INVENTORY SYSTEM Introduction

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1 INVENTORY SYSTEM Introduction Inventory is the stock of any item or resources used in an organization. It is idle resource, but usable. The term stock is generally used to refer to resource in its physical and tangible form, like materials. Thus stock and inventory are identical, although inventory has wider implications. Managing the inventory is the task of maintaining an adequate supply of materials to meet the expected demand pattern for a given financial investment. Thus inventory management has the responsibility to decide upon the optimal materials under given circumstances for procurement of commodities. An inventory system is the set of policies and controls that monitor and evaluate the levels of inventory and then determines that: What level of inventory should be maintained When stock should be reordered How large orders should be placed. The inventory which contributes to be a part of product or which becomes a product is called production inventory or manufacturing inventory. It can be classified into 3 parts, viz: raw materials, semi-finished goods (work-in-progress) and finished goods respectively. Where as in service organization inventory refers to the tangible goods which are to be served or sold. For example, no. of beads, drugs, tonics, serums etc are the inventory of hospitals. An inventory system is basically a process whereby a business keeps track of the goods and material it has available. In its simplest sense it can be done manually by a count at the end of each day. In this way it is possible to keep a record of the goods coming in to the business and goods being sold. However this is only really appropriate for small businesses that do not have a lot of stock. For larger business it is more likely that a computerized system will be required. If the firm is a retail establishment, a customer may look elsewhere to have his or her needs satisfied if the firm does not have the required item in stock when the customer arrives. If the firm is a manufacturer, it must maintain some inventory of raw materials and work-in-process in order to keep the factory running. In addition, it must maintain some supply of finished goods in order to meet demand. Sometimes, a firm may keep larger inventory than is necessary to meet demand and keep the factory running under current conditions of demand. If the firm exists in a volatile environment where demand is dynamic (i.e., rises and falls quickly), an on-hand inventory could be maintained as a buffer against unexpected changes in demand. This buffer inventory also can serve to protect the firm if a supplier fails to deliver at the required time, or if the supplier's quality is found to be substandard upon inspection, either of which would otherwise leave the firm without the necessary raw materials. Other reasons for maintaining an unnecessarily large inventory include buying to take advantage of quantity discounts (i.e., the firm saves by buying in bulk), or ordering more in advance of an impending price increase.

2 86 / PRODUCTION AND OPERATIONS MANAGEMENT Importance of Inventory Inventory management is an important part of a business because inventories are usually the largest expense incurred from business operations. Most companies will use an inventory management system that will track and maintain the inventory required to meet customer demand. It is very difficult to visualize a real work situation, where the required material will be made available at the point where it is used, at the very instant, it becomes necessary. That is what makes the maintenance of inventories a virtual necessity. Viewed that way, inventories could be seen as a 'necessary evil'. The inventory system is more important due to following reasons: It ensures smooth flow of work It ensures proper utilization of labor & machines It helps to provide & maintain good customer service It helps to take advantage of economic purchase order It allows flexibility in production schedule It increases Return on Investment It provides the protection against future uncertainties Reasons for Carrying Inventory Inventory is undesirable, but necessary, because its aim is to absorb the uncertainties of demand some major reasons for carrying inventory in an organization may be: When lead times are longer, stocks have to be maintained for system operation. Demand and lead times are uncertain; and that makes it necessary to build up a safety stock (buffer stocks) in order to ensure that various sub-systems succeed in operating in a somewhat decoupled manner. Cycle stocks may be a means to avail of economies of scale, which in turn will enable the reduction of total system cost, due to ordering, carrying inventory, and backlogging. Technological requirements of batch processing in production, also lead to build up of cycle stocks. Stocks may grow on account of pipeline inventory, or work-in-process inventory, due to the fixed nature of production and transportation rates. This includes materials actually being worked on, or materials which are moving between different work centers, or being in transit en route distribution centers and customers. Inventory Costs In order to maximize the profit of an organization, the management should focus on minimization of inventory cost. The total inventory costs can be classified into following four parts. They are as follows: 1. Ordering cost (setup cost) The cost which is incurred for placing an order of inventory is called ordering cost. Managerial cost, transportation cost and clerical cost to prepare purchase order may be examples of ordering cost. It is also known as procurement cost. This cost is usually assumed to be independent of the quantity ordered, because the fixed cost component is generally more significant than the variable cost component. Therefore, it is expressed as a fixed amount per order for example Rs. 0 per order.. Carrying Cost (Holding cost)

3 INVENTORY SYSTEM CHAPTER The cost which is incurred for keeping stocks in store is called carrying cost or holding cost. Carrying cost comprises cost elements including, security, heat, light, power taxes, cost of blocking, capital i.e interest rate, cost of insurances, storage cost, cost due to obsolescence, leakage, spoilage, pilferage, deterioration, etc. The attributed carrying cost will be obviously higher for perishable items. It is generally expressed as a fraction of the value of the goods stocked per year. For example, if the fraction of carrying charge is 0% per year, and a material worth Rs. 1,000 is kept in inventory for one year, the unit carrying cost will be Rs 00 per item per year. Alternatively it can be expressed as a rand (i.e. countable) value per item. For example, holding cost per item is Rs Material cost (Purchase price cost) The price or amount paid to supplier for buying materials of inventory is called material cost. It inventory is called material cost. It directly affects the cost of production. 4. Stock out cost (Shortage cost) It is the opportunity cost of not having an item in stock, when it is demanded. These types of costs are only incurred when the demand exceeds the supply. This situation may be the result of lost sales e.g., absence of a product in stock in sufficient quantities to meet demand, on account of the company's earlier decision against the production of the item in sufficient quantities to meet expected demand in the near future, or backlogging. It includes cost of back orders, loss of potential profit, lost demand, penalty cost, emergency replenishment, loss of good will etc. Types of Inventory Systems There are types of commonly used inventory system. They are as follows:- 1. Continuous Inventory System (Q/R model) Continuous inventory system is the perpetual inventory system which describes the systems of inventory where information on inventory quantity and availability is updated on a continuous basis as a function of doing business. Generally this is accomplished by connecting the inventory system with order entry and in retail the point of sale system. In this case, book inventory would be exactly the same as, or almost the same, as the real inventory. The basic feature of this inventory system is to manage and maintain balance the use rate of inventory. Under this system, re-ordering level (R) and fixed order quantity (Q) are already predetermined. Whenever the quantity of stock reaches at re-ordering level, then new order for purchasing is placed with fixed order quantity (Q). The order can be placed any time and it depends upon the demand of stock. Therefore withdrawals and additions of stock should be recorded and monitored regularly to make balance of inventory and ensure the re-ordering level. The graphical presentation of continuous inventory system can be seen below: Quantity Current Inventory Q1 Q Q3 ROL Safety Stock LT LT LT Time

4 88 / PRODUCTION AND OPERATIONS MANAGEMENT Assumptions of continuous inventory system Constant and certain demand over the period. Constant and known lead time (Delivery time) Constant purchase price of material. Constant and independent carrying cost and ordering cost. No stock out cost.. Periodic Inventory System Periodic inventory system is an operating doctrine on which ordering level and order quantity are not predetermined; it depends upon the actual time period. It is a system of inventory in which updates are made on a periodic basis. This differs from perpetual inventory systems, where updates are made as seen fit. In a periodic inventory system no effort is made to keep upto-date records of either the inventory or the cost of goods sold. Instead, these amounts are determined only periodically - usually at the end of each year. This physical count determines the amount of inventory appearing in the balance sheet. The cost of goods sold for the entire year then is determined by a short computation. Under this system, inventory is counted in a fixed time period to determine the quantity of inventory for placing an order of inventory. It is period based or time based model, on which it requires higher level of safety stock rather than Q/R model. It is also known as fixed period inventory system. The graphical presentation of periodic inventory system can be seen below: Quantity Q Current Inventory Safety Stock Q Q1 Q3 Place Place Place LT LT LT ROL Stock Out Economic Order Quantity (EOQ) Figure: Periodical Inventory System. The quantity for ordering goods which minimizes the total cost of inventory is called economic order quantity (EOQ). It is the optimal lot size or optimal orders. It is one of the major tools for controlling inventory. Under this, total carrying cost and total ordering cost of inventory remains equal. EOQ only applies where the demand for a product is constant over the year and that each new order is delivered in full when the inventory reaches zero. There is a fixed cost charged for each order placed, regardless of the number of units ordered. There is also a holding or storage cost for each unit held in storage (sometimes expressed as a percentage of the purchase cost of the item). We want to determine the optimal number of units of the product to order so that we minimize the total cost associated with the purchase, delivery and storage of the product. The optimal number of units (i.e. EOQ) depends upon the total demand for the year, ordering cost and carrying cost. So, the required parameters to the solution of EOQ are the total demand for the year, the purchase cost for each item, the fixed cost to place the order and the storage cost for each item per year. Note that the number of times an order is placed will also affect the total cost, however, this number can be determined from the other parameters.

5 Cost Assumptions of EOQ a. Constant and certain demand over the period. b. Constant and known lead time (Delivery time) c. Constant purchase price of material. d. Constant and independent carrying cost and ordering cost. e. No stock out cost. Graphical approach of EOQ INVENTORY SYSTEM CHAPTER Since EOQ is the quantity of materials to be ordered economically. This level of quality total ordering cost & total currying cost becomes equal. So EOQ can be determined graphically at that paint, one ordering cost and currying cost interested each other. Y Total cost Carrying cost(qc/) Ordering cost(ao/q) 0 EOQ Quantity X Figure: Economic order quantity. This figure shows costs on the vertical axis or Y axis and the order quantity on the horizontal or X axis. The straight line which commences at the origin is the carrying cost curve, the total cost of carrying units of inventory. As expected, as we order more on the X axis, the carrying cost line increases in a proportionate manner. The downward sloping curve which commences high on the Y axis and decreases as it approaches the X axis and moves to the right is the ordering cost curve. This curve represents the total ordering cost depending on the size of the order quantity. Obviously the ordering cost will decrease as the order quantity is increased thereby causing there to be fewer orders which need to be made in any particular period of time. The point at which these two curves intersect is the same point which is the minimum of the curve which represents the total cost for the inventory system. Thus the sum of the carrying cost curve and the ordering cost curve is represented by the total cost curve and the minimum point of the total cost curve corresponds to the same point where the carrying cost curve and the ordering cost curve intersect. ABC Inventory System ABC inventory system refers to the priority system of inventory on which total inventories are categorized into three parts. It is based upon the assumption that all inventories are not always equally important for any organization. So some discrimination should be made on total inventory according to their relative importance or priorities.

6 90 / PRODUCTION AND OPERATIONS MANAGEMENT Large manufacturing companies should maintain large numbers of several types of inventories and it is not possible to control all inventories with equal attention. Therefore, the firm should pay more attention to those items which have highest value. In order to pay the optimum level of attention for inventories, they (i.e. inventories) should be prioritized. High investment is made on inventory for accurate production and supply of the product to the consumer. ABC inventory system enables management to reduce the investment to the minimum possible level because a reasonable quantity of 'A' items representing a significant portion of the material cost is purchased. To reduce investment in materials, close control of 'A' items contributes much more than close control of 'C' items. On the other hand, no other inventory control system provides such technique. Management time is saved by this system because this technique requires minimum attention to be paid to some items only. But, other inventory system do not facilitates such separation and in case of any misguidance about the inventory system arises this burden goes to the management and have to spend lots of time on such deviation. ABC inventory system stands for Always Better Control in inventory. Following 3 classifications are made under this inventory system. Group "A" It includes few items with large value. Group "B" It includes moderate items with moderate value. Group "C" It includes more items with small value. Great effort is made on category A items; a small percentage reduction in their inventory costs has a large impact on the organization. In contrast, we often use simpler & lower cost methods to control the inventories of category C. Substantial reduction in inventory cost may produce only small overall saving under this category. Timely inventory records and accuracy are the essential components of inventory management system. It is obvious that actual number of inventory may differ from the number of units reported by the inventory records. In order to understand the theme of ABC system, we need to take an example. For this we assume gold, kerosene and coal in group A, B and C respectively. The procedures of controlling the stock, assuming gold (A), kerosene (B) and coal (C) are given below: Group Percentage of items Percentage of cost A (Gold) B (Kerosene) C (Coal) 10% 0% 70% 70% 0% 10% Let, a store has,000 items of consumption and a monthly expenditure in consumption of Rs. 10,00,000. According to above report, 00 items will have a consumption of Rs. 7,00,000 on gold and 400 items ill account for Rs.,00,000 on kerosene and finally 1,400 items consume material worth Rs. 1,00,000 only on coal. The significance of this analysis is that a very close control is exercised over the items of gold, which account for a high percentage of costs and very little control would be enough for category coal. A tentative weight given to items and value under each group of ABC inventory system can be shown in the figure below:

7 Cum % of value INVENTORY SYSTEM CHAPTER Cummulative % of Item 10% 30% P 10% 0% 70% 0% Q Cummulative % value A 70% B 90% of 100% 70% 10% C 100% Here in the figure above upper triangle PQR refers to the value of inventory and lower triangle OPR refers to the item of inventory. Again, Y A B C Where, Value = Periodic consumption unit cost. Benefit of ABC inventory system is listed below: Inventories are stored to a minimum level. Materials are acquired or produced when needed. Production scheduling and material tracking e.g. how much in hand, the status of outstanding orders, and the schedule for production. Expand short-term capacity or reschedule production and how to reallocate among time periods to stay within capacity limits. Identification of production bottlenecks and capacity problems. Reduction in selling price. Quality service to the customers. O Cumulative % of items X

8 9 / PRODUCTION AND OPERATIONS MANAGEMENT Example 1 From the information given below, draw a ABC inventory charts. Items Annual usage in units Unit cost (Rs) Solution: Calculation of sales value and assignment of ranks Again, Items Annual sales (in units) Unit cost (in Rs.) Sales value (in Rs.) = units rate Rank (as per sales value) Table showing classification of inventory(i.e. product) as per the percentage of sales value and items Rank Sales value Prob. of sales Group probability Items Percent value of each item =100% 10 Group probability Group A B C Now Total Table showing cumulative percentage value of sales and items.

9 Cumulative % of value INVENTORY SYSTEM CHAPTER Cumulative % Item 30% of P 30% 79.1% Q A Cumulative % of value 60% 30% 14.4% B 79.1% 100% 40% 6.5% C 93.5% O R 100% Again, plotting the above cumulative percentage of probability of sales value and items into graph. Y A B C Cumulative % of items 100 X Example From the following information, construct the ABC diagram. Items Units Unit Price (Rs) Items Units Unit Price (Rs) , , , , , , Solution Working out usage (consumption) value and ranking the items in descending order of consumption value. Items Units Unit Price (Rs) Items Units Unit Price (Rs) , , , ,

10 Percentage of Total Inventory Value 94 / PRODUCTION AND OPERATIONS MANAGEMENT ,800 4,000 6, ,900 1, Ranks Item No. Usage percentage The curve can be drawn as follows: ,30 5,700,000 1,00 1, ,165 8,165,54 Cumulative Data , Cumulative Value % Number Cumulative Percentage Quantity (76) (43) A B C (16) (39) Percentage of Total Number of Items Hence items, 11 are classified category A items requiring outmost attention and items 1, 5 and 1 as category B items for moderate control. Other items fall under C category.

11 Max. Inventory INVENTORY SYSTEM CHAPTER Inventory Models Organizations spend lot of money in materials. Material cost represent 0 to 60 percent of the cost of production, even a small saving in material will reflect in profit. Idle scarce material resource is called inventory. Since we invest lot of money in materials and if materials are idle for long time, it is not good for the health of the organization. So it is must to exercise a control over idle scares resources, otherwise lot of money is tied in inventory. Inventory models help to find out the order quantity which minimizes the total costs. Inventory models are classified into following four parts as shown below: Basic inventory model Production run model A Inventory Model when Shortages are permitted Inventory Model with Price Discounts MODEL 1: Basic inventory model (Economic order quantity with instantaneous stock replenishment): The purpose of the study of this model is to find an optimum order quantity (EOQ) denoted by Q* so that the total inventory cost is minimized. It shows the behavior of an inventory system which operates some common assumptions. At the beginning of the cycle time we start with a maximum amount of inventory equal to the order quantity Q. As this amount is used up, the level of inventory drops at a fixed rate equal to the demand rate A. When it reaches a reorder level ROL, enough inventory is available to cover expected demand during the lead time LT. At this, an order is placed equal to Q which arrives at the end of lead time, when the inventory level reaches zero. This amount is placed in stock all at once and the inventory level goes up to its maximum value. This inventory system is characterized in terms of the following assumptions: Assumptions 1. Annual demand (A) rate is constant and known.. Production rate is infinite. 3. Lead time (LT) is constant and known with certainty and independent of demand. 4. Cost per unit (P) is constant. 5. Unit costs of carrying inventory (C) and ordering (O) are known and constant (inventory has one stock points). Graphical representation of the model is shown in figure below. Quality Consumption rate Average inventory Time

12 96 / PRODUCTION AND OPERATIONS MANAGEMENT Let D be the annual demand (units per year) Here, C 0 = Ordering costs per order) Figure: Basic inventory model. C h = Inventory carrying costs per unit per unit year C p = Price per unit Q = Order quantity Q* = Economic order quantity N = Number of orders placed per annum T c = Total cost per annum Annual ordering cost = Number of orders Ordering cost per order = Annual demand Order quantity Ordering cost per order = D Q C0 (i) Annual inventory carrying cost Annual total cost (TC) = Average inventory investment Inventory carrying cost = Max. inventory Minimum inventory Inventory carrying cost = QCh. (ii) = Annual ordering cost + Annual total cost = DC0 Q + QCh To determine economic order quantity (EOQ), Differentiating equation (iii) w.r. t. Q, we get dtc dq = d dq d DC 0 Q 1 0 = DC 0 dq Q 0 = DC 0 DC 0 Q = Ch Q* = + d 1 Q + Ch DC 0 C h dq + Ch Q* is economic order quantity. QC h d dq Q. (iii) (setting equals to zero) Note: If the inventory carrying cost is expressed as a percentage of annual average inventory investment, then Q* = DC 0 CpI where, it is expressed as a percentage of annual inventory investment. The optimal number of orders placed per annum.

13 INVENTORY SYSTEM CHAPTER Annual demand n* = Economic order quantity = D Q* Optimal time interval between two orders Number of working days in a year T* = N* Minimum total yearly inventory cost is given by T cm = DC0 Q 0 + Q*Ch..(iv) Substituting for Q* from Eq n. (iv) T cm = DC 0 DC 0 C h + DC 0 C Ch h T cm = DC 0 C h Example 3 XYZ Company requires 1,000 units of materials annually. If ordering costs are Rs 50 per order, expected lead time is 5 days, unit cost is Rs 5 per unit and annual inventory holding costs are charged at 0% and the company operates 50 days a year, compute EOQ and ROL. Solution Given, We have, Annual demand (D) = 1,000 units Ordering costs (O) = Rs 50 per order Lead time (L) = 5 days Annual holding cost per unit = 0% of unit cost = = Rs 5 Number of working days in a year = 50 days EOQ = = DO C = units per order ROL = Daily demand Lead time + Safety stock Annual demand = Number of working days Lead time + Safety stock = = 40 units MODEL-II Production run model (economic order quantity when stock replenishment is noninstantaneous) This model is applicable when inventory continuously builds up over a period of time after placing an order or when the units are manufactured and used (or sold) at a constant rate. This model is also based on the assumptions in previous model except that of instantaneous replenishment (supply). This is because of the fact that in many situations, the amount ordered is not delivered all at once,

14 Stock level 98 / PRODUCTION AND OPERATIONS MANAGEMENT but available at a finite rate (i.e. given supply rate) per unit of time, i.e. order is either produced internally or will be supplied at some uniform rate, p. This Situation can arise in case of an order being filled by a machine having finite production rate. Graphically it can be illustrated below: Quality Max. inventory level Assumptions 1. Annual demand rate (A) is constant and known.. The production rate (p) per unit time is greater than demand rate (d). 3. Lead time (LT) is constant and known with certainty and independent of demand. 4. Cost per unit (P) is constant. 5. Unit costs of carrying inventory (C) and ordering (O) are known and constant. 6. Supply is continuous and constant until Q units are supplied to stock, then it stops. 7. Production begins immediately after production set-up. Inventory under this situation, builds at the rate of (p - d) units and inventory is maximum at the end of production period t p. Maximum inventory at the end of production run = (p - d) t p (p - d) tp Average inventory = Now, the quantity produced during production period'(q) = P t p tp T tc Time t p = Q p Substituting for t p, Average inventory = (P D) Q P = Q 1 D P Annual inventory carrying cost Annual set-up cost = Average inventory Inventory carrying cost = Q 1 D P Ch = Number of set-up Set-up cost per set-up = D Q C0 Total annual cost (TC) = DC0 Q + Q 1 D P Ch (i)

15 INVENTORY SYSTEM CHAPTER To determine, the economic batch (Lot) size differentiate Eq n. (i) w.r.t. Q, be the first derivative is equals to zero. dtc Q* = where Q* is economic batch quantity. dq = 1 1 D P DC 0 (1 D/P) C h Ch DC0 Q = 0 Note: If the inventory carrying cost is expressed as a percentage of annual inventory investment, then Q* = DC 0 (1 D/P) C h I Optimal total cost is given by TC = D C 0 C h (1 - D/P) Optimal number of production runs where / is expressed in %. Annual demand N* = Economic batch quantity (EBQ) = D Q* Example 4 The annual demand of a product is 100,000 units. The rate of production is 00,000 units per year. The set up cost per production run is Rs 500 and the variable production cost of each item is Rs 10. The annual holding cost per unit is 0% of its value. Find the optimum production lot size and the length of the production run. Solution Given, D = 100,000 P = 00,000 C h = 0. and C 0 = 500 Optimum production lot size is given by Q(EOQ) = = DC 0 P C h P D = 10,000 units Length of the production runs (P N) = Q P = = 0.05 year MODEL III: Inventory Model with Price Discounts Then items are bought in large quantities, the supplier often gives discounts. However, if the material is purchased to take advantage of discount, the average inventory level and so the inventory carrying costs ill increase. Benefits for the purchaser from large orders are lower cost per unit, lower shipping and transportation cost, reduced handling cost and reduction in ordering costs due to less number of orders. These benefits are to be compared with the increase in carrying costs. As the order size or increases, more space should be provided to stock the items. As with other inventory models discussed so far, the overall objective will be to minimize the total variable cost. As the discount quantity increases the material cost decreases, but the carrying cost

16 Inventory cost (Rs) 100 / PRODUCTION AND OPERATIONS MANAGEMENT increases because the order are large. Thus, the major trade-off when considering quantity discount is between the reduced material cost and the increased carrying cost. When we include the cost of the material, the equation for the total annual inventory cost becomes: Total variable cost = Material cost + Ordering cost + Carrying cost Total variable cost = DC + A Q C0 + Q Ch A decision is, therefore, to be taken whether the buyer should stick to economic order quantity or increase the same to take advantage that, at large quantities, the production costs per price are lower (economies of scale) and, hence, part of the savings can be passed on to the customer. TC TC TC Carrying cost Ordering cost O Q(d1) Qoptimal Q(d) Quantity discount Figure: Quantity discount. Example 6 An aircraft company uses rivets at a constant rate of 5,000 per year. Each unit cost Rs 30 and the company personnel estimate that is costs 130 to place an order, and that carrying cost of inventory is 10 percent per year. How frequently should orders be placed? Also determine the optimum size of each order. Solution Given, Aircraft company uses rivets at a constant rate (D) = 500 per year Each unit cost (C*) = Rs 30 Ordering cost per order (C 0) = 130 Carrying cost of inventory is 10% per year Carrying cost (C h) = = Rs 3 EOQ = DC 0 C h = Number of order (N) = 5 times = 465 units Periodic Model In a fixed time period system, inventory is counted only at particular times, such as every week or every month. Counting inventory and placing orders periodically is desirable in situations such as when vendors make routine visits to customers and take orders for their complete line of products, or when buyers want to combine orders to save transportation costs. Other firms

17 INVENTORY SYSTEM CHAPTER operate on a fixed time period to facilitate planning their inventory count; for example, Distributor X calls every two weeks and employees know that all Distributor X s product must be counted. Fixed time period models generate order quantities that vary from period to period, depending on the usage rates. These generally require a higher level of safety stock than a fixed order quantity system. The fixed order quantity system assumes continual tracking of inventory on hand, with an order immediately placed when the reorder point is reached. In contrast, the standard fixed time period models assume that inventory is counted only at the time specified for review. It is possible that some large demand will draw the stock down to zero right after an order is placed. This condition could go unnoticed until the next review period. Then the new order, when placed, still takes time to arrive. Thus, it is possible to be out of stock throughout the entire review period, T, and order lead time, L. Safety stock, therefore, must protect against stockouts during the review period itself as well as during the lead time from order placement to order receipt. Fixed Time Period Model with Safety Stock In a fixed time period system, reorders are placed at the time of review ( T ), and the safety stock that must be reordered is Safety stock = z σ T + L above shows a fixed time period system with a review cycle of T and a constant lead time of L. In this case, demand is randomly distributed about a mean d. The quantity to order, q, is where q = Quantity to be ordered T = The number of days between reviews L = Lead time in days (time between placing an order and receiving it) d = Forecast average daily demand z = Number of standard deviations for a specified service probability σ T + L = Standard deviation of demand over the review and lead time I = Current inventory level (includes items on order) Note: The demand, lead time, review period, and so forth can be any time units such as days, weeks, or years so long as they are consistent throughout the equation.

18 10 / PRODUCTION AND OPERATIONS MANAGEMENT Note: The demand, lead time, review period, and so forth can be any time units such as days, weeks, or years so long as they are consistent throughout the equation. In this model, demand ( d ) can be forecast and revised each review period if desired or the yearly average may be used if appropriate. We assume that demand is normally distributed. The value of z is dependent on the probability of stocking out and can be found using table value of Z attached in the appendix. Example Daily demand for a product is 10 units with a standard deviation of 3 units. The review period is 30 days, and lead time is 14 days. Management has set a policy of satisfying 98 percent of demand from items in stock. At the beginning of this review period, there are 150 units in inventory. How many units should be ordered? SOLUTION The quantity to order is q = d ( T + L ) + z σ T + L I = 10(30+ 14) + z σ T + L + L 150 Before we can complete the solution, we need to find σ T + L and z. To find σ T + L we use the notion, as before, that the standard deviation of a sequence of independent random variables equals the square root of the sum of the variances. Therefore, the standard deviation during the period T + L is the square root of the sum of the variances for each day: The z value for P =.98 is.05. The quantity to order, then, is q = d ( T + L ) + z σ T + L I = 10(30+ 14) +.05(19.90) 150 = 331 units = = 19.9 To ensure a 98 percent probability of not stocking out, order 331 units at this review period WORKED OUT EXAMPLES Example 7 The Prajwal Company requires 1000 units per month through the year at constant rate. If ordering costs are Rs 50 per order, unit cost of item is Rs 5 and annual inventory holding cost are charged at 0%. Then, determine EOQ for the time. Solution Given, Monthly requirements (d) = 1000 units per month Annual requirement of company (D) = = 1,000 Ordering cost per order (C 0) = Rs 50 Carrying or holding cost (C h) = Rs = 5 EOQ = DC 0 C h = = units

19 INVENTORY SYSTEM CHAPTER Price or unit costs are denoted by P or C. Example 8 XYZ company busy in lots of 500 boxes which a 3 months supply. The cost per box is Rs. 15 and the ordering cost is Rs 150. The inventory cost is estimated at 5% of unit value. a. What is the annual cost of he existing inventory policy? b. How much money could be saved by employing the economic order quantity? Solution Given, Ordering cost (C 0) = Rs. 150 Quantity of one order = 500 Annual demand (D) = 500 1/3 =,000 Carrying Cost (C c) = Rs. 15 5% = Rs. 5 a. Total annual cost of existing inventory = D Q C0 + Q Ch = = Rs 6,800 b. Economic order quantity = or Q(EOQ) = DC 0 C h = 1500 Minimal annual cost = D C 0 C h = = Hence saving = 6,850 3,873 = Rs,977. Example 9 Mohan is a confectioner. He buys plastic boxes in bulk and uses them to pack his chocolates. His annual requirement of boxes is 1000, and each box costs him Rs. 30. He has estimated that his ordering costs are m Xs. 10 per order and his carrying costs are 0%. How many boxes should he order at a time so as to minimise his expenses? Solution Given, D = 1000 boxes; C h = Rs. 6 (Rs. 30 x 0.); C 0 = Rs. 10 Now, EOQ = DC 0 C h = = 63 boxes Example 10 If the supplier sells the boxes only in lots of 5, should we buy 50 or 75? Solution In order to make a decision on this problem, we should calculate the total in both cases. The cost of the inventory can be ignored as it remains the same. If Q = 50 Total annual cost (TC) If Q = 75 Total annual cost (TC) = D Q C0 + Q Ch = = D Q C0 + Q Ch = Rs 390

20 104 / PRODUCTION AND OPERATIONS MANAGEMENT = = Rs 385 As the total cost of buying lots of 75 is lesser, we should buy in lots of 75. Example 11 A contractor has to supply 10,000 paper cones per day to a textile unit. He finds that when he starts a production run, he can produce 5,000 paper cone per day. The cost of holding a paper cone in stock for one year is paisa and set up cost of production run is Rs 18. How frequently should the production run be made? Solution Given, Let us assume that the production runs on 300 days in a year D = 10,000 C 0 = Rs 18 C h = 0.0 per year P = 5,000 d = 10,000 Q(EOQ) = = DC 0 P C h P D Frequency of production = = 4 days = 95,000 units Example 1 Yamaha Motors Ltd. has been offered a particular component part it uses according to the following discount pricing schedule provided by the supplier. Order size Unit cost The company uses 700 of the components annually, the annual carrying cost is Rs 14 per unit, and the ordering cost is Rs 75. Determine the amount the firm should order. Solution Given, Annual demand (D) = 700 components Ordering cost (C 0) = Rs 75 Carrying cost (C h) = Rs 14 Discount category Order size Discount Unit cost We have, 1 3 EOQ 1 = DC 0 C n = =

21 Here, EOQ = EOQ 3 = DC 0 C n = DC 0 C n = = = INVENTORY SYSTEM CHAPTER EOQ 1 = belongs to order size 1 199, therefore, optimal quantity is On the other hand, EOQ and EOQ 3 are also respectively which does not belongs to order size and 600 +, so both are disqualified. Hence, 00 and 600 is considered as optimal quantity for second and third category respectively. Thus, Now, EOQ 1 = EOQ = 00 EOQ 3 = 600 Computation of total cost for all categories For, EOQ 1 = and discount price = Rs 65 TC 1 = D Q C 0 + Q1 Ch + Cp D 1 = = Rs 47, For, EOQ = 00 and discount price = Rs 59 TC = D Q C 0 + Q Ch + Cp D = = Rs 43,66.50 For, EOQ 3 = 600 and discount price = Rs 56 TC 3 = D Q C 0 + Q3 Ch + Cp D 3 = = Rs 43,70.83 Decision: Since the minimum total annual cost is Rs 43,6.5 for second category where optimal quantity is 00 units. Therefore, the firm should order 00 quantities for optimality. Some operating parameters to control inventory System 1. Optimum number of orders to be placed in one year (N) D N = EOQ. Cycle Time between two order (CT) CT = No. of days in a year No. of orders = 365 D EOQ = 365 A EOQ

22 106 / PRODUCTION AND OPERATIONS MANAGEMENT 3. ROL (Re-ordering level) = (Daily Average Consumption Average LT) + SS Where, LT = Lead Time SS = Safety Stock or Buffer Stock Or, Maximum Consumption Maximum LT 4. Average stock level = EOQ + SS Or ½ (min. stock + Max. stock) 5. Maximum stock level (M) = SS + ordering quantity Example 13 A company uses annually 50,000 units of an item costing Rs It operates 50 days in a year and procurement times is 10 days. Each order costs Rs. 45 and inventory carrying cost is 15% of the annual average inventory value. If the safety stock is 500 units, find EOQ, Re-order level, minimum, maximum and average inventory levels. Solution Given Co = Rs.45; Ch = 0.15 of 1.0 = Rs. 0.18; D = 50,000. (i) Economic order quantity = = 5,000 units (ii) Average daily demand = = 00 units per day (for 50 days annual working) Lead time = 10 days Average DDLF = 00 x 10 =,000 units Safety stock = 500 units ROL = Safety stock + Average DDLF = 500 +,000 =,500 units Maximum Inventory = ,000 = 5,500 units Minimum Inventory = Safety stock = 500 units Average Inventory = / EOQ = 3,000 units. Theoretical Questions EXERCISE 8 1. What do you mean by inventory? What are the reasons for keeping inventory?. Explain the inventory system. 3. Discuss about the different types of inventory cost. 4. What is ABC inventory system? Describe the benefits of ABC system. 5. What is material handling? Explain its basic principles. 6. Write short notes on: a. Economic order quantity b. Basic inventory model c. Production run inventory model d. Inventory model with price discount

23 Practical Problems 1. What is the percentage change in EOQ when the annual demand is doubled?. What will be the percentage change in EOQ if 50% decreases in the ordering cost and the Holding cost? 3. What will be the % change in EOQ if 40% decrease in the Holding cost and 60% decreases in the ordering cost? 4. What will be the % change in EOQ if 40% decrease in the Holding cost and 60% increase in the ordering cost? 5. Compute total annual cost of inventory if number of order is 0 per order and optimum order quantity is 400 units with holding cost Rs. per unit. 6. A manufactures has to supply his customers with 400 units of his product per year. Shortages are not allowed and storage cost amounts to 1.0 Rs. per unit per year. The setup cost per run is Rs Find the optimal order quantity, optimum number of orders in a year, and cycle time in days. 7. A manufacturing company with stable demand for its product is managing its inventory system for EOQ 30 units. No of order per year are 1 and set up or order preparation cost is Rs. 50 per order. Find what value of the holding cost adjusts this system. Also find the Reorder Level, if lead time is 10 days. 8. A firm requires 18,000 units of raw materials in a period of 6 months. The carrying cost is 0% of cost. The set up cost per set up is Rs. 10. The cost per unit is Rs. 19. The average lead time is 5 days (Assuming 360 days in a year.) Required (a) Economic order quantity (b) Total annual cost (c) No of days between orders (i.e. cycle time) (d) Reorder level 9. A computer shop stocks and sells a mercantile brand of personal computer. It costs the store Rs. 450 each time it Places an order with the manufacturer for the personal computers. The annual cost of carrying the PCs in inventory is Rs The store manager estimates that annual demand for the PCs will be 100 units. Lead time is given as 10 days and assuming working days per annum is 300. Determine: (a) Economic order quantity (b) Re-order point (c) Number of orders 10. A firm consumes 50,000 units of an item per annum, each costing Rs. 10. The order costs are expected to be Rs.40 and inventory carrying costs 10% of the annual average inventory value. Find EOQ. If the company operates 50 days per year, the procurement time is 10 days and the buffer stock is 500 units, find the reorder level and average inventory level.

24 86 / PRODUCTION AND OPERATIONS MANAGEMENT 11. A company demands 50,000 units per quarter. The ordering cost is Rs.40 per order and carrying cost is 10%. The cost price of inventory is Rs. 10 per unit. The lead time for placing order is 5 days. Required: (a) Economic order quantity (EOQ) (b) No of orders (c) Cycle time (d) Total cost of inventory. (e) Reorder level (assuming 300 working days per annum) 1. S and P company requires 1000 units of materials monthly. If ordering costs are Rs. 50 per order, expected lead time is 5 days, unit cost is Rs. 5 per unit and annual inventory holding costs are charged at 0%, and the company operates 50 days a year: compute EOQ and ROL. 13. The production department of a food and beverage company requires 1800 units of raw materials for manufacturing a special food semi-annually. It has been estimated that the cost of placing an order is Rs. 36 and the cost of carrying inventory is 1.5% of the investment in the inventories. The price is Rs. 0 per unit. Purchase manager wishes to determine an ordering policy for the raw material. Recommend the manager in the following aspects: (a) The optimal lot size (b) The optimal reorder time. 14. Star electronics is a dealer of the Sony mobile sets. It has observed that the annual demand is about 768 sets and the annual cost of the holding a mobile set in stock is Rs. 30 where as an order placed for the sets costs Rs.0. Using the information below find: (a) Optimum order quantity. (b) Optimum number of the orders to be placed in a year's time. (c) Total minimum variable cost. 15. What are the main objectives of holding inventory in an organization? What are the major costs involved in holding inventory? The ABC requires 100 units per months though out the year at constant rate. If ordering cost are Rs. 50 per order, unit cost of he item is Rs. 5 and annual inventory holding cost are charged at 30%, then determine the EOQ for the item. 16. Nepal soft drink Co. has a soft drink product which has constant annual demand rate of 3000 cases and cost Rs. 00/case. If ordering cost are Rs. 0 and inventory holding cost are charged at 5%, what is the EOQ for this product? Also determine the cycle time (in days). 17. A manufacturer has to supply his customers with 600 units of his product per year. Shortages are not allowed and the storage cost amounts to Rs. 0.6 per unit per year. The set up cost per run is Rs. 80. Find the optimum run size and the minimum average yearly cost. 18. A manufacturing organization experienced constant annual demand of 10,000 unit of an item which cost Rs. 40 per unit. If the order cost is Rs. 0 per order and the holding cost is

25 INVENTORY SYSTEM CHAPTER % of the item costs determine the economic order quantity and the reorder level with no lead-time known. 19. The local supermarket buys cabbage each day to ensure really fresh produce. Each morning any cabbage that is left from the previous day is sold to a dealer that results it to farmers who use it to feed their animals. This week the supermarket can buy fresh cabbage for Rs 40 per box. The cabbage is sold for rs 100 a box and the dealer that sells old cabbage is willing to pay Rs 15 a box. Past history says that tomorrow s demand for cabbage averages 50 boxes with a standard deviation of 34 boxes. How many boxes of cabbage should the supermarket purchase tomorrow? 0. Next week, super discount airlines has a flight from Kathmandu to New Delhi that will be booked to capacity. The airline knows from past history that an average of 5 customers (with standard deviation of 15) cancel their reservation or do not show for the flight. Revenue from a ticket on the flight is Rs 150. If the flight is overbooked, the airline has a policy of getting the customer on the next available flight and giving the person a free round trip ticket on a future flight. The cost of this free round trip ticket averages Rs 500. Super discount consider the cost of flying the plane from Kathmandu to New Delhi a sunk cost. By how many seats should super discount overbook the flight? 1. KVS Pharmacy fills prescriptions for a popular children s antibiotic, amoxycilin. The daily demand for amoxycilin is normally distributed with a mean of 00 ounces and a standard deviation of 80 ounces. The vendor for the pharmaceutical firm that supplies the drug calls the drug-store s pharmacist every 30 days and checks the inventory of amoxycilin. During a call the druggist indicated the store had 60 ounces of the antibiotic in stock. The lead time to receive an order is four days. Determine the order size that will enables the drugstore to maintain a 99% service level.. Food place market stocks frozen pizzas in a refrigerated display case. The average daily demand for the pizzas is normally distributed, with a mean of 8 pizzas and a standard deviation of.5 pizzas. A vendor for a packaged food distributor checks the market s inventory of frozen foods every 10 days; during a particular visit there were no pizzas in stock. The lead time to receive an order is three days. Determine the order size for this order period that will result in a 98% service level. During the vendor s following visit there were 5 frozen pizzas in stock. What is the order size for the next order period? 3. The Mediterranean Restaurant stocks a red Chilean table wine it purchase from a wine merchant in a nearby city. The daily demand for the wine at the restaurant is normally distributed, with a mean of 18 bottles and a standard deviation of 4 bottles. The wine merchant sends a representative to check the restaurant s wine cellar every 30 days, and during a recent visit there were 5 bottles in stock. The lead time to receive an order is two days. The restaurant manager has requested an order size that will enable him to limit the probability of a stockout to 5%. 4. Daily demand for a product is 10 units with standard deviation of 3 units. The review period is 30 days and lead time is 14 days. Management has set a policy of satisfying 98 percentage of demand from item in stock. At beginning of this review period, there are 150 units in inventory. How many units should be ordered?

26 88 / PRODUCTION AND OPERATIONS MANAGEMENT 5. Wal-Mart would like to develop an inventory ordering policy of a 95% probability of not stocking out. To illustrative your recommended procedure, use as an example the ordering policy for white percale sheet. Demand during white percale sheets is 5000 per year. The store is open 365 days per year. Every two weeks (14 days) inventory is counted and a new order is placed. It takes 10 days for the sheets to be delivered. Standard deviation of demand for the sheets is five per day. There are currently 150 sheets on hand. How many sheets should you order? 6. Acer company is planning to make his monthly (every 30 days) trek to Banglore to pick up supply chips. The trip will take to chip about two days. Before he leaves, he calls for the order to the supply store. He uses chip at an average rate of five per day ( seven days per week) with a standard deviation of one per day. He needs a 98% service probability. If he currently has 35 chip in inventory, how many should he order? What should be the most he will ever have to order? 7. Daily demand for a product is 100 units, with standard deviation of 5 units. The review period is 10 days and the lead time is 6 days. At the time of review there are 50 units in stock. If 98% service probability is desired, how many units should be ordered? 8. University drug pharmaceuticals order its antibiotics every two weeks (14 days) when a sales person visit from one of the pharmaceutical companies. Tetracycline is one of its most prescribed antibiotics, with average daily demand of 000 capsules. The standard deviation of daily demand was derived from examining prescriptions filled over past three months and was found to be 800 capsules. It takes five days for the order arrives. University drug would like to satisfy 99 percentages of prescriptions. The salesperson just arrived and there are currently 5000 capsules in stock. How many capsules should be ordered? 9. After graduation, you decide to go into a partnership in an office supply store that has existed for a number of years. Walking through the store and stock room, you find a great discrepancy in service levels. Some spaces and bins for items are completely empty; others have supplies that are covered with dust and have obviously been there a long time. You decided to take on the project establishing consistent levels of inventory to meet customer demands. Most of your supplies are purchased from just a few distributors that call on your store once every two weeks. You choose, as your first item for study, computer printer paper. You examine the sales record and purchase order and find the demand for the past 1 months was 5000 boxes. Using your calculator you sample some days demand and estimate that the standard deviation of daily demand is 10 boxes. You also search out the figures. Cost per box of paper is Rs 110, Desired service probability of 98 percentage, store is open every day, sales person visits every two weeks and Delivery time following visit is three days. Using your procedure, how many boxes of paper would be ordered if, on the day the salesperson call, 60 boxes on hand? 30. Dave s auto supply custom mixes paint for its customers. The shop performs a weekly inventor count of the main colors that are used for mixing paint. Determine the amount of white paint that should be ordered using the following information. Average weekly demand: 0 gallons, standard deviation of demand 5 gallons/week, desired service probability 98%, current inventory 5 gallons and lead time 1 week.

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