Managing Flow Variability through Safety Inventory

Question 1

• During 2007, Ninentdo’s game system Wii was hard to get due to supply shortages. Analysts estimate that the company was leaving close to $1.3 billion on the table in unmet demand.

techspot. com, December 17, 2007
* Mumbai’s real estate is said to be hot property. However, in the last quarter, sales have dipped so low that builders are getting worried … At the current pace of consumption, it will take two years and four months to exhaust this stock. This is alarming because, a healthy market is supposed to have only an eight month inventory pile-up.

MumbaiMirror.com, February 8, 2011

Answer 1

All examples in the previous slide represent situations where actual demand has exceeded (or fallen short of) supply due to unpredictable variability in demand.

• Although some variation in demand is predictable (e.g. due to trend or seasonality), much of it results from unpredictable random factors called noise.
• Forecasting is an effort to deal with noise. Forecasting might be subjective, i.e. based on judgement and experience, or objective, i.e. based on data analysis.

Question 2

Forecasting:

Answer 2

Forecasts are usually (always) inaccurate (wrong).

• Forecasts should be accompanied by a measure of forecast error such as standard deviation.
• Aggregate forecasts are more accurate than individual forecasts.
• Long-range forecasts are less accurate than short-range forecasts.

Question 3

Postponement (or delayed differentiation) is …

Answer 3

The practice of reorganizing a process in order to delay the differentiation of a generic product to specific end-products closer to the time of sale.

Which of these two processes would you choose for a garment manufacturer that makes blue, green and red T-shirts?

• Process A: colouring fabric, which takes one week, and assembling t-shirt, which also takes one week
• Process B: Assembling t-shirt from white fabric, which takes one week, and then colouring the assembled t-shirt, which also takes one week.

Question 4

Safety Inventory:

Answer 4

Due to demand variability (and the fact that forecasts are always wrong), we will end up having either

• Excess inventories (supply exceeding demand) inventory holding cost and products going obsolete.
• Shortage (demand exceeding supply):
  
  • orders backlogged
  • orders lost

Inventory in excess of average (forecast) demand is called safety inventory

Models exist that prescribe the optimal size of safety inventory to balance the cost of holding inventory against the cost shortage, but it is difficult to estimate the cost of shortage.

Question 5

Service Level Measure:

Answer 5

# * **_Cycle service level:_** the probability that there will be no stockout within a time interval, or equivalently the proportion of time intervals without a stockout.
* **_Fill rate:_** the fraction of total demand satisfied from inventory on hand.

Pr⁡(Stockout) = # intervals with shortage/total # intervals

Cycle Service Level = 1 - Pr⁡(Stockout)

Question 6

A Continuous Review System

Answer 6

Inventory is tracked continuously, and Demand (and potentially lead time) is uncertain.

Two questions:

• How much to order:

Use EOQ model with average demand for R(Q^∗=√(2SR/H))

• When to order:

When the inventory level falls to a pre-specified re-order point.

Our objective is to specify the reorder point for achieving a given cyclic service level

Question 7

Continuous Review System Calculations

Answer 7

We use boldface to represent a random variable.

For a random variable X, we denote its mean (average) by X and its variance (standard deviation) by σ²_X (σ_X).

Uncertain demand is denoted by R units per time unit (day, week, year, etc), and uncertain lead time by L time units (the same unit as R), with corresponding averages R and L, and corresponding variances σ²_R and σ²_L.

When R or L (or both) are random, demand during the lead time will also be random, which we represent by LTD with mean LTD and variance σ²_LTD

Question 8

Safety Inventory Calculations:

Answer 8

ROP = LTD + I_{<span>safety</span>}

• I_safety may be positive or negative.
• The amount of safety inventory depends on the desired service level.

For example, If demand during lead time has a symmetric distribution, i.e. probability that it is larger or smaller than its average is 50%, and *I_{<em>safety</em>} = 0*, we have ROP = LTD and there is a50%chance of stockout and50%** chance of excess inventory.

Question 9

Normal Demand During Lead Time:

Answer 9

Lets assume demand during lead time follows a Normal distribution, i.e LTD _~ Normal (LTD, σ²_LTD)

Question 11

Service Level for Normal Demand

Answer 11

We then have

SL = Pr( LTD ≤ ROP ) = Pr( N(0,1) ≤ z)

where z = I_safety / σ_LTD .

So

SL = Pr(N(0,1)≤z)

for which Normal table can be used.

Question 12

Pros and Cons of Centralization:

Answer 12

• Advantage is reduced safety inventory but there are disadvantages due to longer response time and higher shipment costs.
• A hybrid network, where some parts of the market are served by a centralized facility, whereas others are served by local warehouses, might be optimal.

Question 13

Newsboy Model:

Answer 13

The model characteristics:

• Demand is uncertain (distribution of demand in known)
• Only one opportunity for placing orders
• No initial inventory
• Items left at the end of cycle most be marked down for sale, or even disposed of at a cost.

It is a common situation for items with limited lifecycle, e.g. fashion items, food products, journals and magazines, or Technology items

The situation in the introduction slides attached can be analysed using the Newsboy model

Question 15

Newsboy Model Notations:

Answer 15

In general, let R be the demand random variable, p the unit retail price, c the unit purchase cost, v the markdown price, and Q the size of order. Then

Expected Profit = Expected in-season sales + Expected markdown sales – Purchase Cost

Expected Profit = min⁡(Q,R)×p+max⁡(Q-R,0)×

•In general, let R be the demand random variable, p the unit retail price, c the unit purchase cost, v the markdown price, and Q the size of order. Then

Expected Profit = Expected in-season sales

+ Expected markdown sales – Purchase Cost

Expected Profit = min⁡(Q,R) × p + max⁡(Q-R,0) × v - Q×c

Question 16

A Simpler Way to Find Optimal Q:

Answer 16

• Marginal Benefit: the profit of selling one additional unit: MB =p-c
• Marginal Cost: the cost incurred for each unit left unsold. MC = c-v

Suppose we order Q items.

• The expected marginal benefit from ordering an additional unit is MB×P(R>Q)
• The expected marginal cost from ordering an additional unit is MC×P(R≤Q)

•If expected marginal benefit is larger than or equal to expected marginal cost, it is worth ordering an additional unit

Question 17

Optimality Equation:

Answer 17

The optimal order quantity is the smallest value Q^∗ such that:

P(R≤Q^∗) ≥ MB/(MB+MC)

If the demand distribution is continuous (can take non-integer values), the optimal order quantity is the value of Q^∗ that solves the following equation

P(R≤Q^∗) = MB/(MB+MC)