,1 Manufacturing
Supply chain: chain of activities from raw material to end customer
• MTO – make to order, production –> demand –> consumption
• MTS – make to sell, demand –> production –> consumption
• ATO – assemble to order, production –> demand –> production –> con-
sumption
Supply chain management is the integration of planning for the whole chain
Different types of stock:
• Cycle stock to cover typical demand during a certain period
• Safety stock to cover short-term unpredictable fluctuations in demand
• Seasonal stock to cover long-term predictable demand fluctuations
Models for manufacturing:
• Inventory models when supply chain is coordinated by placing orders
• Queueing and optimization models when the production planning
required capacity considerations
Types of production systems:
• Flow lines for similar items, here the focus is on productivity. Queueing
models can be used for flow line performance analysis.
• Job shops for heterogeneous orders with different routing, due dates etc.
Optimization models can be used for job shop scheduling.
1.1 Queuing models for flow lines
M/M/1 queue: P (N = j) = π(j) = (1 − p)pj
λ λ ni
Tandem of M/M/1 queues: P (Ni = ni , i = 1, ..., V ) = ΠVi=1 (1 − µi )( µi )
ρE[S](1+c2 (S))
M/G/1: E[Wq ] = 2(1−ρ)
ρE[S](c2 (A)+c2 (S))
G/G/1: E[Wq ] = 2(1−ρ)
3
, Approximation of coefficient of variation output process
–> c2d = (1 − ρ2 )c2 (A) + ρ2 c2 (S)
Throughput (rate of production) is not influenced by the variability
Buffers in flow lines:
• Buffers disconnect machines, especially when there is variability. It helps
prevent long queues.
• With finite buffers will slow down the production rate and will lead to
high inventory, this is undesirable because of high stock costs, interest
and depreciation.
In mathematical models, we can assume 2 machines with exponential service
times and finite WIP buffer size between the 2 machines. Machine 1 thus always
produces unless it is blocked. We assume Blocking Before Service (BBS):
a machine can start processing a part only if there is a space available in the
downstream buffer.
• This model is equivalent to M/M/1/N birth-death model with the com-
pletion of machine 1 equal to λ and the state of the system is equal to the
number of items in the buffer + machine 2.
• Buffer size B = 0: throughput of ( µ11 + 1 −1
µ2 )
• Buffer size B = ∞: throughput of (max( µ11 + 1
µ2 ))
−1
= min(µ1 , µ2 )
• Buffer size B in general = P (state > 0) · µ2
To optimise the buffer size:
• The throughput should be monotone in each buffer size.
• Machines in middle should be faster and should have larger buffers.
• Simulation and optimisation: simopt.
Thus, buffers and inventory help cope with variability and modeling helps quan-
tifying the use of buffers. Also: reduction of variability leads to the need for
less inventory because if there is no variability, there is no inventory because
everything is just in time.
So how to reduce variability: reduce buffers to show variability and to be forced
to reduce it. This way you can see where the blocking takes place and you know
where to improve. This way, dependencies are increased to increase influence of
irregularities. This is lean manufacturing.
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