OR MODELS FOR THE PRE-
MASTER IEM
Summary 2023/2024
Abstract
Summary for the course OR Models For The Pre-Master Industrial Engineering &
Management.
Based on the book ‘Operations Research’ by Wayne L. Winston & Slides by L. van der Wegen
(Utwente).
This summary follows the 4 lecture slides on ‘LP and ILP’, ‘DP’, ‘IM’ and ‘Queuing’.
All page numbers in this summary refer to pages in Operations Research by Winston.
Klep, (Julian, Student Utwente)
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
TABLE OF CONTENTS
Lecture 1 – LP & ILP Models .............................................................................................................................4
Time Notation........................................................................................................................................... 4
Example 1 – Giapetto Model – Soldiers & Trains (p.49) ..................................................................................... 4
Graphical Solution To Problem (Example 1) ............................................................................................. 5
Special Cases Of LP Models ................................................................................................................................ 5
Lecture 2 – LP & ILP Models .............................................................................................................................6
Example 2 – Marketing Research Problem ......................................................................................................... 6
Example 3 – Work Scheduling (p.72) .................................................................................................................. 6
Example 4 – Production Scheduling Problem (p.101) ........................................................................................ 7
Assumptions – LP Model........................................................................................................................... 7
Example 5 – Transportation Problem (p.360)..................................................................................................... 8
Example 6 – Production Process Problem (Simple Version) (p.95) .................................................................... 8
Lecture 3 – LP & ILP Models .............................................................................................................................9
Example 6 – Production Process Problem (Extended Version) (p.95) ................................................................ 9
Units In Constraints .................................................................................................................................. 9
Example 7 – Blending Problem (p. 86) .............................................................................................................. 10
Integer Linear Programming ILP ....................................................................................................................... 12
Graphical Clarification ILP ...................................................................................................................... 12
Example 8 – Investment Problem – (PEN) ........................................................................................................ 13
Lecture 4 – LP & ILP Models ........................................................................................................................... 14
Either-Or & If-Then Constraint ......................................................................................................................... 14
Either-Or Constraint (p.488) ................................................................................................................... 14
If-Then Constraint ................................................................................................................................... 16
Example 9 – Fixed Charge (Big M-Constraint) (p.480) ...................................................................................... 17
Lecture 5 – DP Models ................................................................................................................................... 18
Example 10 – Network Problem (p.964) ........................................................................................................... 18
Characteristics Of DP-Applications ......................................................................................................... 18
Example 11 – Fishery Example (p.990) ............................................................................................................. 19
Lecture 6 – DP Models ................................................................................................................................... 20
Example 12 – Production Inventory Problem (p.969) ...................................................................................... 20
Assumptions for LP In DP Models ........................................................................................................... 22
Example 13 – Resource Allocation (p.974) ....................................................................................................... 22
Lecture 7 – DP Models ................................................................................................................................... 25
Example 14 – Wagner-Whitin Method (p.1002)............................................................................................... 25
Example 15 – Silver-Meal Method (p.1003) ..................................................................................................... 26
Example 16 – Equipment Replacement Proplem (p.985) ................................................................................. 27
1
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
Lecture 8 – Inventory Management ............................................................................................................... 28
Background Info + Terminology........................................................................................................................ 28
Basic EOQ Model .............................................................................................................................................. 29
Example 17 – Ordering Cameras (p.854) .......................................................................................................... 30
Nonzero Lead Time ........................................................................................................................................... 30
Definitions .............................................................................................................................................. 30
Effect of NonZero Lead Time .................................................................................................................. 31
Lecture 9a – Inventory Management (Deterministic Demand)....................................................................... 32
Example 18 – EOQ With 2 Quantity Discounts ................................................................................................. 32
Example 19 – EOQ With 3 Quantity Discounts (p.863) ..................................................................................... 32
Continuous Rate EOQ Model ............................................................................................................................ 33
Example 20 – Macho Auto Company (p.867) ................................................................................................... 33
When To Use EOQ Models ...................................................................................................................... 34
Example 21 – Test If EOQ May Be Used ........................................................................................................... 34
Lecture 9b – Inventory Management (Stochastic Demand)............................................................................ 35
Newsboy Problem............................................................................................................................................. 35
Example 22 – Newsboy Problem (p. 881) ......................................................................................................... 36
Lecture 10 – Inventory Management ............................................................................................................. 37
Terminology ............................................................................................................................................ 37
(r,q) In Case Of Backlogging (p.890).................................................................................................................. 37
Demand During Lead Time (p.890) ......................................................................................................... 38
Determine TC(q,r) (Holding Cost, Shortage Cost, Order Cost) ................................................................ 38
Marginal Analysis To Determine Reorder Point ..................................................................................... 40
Example 23 – Disk Stock (p.894) ....................................................................................................................... 41
(r,q) In Case Of lost sales .................................................................................................................................. 42
(s,S) In Case Of Backlogging Or Lost Sales......................................................................................................... 42
Lecture 11 – Inventory Management ............................................................................................................. 43
Service Level Approach ........................................................................................................................... 43
Example 24 – SLM1 & SLM2 ............................................................................................................................... 43
Determining r* And Safety Stock For SLM1 ....................................................................................................... 44
Determining E(Br) ................................................................................................................................... 44
Example 25 – Bads, Inc. (p.903) ........................................................................................................................ 45
Determining r* And Safety Stock Level For SLM2 .............................................................................................. 46
Example 26 – Bads. Inc (2) (p. 906) ................................................................................................................... 46
(R,S) Periodic Review Model ............................................................................................................................. 47
Example 27 – Lowland Appliance ..................................................................................................................... 48
Lecture 12 – Queing ....................................................................................................................................... 49
2
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
M/M/1 .............................................................................................................................................................. 49
Geometric Series Proof Of Theorem ....................................................................................................... 49
Flow Balance Equations .................................................................................................................................... 50
Example 28 – M/M/1 ........................................................................................................................................ 51
Equilibrium Formulas ........................................................................................................................................ 51
Lecture 13 – Queuing ..................................................................................................................................... 52
M/M/1/c ........................................................................................................................................................... 52
Deriving Formula For L ........................................................................................................................... 52
Modeling Arrival And Service Processes (p.1053) ............................................................................................ 53
Lecture 14 – Queuing ..................................................................................................................................... 54
Example 29 – Helpdesk Example (M/M/s) ....................................................................................................... 54
M/M/s Example ................................................................................................................................................ 55
Example 30 - Finite Source Models - Machine Repair Problem........................................................................ 55
3
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
LECTURE 1 – LP & ILP MODELS
TIME NOTATION
t = 0 is beginning of year 1
t = 1 is end of year 1
Be clear: “I consider the inventory at the end of year/month/week t”
EXAMPLE 1 – GIAPETTO MODEL – SOLDIERS & TRAINS (P.49)
2 types of products
2 types of labor
Problem: What production plan maximizes weekly profit?
o Decision variables
𝑋 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑜𝑙𝑑𝑖𝑒𝑟𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑒𝑎𝑐ℎ 𝑤𝑒𝑒𝑘
𝑋 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑎𝑖𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑒𝑎𝑐ℎ 𝑤𝑒𝑒𝑘
o Objective function
𝑀𝑎𝑥 𝑧 = 3𝑋 + 2𝑋
o Restrictions
2𝑋 + 𝑋 ≤ 100 (𝑓𝑖𝑛𝑖𝑠ℎ𝑖𝑛𝑔)
𝑋 + 𝑋 ≤ 80 (𝐶𝑎𝑟𝑝𝑒𝑛𝑡𝑟𝑦)
𝑋 ≤ 40 (𝑑𝑒𝑚𝑎𝑛𝑑)
𝑋 ≥ 0, 𝑋 ≥ 0 (𝑠𝑖𝑔𝑛 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑠)
4
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
GRAPHICAL SOLUTION TO PROBLEM (EXAMPLE 1)
𝑧 = 180
𝑋 = 20 ∩ 𝑋 = 60
SPECIAL CASES OF LP MODELS
LP has alternative optimal solutions
o Isoprofit line parallel to a restriction
LP is infeasible
o Constraints are too restrictive
LP is unbounded
o Constraints are missing
5
MASTER IEM
Summary 2023/2024
Abstract
Summary for the course OR Models For The Pre-Master Industrial Engineering &
Management.
Based on the book ‘Operations Research’ by Wayne L. Winston & Slides by L. van der Wegen
(Utwente).
This summary follows the 4 lecture slides on ‘LP and ILP’, ‘DP’, ‘IM’ and ‘Queuing’.
All page numbers in this summary refer to pages in Operations Research by Winston.
Klep, (Julian, Student Utwente)
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
TABLE OF CONTENTS
Lecture 1 – LP & ILP Models .............................................................................................................................4
Time Notation........................................................................................................................................... 4
Example 1 – Giapetto Model – Soldiers & Trains (p.49) ..................................................................................... 4
Graphical Solution To Problem (Example 1) ............................................................................................. 5
Special Cases Of LP Models ................................................................................................................................ 5
Lecture 2 – LP & ILP Models .............................................................................................................................6
Example 2 – Marketing Research Problem ......................................................................................................... 6
Example 3 – Work Scheduling (p.72) .................................................................................................................. 6
Example 4 – Production Scheduling Problem (p.101) ........................................................................................ 7
Assumptions – LP Model........................................................................................................................... 7
Example 5 – Transportation Problem (p.360)..................................................................................................... 8
Example 6 – Production Process Problem (Simple Version) (p.95) .................................................................... 8
Lecture 3 – LP & ILP Models .............................................................................................................................9
Example 6 – Production Process Problem (Extended Version) (p.95) ................................................................ 9
Units In Constraints .................................................................................................................................. 9
Example 7 – Blending Problem (p. 86) .............................................................................................................. 10
Integer Linear Programming ILP ....................................................................................................................... 12
Graphical Clarification ILP ...................................................................................................................... 12
Example 8 – Investment Problem – (PEN) ........................................................................................................ 13
Lecture 4 – LP & ILP Models ........................................................................................................................... 14
Either-Or & If-Then Constraint ......................................................................................................................... 14
Either-Or Constraint (p.488) ................................................................................................................... 14
If-Then Constraint ................................................................................................................................... 16
Example 9 – Fixed Charge (Big M-Constraint) (p.480) ...................................................................................... 17
Lecture 5 – DP Models ................................................................................................................................... 18
Example 10 – Network Problem (p.964) ........................................................................................................... 18
Characteristics Of DP-Applications ......................................................................................................... 18
Example 11 – Fishery Example (p.990) ............................................................................................................. 19
Lecture 6 – DP Models ................................................................................................................................... 20
Example 12 – Production Inventory Problem (p.969) ...................................................................................... 20
Assumptions for LP In DP Models ........................................................................................................... 22
Example 13 – Resource Allocation (p.974) ....................................................................................................... 22
Lecture 7 – DP Models ................................................................................................................................... 25
Example 14 – Wagner-Whitin Method (p.1002)............................................................................................... 25
Example 15 – Silver-Meal Method (p.1003) ..................................................................................................... 26
Example 16 – Equipment Replacement Proplem (p.985) ................................................................................. 27
1
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
Lecture 8 – Inventory Management ............................................................................................................... 28
Background Info + Terminology........................................................................................................................ 28
Basic EOQ Model .............................................................................................................................................. 29
Example 17 – Ordering Cameras (p.854) .......................................................................................................... 30
Nonzero Lead Time ........................................................................................................................................... 30
Definitions .............................................................................................................................................. 30
Effect of NonZero Lead Time .................................................................................................................. 31
Lecture 9a – Inventory Management (Deterministic Demand)....................................................................... 32
Example 18 – EOQ With 2 Quantity Discounts ................................................................................................. 32
Example 19 – EOQ With 3 Quantity Discounts (p.863) ..................................................................................... 32
Continuous Rate EOQ Model ............................................................................................................................ 33
Example 20 – Macho Auto Company (p.867) ................................................................................................... 33
When To Use EOQ Models ...................................................................................................................... 34
Example 21 – Test If EOQ May Be Used ........................................................................................................... 34
Lecture 9b – Inventory Management (Stochastic Demand)............................................................................ 35
Newsboy Problem............................................................................................................................................. 35
Example 22 – Newsboy Problem (p. 881) ......................................................................................................... 36
Lecture 10 – Inventory Management ............................................................................................................. 37
Terminology ............................................................................................................................................ 37
(r,q) In Case Of Backlogging (p.890).................................................................................................................. 37
Demand During Lead Time (p.890) ......................................................................................................... 38
Determine TC(q,r) (Holding Cost, Shortage Cost, Order Cost) ................................................................ 38
Marginal Analysis To Determine Reorder Point ..................................................................................... 40
Example 23 – Disk Stock (p.894) ....................................................................................................................... 41
(r,q) In Case Of lost sales .................................................................................................................................. 42
(s,S) In Case Of Backlogging Or Lost Sales......................................................................................................... 42
Lecture 11 – Inventory Management ............................................................................................................. 43
Service Level Approach ........................................................................................................................... 43
Example 24 – SLM1 & SLM2 ............................................................................................................................... 43
Determining r* And Safety Stock For SLM1 ....................................................................................................... 44
Determining E(Br) ................................................................................................................................... 44
Example 25 – Bads, Inc. (p.903) ........................................................................................................................ 45
Determining r* And Safety Stock Level For SLM2 .............................................................................................. 46
Example 26 – Bads. Inc (2) (p. 906) ................................................................................................................... 46
(R,S) Periodic Review Model ............................................................................................................................. 47
Example 27 – Lowland Appliance ..................................................................................................................... 48
Lecture 12 – Queing ....................................................................................................................................... 49
2
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
M/M/1 .............................................................................................................................................................. 49
Geometric Series Proof Of Theorem ....................................................................................................... 49
Flow Balance Equations .................................................................................................................................... 50
Example 28 – M/M/1 ........................................................................................................................................ 51
Equilibrium Formulas ........................................................................................................................................ 51
Lecture 13 – Queuing ..................................................................................................................................... 52
M/M/1/c ........................................................................................................................................................... 52
Deriving Formula For L ........................................................................................................................... 52
Modeling Arrival And Service Processes (p.1053) ............................................................................................ 53
Lecture 14 – Queuing ..................................................................................................................................... 54
Example 29 – Helpdesk Example (M/M/s) ....................................................................................................... 54
M/M/s Example ................................................................................................................................................ 55
Example 30 - Finite Source Models - Machine Repair Problem........................................................................ 55
3
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
LECTURE 1 – LP & ILP MODELS
TIME NOTATION
t = 0 is beginning of year 1
t = 1 is end of year 1
Be clear: “I consider the inventory at the end of year/month/week t”
EXAMPLE 1 – GIAPETTO MODEL – SOLDIERS & TRAINS (P.49)
2 types of products
2 types of labor
Problem: What production plan maximizes weekly profit?
o Decision variables
𝑋 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑜𝑙𝑑𝑖𝑒𝑟𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑒𝑎𝑐ℎ 𝑤𝑒𝑒𝑘
𝑋 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑎𝑖𝑛𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑒𝑎𝑐ℎ 𝑤𝑒𝑒𝑘
o Objective function
𝑀𝑎𝑥 𝑧 = 3𝑋 + 2𝑋
o Restrictions
2𝑋 + 𝑋 ≤ 100 (𝑓𝑖𝑛𝑖𝑠ℎ𝑖𝑛𝑔)
𝑋 + 𝑋 ≤ 80 (𝐶𝑎𝑟𝑝𝑒𝑛𝑡𝑟𝑦)
𝑋 ≤ 40 (𝑑𝑒𝑚𝑎𝑛𝑑)
𝑋 ≥ 0, 𝑋 ≥ 0 (𝑠𝑖𝑔𝑛 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑠)
4
, Julian Klep
OR Models for Pre-Master IEM
Winston Ch. 1, 3, 9, 15, 16, 18, 20
GRAPHICAL SOLUTION TO PROBLEM (EXAMPLE 1)
𝑧 = 180
𝑋 = 20 ∩ 𝑋 = 60
SPECIAL CASES OF LP MODELS
LP has alternative optimal solutions
o Isoprofit line parallel to a restriction
LP is infeasible
o Constraints are too restrictive
LP is unbounded
o Constraints are missing
5
