W13 service inventory (continued)

  INVENTORI

PROBABILISTIK

  Inventory Models

• Economic Order Quantity (EOQ)
• Special Inventory Models

With Quantity Discounts

  Planned Shortages

• Demand Uncertainty - Safety Stocks
• Inventory Control Systems Continuous-Review (Q,r) Periodic-Review (order-up-to)
• Single Period Inventory Model

  Ketidakpastian Ketidakpastian dalam inventori ● Demand :

  1. unused stock 2. stockout/ shortage ● Cost ● Lead time:

  1. unused stock 2. stockout/ shortage ● Supplied quantity

  

Reorder Point dengan Safety Stock

vel le y tor ven In

Reorder point Safety stock

  LT LT Time Model Persediaan dengan Demand Probabilistik dan LT ≠ 0 dan tetap ● Adanya LT membuat perlunya ditentukan

REORDER POINT: titik dimana pemesanan

harus dilakukan ● Demand probabilistik (Distribusi Normal) membuat terdapat kemungkinan persediaan habis sedangkan pesanan belum datang

  ● Untuk mengatasi dibuat SAFETY STOCK

  Demand Probabilistik ● Reorder Point besarnya sama dengan demand selama lead time: ROP = D

  ×LT

● Contoh: jika demand per tahun 10.000 unit; lead

time pemesanan selama 1 minggu; maka:

   ROP = demand selama 1 minggu  ROP = 1/52 x 10.000 = 192,3 ~ 193  Artinya jika persediaan mencapai 193 unit maka pemesanan harus dilakukan

● Reorder point tersebut belum memperhitungkan

besarnya Safety Stock

  

Demand selama Lead Time

all demand met shortages

  ROP LT ×D

  

Demand Probabilistik

● Safety stock dibuat untuk mengurangi kemungkinan out of stock (shortage)

● Dipengaruhi oleh lead time dan variansi demand

● Jika D adalah demand per unit waktu dan

   adalah standard deviasi, maka demand selama lead time adalah LT

  ×D, variansi demand selama lead time adalah 

  2 ×LT dengan standard deviasi adalah ( 

  2 ×LT)

  1/2 ● Safety stock ditentukan dengan perhitungan:

  SS = Z × Standard deviasi demand selama LT

  Demand Probabilistik (Uncertainty in Demand)

Keputusan persediaan yang harus dibuat adalah:

● Lot (jumlah) pesanan:

  D RC 2 

  Q 

  HC ● Saat pemesanan kembali:

  ROP  D  LT  Z   LT 

     

  Z

  1.70E-03

  1.18E-03

  1.22E-03

  1.26E-03

  1.31E-03

  1.35E-03

  3.0

  1.40E-03

  1.44E-03

  1.49E-03

  1.54E-03

  1.59E-03

  1.64E-03

  1.75E-03

  1.11E-03

  1.81E-03

  1.87E-03

  2.9

  1.93E-03

  1.99E-03

  2.05E-03

  2.12E-03

  2.19E-03

  2.26E-03

  2.33E-03

  2.40E-03

  2.48E-03

  2.56E-03

  1.14E-03

  1.07E-03

  2.64E-03

  6.64E-04

  4.34E-04

  4.50E-04

  4.67E-04

  4.84E-04

  3.3

  5.01E-04

  5.19E-04

  5.38E-04

  5.57E-04

  5.77E-04

  5.98E-04

  6.19E-04

  6.41E-04

  6.87E-04

  1.04E-03

  3.2

  7.11E-04

  7.36E-04

  7.62E-04

  7.89E-04

  8.16E-04

  8.45E-04

  8.74E-04

  9.04E-04

  9.35E-04

  9.68E-04

  3.1

  1.00E-03

  2.8

  2.72E-03

  4.04E-04

  9.14E-03

  6.76E-03

  6.95E-03

  7.14E-03

  7.34E-03

  7.55E-03

  7.76E-03

  7.98E-03

  8.20E-03

  2.4

  8.42E-03

  8.66E-03

  8.89E-03

  9.39E-03

  6.39E-03

  9.64E-03

  9.90E-03

  1.02E-02

  1.04E-02

  1.07E-02

  2.3

  1.10E-02

  1.13E-02

  1.16E-02

  1.19E-02

  1.22E-02

  1.26E-02

  1.29E-02

  6.57E-03

  2.5

  2.80E-03

  4.15E-03

  2.89E-03

  2.98E-03

  3.07E-03

  3.17E-03

  3.26E-03

  3.36E-03

  3.47E-03

  2.7

  3.57E-03

  3.68E-03

  3.79E-03

  3.91E-03

  4.02E-03

  4.27E-03

  6.21E-03

  4.40E-03

  4.53E-03

  4.66E-03

  2.6

  4.80E-03

  4.94E-03

  5.09E-03

  5.23E-03

  5.39E-03

  5.54E-03

  5.70E-03

  5.87E-03

  6.04E-03

  4.19E-04

  3.90E-04

  1.36E-02

  2.17E-05

  4.2

  1.40E-05

  1.47E-05

  1.53E-05

  1.60E-05

  1.67E-05

  1.75E-05

  1.82E-05

  1.91E-05

  1.99E-05

  2.08E-05

  4.1

  2.26E-05

  1.29E-05

  2.36E-05

  2.47E-05

  2.57E-05

  2.68E-05

  2.80E-05

  2.92E-05

  3.05E-05

  3.18E-05

  4.0

  3.32E-05

  3.46E-05

  3.61E-05

  3.76E-05

  1.34E-05

  1.23E-05

  4.09E-05

  6.28E-06

  3.62E-06

  3.79E-06

  3.97E-06

  4.16E-06

  4.35E-06

  4.56E-06

  4.77E-06

  5.00E-06

  5.23E-06

  5.48E-06

  4.4

  5.73E-06

  6.00E-06

  6.57E-06

  1.18E-05

  6.88E-06

  7.20E-06

  7.53E-06

  7.88E-06

  8.24E-06

  8.62E-06

  4.3

  9.01E-06

  9.43E-06

  9.86E-06

  1.03E-05

  1.08E-05

  1.13E-05

  3.92E-05

  4.26E-05

  3.76E-04

  2.33E-04

  1.47E-04

  1.53E-04

  1.59E-04

  3.6

  1.66E-04

  1.72E-04

  1.79E-04

  1.86E-04

  1.93E-04

  2.00E-04

  2.08E-04

  2.16E-04

  2.24E-04

  3.5

  1.36E-04

  2.42E-04

  2.51E-04

  2.60E-04

  2.70E-04

  2.80E-04

  2.91E-04

  3.02E-04

  3.13E-04

  3.25E-04

  3.37E-04

  3.4

  3.50E-04

  3.63E-04

  1.42E-04

  1.31E-04

  4.44E-05

  3.8

  4.63E-05

  4.82E-05

  3.9

  5.03E-05

  5.24E-05

  5.46E-05

  5.68E-05

  5.92E-05

  6.17E-05

  6.42E-05

  6.69E-05

  6.96E-05

  7.25E-05

  7.55E-05

  1.26E-04

  7.85E-05

  8.18E-05

  8.51E-05

  8.86E-05

  9.21E-05

  9.59E-05

  9.97E-05

  1.04E-04

  1.08E-04

  3.7

  1.12E-04

  1.17E-04

  1.21E-04

  1.32E-02

  1.39E-02

  0.01

  2.68E-01

  2.33E-01

  2.36E-01

  2.39E-01

  2.42E-01

  0.7

  2.45E-01

  2.48E-01

  2.51E-01

  2.55E-01

  2.58E-01

  2.61E-01

  2.64E-01

  2.71E-01

  2.27E-01

  2.74E-01

  0.6

  2.78E-01

  2.81E-01

  2.84E-01

  2.88E-01

  2.91E-01

  2.95E-01

  2.98E-01

  3.02E-01

  3.05E-01

  3.09E-01

  0.5

  2.30E-01

  2.24E-01

  3.16E-01

  1.84E-01

  1.52E-01

  1.56E-01 1.5 39E01

  1.59E-01

  1.0

  1.61E-01

  1.64E-01

  1.66E-01

  1.69E-01

  1.71E-01

  1.74E-01

  1.76E-01

  1.79E-01

  1.81E-01

  0.9

  2.21E-01

  1.87E-01

  1.89E-01

  1.92E-01

  1.95E-01

  1.98E-01

  2.01E-01

  2.03E-01

  2.06E-01

  2.09E-01

  2.12E-01

  0.8

  2.15E-01

  2.18E-01

  3.12E-01

  3.19E-01

  1.47E-01

  4.80E-01

  4.36E-01

  4.40E-01

  4.44E-01

  4.48E-01

  4.52E-01

  4.56E-01

  4.60E-01

  0.1

  4.64E-01

  4.68E-01

  4.72E-01

  4.76E-01

  4.84E-01

  4.29E-01

  4.88E-01

  4.92E-01

  4.96E-01

  5.00E-01

  0.0

  0.09

  0.08

  0.07

  0.06

  0.05

  0.04

  0.03

  

0.02

  4.33E-01

  4.25E-01

  3.23E-01

  3.71E-01

  3.26E-01

  3.30E-01

  3.34E-01

  3.37E-01

  3.41E-01

  3.45E-01

  0.4

  3.48E-01

  3.52E-01

  3.56E-01

  3.59E-01

  3.63E-01

  3.67E-01

  3.75E-01

  0.2

  3.78E-01

  3.82E-01

  0.3

  3.86E-01

  3.90E-01

  3.94E-01

  3.97E-01

  4.01E-01

  4.05E-01

  4.09E-01

  4.13E-01

  4.17E-01

  4.21E-01

  1.49E-01

  1.45E-01

  2.2

  3.67E-02

  1.9

  2.94E-02

  3.01E-02

  3.07E-02

  3.14E-02

  3.22E-02

  3.29E-02

  3.36E-02

  3.44E-02

  3.52E-02

  3.59E-02

  1.8

  3.75E-02

  2.81E-02

  3.84E-02

  3.92E-02

  4.01E-02

  4.09E-02

  4.18E-02

  4.27E-02

  4.36E-02

  4.46E-02

  1.7

  4.55E-02

  4.65E-02

  4.75E-02

  4.85E-02

  2.87E-02

  2.74E-02

  5.05E-02

  1.92E-02

  1.43E-02

  1.46E-02

  1.50E-02

  1.54E-02

  1.58E-02

  1.62E-02

  1.66E-02

  1.70E-02

  1.74E-02

  1.79E-02

  2.1

  1.83E-02

  1.88E-02

  1.97E-02

  2.68E-02

  2.02E-02

  2.07E-02

  2.12E-02

  2.17E-02

  2.22E-02

  2.28E-02

  2.0

  2.33E-02

  2.39E-02

  2.44E-02

  2.50E-02

  2.56E-02

  2.62E-02

  4.95E-02

  5.16E-02

  1.42E-01

  1.15E-01

  9.34E-02

  9.51E-02

  9.68E-02

  1.3

  9.85E-02

  1.00E-01

  1.02E-01

  1.04E-01

  1.06E-01

  1.08E-01

  1.09E-01

  1.11E-01

  1.13E-01

  1.2

  9.01E-02

  1.17E-01

  1.19E-01

  1.21E-01

  1.23E-01

  1.25E-01

  1.27E-01

  1.29E-01

  1.31E-01

  1.34E-01

  1.36E-01

  1.1

  1.38E-01

  1.40E-01

  9.18E-02

  8.85E-02

  5.26E-02

  1.5

  5.37E-02

  5.48E-02

  1.6

  5.59E-02

  5.71E-02

  5.82E-02

  5.94E-02

  6.06E-02

  6.18E-02

  6.30E-02

  6.43E-02

  6.55E-02

  6.68E-02

  6.81E-02

  8.69E-02

  6.94E-02

  7.08E-02

  7.21E-02

  7.35E-02

  7.49E-02

  7.64E-02

  7.78E-02

  7.93E-02

  8.08E-02

  1.4

  8.23E-02

  8.38E-02

  8.53E-02

  Probabilitas terjadi stockout = 0.0495 Z=1.65

Penentuan Nilai Z

  

Service level Stock Out Z value

Probability

  0.90

  0.10

  1.28

  0.95

  0.05

  1.65

  0.98

  0.02

  2.05

  0.99

  0.01

  2.33 0.9986 0.0014

  3.75

  

Contoh

Permintaan sebuah item berdistribusi normal dengan

rata-rata 1000 unit per minggu dan standard deviasi

200 unit. Harga item $10 per unit dan ongkos pesan

$100. Ongkos simpan ditetapkan sebesar 30% dari

nilai inventori per tahun dan lead time tetap selama 3

minggu. Tentukan kebijakan inventori jika diinginkan

service level 95%, dan berapakah ongkos untuk safety

stock-nya

  D = 1000 per minggu ( =200)

  UC = $10 per unit RC = $ 100 per pesan HC = 0.3 x $10 = $3 per unit per tahun LT = 3 minggu

  Contoh 1862 unit

•        

  3 100 52 1000

  2

  2

  HC D RC Q 3568 unit 568 3000

  3 200 64 . 1 1000

  3        

       LT Z D LT ROP 

   service level 95%, Z=1.64 (Lihat Tabel Distribusi Normal) Ongkos ekspektasi safety stock:

  Holding cost stock Safety  

  

Lead Time Probabilistik

(Uncertainty in Lead Time)

● LT lebih pendek maka akan muncul unused stock, namun jika LT lebih panjang maka muncul shortage

● Probabilitas shortage adalah probabilitas bahwa

demand selama lead time lebih besar daripada

reorder level, sehingga,

   Prob LT  D  ROP Service level

   

ROP

     Prob LT   

  

Contoh

Lead time untuk pemesanan sebuah produk

berdistribusi Normal dengan mean 8 minggu dan

standard deviasi 2 minggu. Jika permintaan konstan

sebesar 100 unit per minggu, berapakah kebijakan

pemesanan yang memberikan suatu service level

siklus 95%

  ROP   Prob LT   .

  95   D

   

Dari Tabel Normal, untuk probabilitas 95%  Z=1.64,

sehingga LT = 8 + (1.64x2) = 11.3 minggu (ROP=1130

  

D & LT Probabilistik

Jika demand mempunyai rata-rata D dan standard

   deviasi

  D dan, lead time mempunyai rata-rata LT dan standard deviasi LT

  D maka, demand selama lead time LT

  ×D dan standard deviasi

  2

  2

  2

  

Contoh (Uncertain in both LT dan D)

Permintaan sebuah produk berdistribusi normal dengan rata-rata 400 unit per bulan dan standar deviasi 30 unit per bulan. Lead time juga berdistribusi normal dengan rata-rata 2 bulan dan standar deviasi 0.5 bulan. Berapakah ROP yang memberikan service level 95%? Berapakah jumlah pemesanan kembali jika ongkos pesan $400 dan ongkos simpan $10 per unit per bulan?

  D = 400 unit per bulan  = 30 unit

  D LT = 2 bulan Contoh (Uncertain in both LT dan D) Demand selama LT = LT x D = 800 unit Standard deviasi demand selama lead time: Untuk service level 95%, Safety stock = 1.64 x 204.45 = 335 unit maka, ROP = LT x D + SS

  = 800 + 335 = 1135 unit         unit 204 45 .

  400 5 .

  30

  

2

  2

  2

  2

  2

  2

  2 

          LT D D LT

    Contoh (Uncertain in both LT dan D) Ukuran pemesanan optimal (ekonomis):

   RC  D

  2

• Q  HC

  2  400  400  10  179 unit Klasifikasi Inventori: ABC

• Manajemen persediaan sering kali harus dibedakan menurut karakteristik masing-masing item
• Salah satu klasifikasi yang umum digunakan pada manajemen persediaan adalah sistem ABC dimana item- item dikelompokkan menjadi tiga kelas
• Pembagian kelas ini didasarkan atas tingkat kepentingan masing-masing item

  Karakteristik A B C Persentase nilai 75 - 80%

  10

  5

• – 15% – 10% Persentase jumlah item

  15

  20

  60

• – 20% – 25% – 65%

Cara Melakukan Klasifikasi

  1. Tabulasikan nama, harga per unit, dan jumlah unit yang dikonsumsi per tahun.

  2. Kalikan harga per unit dengan jumlah unit yang dipakai selama setahun untuk mendapatkan nilai rupiah konsumsi setahun dari masing-masing item.

  3. Jumlahkan nilai rupiah tahunan untuk keseluruhan item dan

hitung persentase pemakaian tahunan untuk tiap-tiap item.

  4. Sorting (urutkan) item-item mulai dari yang konsumsi rupiah tahunannya besar.

  5. Buat klasifikasi ABC dengan aturan mendekati yang di atas.

  Contoh Klasifikasi ABC

Nama Rp/unit kons/th Rp/th %Rp/th

  A 100,000 300 30,000,000

  10.08 B 1,000,000 200 200,000,000

  67.23 C 50,000 30 1,500,000

  0.50 D 20,000 80 1,600,000

  0.54 E 10,000 700 7,000,000

  2.35 F 150,000 350 52,500,000

  17.65 G 90,000 20 1,800,000

  0.61 H 25,000 80 2,000,000

  0.67 I 5,000 100 500,000

  0.17 J 2,000 300 600,000

  0.20 297,500,000 100 Contoh Klasifikasi ABC Tabel Perhitungan untuk klasifikasi ABC

Nama Rp/unit kons/th Rp/th %Rp/th Kumul.%Rp/th

  B 1,000,000 200 200,000,000 67.23 67.23 (A) F 150,000 350 52,500,000 17.65 84.87 (A) A 100,000 300 30,000,000 10.08 94.96 (B) E 10,000 700 7,000,000

  2.35 97.31 (B) H 25,000 80 2,000,000 0.67 97.98 (B) G 90,000 20 1,800,000 0.61 98.59 (C) D 20,000 80 1,600,000 0.54 99.13 (C) C 50,000 30 1,500,000 0.50 99.63 (C) J 2,000 300 600,000

  0.20 99.83 (C) I 5,000 100 500,000 0.17 100.00 (C) 297,500,000 100 Betting on Uncertain Demand: Newsvendor Model Single Period Inventory Model

The Newsboy Model: an Example

  Mr. Tan, a retiree, sells the local newspaper at a Bus terminal. At 6:00 am, he meets the news

truck and buys # of the paper at $4.0 and then

sells at $8.0. At noon he throws the unsold and goes home for a nap.

  If average daily demand is 50 and he buys just 50 copies daily, then is the average daily profit =50*4 =$200?

Betting on Uncertain Demand

• • You must take a firm bet (how much stock to

  order) before some random event occurs (demand) and then you learn that you either bet too much or too little
• More examples: Products for the Christmas season; Nokia’s new set, winter coats, New- Year Flowers, …

Bossini -- Winter Clothes • Season: Dec. – Jan./Feb

• Purchase of key materials (fabrics, dyeing/printing, …) takes long times (upto 90 days)
• Into the selling season, it is too late!

  Hong Kong Seattle Denver

  Case: Sport Obermeyer

The SO Supply Chain

  Shell Fabric Subcontractors Lining Fabric Insulation mat.

  Cut/Sew Distr Ctr Retailers Snaps Zippers Others O’Neill’s Hammer 3/2 wetsuit

  

Hammer 3/2 timeline and economics

Generate forecast

  Economics: of demand and

  Each suit sells for to TEC p = $180

• submit an order

  Spring selling season

• TEC charges

  c = $110 per suit

• Discounted suits

  Nov Dec Jan Feb Mar Apr May Jun Jul Aug sell for v = $90

  Receive order Left over from TEC at the units are end of the discounted month

• The “too much/too little problem”:

• – Order too much and inventory is left over at the end of the season

  – Order too little and sales are lost.

Newsvendor model implementation steps

• Gather economic inputs:

• – Selling price, production/procurement cost, salvage value

  of inventory
• Generate a demand model:
• – Use empirical demand distribution or choose a standard distribution function to represent demand, e.g. the normal distribution, the Poisson distribution.
• Choose an objective:
• – e.g. maximize expected profit or satisfy a fill rate constraint.
• Choose a quantity to order.

  The Newsvendor Model: Develop a Forecast Just one approach Historical forecast performance at O’Neill 7000 6000

  .

  5000 d

  4000 an em d

  3000 al u ct A

  2000 1000 1000 2000 3000 4000 5000 6000 7000 Forecast

Empirical distribution of forecast accuracy Empirical distribution function for the historical A/F ratios

  1.19 ZEN 2MM S/S FULL 680 453 227

  0.56 EVO 4/3 440 623 -183

  1.42 ZEN FL 3/2 450 365

  85

  0.81 HEAT 4/3 460 450

  10

  0.98 ZEN-ZIP 2MM FULL 470 116 354

  0.25 HEAT 3/2 500 635 -135

  1.27 WMS EPIC 3/2 610 830 -220

  1.36 WMS ELITE 3/2 650 364 286

  0.56 ZEN-ZIP 3/2 660 788 -128

  0.67 EPIC 2MM S/S FULL 740 607 133

  0.80 HEATWAVE 4/3 430 274 156

  0.82 EPIC 4/3 1020 732 288

  0.72 WMS EPIC 4/3 1060 1552 -492

  1.46 JR HAMMER 3/2 1220 721 499

  0.59 HAMMER 3/2 1300 1696 -396

  1.30 HAMMER S/S FULL 1490 1832 -342

  1.23 EPIC 3/2 2190 3504 -1314

  1.60 ZEN 3/2 3190 1195 1995

  0.37 ZEN-ZIP 4/3 3810 3289 521

  0.86 WMS HAMMER 3/2 FULL 6490 3673 2817

  0.57

  0.64 ZEN 4/3 430 239 191

  0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

  100%

  83

  0.00

  0.25

  0.50

  0.75

  1.00

  1.25

  1.50

  1.75 A/F ratio P roba bi li ty

  Product description Forecast Actual demand Error* A/F Ratio** JR ZEN FL 3/2 90 140 -50

  1.56 EPIC 5/3 W/HD 120

  37

  1.50 WMS EPIC 2MM FULL 390 311

  0.69 JR ZEN 3/2 140 143 -3

  1.02 WMS ZEN-ZIP 4/3 170 163

  7

  0.96 HEATWAVE 3/2 170 212 -42

  1.25 JR EPIC 3/2 180 175

  5

  0.97 WMS ZEN 3/2 180 195 -15

  1.08 ZEN-ZIP 5/4/3 W/HOOD 270 317 -47

  1.17 WMS EPIC 5/3 W/HD 320 369 -49

  1.15 EVO 3/2 380 587 -207

  1.54 JR EPIC 4/3 380 571 -191

  79

Normal distribution tutorial

• All normal distributions are characterized by two parameters, mean = m and standard deviation = 
• All normal distributions are related to the standard normal that has mean = 0 and standard deviation = 1.
• For example:
• – Let Q be the order quantity, and (m, ) the parameters of the normal demand forecast.
• – Prob{demand is Q or lower} = Prob{the outcome of a standard normal is z or lower}, where
• – (The above are two ways to write the same equation, the first allows you to calculate z from Q and the second lets you calculate Q from z.)
• – Look up Prob{the outcome of a standard normal is z or lower} in the Standard Normal Distribution Function Table.

  

or

Q

z Q z

m m 

       

• 0.0020 0.0040 0.0060 0.0080 0.0100
• 100 -75 -50 -25

  0.25

  25 100 125     

  1

  Rescale the x and y axes by dividing by

  Center the distribution over 0 by subtracting the mean

  Start with m= 100, = 25, Q = 125

  0.40

  0.35

  0.30

  0.20

  0.0120 0.0140 0.0160 0.0180

  0.15

  0.10

  0.05

  50 75 100

  25

  0.01 0.012 0.014 0.016 0.018

  50 75 100 125 150 175 200 0.002 0.004 0.006 0.008

  25

   Q m z

  Using historical A/F ratios to choose a Normal distribution for the demand forecast

• Start with an initial forecast generated from hunches, guesses, etc.
• – O’Neill’s initial forecast for the Hammer 3/2 = 3200 units.
• Evaluate the A/F ratios of the historical data:

  1. Why not just order/buy 3200 units? It is Actual demand A/F ratio

   the most likely outcome!

  Forecast

  2. Forecasts always are biased, so order less than 3200

  3. Gross margin is 40%, should order

• Set the mean of the normal distribution to

  more, if is a hit Expected actual demand  Expected A/F ratio  Forecast

• Set the standard deviation of the normal distribution to

  Standard deviation of actual demand  

   Standard deviation of A/F ratios Forecast

Empirical distribution of forecast accuracy Empirical distribution function for the historical A/F ratios

  1.19 ZEN 2MM S/S FULL 680 453 227

  0.56 EVO 4/3 440 623 -183

  1.42 ZEN FL 3/2 450 365

  85

  0.81 HEAT 4/3 460 450

  10

  0.98 ZEN-ZIP 2MM FULL 470 116 354

  0.25 HEAT 3/2 500 635 -135

  1.27 WMS EPIC 3/2 610 830 -220

  1.36 WMS ELITE 3/2 650 364 286

  0.56 ZEN-ZIP 3/2 660 788 -128

  0.67 EPIC 2MM S/S FULL 740 607 133

  0.80 HEATWAVE 4/3 430 274 156

  0.82 EPIC 4/3 1020 732 288

  0.72 WMS EPIC 4/3 1060 1552 -492

  1.46 JR HAMMER 3/2 1220 721 499

  0.59 HAMMER 3/2 1300 1696 -396

  1.30 HAMMER S/S FULL 1490 1832 -342

  1.23 EPIC 3/2 2190 3504 -1314

  1.60 ZEN 3/2 3190 1195 1995

  0.37 ZEN-ZIP 4/3 3810 3289 521

  0.86 WMS HAMMER 3/2 FULL 6490 3673 2817

  0.57

  0.64 ZEN 4/3 430 239 191

  0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

  100%

  83

  0.00

  0.25

  0.50

  0.75

  1.00

  1.25

  1.50

  1.75 A/F ratio P roba bi li ty

  Product description Forecast Actual demand Error* A/F Ratio** JR ZEN FL 3/2 90 140 -50

  1.56 EPIC 5/3 W/HD 120

  37

  1.50 WMS EPIC 2MM FULL 390 311

  0.69 JR ZEN 3/2 140 143 -3

  1.02 WMS ZEN-ZIP 4/3 170 163

  7

  0.96 HEATWAVE 3/2 170 212 -42

  1.25 JR EPIC 3/2 180 175

  5

  0.97 WMS ZEN 3/2 180 195 -15

  1.08 ZEN-ZIP 5/4/3 W/HOOD 270 317 -47

  1.17 WMS EPIC 5/3 W/HD 320 369 -49

  1.15 EVO 3/2 380 587 -207

  1.54 JR EPIC 4/3 380 571 -191

  79

Table 11.2

  

Product description Forecast Actual demand A/F Ratio* Rank Percentile**

ZEN-ZIP 2MM FULL 470 116 0.25 1 3.0% ZEN 3/2 3190 1195 0.37 2 6.1% ZEN 4/3 430 239 0.56 3 9.1% WMS ELITE 3/2 650 364 0.56 4 12.1% WMS HAMMER 3/2 FULL 6490 3673 0.57 5 15.2% JR HAMMER 3/2 1220 721 0.59 6 18.2% HEATWAVE 4/3 430 274 0.64 7 21.2% ZEN 2MM S/S FULL 680 453 0.67 8 24.2% EPIC 5/3 W/HD 120

  83

  0.69 9 27.3% EPIC 4/3 1020 732 0.72 10 30.3% WMS EPIC 2MM FULL 390 311 0.80 11 33.3% ZEN FL 3/2 450 365 0.81 12 36.4% EPIC 2MM S/S FULL 740 607 0.82 13 39.4% ZEN-ZIP 4/3 3810 3289 0.86 14 42.4% WMS ZEN-ZIP 4/3 170 163 0.96 15 45.5% JR EPIC 3/2 180 175 0.97 16 48.5% HEAT 4/3 460 450 0.98 17 51.5% JR ZEN 3/2 140 143 1.02 18 54.5% WMS ZEN 3/2 180 195 1.08 19 57.6% WMS EPIC 5/3 W/HD 320 369 1.15 20 60.6% ZEN-ZIP 5/4/3 W/HOOD 270 317 1.17 21 63.6% ZEN-ZIP 3/2 660 788 1.19 22 66.7% HAMMER S/S FULL 1490 1832 1.23 23 69.7% HEATWAVE 3/2 170 212 1.25 24 72.7% HEAT 3/2 500 635 1.27 25 75.8% HAMMER 3/2 1300 1696 1.30 26 78.8% WMS EPIC 3/2 610 830 1.36 27 81.8% EVO 4/3 440 623 1.42 28 84.8% WMS EPIC 4/3 1060 1552 1.46 29 87.9% JR EPIC 4/3 380 571 1.50 30 90.9% EVO 3/2 380 587 1.54 31 93.9% JR ZEN FL 3/2

  90 140

  1.56 32 97.0%

If the coming year is a similar to the last year, i.e., the forecasting errors are similar, then

• There is a 3% chance that demand will be 800 units or fewer (0.25*3200)
• There is a 90.9% chance demand is 150%

  of the forecast or lower (or 1.5*3200 = 4,800)

O’Neill’s Hammer 3/2 normal distribution forecast

 9975 3192 3200   . demand actual Expected

  369 1181 3200    . demand actual of deviation Standard

  Product description Forecast Actual demand Error A/F Ratio JR ZEN FL 3/2 90 140 -50 1.5556 EPIC 5/3 W/HD 120

  83 37 0.6917 JR ZEN 3/2 140 143 -3 1.0214 WMS ZEN-ZIP 4/3 170 156 14 0.9176

  … … … … …

  ZEN 3/2 3190 1195 1995 0.3746 ZEN-ZIP 4/3 3810 3289 521 0.8633 WMS HAMMER 3/2 FULL 6490 3673 2817 0.5659 Average

  0.9975 Standard deviation 0.3690

• O’Neill should choose a normal distribution with mean 3192 and standard deviation 1181 to represent demand for the Hammer 3/2 during the

Empirical vs normal demand distribution

  0.00

  0.10

  0.20

  0.30

  0.40

  0.50

  0.60

  0.70

  0.80

  0.90

  1.00 1000 2000 3000 4000 5000 6000

  P ro b a b il it y .

Quantity

  The Newsvendor Model: The order quantity that maximizes expected profit

  “Too much” and “too little” costs

• C o = overage cost

• – The cost of ordering one more unit than what you would have ordered

  had you known demand.
• – In other words, suppose you had left over inventory (i.e., you over ordered). C

  o is the increase in profit you would have enjoyed had you ordered one fewer unit.

• – For the Hammer 3/2 C o = Cost – Salvage value = c – v = 110 – 90 = 20
• C u = underage cost
>– The cost of ordering one fewer unit than what you would have ordered had you known demand.

• – In other words, suppose you had lost sales (i.e., you under ordered). C u

  is the increase in profit you would have enjoyed had you ordered one

  more unit.
• – For the Hammer 3/2 C u = Price – Cost = p – c = 180 – 110 = 70

  Balancing the risk and benefit of ordering a unit

• Ordering one more unit increases the chance of overage …

  th (+1) unit = C x F(Q)

• – Expected loss on the Q o
• – F(Q) = Distribution function of demand = Prob{Demand <= Q)
• … but the benefit/gain of ordering one more unit is the reduction in the chance of underage:

  th (+1) unit = C x (1-F(Q))

• – Expected gain on the Q u

  80

   70 As more units are ordered, .

  Expected gain

  60 the expected benefit from ss lo

  50 r ordering one unit decreases o in

  40 a while the expected loss of g d

  30 te

  Expected loss c ordering one more unit e p

  20 x E increases.

  10

  Newsvendor expected profit maximizing order quantity

• To maximize expected profit order Q units so that the expected loss on the Q

  th unit equals the expected gain on the

  Q th unit:

• Rearrange terms in the above equation ->
• The ratio C
• C

  u / (C o

  u ) is called the critical ratio.

      Q F C Q F C u o

      ( 1 ) u o u

  C C C Q F   ) (

• Hence, to maximize profit, choose Q such that we don’t have lost sales (i.e., demand is Q or lower) with a probability that equals the critical ratio
• Inputs:

• – Empirical distribution function table; p = 180; c = 110; v = 90; C
• Evaluate the critical ratio:
• Lookup 0.7778 in the empirical distribution function table

• – If the critical ratio falls between two values in the table, choose the one that

  leads to the greater order quantity (choose 0.788 which corresponds to A/F ratio 1.3)

  Product description Forecast Actual demand A/F Ratio Rank Percentile … … … … … …

  HEATWAVE 3/2 170 212

  1.25 24 72.7% HEAT 3/2 500 635 1.27 25 75.8% HAMMER 3/2 1300 1696 1.30 26 78.8%

  … … … … … … Finding the Hammer 3/2’s expected profit maximizing order quantity with the empirical distribution function

  u = 180-110 = 70; C o

  = 110-90 =20

  70

  20

  70   

  

u o

u

  C C C A round-up

• Convert A/F ratio into the order quantity 7778 .

  Hammer 3/2’s expected profit maximizing order quantity using the normal distribution = 180-110 = 70; C = 110-90 =20; critical ratio =

• Inputs: p = 180; c = 110; v = 90; C

  

u o

0.7778; mean = m = 3192; standard deviation =  = 1181

• Look up critical ratio in the Standard Normal Distribution Function Table:

  z

  0.01

  0.02

  0.03

  0.04

  0.05

  0.06

  0.07

  0.08

  0.09

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

• – If the critical ratio falls between two values in the table, choose the greater z- statistic
• – Choose z = 0.77

  Q  m  z  