*3.2. Basic Mathematical Notation of the Problem*

Prior to formulating the problem of robust job scheduling under uncertainty in the job-shop system, we need to define the elements of the production process:

• Set *M* is a set of *m* machines (workstations) processing jobs:

$$M = \{M\_1, M\_2, \dots, M\_m\}.\tag{1}$$

• Set *J* is a set of *n* jobs (tasks) to process

$$J = \{I\_1, I\_2, \dots, I\_n\}.\tag{2}$$

Processing job *Ji* on machine *Mj* constitutes an operation, which is called operation *j* of job *i* in the following. Therefore, it is necessary to define:

• *MO*—a matrix of *m* columns and *n* rows describing the technology (the job order):

$$MO = \begin{vmatrix} o\_{ij} \end{vmatrix}\_{\prime} \tag{3}$$

where *oij*—the order position of the operation *j* of job *i*, which is *oij* = 0 when the job is not processed on the machine; and *oij* = {1, ... , *m*}, when it is.

• Matrix *PT*—a matrix describing processing times of operations:

$$PT = \begin{bmatrix} pt\_{ij} \end{bmatrix} \tag{4}$$

where *ptij*—the processing time of operations *j* of job *i;* for each *oij* = 0, *ptij* = 0.

• Set *FTMl* of potential machine failure times:

$$FT\_{\rm MI} = \left\{ f t\_{\rm MI\_1}, f t\_{\rm MI\_2}, \dots, f t\_{\rm pll\_z} \right\}, \ l \in \text{<1;} \ m >,\tag{5}$$

where *f tMlz*—time to failure of the machine *l*; where *z* is a natural number representing the *z*-th machine failure.

• Set *TBMl* of time buffers to include in the nominal schedule (for machine *l*) to obtain a robust schedule:

$$TB\_{MI} = \left\{ tb\_{Ml\_1}, tb\_{Ml\_2}, \dots, tb\_{Ml\_z} \right\} \tag{6}$$

where *tbMlz*—the size of the time buffer in the schedule at the failure time *f tMlz* ; where for *f tMlz* - 0, *tbMlz* -0.
