Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
LECTURE 8 – INVENTORY MANAGEMENT
BACKGROUND INFO + TERMINOLOGY
There are 2 ways of approaching models, deterministic and stochastic models
o (Ch. 15 & 16 respectively)
Deterministic models (based on real input data, absence of randomness)
o Constant demand (continuous review)
Economic Order Quantity (ch. 15.2)
EOQ with quantity discounts (ch. 15.3)
Continuous rate EOQ/EBQ (ch. 15.4)
EOQ with backorders allowed (ch. 15.5) (During this year 23/24, this is not asked in
exam, so it is also not covered in this summary)
o Nonconstant demand (periodic review)
Wagner-Whitin, Silver-Meal (ch. 18.7) (Discussed earlier)
Stochastic models (includes randomness and uncertainty)
o Single-period inventory models
Newsboy problem
o Continuous review models
Cost based (r,q) and (s, S) models
Service level based
o Periodic review models
(R, S) models
Terminology
o 𝒒 = 𝒐𝒓𝒅𝒆𝒓 𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚
o 𝑲 = 𝒐𝒓𝒅𝒆𝒓𝒊𝒏𝒈 𝒄𝒐𝒔𝒕𝒔
o 𝒉 = 𝒉𝒐𝒍𝒅𝒊𝒏𝒈 𝒄𝒐𝒔𝒕𝒔
o 𝑫 = 𝒅𝒆𝒎𝒂𝒏𝒅
o 𝒓 = 𝒓𝒆𝒐𝒓𝒅𝒆𝒓 𝒑𝒐𝒊𝒏𝒕
o 𝑬𝑶𝑸 = 𝒆𝒄𝒐𝒏𝒐𝒎𝒊𝒄 𝒐𝒓𝒅𝒆𝒓 𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚
o 𝑳 = 𝒍𝒆𝒂𝒅 𝒕𝒊𝒎𝒆
28
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
BASIC EOQ MODEL
For any order, ordering and setup cost K
Cost per unit per year of inventory is h
o Given directly or as percentage of purchasing price
Unit purchase price p
Assumptions
o Repetitive ordering
o Constant demand
o Continuous ordering
o Zero lead time (instantaneous delivery)
Find q that minimizes total annual cost (q*)
𝑻𝑪(𝒒) = 𝒐𝒓𝒅𝒆𝒓𝒊𝒏𝒈 𝒄𝒐𝒔𝒕
+ 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒊𝒏𝒈 𝒄𝒐𝒔𝒕
+ 𝒉𝒐𝒍𝒅𝒊𝒏𝒈 𝒄𝒐𝒔𝒕
𝑂𝑟𝑑𝑒𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = 𝐾 ⋅
𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = 𝑝 ⋅ 𝐷
⋅
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = → 𝑜𝑟𝑑𝑒𝑟𝑠 𝑒𝑎𝑐ℎ 𝑦𝑒𝑎𝑟
𝐶𝑦𝑐𝑙𝑒 𝑙𝑒𝑛𝑔𝑡ℎ =
𝑇𝐶(𝑞) = + 𝑝𝐷 +
𝑇𝐶 (𝑞) = − = 0 𝒊𝒇
2𝐾𝐷
𝑞 = 𝑞∗ = = 𝐸𝑂𝑄
ℎ
29
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
EXAMPLE 17 – ORDERIN G CAMERAS (P.854)
Deterministic – stochastic model
Given is the following
o Demand D = 10000 cameras per year
o Ordering cost K = $5
o Cost price = $100
o Capital opportunity cost hd = 20%
Calculations
o ℎ = 𝑝 ⋅ ℎ = 100 ⋅ 0.20 =
20 (ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡)
⋅ ⋅
o 𝐸𝑂𝑄 = = = √5000 =
70
70.71 =
71
Remarks
o 𝐸𝑂𝑄 → ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = 𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡
o 𝑇𝐶(𝑞) is reasonable flat around the EOQ
o Small variations in K, D, h have little effect on EOQ
NONZERO LEAD TIME
DEFINITION S
Safety stock is inventory buffer against uncertainty in demand and supply (Stochastic)
o Actual demand deviates from forecast
o Timing of supply shows unpredictable variation (Lead Time variation)
o Quantity of supply shows unpredictable variation
On-Hand stock or physical stock always ≥ 0 [𝑂𝐻𝐼(𝑡)]
Stockout
o Event that D > OHI
Backorders or backlog
o Amount of unfilled demand waiting to be filled when replenishment order arrives [𝐵(𝑡)]
Net stock or inventory
o 𝐼(𝑡) = 𝑂𝐻𝐼(𝑡) − 𝐵(𝑡)
o In deterministic cases
There are no shortages, so 𝐼(𝑡) = 𝑂𝐻𝐼(𝑡)
Economic inventory = 𝑂𝐻𝐼(𝑡) + On-order inventory
30
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
LECTURE 8 – INVENTORY MANAGEMENT
BACKGROUND INFO + TERMINOLOGY
There are 2 ways of approaching models, deterministic and stochastic models
o (Ch. 15 & 16 respectively)
Deterministic models (based on real input data, absence of randomness)
o Constant demand (continuous review)
Economic Order Quantity (ch. 15.2)
EOQ with quantity discounts (ch. 15.3)
Continuous rate EOQ/EBQ (ch. 15.4)
EOQ with backorders allowed (ch. 15.5) (During this year 23/24, this is not asked in
exam, so it is also not covered in this summary)
o Nonconstant demand (periodic review)
Wagner-Whitin, Silver-Meal (ch. 18.7) (Discussed earlier)
Stochastic models (includes randomness and uncertainty)
o Single-period inventory models
Newsboy problem
o Continuous review models
Cost based (r,q) and (s, S) models
Service level based
o Periodic review models
(R, S) models
Terminology
o 𝒒 = 𝒐𝒓𝒅𝒆𝒓 𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚
o 𝑲 = 𝒐𝒓𝒅𝒆𝒓𝒊𝒏𝒈 𝒄𝒐𝒔𝒕𝒔
o 𝒉 = 𝒉𝒐𝒍𝒅𝒊𝒏𝒈 𝒄𝒐𝒔𝒕𝒔
o 𝑫 = 𝒅𝒆𝒎𝒂𝒏𝒅
o 𝒓 = 𝒓𝒆𝒐𝒓𝒅𝒆𝒓 𝒑𝒐𝒊𝒏𝒕
o 𝑬𝑶𝑸 = 𝒆𝒄𝒐𝒏𝒐𝒎𝒊𝒄 𝒐𝒓𝒅𝒆𝒓 𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚
o 𝑳 = 𝒍𝒆𝒂𝒅 𝒕𝒊𝒎𝒆
28
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
BASIC EOQ MODEL
For any order, ordering and setup cost K
Cost per unit per year of inventory is h
o Given directly or as percentage of purchasing price
Unit purchase price p
Assumptions
o Repetitive ordering
o Constant demand
o Continuous ordering
o Zero lead time (instantaneous delivery)
Find q that minimizes total annual cost (q*)
𝑻𝑪(𝒒) = 𝒐𝒓𝒅𝒆𝒓𝒊𝒏𝒈 𝒄𝒐𝒔𝒕
+ 𝒑𝒖𝒓𝒄𝒉𝒂𝒔𝒊𝒏𝒈 𝒄𝒐𝒔𝒕
+ 𝒉𝒐𝒍𝒅𝒊𝒏𝒈 𝒄𝒐𝒔𝒕
𝑂𝑟𝑑𝑒𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = 𝐾 ⋅
𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = 𝑝 ⋅ 𝐷
⋅
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = → 𝑜𝑟𝑑𝑒𝑟𝑠 𝑒𝑎𝑐ℎ 𝑦𝑒𝑎𝑟
𝐶𝑦𝑐𝑙𝑒 𝑙𝑒𝑛𝑔𝑡ℎ =
𝑇𝐶(𝑞) = + 𝑝𝐷 +
𝑇𝐶 (𝑞) = − = 0 𝒊𝒇
2𝐾𝐷
𝑞 = 𝑞∗ = = 𝐸𝑂𝑄
ℎ
29
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
EXAMPLE 17 – ORDERIN G CAMERAS (P.854)
Deterministic – stochastic model
Given is the following
o Demand D = 10000 cameras per year
o Ordering cost K = $5
o Cost price = $100
o Capital opportunity cost hd = 20%
Calculations
o ℎ = 𝑝 ⋅ ℎ = 100 ⋅ 0.20 =
20 (ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡)
⋅ ⋅
o 𝐸𝑂𝑄 = = = √5000 =
70
70.71 =
71
Remarks
o 𝐸𝑂𝑄 → ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = 𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡
o 𝑇𝐶(𝑞) is reasonable flat around the EOQ
o Small variations in K, D, h have little effect on EOQ
NONZERO LEAD TIME
DEFINITION S
Safety stock is inventory buffer against uncertainty in demand and supply (Stochastic)
o Actual demand deviates from forecast
o Timing of supply shows unpredictable variation (Lead Time variation)
o Quantity of supply shows unpredictable variation
On-Hand stock or physical stock always ≥ 0 [𝑂𝐻𝐼(𝑡)]
Stockout
o Event that D > OHI
Backorders or backlog
o Amount of unfilled demand waiting to be filled when replenishment order arrives [𝐵(𝑡)]
Net stock or inventory
o 𝐼(𝑡) = 𝑂𝐻𝐼(𝑡) − 𝐵(𝑡)
o In deterministic cases
There are no shortages, so 𝐼(𝑡) = 𝑂𝐻𝐼(𝑡)
Economic inventory = 𝑂𝐻𝐼(𝑡) + On-order inventory
30
