The notations used in Chapter 3.5 are summarized in Table 3.3. A flexible manufacturing system (FMS) is a collection of machines, linked by an automated materials handling system and directed by a central computer. Different part types are produced in the system and each part type has a finite number of operations.
An operation, oj, is defined by its processing time on a machine and by the set of cutting tools required. In FMSs, generally more than one machine can perform certain operations. In Figure 3.1, for example, each machine can perform milling operations. This provides routing flexibility.
The set of all operations, which can be performed on any machine in a particular group of machines, is called operation type, oth. An operation type is an aggregated set of operations.
In the example of Figure 3.1, the drilling operations can only be performed on machine M1.
Therefore this operation is also an operation type (ot1). Milling can be done on machines M1 and M2. Therefore the milling operations of these two machines can be aggregated into another operation type (ot2).
To analyze the capacity of an FMS, the available capacity of every combination of the operation types has to be known. A specific combination of different operation types is called an operation type set, Sk.
The tooling of a machine determines the operations that the machine can perform.
Operations are aggregated into operation types. Therefore a machine can perform a set of operation types. An operation type set assignment parameter, zkm, specifies the operation type sets assigned to machine m. If zkm=1, then machine m can perform all operations belonging to operation type set k.
The calculation of the capacity of each operation type set is based on machine capacity.
The capacity of a machine, cm, is expressed in capacity units (CUs) over a period of, for example, a shift or two, or a day, or a week. The capacity unit is a normalized measure of the available capacity for the period examined. For example, 1 CU is equal to 8 hours, if the production capacity of one 8 hour long shift is to be examined.
An upper capacity bound of a particular operation type set k, uk, is the maximum amount of capacity available for operation type set k. It is calculated as the sum of the CUs of those machines that are capable of performing any and all operations belonging to that operation type set, that is,
{ ∑ } ∑
= ′′
∈ ′′
′′
=
⋅
=
′′
M m
m k m S
S k
k c z k K
u
k
k 1
, , 1K
(3.1) In equation (3.1), Sk′′ is the set of all operation type sets that contain any of the operation types of operation type set Sk. For each operation type set k, it is checked to see if the corresponding Sk′′ sets can be found on any of the machines. If they can, then the corresponding machine capacities are considered at the calculation.
In each column of the matrix defined by zkm, only one element is equal to 1, since only one operation type set can be assigned to a machine m. Therefore, the capacity of each machine can be considered only once in the summation in (3.1). But the capacity of machine
m is considered in the summation only if the operation type of that machine is an element of the set Sk′′, that is, for which Sk′′∈Sk′′ and zk′′m =1.
Table 3.3 Summary of notation of Chapter 3
A lower capacity bound of an operation type set k, lk, is the minimum amount of unassigned capacity for operation type set k that is available only for the operations that belong to that operation type set. It is calculated as the sum of the CUs of those machines that are capable of performing only those operations belonging to that particular operation type set.
{ }
c z k Kl
M m
m k m S
S k k
k k
, , 1
1
= K
⋅
=
∑ ∑
= ′
∈ ′
′ ′ (3.2)
In equation (3.2), S′k is the set of those operation type sets that contain only operation types belonging to Sk. For each operation type set k it is checked to see if the corresponding
Subscripts:
i – index of part type (1, ...,I), h – index of operation type (1, ...,H),
k – index of a set of operation types (1, ...,K),
k′ – index of a subset of a set of operation types (1, ...,K′), k″ – index of a subset of the set of all operation types (1, ...,K″), m – index of machines (1, ...,M).
Parameters:
oj – operation j, oth – operation type h, Sk – operation type set k,
S′k – set of operation type sets that contain only operation types belonging to Sk, S″k – set of operation type sets that contain any of the operation types of
operation type set Sk,
cm – production capacity of machine m,
zkm – operation type set assignment parameter. It is equal to 1 if operation type set k is assigned to machine m, and it is equal to 0 otherwise,
uk – upper capacity bound of operation type set k, lk – lower capacity bound of operation type set k, α – acceptable ratio of capacity under-utilization, β – acceptable ratio of capacity over-utilization, pji – processing time of operation j of a part of type i,
pthi – processing time of all operations of operation type h of type i,
pski – processing time of all operations of operation type set k of a part of type i, rth – capacity requirement of operation type h,
rsk – capacity requirement of operation type set k, wi – weight of part type i,
∆rth
– – feasible decrease of the capacity requirement of operation type h,
∆rth
+ – feasible increase of the capacity requirement of operation type h,
∆cm
– – feasible decrease of the capacity of machine m,
∆cm
+ – feasible increase of the capacity of machine m, wi – weight of a part type in the objective function.
Variables:
xi – production requirements of part type i.
34 Sk′ sets can be found on any of the machines. If they can, then the corresponding machine capacities are considered at the calculation. The capacities of those machines need to be summed, which contain any of the operation type sets of S′k, that is, for which Sk′ ∈Sk′ and
=1
′m
zk .
Some details about these bounds are as follows. If each machine that can perform any of the operation types of a specific operation type set k can perform operation types not belonging to operation type set k, then the lower capacity bound of operation type set k is equal to zero. For example, if each machine can perform all operation types, then the lower capacity bound of all but one operation type sets are equal to zero. In this case, the only non-zero lower capacity bound will belong to the operation type set that contains all operations types. For this operation type set, the lower and upper capacity bounds always coincide, and are non-zero.
The available capacity per period for an operation type set is a range defined by the upper and lower capacity bounds. A necessary condition of capacity availability is that the capacity requirements from all operation type sets must be less than their corresponding upper bounds.
When all operations have been assigned to machines and the workload is less than the lower capacity bound of any operation type set, then there is machine idle time.
In real manufacturing systems, production managers may need to work around a certain amount of lack of capacity. To supplement capacity, management may consider overtime, subcontracting, or other possible capacity adjustments. The size of acceptable capacity over-utilization, β, is expressed as a percentage of total capacity. The capacity increased by acceptable over-utilization is called the extended capacity upper bound.
Also, production managers are generally resigned to a certain amount of idle capacity.
Idle capacity is either planned and serves as buffer capacity to absorb the effect of unexpected events (i.e., machine breakdowns, tool breakages, quality problems, expected or unexpected rush orders) or it is a consequence of scheduling constraints. The size of acceptable capacity under-utilization, α, is expressed as a percentage of total capacity. The capacity decreased by the acceptable idle time is called the extended capacity lower bound.
The upper and lower capacity bounds are necessary but not sufficient conditions for the feasibility of a production plan. The role of the extended upper and lower capacity bounds is to provide a capacity reserve for those phenomena not considered in the aggregate planning phase. The extended capacity bounds allow a link between the aggregate and the operational planning and control levels.
The available capacity range and the extended capacity range are information about capacity that can be used for production planning. To analyze the utilization of a production system, the capacity requirements of the operation types and operation type sets should also be known. This data is based on the processing times of the individual operations.
The processing time of operation j of a part of type i, pij, is expressed in terms of CUs, rather than in hours or minutes. The processing time of operation type h of part type i, pthi, is the sum of the processing times of all of those operations that belong to operation type h, that is,
{ ∑ }
∈
=
h j ot o j
ji
hi p
pt (3.3)
The processing time of operation type set k of one part of type i, pski, is the sum of the processing times of all of those operation types that belong to operation type set k, that is,
{ ∑ }
∈
=
k
h S
ot h
hi
ki pt
ps
(3.4)
If the production requirements of part type i, xi, are known, then the capacity requirements of operation type h can be calculated as the sum of the processing times of all of those operations that belong to operation type h, that is,
{ ∑ }
∑
= ∈
=
h j ot o j
i ij I
i
h p x
rt
1 (3.5)
The capacity requirements of operation type set k is the sum of the capacity requirements of all of those operation types that belong to operation type set k, that is,
{ ∑ }
∈
=
k
h S
ot h
h
k rt
rs