Lecture 1

M/M/1 exponential/gemiddelde tijden
M/D/1 fixed/vaste tijden

λ = average number of arrivals per hour ρ = utilisation rate µ = service rate

Propability of more than k customers in the system Pn>k= λµk+1= ρk+1
Probability of exactly n customers in the system P0= 1-λµ =1-ρ Pk= Pn>k-1- Pn>k= λµk-λµk+1= 1-λµ λµ k=1-ρ ρk

Lecture 2

Interarrival time =tijd tusen 2 arrivals
Throughput time = tijd tussen binnenkomst van het systeem en verlaten van het systeem
Design capacity = theoretical maximum output of a system/procs in a given period
Effective capacity = capacity that can be expected given the product mix, methods of scheduling, maintenance and standards of quality.
Sufficiency design capacity = suffient als het systeem het aantal klanten aankan, anders insuffient.
Bottleneck = Process that limits the output of a system (smallest design capacity). If the system has no bottleneck, then the arrival rate is defind as bottleneck.
Departure rate (throughput) = number of products/customers that leave the system per time unit. If arrival process is bottleneck, then: departure rate = arrival rate!
Utilisation = fraction of time a machine is used for production. n = number of parallel servers ρ = utilisation rate
Utilisation rate = operating timetotal time=actual outputdesign capacity ρ=λnµ
Productive utilisation rate = equal to utilisation but excludes set-up times
Work-in-progress = number of products that entered the system but are no yet finished
W = throughput time L=λW (Little’s equition) ρi = utilisation rate of process i
Xi = batch size at process i i=1nρiXi

Lecture 3

L = learning rate T2N=LTN
TN = time needed to produce product N
N = product number TN= T1NlogLlog2
T1 = time needed for product 1

Available inventory ( projected on hand) = inventory is availeble at the beginning of period.
Lead time = time between recognition of need and receiving of order.
Time-phased product structure = bill of material with lead times for each component.
Net requirements = shortage of product
Planned order receipts = when order needs to be received.
Planned order releases = when planned order receipts need to be ordered.

Lot for lot ordering = amount ordered is amount needed
MOQ =Minimum order quantity = amount ordered has minimum value MOQ
FOQ = Fixed order quantity = amount ordered is multiple of fixed quantity FOQ

Lecture 4
Economic order quantity
D = annual demand
Q = number of units per order DQ=# orders per jaa r bestellen
S = ordering of setup costs per order
Annual ordering or setup cost: DQ×S
H = holding cost per unit per year Q2=average inventory
Annual holding cost: Q2×H
P = purchasing cost per unit
D = annual demand
Annual purchasing cost: P×D
Total = cost: TC= DQ×S + Q2×H + PD
Optimal order quantity: Q*=2DSH note: is always rounded to the nearest integer
Q* = optimal order quantity = Economic order quantity
Q* = bij Economic production quantity is het de optimal production quantity!!!
W = number of working days per year
N = expected number of orders per year N=DQ*
T = expected working time between orders T=WN
ROP = when to order? = Reorder point = wat nog in voorraad is als er besteld moet worden d = daily demand d=DW
L = lead time in days ROP=d×L

Economic production quantity
Inventory holding cost: maximum inventory2×H p = daily production rate = eenheden die per dag geproduceerd worden t = length of production run in days u = daily demand rate = eenheden per dag uit voorraad
Maximum inventory = pt-ut=Q-uQp=Q1-up= Qp(p-u)
Note: maximum inventory is always rounded up!!
Inventory holding costs = maximum inventory2×H=Q2p(p-u)×H
Note: for the inventory cost the maximum inventory is not rounded.
S = setup cost per order
Q = production quantity
Setup cost = DQ×S
Total cost TC=Q2pp-u×H+DQ×S
Optimal production quantity: 12pp-u×H-DQ*2×S=0→ Q *=2DSH×pp-u
Quality discount model
I = percentage of unit price
P = unit price
Holding cost H=I×P
Annual setup cost DQ×S
Purchasing cost: P×D
Total cost TC= DQ×S + Q2×IP + PD
Optimal order quantity: Q*=2DSH

Lecture 5

λ = failure rate (hoevaak iets kapot gaat) t = tijdseenheid reliability = e-λt
MTBF = mean time between failures
MDT = mean down time
MTTR = mean time to repair (if no delays then MTTR=MDT)
Failure rate = λ=1MTBF
A = availability
Availability = A=MTBFMTBF+MDT
Serial systems
Failure rate = i=1nfri=fr1+fr2+fr3+fr4+fr5 (fr =λ)
Reliability = i=1ne-fr×t=e-fr1+fr2+fr3+fr4+fr5t=e-fr1×t×e-fr2×t×e-fr3×t×e-fr4×t×e-fr5×t
Availability = 1fr1fr+MDT= 1fr11fr1+MDT×1fr21fr2+MDT×1fr31fr3+MDT×1fr41fr4+MDT×1fr51fr5+MDT

Parallel systems
Maintenance = a series of actions to be taken with the intention to retain a machine part or product in, or restore it to, a state in which it can perform it’s intended function.
Corrective maintenance = aim is to restore tot functional state (failure based maintenance).
Preventive maintenance = aim is to retain in functional state (time based maintenance, usage based maintenance and condition based maintenance).
Lecture 6
Linear programming = mathematical technique used to find the optimal value for the objective.
Range of variables
Positive >0
Negative <0
Non-negative ≥0
Non-positive ≤0
Integer Only round numbers
Real All numbers
Binary 0 of 1

Linear programming + Integer programming
Uitgelegd in voorbeelden op de slides in lecture 6.

Lecture 7
Alleen duidelijk aan de hand van de voorbeelden op de slides.

Lecture 8

Activity on node (AON) = activieteiten weergegeven in de bolletjes
Activity on arrow (AOA)= ativiteiten weergegeven op de pijltjes tussen de bolletjes
Critical path = longest path through the network
Forward pass = earliest start and the earliest finish
Backward pass = laatste moment dat een activiteit nog kan starten zodat het op tijd klaar is. Latest start and latest finish
Om het critical path te vinden moet je bij elke activiteit de begintijd en de eindtijd neerzetten, en daar tussenin de tijd die het proces duurt. Op deze manier kom je op de eindtijd, dit geeft aan hoelang het proces duurt. Van hieruit kun je terugrekenen op dezelfde manier. Hierbij zijn er een bepaald aantal activiteiten die bij het terugrekenen en het heenrekenen dezelfde getallen hebben. Deze activiteiten vormen het critical path.

Crash cost per week = crash cost-normal costnormal time-crash time
Bij project crashing verkort je het critical path door activiteiten sneller te laten verlopen. Je veranderd de goedkoopste activiteit op het critical path. Voor voorbeeld zie slides van lecture 7.

Sequencing jobs = short term scheduling.
4 soorten van priority rules:

Johnson’s rule = minimise processing time of jobs and minimise total idle time of machines.
Doe het op deze manier:

Lecture 9
Factor rating method = multiply scores by weights for each factor and total. Select location with maximum score. Dus gewoon weight x score en dan alle factoren bij elkaar optellen.

Cost-volume analysis
Bij deze methode moet je een break-even point uitrekenen door formules aan elkaar gelijk te stellen.
Fixed costs France 30000, variable costs per unit 75.
Fixed costs Germany 60000, variable costs per unit 45.
Dus: 30000+75x=60000+45xx=1000 dus vanaf 1000 units is germany voordeliger.

Center-of-gravity method
X-coordinate: Cx=idixWiiWi y-coordinate: Cy=idiyWiiWi dix= x coordinate of location i
Wi= volume o goods moved to or from location i diy=y coordinate of location i

Cost Analyses