Composite Schedules of Manufacturing Cells and Their Timed Petri Net Models
چکیده
Composite schedules of manufacturing cells and their timed Petri net models 2991 are quite satisfactory [2]. In particular, the throughput (or the cycle time) of a timed net model can easily be determined from the structure of a net [7]. This paper presents a systematic method of generating all composite schedules for a given cell. Each of such composite schedules is modeled by a timed Petri net, and its performance is determined by invariant analysis of the derived net model. Simple examples of manufacturing cells are used as an illustration of the proposed approach. 2. SIMPLE SCHEDULES Simple schedules of robotic cells are schedules in which exactly one part enters and one leaves the cell in each cycle (although the part which leaves the cell may not be the same as the one which enters the cell). It is known [4] that for a cell with m machines, there are m! different simple schedules. For m = 3 (a sketch of a 3–machine cell is shown in Fig.1) there are six simple schedules, which are denoted here as A, B, C, D, E and F. M3 M1 M2 M3 In Out Robot Fig.1. Layout of a three–machine cell. Assuming, for simplicity, that each part follows the same path from the input (In) to machine–1 (M1), to machine–2 (M2), to machine–3 (M3), and finally to the output of the cell (Out), the simple schedules can be described by the following sequences of cell configurations, where each configuration corresponds to a distribution of parts among the machines of the cell (when the robot does not carry a part); more specifically, for an m–machine cell, each configuration is described by an m–tuple of machine descriptions: (k1, k2, ..., km) where each machine description ki is “1” if the machine Mi is loaded with a part in this configuration, and otherwise is “0” (in the case of multiple machines performing exactly the same operations, the values describing each multi–machine station would assume the values from “0” to “n” where n is the number of identical machines). The six simple schedules for a 3–machine cell are: A: (0, 0, 0)→ (1, 0, 0)→ (0, 1, 0)→ (0, 0, 1)→ (0, 0, 0) B: (0, 0, 1)→ (1, 0, 1)→ (0, 1, 1)→ (0, 1, 0)→ (0, 0, 1) C: (0, 0, 1)→ (1, 0, 1)→ (1, 0, 0)→ (0, 1, 0)→ (0, 0, 1) D: (0, 1, 0)→ (1, 1, 0)→ (1, 0, 1)→ (1, 0, 0)→ (0, 1, 0) E: (0, 1, 0)→ (1, 1, 0)→ (1, 0, 1)→ (1, 0, 0)→ (0, 1, 0) F: (0, 1, 1)→ (1, 1, 1)→ (1, 1, 0)→ (1, 0, 1)→ (0, 1, 1) Each change of configurations corresponds to a part moving from one machine to another, from the input to the first machine, or from the last machine to the output; all schedules uniformly begin by moving a (new) part from the input to the first machine. The simple schedules can be generated systematically by tracing all possible “passages” of parts through the cell [8]. These passages are correspond to the following changes of cell configurations: • a configuration (k1, ..., ki, ki+1, ..., km) derives a configuration (k1, ..., ki−1, ki+1+1, ...km) if and only if the value of ki is “1” and the value of ki+1 is “0”, i = 1, ...,m− 1; • a configuration (k1, k2, ..., 1) always derives a configuration (k1, k2, ..., 0) (this derivation corresponds to moving a part from the last machine Mm to the output of the cell), • it is assumed that each schedule begins by moving a (new) part from the input to the machine M1, so the first derivation is always from (0, k2, ...km) to (1, k2, ..., km), • for a cell with m machines, the length of all simple schedules is equal to m + 1 (it corresponds to a passage of a part, although not necessarily the same, from the input, through all machines of the cell, to the output). Since parts are transported from one machine (or input) to another (or output) by the robot, the sequences of robot’s actions can easily be derived from the sequences of configurations by “implementing” the moves of parts corresponding to changes of consecutive configurations. For example, schedule A begins be transporting a part from the input to M1 and loading it; when the first operation is finished, the robot unloads M1, moves the part to M2 and loads it there, and so on. The sequences of robot actions are as follows (the robot moves from X to Y are denoted by X ⇒ Y if the robot carries a part and by X → Y otherwise): A: In⇒M1 ⇒M2 ⇒M3 ⇒ Out→ In B: In⇒M1 ⇒M2 →M3 ⇒ Out→M2 ⇒M3 → In C: In⇒M1 →M3 ⇒ Out→M1 ⇒M2 ⇒M3 → In D: In⇒M1 →M2 ⇒M3 →M1 ⇒M2 →M3 ⇒ Out → In E: In⇒M1 →M2 ⇒M3 ⇒ Out→M1 ⇒M2 → In F: In⇒M1 →M3 ⇒ Out→M2 ⇒M3 →M1 ⇒M2 → In Composite schedules of manufacturing cells and their timed Petri net models 2992 3. COMPOSITE SCHEDULES For composite schedules, several parts enter and leave the cell in each cycle. Models of composite schedules can be regarded as interleaved compositions of simple schedules, corresponding to processing of consecutive parts which enter the cell within one cycle, and which move from one machine to another in consecutive steps of the schedule. A systematic generation of composite schedules can be obtained by a simple extension of the procedure used for generation of simple schedules. Composite schedules which process n parts withing a single cycle are called simply n–schedules. For composite schedules, the configurations are extended by a hypothetical “input container”, which initially contains exactly n parts for each n–schedule, and which must be emptied during the schedule. This container is represented by an additional element of configurations, for example, the initial element, separated from the remaining machine descriptions by a colon (rather than of a comma). A typical configuration for description of composite schedules of an m–machine cell is thus: (k0 : k1, k2, ..., km) The (revised) rules describing changes of configurations are: • a configuration (k0 : k1, ..., ki, ki+1, ..., km) derives a configuration (k0 : k1, ..., ki − 1, ki+1 + 1, ...km) if and only if the value of ki is “1” and the value of ki+1 is “0”, k = 1, ...,m− 1; • a configuration (k0 : k1, k2, ..., 1) always derives a configuration (k0 : k1, k2, ..., 0) (this derivation corresponds to moving a part from the last machine Mm to the output of a cell), • a configuration (k0 : 0, k2, ..., km) derives a configuration (k0 − 1 : 1, k2, ..., km) if and only if the value of k0 is greater than 0 (this derivation corresponds to moving a part from the input to the first machine M1), • it is assumed that each schedule begins by moving a (new) part from the input to the machine M1, so the first derivation is always (k0 : 0, k2, ...km) to (k0 − 1 : 1, k2, ..., km), • for a cell with m machines, the length of all n– schedules is equal to n ∗ (m+ 1). For a 3–machine cell, there are 34 different 2– schedules, including 6 schedules which are just simple schedules repeated twice. All 34 2–schedules can be systematically derived by repeatedly applying the rules to the four initial configurations of the cell. For the initial configuration (0,0,0), there are five 2– schedules: (5) (0:0,0,0) ? (0:0,0,1) ? (0:0,1,0) ? (0:1,0,0) ? (1:0,0,0) (4) (0:0,0,0) ? (0:0,0,1) ? (0:0,1,0) ? (0:0,1,1) (3) (0:0,0,0) ? (0:0,0,1) ? (0:0,1,0) ? (0:1,0,0) (0:1,0,1) H H Hj (1:0,0,1) X X X X Xz (2) (0:0,0,0) ? (0:0,0,1) ? (0:0,1,0) ? (0:0,1,1) (1) (0:0,0,0) ? (0:0,0,1) ? (0:0,1,0) ? (0:1,0,0) (0:1,0,1)