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**Topic 3 Operations Management (2) Inventory Models**

Prof. Upendra Kachru

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**WHY STUDY INVENTORY? All organizations have inventory**

Can be a sizable organizational asset Influences sales (revenue generation and customer relations) Influences production/ operations costs Large amounts reduces ROI Costs of having inventory Frequently the largest single expenditure Excesses can result in losses Prof. Upendra Kachru

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WHAT IS INVENTORY? Inventory is the stock of any item or resource used in an organization. Prof. Upendra Kachru

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**Inventory Categories Working Stock Safety Stock Anticipation Stock**

Pipeline Stock Decoupling Stock Psychic Stock Prof. Upendra Kachru 4

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**Functions of Inventory**

To protect against variations (fluctuations) in demand and supply To take advantage of batches and longer production runs To provide flexibility to allow changes in production plans in view of changes in demand etc. To facilitate intermittent production To take advantage of price discounts by bulk purchases To meet anticipated and current demand Prof. Upendra Kachru

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**Functions of Inventory**

Excess Demand from Inventory J F M A M J J A S O N D Demand Capacity Requirement without Inventory Capacity Requirement with Inventory Excess Production to Inventory To smooth production requirements from seasonality

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**Functions of Inventory**

Department 1 Department 2 Department 3 Department 4 Interstage Inventory D1 Interstage Inventory D2 Interstage Inventory D3 To decouple different components of the internal inventory-distribution system.

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**Raw Materials & Supplies**

(Purchasing) Supply INVENTORY FUNCTIONS Safety Stock In-Process Goods (Production) Decoupling Stock Finished Goods (Production) Anticipation Stock To meet anticipated demand (Marketing) Demand Psychic Stock Prof. Upendra Kachru 8

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**Significance of Inventory**

Functional Area Functional Responsibility Inventory Goal Inventory Inclination Marketing Sell the Product Maximize customer service High Production Make the Product Efficient lot sizes Purchasing Buy required materials Low cost per unit Finance Provide working capital Efficient use of capital Low Engineering Design the product Avoid obsolescence Prof. Upendra Kachru

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**Typical Response Area Marketing / Sales Production Purchasing Finance**

revenue generation customer relations Typical Response I can’t sell without adequate stocks. I can’t keep our customers if we continue to stockout and there is not sufficient product variety If I can produce larger lot sizes, I can reduce per unit cost and function efficiency. I can reduce our per unit cost if I buy large quantities in bulk. Where am I going to get the funds to pay for the inventory? The levels should be lower. I am out of space. I can’t fit anything else in the building. Area Marketing / Sales Production Purchasing Finance Warehousing Production efficiency cost of operations cost of operations Finance liquidity return on investment Logistics Handling Capacity Space Prof. Upendra Kachru

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**Inventory Objectives Balancing Objectives 1. Provide customer service**

2. Support plant efficiency 3. Minimize inventory investment Maximize Customer Service Minimize Inventory Investment Operating Efficiency Prof. Upendra Kachru 11

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**Inventory – PLANNING & CONTROL**

CONSTRAINTS Mgt. policies Working capital Space Plant capacity INPUTS OUTPUTS OPERATIONS PLANNING Forecasts Demand rates Production rates Stock-on-hand Backorders Lead times Product structures DECISION RULES 1. What to order? 2. When to order? 3. How much? 4. From whom? INVENTORY PLANNING AND CONTROL Purchase Order / Set-up Holding Stockout COSTS Prof. Upendra Kachru

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Inventory Costs Inventory Costs are additive Prof. Upendra Kachru

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**Inventory Costs Holding (or carrying) costs**

Ordering Costs/ Setup (or production change) costs Shortage or Stock-out Costs Inventory Costs Prof. Upendra Kachru

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**Stock-out Costs External Shortage**

1. Present Profit Loss (potential sales) 2. Backorder Costs 3. Future Profit Loss (goodwill erosion) Internal Shortage 1. Lost Production (idle people / machines) 2. Substitute Cost (alternate) 3. Overtime / Extra Shift Cost 4. Delay Project Completion Date Prof. Upendra Kachru

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Inventory Metrics Average Inventory Investment: The rupee value of a company’s average level of inventory is one of the most common measures of inventory. Inventory Turnover Ratio: It is a ratio that measures how many times during a year the inventory turns around. Inventory turnover = annual cost of goods sold/average inventory investment Prof. Upendra Kachru

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Inventory Metrics Days of Inventory: This measure is an indication of approximately how many days of sales can be supplied solely from inventory. Days of inventory = avg. inventory investment/ (annual cost of gods sold/days per year) Days of inventory = days per year/ inventory turnover rate Prof. Upendra Kachru

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**Inventory Control by Classification Systems**

The inventory of a medium sized business organization would comprise thousands of items, each item with different usage, price, lead time and specifications. There could be different procurement and technical problems associated with different items. In order to escape this quagmire many selective inventory management techniques are used. Prof. Upendra Kachru

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**Vilfredo Pareto’s 80-20 rule.**

The ABC classification is based on focusing efforts where the payoff is highest; i.e. high-value, high-usage items must be tracked carefully and continuously. Typically only 20 percent of all the items account for 80 percent of the total rupee usage, while the remaining 80 percent of the items typically account for remaining 20 percent of the rupee value. The large value items constitute only 20 percent, the ABC analysis makes the task relatively easier. Prof. Upendra Kachru

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**TYPICAL ABC INVENTORY ANALYSIS**

20 40 60 80 100 120 C B A PERCENT OF TOTAL DOLLAR USAGE PERCENT OF TOTAL ITEMS A = HIGH VALUE ITEMS B = MEDIUM VALUE ITEMS C = LOW VALUE ITEMS Prof. Upendra Kachru

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**TYPICAL ABC INVENTORY ANALYSIS**

80 PERCENT OF RUPEE VALUE 60 40 B 20 C 20 PERCENT OF ITEMS 40 60 Prof. Upendra Kachru

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**RELATIVE ANALYSIS OF ABC CLASSIFICATIONS**

Item Degree of Type of Records Lot Sizes Frequency of Size of Safety Control Review Stocks A Tight Accurate / Complete Low Continuous Small B Moderate Good Medium Occasional Moderate C Loose Simple Large Infrequent Large Prof. Upendra Kachru

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**ABC EXCEPTIONS 1. Difficult Procurement Items 2. Short Shelf Life**

3. Large Storage Space Requirements 4. Item’s Operational Criticality 5. Likelihood of Theft 6. Difficult Forecast Items Prof. Upendra Kachru

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**Other Classification Systems**

Title Basis Main Uses ABC (Level of Usage) Value of consumption raw material components and work-in progress inventories HML (High, medium, low usage) Unit price of the material Mainly to control purchase. FSND (Fast, Slow moving, Non moving, Dead ) Consumption pattern of the component Control obsolescence. SDE (Scarce, difficult, easy to obtain items) Problems faced in procurement Lead time analysis and purchasing strategies GOLF (Government, Ordinary, Local, Foreign) Source of the material Procurement strategies VED (Vital, Essential, (Desirable) Criticality of the component To determine the stocking levels of spare parts. SOS (Seasonal, Off-seasonal) Nature of suppliers Seasonal items like agriculture products XYZ ( Value of Stock) Value of items in storage To review the inventories and their use scheduled intervals. Prof. Upendra Kachru

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INVENTORY MODELS E(1) Independent Demand Dependent Demand subassemblies, raw materials, etc) Finished product Component parts Inventory systems are predicated on whether demand is derived from an end item or is related to the item itself. There are two types of models that are used in the case of independent demand: Single Period Models, and Multiple Period Models. Prof. Upendra Kachru

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SINGLE PERIOD MODELS Prof. Upendra Kachru

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**Single-Period Inventory Model**

Single-Period Inventory Models are a special case of periodic inventory systems. One time purchasing decision (Example: vendor selling food at Siababa temple) Seeks to balance the costs of inventory overstock and under stock It is used for a wide variety of service and manufacturing applications Prof. Upendra Kachru 7 27

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**Problem and Solution F(0.9)= ŷ+1.282σ f(z) z z***

After prayers at the Siababa temple on Thursdays, people go to a vendor to eat food. The vendor has collected data over a few months that show, on an average, 100 meals were sold with a standard deviation of 10 meals. If our vendor wants to be 90 percent sure of not running out of food each Thursday, how many meals should he prepare? If we assume that the distribution is normal and the vendor prepared food for exactly 100 persons, the risk of food running out would be 50 percent. The demand would be expected to be less than 100 meals 50 percent of the time, and greater than 100 the other 50 percent. To be 90 percent sure of not falling short, he needs to prepare more food. From the “standard normal distribution“, we can find out that he needs to have additional food to cover standard deviations. In order to ensure that he is 90 percent sure having sufficient food: The number extra food required would be x 100 = 128.2, or 129 meals. Prof. Upendra Kachru

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**Single-Period Inventory Model**

If Co = Cost per unit of demand overage, Cu = Cost per unit of demand underage, The probability that the unit will be sold is ‘P’; Here (1-P) is the probability of the service/ product not being sold. The expected marginal cost equation can be represented as: P * Co < (1-P) * Cu Solving for P, we obtain P < [Cu / (Co +Cu)] Prof. Upendra Kachru

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**The Classical Newsvendor’s Problem**

A newspaper vendor is faced with the problem of deciding how many newspapers to order daily so as to maximize the daily profit. Daily demand (d) for newspapers is a random variable. No reordering is possible during a day, If the newsvendor orders fewer papers than customers demand he or she will lose the opportunity to sell some papers. If supply exceeds demand, the vendor will be stuck with papers which cannot be sold. Prof. Upendra Kachru

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Demand Data Based on observations over several weeks, the vendor has established the following probability distribution of daily demand: The vendor purchases daily papers at Rs.2 and sells them at Rs. 5 apiece. Leftover papers are valueless and are discarded (i.e. no salvage value). Demand d Probability P(d) Cumulative Prob. F(d) = P(D d) 35 or less 36 37 38 39 40 41 42 43 44 45 46 or more 0.00 0.05 0.07 0.08 0.15 0.20 0.10 0.03 0.02 0.12 0.35 0.50 0.70 0.85 0.95 0.98 1.00 Prof. Upendra Kachru

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The vendor identifies two penalty costs which he/she will incur, regardless of his/her decision: Cost of Overage CO = Purchase Price - Salvage Value = c - s For each paper overstocked the newsvendor incurs a penalty cost of: CO = Rs – Rs.0.00 = Rs Cost of Underage CU = Selling Price - Purchase Price = p - c For each paper understocked the newsvendor incurs a penalty (opportunity) cost of: CU = Rs – Rs = Rs. 3.00 Prof. Upendra Kachru

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**Consider the decisions: **

Assume that there is already a policy in place to order a certain number of papers daily, say 38. Consider the decisions: D1 : Continue the present policy: Stock 38 papers. D2 : Order one more paper: Stock 39 papers. The possible events are: E1 : The 39th paper sells (i.e. demand 39 = demand > 38). E2 : The 39th paper does not sell (i.e. demand 39 = demand 38). Prof. Upendra Kachru

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**To Stock or Not to Stock! The expected payoff is:**

Item 39 will not sell on a given day only if demand on that day is for 38 or fewer items: P(D 38) = F(38) = 0.20. The probability that an item will not sell is the cumulative probability associated with the previous item. Item 39 will sell on a given day only if demand on that day is for 39 or more items: P(D 39) = 1 - P(D 38) = 1 - F(38) = = 0.80. The expected payoff is: Rs. 3(0.8) + (- Rs. 2)(0.2) = Rs. 2. This implies an increase in profit of Rs as compared to the alternative decision which has a payoff of Rs He should stock the 39th paper. Prof. Upendra Kachru

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**Inventory Management Game**

Product Price Sell Don’t Sell (Overage) Stockout (Underage) Burger 18.00 7.00 9.00 11.00 Pizza 23.00 12.00 17.00 Patties 8.00 4.00 6.00 Samosa 2.00 3.00 Sandwich 10.00 5.00 Pastry 3.50 Hotdog Prof. Upendra Kachru

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**MULTI PERIOD INVENTORY MODELS**

Prof. Upendra Kachru

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**Multi-Period Inventory Models**

Fixed-Order Quantity Models: Event triggered (Example: running out of stock) Fixed-Time Period Models: Time triggered (Example: Monthly sales call by sales representative) Time T1 T2 Inventory Level ‘Q’ Prof. Upendra Kachru Prof. Upendra Kachru

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**Fixed order Quantity and Fixed-Time Period Differences**

Feature Fixed-order quantity Model Fixed-Time Period Model Order quantity The same amount ordered each time Quantity varies each time order is placed When to place order Reorder point when inventory position dips to a predetermined level Reorder when the review period arrives Record keeping Each time a withdrawal or addition is made Counted only at review period. Size of inventory Less than fixed-time period model Larger than fixed-order quantity model Time to maintain Higher due to perpetual record keeping Type of items Higher-priced, critical, or important items. Prof. Upendra Kachru

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**Fixed Order Quantity Models**

Economic Order Quantity (EOQ) models, due to simplicity and versatility, are fixed order quantity models used for material planning. When independent demand is the most important issue, the EOQ model provides a solution to the problem. Prof. Upendra Kachru

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**The Inventory Cycle Q Usage rate Lead time**

The inventory cycle determines when an order should be placed and how much should be ordered so as to minimize average annual variable costs. Profile of Inventory Level Over Time Q Usage rate Quantity on hand Reorder point Place order Lead time Receive order Time Receive order Prof. Upendra Kachru

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**The EOQ Model The basic assumptions in the EOQ Model are as follows:**

The rate of demand for the item is deterministic and is a constant ‘D’ units per annum independent of time. Lead time is zero or constant and it is independent of both demand as well as the quantity ordered. Price per unit of product is constant Inventory holding cost is based on average inventory Ordering or setup costs are constant 41 Prof. Upendra Kachru Prof. Upendra Kachru

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**COST MINIMIZATION GOAL**

By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs average inventory level: The holding cost per unit: The setup cost per unit: The production cost per unit: Total Cost C O S T Holding Costs QOPT Annual Cost of Items (DC) Ordering Costs Order Quantity (Q) 42 Prof. Upendra Kachru 11 42

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**BASIC FIXED-ORDER QUANTITY (EOQ) MODEL FORMULA**

TC=Total annual cost D =Demand P =Cost per unit Q =Order quantity A =Cost of placing an order or setup cost R =Reorder point L =Lead time H = v*r =Annual holding and storage cost per unit of inventory Total Annual = Cost Annual Purchase Cost Annual Ordering Cost Annual Holding Cost + + TC = P*D + D*A / Q + Q*v*r / 2 Prof. Upendra Kachru 12 43

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**The EOQ We also need a reorder point to tell us when to place an order**

Prof. Upendra Kachru 13 44

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EOQ Model Problem A company, for one of its class ‘A’ items, placed 8 orders each for a lot of 150 numbers, in a year. Given that the ordering cost is Rs. 5,400.00, the inventory holding cost is 40 percent, and the cost per unit is Rs Find out if the company is making a loss in not using the EOQ Model for order quantity policies. What are your recommendations for ordering the item in the future? And what should be the reorder level, if the lead time to deliver the item is 6 months? ‘D’ = Annual demand = 8*150 = 1200 units ‘v’ = Unit purchase cost = Rs ‘A’ = Ordering Cost = Rs ‘r’ = Holding Cost = 40% Prof. Upendra Kachru

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**QEOQ = √ (2*A*D /r*v) = 900 units.**

TC= = √ 2*5400*1200*0.40*40 Using the Economic Order Equation: QEOQ = √ (2*A*D /r*v) = 900 units. Minimum Total Annual Cost (TC) = √ 2*A*D*r*v = Rs. 14,400.00 The Total annual Cost under the present system = Rs. 45,000.00 The loss to the company = Rs. 45,000 – Rs. 14,400 = Rs. 30,600.00 Reorder Level = Ro = L*D = (6/12)* 1200 = 600 units The company should place orders for economic lot sizes of 900 units in each order. It should have a reorder level at 600 units. = Rs. (1200*5400/ *40*150/2) = Rs. (43, ) Prof. Upendra Kachru

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**Total Costs with Purchasing Cost**

EOQ TC with Purchasing Cost TC without Purchasing Cost Purchasing Cost Quantity Adding Purchasing cost doesn’t change EOQ Prof. Upendra Kachru

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**Total Cost with Constant Carrying Costs**

OC EOQ Quantity Total Cost TCa TCc TCb Decreasing Price CC a,b,c Prof. Upendra Kachru

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Problem Novelty Ltd carries a wide assortment of items for its customers. One item, Gaylook, is very popular. Desirous of keeping its inventory under control, a decision is taken to order only the optimal economic quantity, for this item, each time. You have the following information. Make your recommendations: Annual demand : 1,60,000 units Price per unit : Rs.20 Carrying cost : Re.1 per unit or 5 per cent Cost per order : Rs. 50 Determine the optimal economic quantity. Prof. Upendra Kachru

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Solution Order per year Size Average inventory Carrying cost (Re.1) Ordering cost (Rs.50 per order) Total cost per year 1 1,60,000 80,000 50 10 16,000 8,000 500 8,500 40 4,000 1,000 5,000 80 2,000 100 1,600 800 5,800 The optimum economic quantity (lot size) for this item is 4,000 numbers. Prof. Upendra Kachru

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What-if Show that changing the order quantity by a small amount has very little effect on the cost. Prof. Upendra Kachru

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Quantity Discounts Quantity discounts, which are price incentives to purchase large quantities, create pressure to maintain a large inventory. For any per-unit price level, P, the total cost is: Total annual cost = Annual holding cost + Annual ordering or setup cost + Annual cost of materials C = (H) (A) + PD Q 2 D Prof. Upendra Kachru

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**Total cost curves with purchased materials added**

Quantity Discounts C for P = Rs.4.00 C for P = Rs.3.50 C for P = Rs.3.00 PD for P = Rs.4.00 P = Rs.3.50 P = Rs.3.00 EOQ 4.00 EOQ 3.50 EOQ 3.00 First price break Second price break Total cost (Rupees) Purchase quantity (Q) Total cost curves with purchased materials added EOQs and price break quantities Prof. Upendra Kachru

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**Finding Q with Quantity Discounts**

Step 1. Beginning with the lowest price, calculate the EOQ for each price level until a feasible EOQ is found. It is feasible if it lies in the range corresponding to its price. Step 2. If the first feasible EOQ found is for the lowest price level, this quantity is the best lot size. Otherwise, calculate the total cost for the first feasible EOQ and for the larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal. Prof. Upendra Kachru

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Problem A supplier for Apollo Hospital has introduced quantity discounts to encourage larger order quantities of a special catheter. The price schedule is: Order Quantity Price per Unit 0 – 299 Rs 300 – 499 Rs 500 or more Rs Annual demand (D) = 936 units Ordering cost (A) = Rs. 45 Holding cost (H) = rv = 25% of unit price Prof. Upendra Kachru

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**Step 1: Start with lowest price level:**

EOQ = 2DS H 2(936)(45) 0.25(57.00) = = 77 units Not feasible EOQ = 2DS H 2(936)(45) 0.25(58.80) = = 76 units Not feasible EOQ = 2DS H 2(936)(45) 0.25(60.00) = = 75 units Feasible This quantity is feasible because it lies in the range corresponding to its price. Prof. Upendra Kachru

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Step 2: The first feasible EOQ of 75 does not correspond to the lowest price level. Hence, we must compare its total cost with the price break quantities (300 and 500 units) at the lower price levels (Rs and Rs.57.00): C = (rv) (A) + PD Q 2 D C75 = [(0.25)(Rs )] (Rs. 45) + Rs (936) 75 2 936 C75 = Rs. 57,284 C300 = [(0.25)(Rs )] (Rs. 45) + Rs (936) 300 2 936 = Rs. 57,382 C500 = [(0.25)(Rs.57.00)] (Rs.45) + Rs (936) 500 2 936 The best purchase quantity is 500 units, which qualifies for the deepest discount. = Rs. 56,999 Prof. Upendra Kachru

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© 2007 Pearson Education Decision Point: If the price per unit for the range of 300 to 499 units is reduced to Rs , the best decision is to order 300 catheters. Annual Demand 936 Ordering Cost Rs Holding Cost 25% Price EOQ Inventory Cost Order Cost Purchase Cost Total Cost Rs 75 Rs Rs. 56,160 Rs. 57,284 Rs 300 Rs. 2175 Rs Rs. 54,288 Rs. 56,603 Rs 500 Rs. 3563 Rs Rs. 53,352 Rs. 56,999 This shows that the decision is sensitive to the price schedule. A reduction of slightly more than 1 percent is enough to make the difference in this example. Prof. Upendra Kachru

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**QUANTITY DISCOUNTS Advantages Disadvantages**

Lower unit cost Higher holding costs Lower ordering costs Larger inventory investment Fewer stockouts Older stock Price increase hedge Slow inventory turnover

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**Fixed-Time Period Models**

In many retail merchandising systems, a fixed- time period system is used. Sales people make routine visits to customers and take orders. Inventory, therefore, is counted only at particular times. Fixed-time period models generate order quantities that vary from period to period, depending on the usage rates. Prof. Upendra Kachru

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Fixed Period Model Prof. Upendra Kachru

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**T = Time between orders I = Existing Inventory L = Lead Time**

Q = Order Size Total Annual Cost = Purchase Cost + Ordering Cost + Holding Cost

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Prof. Upendra Kachru

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**Accounting for Safety Stock:**

Order Quantity = Average demand over the vulnerable period + safety stock - Inventory currently on hand Accounting for Safety Stock: Where z = Number of standard deviations for a specified service probability σT + L= Standard deviation of demand over the review and lead time Prof. Upendra Kachru

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**Fixed Time Period System - Advantages**

Fewer orders are placed Purchase discounts more likely Lower shipping and freight costs Prof. Upendra Kachru

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**Requires storage space Incurs taxes Requires insurance **

Consumes capital Requires storage space Incurs taxes Requires insurance Can become lost, stolen, damaged, outdated, or obsolete Must be counted, sorted, verified, stored, retrieved, moved, issued, and protected Prof. Upendra Kachru 66

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**Classical Inventory Prblems**

Ever - increasing storage space needs Slow-moving materials Disposition of scrap, obsolete, & surplus materials Transaction recording errors Misplaced materials Prof. Upendra Kachru

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**RAW MATERIALS INVENTORY PROFILE**

Excess Stock Surplus / Idle Working Stock Safety Stock Outputs Inputs Nonproductive Productive Prof. Upendra Kachru

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**IN-PROCESS INVENTORY PROFILE**

Unreleased Orders Orders in Transit Orders in Temporary Storage Orders Waiting to be Worked Orders Being Inspected Orders Being Worked Backlog Nonproductive Productive Outputs Inputs Finished Goods In-Process Inventory Prof. Upendra Kachru

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**Inventory System Improvement**

1. Standardize Stock Items 2. Reduce Lead Times 3. Reduce Cycle Times 4. Use Fewer Suppliers 5. Inform Suppliers of Expected Demand 6. Contract for Minimum Annual Purchases 7. Buy on Consignment 8. Consider Transportation Costs 9. Order Economical Quantities 10. Control Access to Storage Areas 11. Obtain Better Forecasts Prof. Upendra Kachru

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**Inventory System Improvement**

12. Dispose of Excess Stock 13. Improve Record Accuracy (cycle count) 14. Improve Capacity Planning 15. Minimize Setup Times 16. Simplify Product Structures 17. Multishift operations 18. Continuous Improvement Prof. Upendra Kachru

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