University of Illinois at Chicago
College of Business Administration
Department of Information & Decision Sciences

IDS 355:   Operations Management I
Textbook:   Render & Heizer, 2nd ed.   (RH)
Prof. Stanley L. Sclove

Notes to Accompany RH Chapter 9: Inventory for Independent Demand
These notes Copyright © 1998 Stanley Louis Sclove

Chapter Outline

9.1. Functions of Inventory
9.2. Inventory Management
9.3. Inventory Models
9.4. Inventory Models for Independent Demand
9.5. Probabilistic Models with Constant Lead Time
9.6. Fixed-Period Systems

The "ven" part of the word "inventory" refers to motion (coming and going); so "inventory" literally means the amount of goods coming in. In the business context it refers to goods on hand, waiting to be used or sold.

9.1. Functions of Inventory

The ideal would be to get just enough goods there just when you need them, so you don't have to use up space storing extra items. This ideal is called "just-in-time" inventory control. Since it has usually proved impossible or impractical to attain this ideal, the costs defined in this chapter are pertinent.

9.2. Inventory Management

9.3. Inventory Models

9.3.1. Independent versus Dependent Demand

Chapter 9 considers independent demand; Chapter 10, dependent demand.

9.3.2. Holding, Ordering and Setup Costs

Holding or carrying cost is the cost of maintaining inventory, including the cost of space, heat and electricity, insurance and loss of interest if the money were in the bank instead of tied up in inventory. The cost of holding one unit for one year is denoted by H. The constant H is often expressed as a proportion I of the unit price P:   H = IP. The value of I in several examples in the book is .26 (26%), .40 (40%) and 1.00 (100%).

The Ordering or setup cost is the cost of placing an order; it is denoted by S.

9.4. Inventory Models for Independent Demand   pp. 311ff.

Two basic questions:   When to order and how much to order.

9.4.1. The Basic Economic Order Quantity (EOQ) Model

CALCULATING ANNUAL ORDERING AND HOLDING COSTS

A cost formula is developed. It consists of annual ordering cost plus annual holding cost. The quantity ordered is denoted by Q. The larger Q is, the smaller the annual ordering cost, but the larger the annual holding cost, so there is a trade-off.

9.4.2. Minimize Costs

SEEKING THE BEST REORDER POLICY

Q = number of pieces (units) per order

Q* = Optimum number of pieces per order   (Q* is the "EOQ".)

D = Annual demand in units for the inventory item

N = Number of orders placed per year   (N = D/Q)

S = Setup or ordering cost for each order

H = Holding or carrying cost per unit per year

A spreadsheet can be used. Since D = NQ, Q = D/N and N = D/Q, so a first column can be either a range of values of Q or a range of values of N.     Personally, I would use N, since it is easier to guess a suitable range for N than for Q. Subsequent columns contain the order cost, the holding cost, and their sum.

THE ECONOMIC ORDER QUANTITY MODEL

TC, the Total annual cost, is the sum of the Annual Holding and Order Costs:

TC   =   Annual setup cost + Annual holding cost.

1. Annual setup cost = NS = (D/Q)S

2. Annual holding cost = (Average inventory level)(Holding cost per unit per year) = (Q/2)(H)

3. The TC function takes the form a/Q + bQ + c with a, b, c>0. The constants a = DS and b = H/2 . Consequenty, the following fact can be used in this problem (and related ones): A function a/x + bx + c with a > 0 and b > 0 is minimized at x* = (a/b)1/2. Also, at x*, the two terms are equal:   a/x* = bx*.

This fact is used to derive the expression for the EOQ, Q*:   f(x) = f(Q) = a/Q + bQ; a/Q* = bQ*; multiplying by Q*, a = bQ*2, so Q* = (a/b)1/2 =   (2DS/H)1/2. The optimal number N* of times to order, is N* = D/Q*.

9.4.3. Reorder Points     p. 316

The lead time is the time between when an order is placed and when it arrives. One must allow for lead-time demand. The reorder point (ROP) is the inventory level at which an order is placed: The inventory is monitored, and when the inventory drops to the ROP, the order is placed. The ROP is the number of units which will be needed during the lead time: ROP = (Demand per day)(Lead time in days for the order to arrive) = d x L .

The equation for ROP assumes that demand is uniform and constant.   When this is not the case, extra stock, the safety stock, should be added.

9.4.4. Production Order Quantity Model

In a production environment, assume that production is at the rate p units per day and demand is at the rate d units per day, where p > d. The production order quantity (number of units per production run) is denoted by Qp. The same math as above applies to the determination of the optimal value Qp*. The holding cost is (H)(ave.inv.);   ave.inv. = (max inv.)/2; and it can be shown that max.inv = Q[(p-d)/p], i.e., Q(1-d/p). The total production cost is   Setup cost + Holding cost = a/Q + bQ, with
a = DS and b = H[1 - (d/p)]/2,

where
d = daily demand rate (usage rate)   and
p = daily production rate.

The optimal daily production rate Qp*, is   (a/b)1/2 = ( 2DS/{H[1 - (d/p)]})1/2 .

9.4.5. Quantity Discount Models     p. 320

The annual purchase cost APC is DP, where D is the annual demand and P is the unit price. A supplier may offer a discount, i.e., a unit price Pdiscount that is substantially less than the usual price P if the buyer buys a large quantity a one time. This may or may not be advantageous to the buyer. The total cost function is TC = AIC + APC, annual inventory cost plus annual purchase cost. You must compute the Total Cost when taking advantage of the discount to see whether it is really the less expensive option.

9.5. Probabilistic Models with Constant Lead Time

Probabilistic demand is incorporated into the evaluation of various alternative inventory control schemes. Two main things to focus on are: reorder point/reorder quantity models and (relatively simple) one-period inventory models.

The basic question is: How much inventory should a firm hold to provide reasonable protection against uncertainty? There is uncertainty in (1) the demand and (2) the lead-time (time between placing and receiving the order).

THE REORDER POINT/REORDER QUANTITY MODEL

To review the standard (r,Q) model, note that the symbol r denotes the reorder level; Q denotes the reorder quantity. The symbol ROP is used for r in RH.

The reorder quantity Q is determined as the EOQ.

The reorder level r would be set equal to the lead-time demand if demand were known with certainty. If demand is uncertain, the approach is to choose r large enough so that the stockout probability is small enough.
CHOICE OF r
If r is too small, stockouts can occur; if; if r is too large, too much safety stock is carried.
CHOICE OF r; UNIFORM LEAD-TIME DEMAND
The idea of stockout probability, p(s), helps define r. First a uniform lead-time demand is investigated.
SELECTING A PROBABILITY OF STOCKING OUT
AVERAGE STOCKOUTS PER YEAR This is dependent upon the number of orders per year as wellas the probability of a stockout per order.

THE BINOMIAL DISTRIBUTION AND THE NUMBER OF STOCKOUTS The binomial distribution can be used to calculate the probability of any given number of stockouts.

EFFECT OF ORDER SIZE ON STOCKOUTS As order quantity (Q) increases, the average number of stockouts per year decreases.
CHOICE of r: NORMAL LEAD-TIME DEMAND
The normal approximation of lead time is investigated. Changing the lead-time distribution changes the relation between r and p(s).
EXPECTED ANNUAL COST OF SAFETY STOCK
Associated with various choices for r will be a safety stock and a related expected annual holding cost.
USING SIMULATION TO CHOOSE r and Q
Simulation can be used on inventory problems.

9.6. Fixed-Period Systems

A firm might reorder at fixed times, e.g., once a month, rather than when the stock drops to the ROP. This fixed-period system avoids continuous monitoring of stock but can result in stockouts.
latest revision 4-July-1998