Chapter 3
The Robustness of Multi-Purpose Machines
Workshop Configuration
54 Flexibility and Robustness in Scheduling
3.2.1. Modeling the workshop
3.2.1.1. Production resources
A photolithography workshop is composed of m parallel machines that are partially multi-functional: each machine can process a subset of product types. This is due to the fact that processing a product type requires equipping the machine with a resin, which is available in limited quantity. This limitation is the origin of the qualification constraints: a machine is said to be qualified for a product type if it is equipped with the required resin, otherwise it is not qualified. Products of the same type require the same resin, and are considered identical. The number of product types is denotedn. The set of all the qualifications of the machines is the workshop configuration. This configuration is the result of a decision-making process aiming at reaching a loading ratio equal to 100%. Furthermore, this choice must be compliant with technological constraints: an old-generation machine may not be able to process a new product type (even with the necessary resin), or some different resins may not be installed on the same machine because of incompatibility. For these reasons, the photolithography workshop is a typical example of a set of multi-purpose machines (see [BRU 97]).
In the most general version of the model, the processing speed of machine j for the products of typei is denoted v(i, j), the machines are said to be unrelated.
With the notation introduced in Chapter 1, we have α1 = R. Furthermore, v(i, j) is the quantity of products of type i that machine j can process per unit of time.
It is necessarily non-negative, andv(i, j) = 0 models a technological constraint. It must not be confused with a qualification constraint, as qualification is the result of a choice: whenv(i, j) = 0, there is no choice to be made as machine j cannot process any product of typei. Qualification arises only if v(i, j) > 0.
Preemption and splitting are allowed, so a product type may be processed on several machines simultaneously provided, however, that these machines are qualified for the given product type. Thus, this scheduling problem can be viewed as an allocation problem since the starting times of the products have no impact on the makespan. The set of machines[1, m] is denoted J, and the set of product types [1, n] is denoted I. The qualification matrix Q is an n by m binary matrix such that Q(i, j) = 1 if and only if machine j is qualified for product type i, otherwise Q(i, j) is zero. Each column ofQ models the qualification of a machine, each row of Q models the machines qualified for a product type. The processing speed matrixv has the same dimensions asQ.
The Robustness of Multi-Purpose Machines Workshop Configuration 55
Provided thatv(i, j) = 0 implies Q(i, j) = 0 for all i and j, matrix Qvis defined byQv(i, j) = Q(i, j)× v(i, j) (∀i ∈ I) (∀j ∈ J). This matrix models the machines’
processing speed only for the product types for which they are qualified.
A qualification matrix is said to be admissible if and only if it has neither a zero-row nor a zero-column. Indeed, a zero-row shows that products which exist cannot be processed by the workshop. Besides, a zero-column shows a machine that is not qualified for any product type: as a consequence, it should not be considered as a resource for the problem. In the rest of this chapter, only admissible qualification matrices are considered. Moreover, the technological constraints are also assumed to always be such that admissible qualification matrices can be built.
3.2.1.2. Modeling the workshop demand
The products that should be processed the same way in the workshop define a product type. The workshop demand is a column vector denoted N , having n elements. The non-negative real N (i) is the total amount of products of type i to be processed by the workshop. N (i) is not necessarily an integer, as N (i) can be the result of a mean calculation. Demand N is admissible if and only if N (i) is non-negative for alli in I. In the rest of this chapter, only admissible demands are considered.
The following example withn = 3 product types and m = 4 machines is to be used later to illustrate the results presented in this chapter.
Q =
⎡
⎢⎣
1 1 0 0
1 1 1 0
0 1 0 1
⎤
⎥⎦ v =
⎡
⎢⎣
1 2 0.5 1.2
1 2 0.5 0
0 2 0.5 1.2
⎤
⎥⎦
so,
Qv=
⎡
⎢⎣
1 2 0 0
1 2 0.5 0
0 2 0 1.2
⎤
⎥⎦ N =
⎡
⎢⎣ 1 2 1.7
⎤
⎥⎦
3.2.2. Modeling disturbances on the data
The context of semi-conductor manufacturing is highly uncertain. New product types are launched very often because of regular advances in new technologies, but launching new production processes often generates a lot of scrap at the beginning.
56 Flexibility and Robustness in Scheduling
This is the reason why semi-conductors (microprocessors, memory modules) are often very expensive when they appear on the market, and why they become so cheap when the technologies involved in the processing are well mastered. Scrap is not only an economic loss, it also generates disturbances in production planning. If the photolithography workshop is involved at several stages in the production process of semi-conductors, it is said to be reentrant. Furthermore, the workshop we have been focusing on is part of a manufacturing center dedicated to mass production, as well as to research and development. For that reason, the photolithography workshop produces a great variety of product types which also have very different volumes, so that its actual demand cannot be accurately predicted in advance. On the other hand, the machine qualification process is updated monthly, and must ensure that the chosen configuration fits the demand for the entire month. The configuration cannot be updated dynamically to fit the actual demand as qualification is a very time-consuming process (it requires the machines to be stopped in order to perform tests and setups).
Thus, the machine configuration process aims at choosing a configuration that leads to the highest loading ratio for a large set of possible demands.
Disturbances affecting upstream workshops may have different kinds of impacts on the photolithography workshop. If a machine breakdown in an upstream workshop leads to a decrease of the workshop demand, the demand mix may also be changed.
In the rest of this chapter, the photolithography workshop forecast demand is denoted Nref, and the actual demand is denotedN . The actual demand N differs from Nrefby a quantity denoteddN , due to disturbances in upstream workshops. More precisely, it can be written thatN = Nref + dN , where dN can have negative entries. Thus, dN (i) < 0 means that the quantity of products of type i is less than expected.
However, as the actual demand is non-negative,Nref(i) + dN (i) must be positive or zero for eachi.
The problem of determining the photolithography workshop configuration under uncertain demand can be stated using the formalism adopted in Chapter 1.
The problem P is to find the best possible configuration for the photolithography workshop. The uncertainties are supposed to impact the demand. Such a demand is a scenario denotedI in Chapter 1. The configuration Q corresponds to a solution S. The performance of S on the instanceI, denoted zI(S) is the loading ratio of the machines in the photolithography workshop. Providing performance guarantees is also a key feature here, as the loading ratio is expected to remain equal to 100%.
To do so, the robustness criterion denotedR4 in Chapter 1 is to be minimized, as it assesses the absolute deviation from a fixed level˜z.
The Robustness of Multi-Purpose Machines Workshop Configuration 57
3.2.3. Performance versus robustness: load balance and stability radius 3.2.3.1. Performance criterion for a configuration
A manager’s attention must often be focused on the photolithography workshop for several reasons. As new technologies frequently appear in the field, photolithography machines are not only subject to rapid obsolescence, but are also very expensive.
That is the reason why this workshop often has a low production capacity, compared to other workshops. As a consequence, the photolithography workshop is often a bottleneck in the production flow, being responsible for WIP (Work In Progress) waves that cause disorder in the whole production process by spreading along the production flow. Thus, the workshop configuration is expected to ensure that all the machines have the highest loading ratio. More precisely, the configuration performance is its ability to guarantee the existence of a balanced load plan which meets the demand;
the load plan will be called production plan in this chapter.
3.2.3.2. Robustness
The configuration Qv is expected to ensure that the machines’ loading ratio is equal to 100% for the actual demandN = Nref + dN . This should hold on a “neighborhood” of Nref. The performance guarantee offered by Qv around Nref is measured using the stability radius [SOT 98]. The stability radius assesses the minimum magnitude of a disturbance vector dN being such that a balanced production plan cannot be found for N . Thus, this work can be classified among proactive approaches.