq 1 + (1 B)S we obtain the expected cost per unit time TC (Q 1 S) = C 2 D + C 1 V σ 2 + C 1 v (µq 1 BS) 2 µq 1 + (1 B)S 2[µQ 1 + (1 B)S] [µq 1 + (1 B)S] C 3 SD + P(1 B)SD + C 3 BS 2 + 2[µQ + (1 B)S] µq + (1 B)S [2µQ 1 + (1 B)S] ---(3.5) To transform the above equation into the expected length of the review period µq 1 BS µq 1 + (1 B)S and fill rate, F is represented = µq 1 + (1 +B)S and T = D In each function of the equation of expected cost per unit time, this now gives us the actual total cost and the fill rate. We obtain the expected total cost TC (T,F) = C 2 C 1 v σ 2 + DTF 2 +C B BDT (1 F) 2 T 2DT (P + C 3 ) D (1 B) (1 F) -----(3.6) Differentiating equation (3.6) partially with respect to T and F and equating each to Zero to obtain the minimum cost and fill rate TC = - C 2 - C 1 vσ 2 + C 1 vdf 2 + C B BD (1 F) 2 =0 F T 2 2DT TC = C 1 DTF C B BDT (1 F) (P +C 3 ) (D) (1-B) = 0 F

31 31 and finally to obtain the optimal values of T and F, the two equations are solve simultaneously T = C 1 v + C B B 2C 2 D - C 3 + P (1 B) 2 + δ 2 D 2 C B B C 1 v (C 1 v + C B B) C 1 v -----(3.7) F = C B BT + (P + C 3 ) (1 B) (C 1 v + C B B)T (3.8) Jonah et al (2007) The solution to equation (3.7) and (3.8) exist if and only if β 1 + C 3 (2C 2 D + C 1 Vδ 2 ) C 1 V D 2 C C 3 = β * P P 2 D 2 P (3.9) This theorems are presented in our notation To obtain the quantity to be backordered we multiply the review period T with the demand D, subtract the fraction that is totally lost (1 β) S and divide through by the bias factor µ That gives us Q = T D (1 β)s µ (3.10) And to obtain the maximum stock out we multiply the review period T with the demand D and the converse of t he fill rate (1- F) S = T D (1 F) (3.11)

32 Determination of Re-order Point The foregoing seeks here to develop a re-order point in an inventory system where a deterministic inventory is assumed and deterioration effect is experienced, and where demand is partially backlogged. The re-order point determination is necessary in view of the fact that there needs to be a balance point between shortage and holding cost of inventory. This analysis is predicated on the fact that the re-order point can be established almost independent of order quantity but with the parameters Maximum Expected Cost Approach Some researches have been made in this area establishing some approaches to this effect. Some of these are the maximum expected cost and the service level approaches. The customer service level is described as the percentage of orders filled from stock on hand which is also called the fill rate. This together with its counterpart: the stock-out rate equals 100%. A service level of 0.98 means that customer orders would be filled 98% with a s tock out of 0.2 (2%). One of the equations to obtain service level is that given by Irwin [37]. SS =Z LT (SD) 2 + SS R.O.P = D (LT) + SS (3.12)

33 33 Where SS Z LT SD = Safety stock = Value from normal distribution table = Lead time = Standard deviation of demand R.O.P = Re-order point Another equation relevant in solving for re-order point is that given by Hamid et al [40]: Where R = D L + Z k δ L (3.13) R = Reorder point D L = Average demand Z k = Value associated with the desired service level K during the lead time δ L = Standard deviation of demand during lead time The Service Level Approach To adopt the service level approach of Weyne [36]. If a service level approach of x% is desired in inventory decision making then: (1 x%) = δ L NL δ L Q 1 r E (L) = NL r E (L) = Q 1 (1-x%) δ L δ L NL r E (L) = [µq1 + (1 B)S] (1 x%) δ L δ L (3.14)

34 34 Where X NL R = desired service level = Normal loss function obtained from normal loss function table = re-order point E(L) = Expectation of lead time demand. All other notation remains the same as of partial backorder model.

35 35 CHAPTER Application And Analysis of Results The previous chapter dealt with development of models, theories and formulae for managing inventories in cases where yield is random or uncertain and where stock-out is likely to occur within the lead time. Having been able to determine the length of the inventory period (T), the fill rate (F), partial back-ordering inventory rate (β) the re-order point (R). A numerical application of these theories so far obtained will be tested using data collected from the Champion Breweries Nig. Plc, Aka Industrial Layout, Uyo Akwa Ibom State. 4.2 Estimation of Review Period, Fill Rate, Quantity and Stock-out As mentioned in the methodology we obtained data of the following on Plain Sorghum. Demand (D) - 300,000 tons Set up cost (C 2 ) - N100,000/replenishment Variable cost (V) - N1000/ton Holding cost (C 1 ) - N900/ton Shortage cost (C 3 ) - N500/ton Backordering cost (C B ) - N300/ton Profit (P) - N7,000/ton Seeing that these figures a large we may reduce all by a scale of X10 2 for easy calculation therefore we use the following figure

36 36 D = 3000, C 2 = 100, v = 10, C 3 = 5, P = 70, C B = 3, C 1 = 9 We also assume a variance of O and a bias µ of I To start with establish the lower bound for β using equation (9) β 1 + C 3 (2C 2 D + C 1 Vδ 2 ) C 1 V D 2 C C 3 P P 2 D 2 P =β * 5 (2x1000x300)9x x x β 1 + β = This is the lower bound for the partial back-ordering rate which we wish to establish. Then proceed to solve for the review period and the fill rate using equations (3.7) and (3.8) T is given as T = 0.5 T = C 1 V x C β B 2C 2 D (C 2 + P (1 β) 2 ) + δ 2 D 2 C B β C 1 V (C 1 V + C β B )C 1 V 9x x x1000x ( x3x x100 (9x x x100 And F is given as F = (C B BT + (P + C 3 ) (1 B) C 1 v + C B B)T F = 3x (70 + 5) ( ) 9x x0.9489) F = = 2.3% S = T D (1- F)

37 37 S = 0.5 x 3000 ( ) S = 1, Q = T D (1 β)s µ Q = 0.5x3000 ( ) = :. Q = x 10 2 = tons Q is the quantity of Plain sorghum that should be ordered. 4.3 Estimation of Re-order Point To find the re-order point we assume one month (L) (4 weeks) lead time Demand E (D) = 3000 Demand variances δ of 20. To calculate the expectation of lead time E (L) = L X E (D) 4 = 52 x 3000 = tons Variance of lead time δ L = L x δ D = 4/52 x20 = = 5.55 To check the re-order point now.

38 38 E (L) = tons δ D = 5.55 Q = tons S = tons F = = 2.3% since F = 2.3% we need a service level of 100% -2.3% = = 97.7% suitable normal loss function of and u = 0.5 r E(L) = µq + (1- B)S(1 x%) δ D δ L NL 0.5x ( ) (0.023) r = = tons 4.4 The Implication of Biennial Orderings From the forgoing, when the stock reduces to 588 tons, tons of plain sorghum should be ordered. This is expected to fall twice in one production year. The bases of the study on inventory are to reduce cost which includes reduction of losses. Thus, study on back ordering is targeted towards reducing losses that arise from deterioration, spoilage and damages in stocking. Therefore, ordering for a shorter period of time eliminates or reduces this problem. However, other advantages of shorter term ordering do exist. By placing orders biennially the holding cost of the stock for half a year has been eliminated, the haulage cost and ordering cost remain. However, the last two are minimal compared to the holding cost. As defined in the definition of

39 39 terms (chapter 2); Holding cost is the cost of holding items in inventory and comprises of rents, insurance and opportunity cost of tied up capital Rents The size of a warehouse would be dependent on the level of inventory expected by the company. Consequently, investment on the warehouse would be dependent on the level of stock. The lower the stock level, the lower the size of warehouse and therefore the lower the price to be paid for the warehouse Insurance The second investment on holding stock is insurance. Property insurance is that granted to cover business against a wide variety of liability and property damage or losses. Commercial property policies cover the building occupied by a business and such items are furniture, fixtures, machinery and inventories of a business. In the immediate situation, if a 20% or 15% insurance policy is undertaken on the N18m worth of stock as ordered by Champion Breweries Plc the resulting amount would be huge compared to that which would have been done on half the same quantity. Realizing that the quantity normally ordered annually is not usually exhausted but rather attracts a lot of spoilage. Furthermore looking at Papachristos et al (31), in the figures estimated in numerical examples, the holding cost was put at 2.3%. If we estimate same in

40 40 our case it would be found that the holding cost on N9m worth of plain sorghum would be about Two Hundred Thousand Naira (N200,000.00) whereas information obtained form the production manager of the company shows that cost on bringing down the product of each order fall within the range of Fifty Thousand Naira (N50,000.00) eliminating a lot of costs Opportunity Cost Retaining large stocks in inventory meant tying down money that could have been used in other areas in the company. As we already know, opportunity cost is the cost of meeting one need at the expense of the other. Therefore keeping large stock is certainly at the expense of other well meaning needs of the company. It therefore becomes more profitable to stock smaller quantities and also place orders for smaller quantities.

41 41 CHAPTER FIVE 5.0 CONCLUSION AND RECOMMENDATIONS 5.1 Conclusion In this research work the problems of shortages, yield uncertainty or yield randomness were considered as major inventory problems. It was found that in attempt to solve this problem and obtain acceptable lot size models other researches have employed different models, including, lost sale backordering and partial backorder. The partial backordering model of Jonah and Chukwu was adopted and reviewed. This enabled the obtaining of models for review period (T), the fill rate (F), the stock out (S) and consequently the order quantity (Q). We then endeavoured to use the length of the inventory review period and the fill rate as decision variable, and with values obtained from the Champion Breweries Plc. Aka Industrial Layout Uyo Akwa Ibom State we solved for the order quantity of plain sorghum. Initially the company used to order six hundred tones ( tons) of plain sorghum once per year, but due to spoilages and general depreciation, not all are used. Another problem posed was that of holding cost. To circumvent these problems a working parameter of three hundred thousand tons of plain sorghum was chosen; this gave an inventory review period (T) of 0.5 which meant order could be placed every six months. The order quantity (Q) for this period was found to tons. Choosing a four

42 42 weeks (4wk) lead time it was found that orders should be placed when stock dropped to 58,746 tons. By this shortages are minimized, and keeping extremely large stock is avoided. 5.2 Recommendations Many companies are still operating today without any inventory control at all not even the early generation first-in-first-out (FIFO), hence inconsistencies of production are experienced. We are therefore by this encouraging all recognized companies to; (1) Inculcate inventory control systems in their operations for a more successful operation (2) Companies that have existing inventory system should attempt to implement the partial backordering system to enable the minimization of shortages and losses. (3) That seminars should be held in companies and Government parastatals to give orientation to storekeepers and administrative officers on the importance of inventory control. (4) That inventory course be introduced to most faculties of the university.

43 Recommendations for Further Research 1 Development of a software programme for managing the partial back order inventory model. 2 Employing analytical method by estimating different values of quantity, set-up cost and other cost in the study of partial back-ordering inventory model. 3 Solving for a situation where demand is gradually exponential.

44 44 REFERENCES [1] Black, C. D, (2004), Optimal Inventory control in cardboard box producing factories: M.Sc. Thesis A case study, pp [2] Jonah, U. A. and Chukwu, W. I. E. (2007), Inventory modeling: A new look, International Journal of Physical sciences, Vol. 4, No. 3. [3] Monks, G. J. (1996), Operations management, Schaum outlines. Mcgraw Hill publishers, p , [4] Haris, T. E., Marschak, J. and Arrow, K. J. (1951), Optimal inventory policy econometrical, Vol. 19, Isue 3, pp [5] Oluleye, A.E., Oladeji, O. & Agholor, D. I. (2001), Inventory system modeling: A case of an automotive battery manufacture, Nigeria Journal of Engineering Management (NJEM), Vol. 2, No. 1, p [6] Aderibam A.A. Kareem B. and Ogudengbe, T. I. (2003), A model for progressive inventory management for job shops. Nigeria Journal of Engineering Management (NJEM), Vol. 4, No. 1, pp [7] Giri, B.C. and Yun, W. Y. (2005), Optimal lot sizing for an unreliable production system under partial backlogging and at most two failures in a production cycle, Intl Journal of Production Economics, Vol. 95, pp [8] McLachlin, R. (1997). Management initiative and Manufacturing Journal of Operation Management Vol. 15 pp [9] Wang, S. P. (2002), An inventory replenishment policy for deteriorating with shortages and partial backlogging, computers and operations research, Vol. 29,pp [10] Faaland, B. H.; Schmitt, T. G.; and Arreola-Risa, A. (2004), Economic lot scheduling with cost sales and set up times, Institute of Industrial Engineers (IIE) Transactions. Vol. 4, pp 1-3. [11] Netessine, S.; Rudi, N.; and Wang, Y. (2006), Inventory competition and incentives to backorder, Institute of Industrial Engineers Transaction. Vol. 6 pp, 1-3. [12] Ezema, I. (2002), Production inventory control for job shops. An M. Eng Thesis, University of Nigeria, Nsukka. pp, 5-11.