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Topic 6. INVENTORY MANAGEMENT. I. Introduction. What is inventory? stored resource used to satisfy current or future demand Types of Inventories: Raw Materials/Components In-Process Goods (WIP) Finished Goods Supplies. Introduction. Inventory Related Costs:

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I introduction
I. Introduction

  • What is inventory?

    • stored resource used to satisfy current or future demand

  • Types of Inventories:

    • Raw Materials/Components

    • In-Process Goods (WIP)

    • Finished Goods

    • Supplies


Introduction
Introduction

  • Inventory Related Costs:

    • Holding Cost -- cost to carry a unit in inventory for a length of time (annual), includes interest opportunity cost, insurance, taxes, depreciation, obsolescence, deterioration, May be expressed as a percentage of unit price or as a dollar amount per unit


Introduction1
Introduction

  • Inventory Related Costs (continued):

    • Order Cost -- Cost of ordering and receiving inventory, Include determining how much is needed, preparing invoices, shipping costs, inspecting goods upon receipt for quantity and quality, Generally expressed as a fixed dollar amount, regardless of order size

    • Inventory may also influence purchasing cost

      • Inventory is costly


Introduction2
Introduction

  • Inventory Related Costs (continued):

    • Shortage Cost-- result when demand exceeds the inventory on hand, Include the opportunity cost of not making a sales, loss of customer goodwill, late charges, and in the case of internal customers, the cost of lost production or downtime, difficult to measure, thus may have be subjectively estimated


Introduction3
Introduction

  • Why Hold Inventories?

    • Meet anticipated demand

      • Lead time – the time period between place an order until receive the order

      • Average lead time demand is considered as anticipate demand

    • Protect against stock-out

      • Safety stock – more than average lead time demand inventory


Introduction4
Introduction

  • Why Hold Inventories (continued)?

    • De-couple successive operations - separate production from distribution

      • Wine production and inventory

    • Smooth production process

      • Snowmobile production and inventory

    • Buy/Produce in economic lot sizes - take advantage of quantity discounts

    • Hedge against price increases


Introduction5
Introduction

  • JIT Inventory – minimum inventory needed to keep a system running, small lot sizes

    • Advantages

      • lower inventory costs

      • easy to identify problems and potential problems

    • Disadvantages

      • requires accurate timing and cooperation

      • breakdowns stop everything


Introduction6

High

A

Annual

$ volume

of items

B

C

Low

Few

Many

Number of Items

Introduction

  • Inventory Classification

    • Identify important

      Items and more inventory

      control on important items

    • Measure of importance:

    • ABC analysis:

      • A = 70-80% of total inventory value, but only 15% of items

      • B = 15-25% of total inventory value, but 30% of items

      • C = 5% of total inventory value, but 55% of items


Introduction7

0

214800 232087768

Introduction

  • Monitor Inventory

    • As important as demand forecast for decision making

    • Universal Product Code - Bar code printed on a label that hasinformation about the item to which it is attached

    • Cycle counting: taking physical counts of items and reconciling with records on a continual rotating basis, regular inventory audits, ABC approach


Introduction8
Introduction

  • Inventory Systems

    • Objective: minimize annual total inventory cost and maintain satisfied service level.

      • service level: probability of no shortage

        Total Inventory Cost is not Inventory Cost

      • Annual total inventory cost (TC) = annual product cost + annual inventory cost

      • Annual product cost = annual demand * unit price

      • Annual inventory cost = annual holding cost + annual setup (order) cost + annual shortage cost


Introduction9
Introduction

  • Possible performance measures

    • customer satisfaction

      • number of backorders/lost sales

      • number of customer complaints

    • inventory turnover

      • ratio of annual cost of goods sold to average inventory investment

    • days of inventory

      • expected number of days of sales that can be supplied from existing inventory


Introduction10
Introduction

  • Requirements for Effective Inventory Management :

    • A system to keep track of the inventory on hand and on order

    • A classification system for inventory items

    • A reliable forecast of demand that includes an measure of forecast error

    • Reasonable estimates of inventory holding costs, ordering costs, and shortage costs

    • Knowledge of lead times and lead time variability


Introduction11
Introduction

  • 1. Continuous (Perpetual) Review System: (event-triggered)

    • Monitor the inventory level all the time, order a fixed quantity (Q) when the inventory level drops to the reorder point (ROP)

    • Calculate: Q and ROP

      • Re-Order Point (ROP) – an inventory level when actual inventory drops to it will trigger an activity of re-order.


Introduction12
Introduction

  • 2. Periodic Review System: (time-triggered)

    • Place an order every fixed period T. Each time bring the current inventory to a target level M

    • Calculate: T and M

  • 3. Advantages and Disadvantages?


Introduction13
Introduction

  • Dependent and Independent Demand:

    • Dependent demand: derived demand, lumpy (subassemblies and components)

      • cars

    • Independent demand: from customer side, smooth (end items and finished goods)

      • tires


Ii inventory models on order quantity
II. Inventory Models On Order Quantity

  • Model Basics (consider as annual)

    • Total Cost (TC) = Product Cost + Inventory Cost

      Inventory Cost = Holding Cost

      + Setup (Order) Cost + Shortage Cost

      TC = Product Cost + Holding Cost

      + Setup (Order) Cost + Shortage Cost


Inventory models on order quantity
Inventory Models On Order Quantity

  • Product Cost = Annual Demand * Unit Price

  • Holding Cost = average inventory level * Holding Cost per unit per year

  • Ordering Cost = # of orders * Setup Cost per order

    • # of orders = annual demand / order quantity

  • Shortage Cost = Shortage Cost per unit

    * average # of shortage per year

    Best Order Quantity = a quantity that minimizes TC


Inventory models on order quantity1
Inventory Models On Order Quantity

EOQ Model (Economic Order Quantity), Fixed-Order-Quantity Model

  • Assumptions

    • There is one product type

    • Demand is known and constant

    • Lead time is known and constant

    • Receipt of inventory is instantaneous (one batch, same time)

    • Shortage is not allowed


Eoq model continued
EOQ Model (continued)

Q

Reorder

point

Place

order

Receive

order

Receive

order

Place

order

Receive

order

Lead time


Eoq model continued1
EOQ Model (continued)

  • Notation and Terminology

    • Q = order quantity(# of pieces per order)

    • Q0 = Economic Order Quantity (EOQ)

    • D = demand for the time period considered (units per year)

    • S = setup/order cost ($ per order)

    • H= holding cost per unit per year ($ per unit per year)

      • in general proportional to the price, H = I*P


Eoq model continued2
EOQ Model (continued)

  • Notation and Terminology (continued)

    • I = Interest rate (expanded) (% per year)

    • P= unit price ($ per unit)

    • IC = inventory cost = setup cost + holding cost

    • TC = IC+ product cost

      Find Out EOQ


Eoq model continued3
EOQ Model (continued)

  • Average Inventory Level =

  • Holding Cost =

  • Number of orders per year =

  • Setup (Order) Cost =

  • Shortage Cost = 0, why?


Eoq model continued4
EOQ Model (continued)

  • Product Cost =

  • IC =

  • Total Cost (TC) =

  • Minimize TC Minimize IC, why?


Eoq model continued5
EOQ Model (continued)

  • Observation: at the best order quantity EOQ (Q0),

    holding cost = setup cost

    Solve Q0, we have


Eoq model continued6
EOQ Model (continued)

The Inventory Cost Curve is U-Shaped

Annual Cost

Annual

Carrying Costs

Annual

Ordering Costs

QO

(EOQ)

Order Quantity (Q)


Eoq model continued7
EOQ Model (continued)

  • Example:

    Annual demand = 10,000 unit/year, ordering cost = $50/order, unit cost (price) = $4/unit, expanded interest rate = 25%/year. EOQ? TC at EOQ?


Eoq model continued8
EOQ Model (continued)

  • Sensitivity of IC with related to Q

    -- Example (continued)


Eoq model continued9
EOQ Model (continued)

  • Conclusion:

    • 1. Inventory cost curve is flat around EOQ

    • 2. Flatter when Q increases than when Q decreases from EOQ

  • Thinking Challenge:

    • If the order quantity Q = 2*EOQ, by how much IC will increase?


Eoq model continued10
EOQ Model (continued)

  • Sensitivity of EOQ with related to D, H, S, P, I

    • 1. Insensitive to parameter change

    • 2. Directions?


Epq model
EPQ Model

EPQ (Economic Production Quantity) Model: Fixed Order Quantity Model with Incremental Replenishment

  • Problem description:

  • Assumptions

    • There is one product type

    • Demand is known and constant

    • Receipt of inventory is gradual and at a constant replenishment (production) rate

    • Shortage is not allowed


Epq model continued
EPQ Model (continued)

Production rate

- usage rate

Q

Quantity

on hand

Usage

rate

Reorder

point

Time

Start to

produce

Start to

produce

Finish

production

Production run length


Epq model continued1
EPQ Model (continued)

  • Notation and Terminology

    • Qp = production quantity(# of pieces/production run)

    • Qp0 = Best production quantity (EPQ)

    • p = daily production rate (units per day)

    • d = daily demand rate (units per day)

    • D = demand rate (units per year)

    • S = production setup (order) cost($ per setup)

    • H = holding cost per unit per year (again H = I*P in general)

    • T= production run length = Q/p


Epq model continued2
EPQ Model (continued)

  • Maximum Inventory Level =

  • Average Inventory Level =

  • Annual Holding Cost =


Epq model continued3
EPQ Model (continued)

  • Number of production runs per year =

  • Order Cost =

  • IC =

  • TC =

  • Minimize TC Minimize IC, why?


Epq model continued4
EPQ Model (continued)

  • Observation: at EPO,

    holding cost = setup cost

  • Best Production Quantity (EPQ) formula:


Epq model continued5
EPQ Model (continued)

  • Remarks: EPQ > EOQ (why?)

  • Example: D=2000 unit/year, S=$5/setup, H=$0.4/unit/year, p=100 unit/day, 200 working days/year. Find the best production batch size and the # of production runs/year.


Eoq with discount
EOQ with discount

EOQ with Discount Model:

  • Assumptions: same as with EOQ, plus discount on all units

  • Terminology

    • Price breaks: the smallest order quantity to receive a discount price

    • Feasibility: the order quantity matching the claimed price is feasible, otherwise infeasible.


Eoq with discount continued
EOQ with discount (continued)

  • Example:

    Order Price

    0-399 $2.1/unit

    400-699 $2.0

    Great equal 700 $1.9

  • Idea is to compare TC curves under different prices - why TC?


Eoq with discount continued1

Total Cost Curve

for Price 1

Total Cost Curve

for Price 2

Total Cost Curve

for Price 3

Order Quantity

EOQ with discount (continued)

$ cost

400

700


Eoq with discount continued2

Total Cost Curve

for Price 1

Total Cost Curve

for Price 2

Total Cost Curve

for Price 3

Order Quantity

EOQ with discount (continued)

$ cost

400

700


Eoq with discount continued3
EOQ with discount (continued)

  • Observations:

    • EOQ with a lower price, if feasible, is better than any order quantity with the same or higher price.

    • Potential best order quantity: cheapest feasible EOQ, price breaks associated with lower prices.


Eoq with discount continued4
EOQ with discount (continued)

  • Solution Procedure:

    • 1. Find the feasible EOQ with cheapest possible price.

    • 2. Calculate TCs of the EOQ (from Step 1) and price breaks above EOQ.

    • 3. Pick the order quantity with lowest TC


Eoq with discount continued5
EOQ with discount (continued)

  • Example (continued) Annual demand = 10,000 unit/year, order cost = $5.5/order. Assuming holding costs are proportional to unit prices and annual interest rate = 20%. Find the best order quantity.


Iii models on reorder points when to order
III. Models on Reorder Points - When to Order?

  • Models on Reorder Points - When to Order?

    • Find ROP (Re-Order Point)

  • ROP depends on:

    • Lead Time: time between placing and receiving an order

    • Demand Distribution: how uncertain

    • Desired Service Level: probability of no shortage = 1-P(s), where P(s) = probability of shortage


Models on reorder points when to order continued
Models on Reorder Points - When to Order ? (continued)

  • Constant Demand Rate:

    • Constant daily demand rate = d, Lead time = L days

      ROP = d * L = Lead time demand

  • Remark:

    • no uncertainty in demand

    • service level = 100%

    • safety stock = 0


Models on reorder points when to order continued1
Models on Reorder Points - When to Order ? (continued)

  • Variable Demand with Stable Average Rate

  • How continuous review system works?

    • Lead time demand: demand during the lead time

    • ROP Lead time demand ==>

    • ROP < Lead time demand ==>

    • ROP = Average lead time demand + Safety Stock = m + SS


Models on reorder points when to order continued2
Models on Reorder Points - When to Order ? (continued)

  • Remarks:

    • Higher the desired service level --->

    • More uncertain the demand --->

  • Two methods to determine the SS


Models on reorder points when to order continued3
Models on Reorder Points - When to Order ? (continued)

  • 1. Determine SS and ROP based on shortage cost inf. (if available)

    • SS increases  Holding cost ? Shortage cost ?

    • Best SS minimizes total inventory cost


Models on reorder points when to order continued4
Models on Reorder Points - When to Order ? (continued)

  • 1. Determine SS and ROP based on shortage cost inf. (continued)

    -- Example: Consider a light switch carried by Litely. Litely sells 1,350 of these switches per year, and places order for 300 of these switches at a time. The carrying cost per unit per year is calculated as $5 while the stock out cost is estimated at $6 ($3 lost profit per switch and another $3 lost in goodwill, or future sales loss). Find the best SS level and ROP for Litely.


Models on reorder points when to order continued5
Models on Reorder Points - When to Order ? (continued)

Determine SS and ROP based on shortage cost inf. (continued)

  • 1. Determine SS and ROP based on demand inf. during each lead time period:


Models on reorder points when to order continued6
Models on Reorder Points - When to Order ? (continued)

  • Determine SS and ROP based on shortage cost inf. (continued)

  • If SS = 0, ROP = m = 15 switches


Models on reorder points when to order continued7
Models on Reorder Points - When to Order ? (continued)

Determine SS and ROP based on shortage cost inf. (continued)

  • # of orders per year =

  • For no safety stock, Litely has the following shortage table. Why?


Models on reorder points when to order continued8
Models on Reorder Points - When to Order ? (continued)

  • Determine SS and ROP based on shortage cost inf. (continued)

  • Determine the best SS in following table


Models on reorder points when to order continued9
Models on Reorder Points - When to Order ? (continued)

  • 2. Determine ROP and SS based on lead time demand distribution and desired service level:


Models on reorder points when to order continued10
Models on Reorder Points - When to Order ? (continued)

  • Case 1. Empirical Lead time demand distribution

  • -- Example:


Models on reorder points when to order continued11
Models on Reorder Points - When to Order ? (continued)

  • Find R and SS to achieve the service level of 85% and 95%, respectively.


Models on reorder points when to order continued12
Models on Reorder Points - When to Order ? (continued)

  • Case 2. Lead time demand is Normally distributed with (m, )

  • SS = , ROP = m + SS, z = single tail normal score of desired service level.

    ( is the standard deviation)

  • Example:

  • Lead time demand is Normally distributed with mean = 4 and standard deviation = 3. Find ROP and SS to achieve the service level of 85% and 95%, respectively.



Homework additional problems
Homework (Additional problems) Problem)

  • Problem 1: A toy manufacturer uses approximately 36,000 silicon chips annually. The chips are used at a steady rate during the 240 days the plant operates. Annual holding cost is 50 cents per chip, and ordering cost (per order) is $25/order. Assume that each of their orders comes in one batch. Determine:

    • a. .the best order quantity

    • b. demonstrate that your order quantity is optimal by showing that annual ordering costs = annual holding costs

    • c. the average inventory level

    • d. the number of orders per year

    • e. the number of working days between orders (Hint: days between orders = # days in a year / # of orders per year. Why?)


Homework additional problems1
Homework (Additional problems) Problem)

  • Problem 2. The Dine Corporation is both a producer and a user of brass couplings. The firm operates 200 days a year and uses the couplings at a steady rate of 50 per day. Couplings can be produced at a rate of 150 per day. Inventory holding cost is estimated at $5 per unit per year. Machine setup costs are $40 per production run. Determine:

    • a. the best production run size

    • b. demonstrate that your production run size is optimal by showing that annual set up costs = annual holding costs (Hint: find the formula of holding and setup cost for EPQ model in my lecture note.)

    • c. the maximum inventory level (Hint: find the formula in the derivation of EPQ)

    • d. the number of production runs per year

    • e. the cycle time and the production time within each cycle (Hint: cycle time is given by Q/d and production time is given by Q/p. Why? Think before using the formula)


Homework additional problems2
Homework (Additional problems) Problem)

  • Problem 3

  • A small manufacturing firm used roughly 3,400 pounds of chemical dye each year. Currently the firm purchases 300 pounds per order and pays $3 per pound. The supplier has just announced that orders of 1,000 pounds or more will be filled at a price of $2.5 per pound. The manufacturing firm incurs a cost of $100 each time it submits an order and assigns an annual holding cost of 20% of the purchase price per pound.

    • a. determine the best order size that minimizes the total cost

    • b. if the supplier offered the discount at 2,500 pounds instead of at 1,000 pounds, what order size would minimize total cost?


Homework additional problems3
Homework (Additional problems) Problem)

  • Problem 4: A product is ordered four times every year. Inventory carrying cost is $20 per unit per year, and the cost of shortage for each unit is $40. Given the following demand probabilities during the reorder period


Homework additional problems4
Homework (Additional problems) Problem)

  • Problem 4 (continued)

    • a) What is the average lead time demand?

    • b) What would be the reorder point without safety stock?

    • c) What would be the probabilities of the following shortage levels if the company uses the reorder point without safety stock?


Homework additional problems5
Homework (Additional problems) Problem)

  • Problem 4 (continued)

    • d) Follow the Litely example in my lecture to find out the best safety stock level to minimize the total cost.

    • e) What is the reorder point to achieve the 95% service level? What is the associated safety stock? (Hint: you need to follow the example in my lecture note under Case 1)