4.2 Example Problem for MILP Demonstration We now present a small 4/FF2/C MILP formu- lation. In this instance, we will assume that travel times and sequence-independent setup times are negli- gible (so, = 0 for all i and ;). Further, we will as- sume that all jobs are available to be processed at the same time (so, r, can be set to 0 for all i). The baseline processing times and their corresponding times for the faster machines are shown in Table 1. The other known values that complete the problem are provided in the following list: M,3 9,1 = 1.00.83 - 1.20.531 = 1.40 M=2 51. - 1.00.5 = 1.15 Q = 1000 . . . Even though this example problem is small, the MILP formulation is lengthy. Therefore, due to arti- cle-size limitations, the MILP formulation for the ex- ample is not included in this work. The interested reader should find, in an attempt to generate the MILP for this problem. the following numbers of constraints: 4 for Set (2): 8 for Set (3): 8 for Set (4): 60 for Sets (5) and (6), combined: 20 binary integer variable decla- rations for Set (7): 12 binary integer variable declara- tions for Set (8); Set (9), non-negativity, generally needs not be explicitly stated for most solver packages; and 4 for Set (10). The interested reader is also en- couraged to contact the authors for a copy of the MILP formulation for this example. This model was converted for input to Hyper- LINDO/PC v.6.01 (henceforth, referred to as LINDO). The solution for this model was obtained by LINDO in under 10 seconds with an Intel Pen- tium48 processor (3.00 GHz) The optimal makespan is 14.57 and, to be succinct, only the com solution output will be shown: machine. 2. Consider the example problem, which is given in Section 4.2 of Article 4 (Santos and Rou 2007). Suppose that the stages in the example problem have been swapped. That is, machines 1 and 2 of Stage 2 becomes machines 1 and 2 of Stage 1, and machines 1, 2 and 3 of Stage 1 becomes machines 1, 2 and 3 of Stage 2 in the new example problem. Then, the optimal schedule obtained by solving the MILP model given in Section 4.1 is as: Stage 1 Machine 1: Job 2 - Job 1 Machine 2: Job 4 - Job 3 Stage 2 Machine 1: Job 2 Machine 2: Job 4 - Job 3 Machine 3: Job (a) Draw the Gantt chart of the optimal schedule. (b) Using the optimal schedule, determine the values of all decision variables, and write each constraint explicitly. (c) When you compare the optimal schedules of the example problem given in the article and the new example described above, what do you observe? 4.2 Example Problem for MILP Demonstration We now present a small 4/FF2/C MILP formu- lation. In this instance, we will assume that travel times and sequence-independent setup times are negli- gible (so, = 0 for all i and ;). Further, we will as- sume that all jobs are available to be processed at the same time (so, r, can be set to 0 for all i). The baseline processing times and their corresponding times for the faster machines are shown in Table 1. The other known values that complete the problem are provided in the following list: M,3 9,1 = 1.00.83 - 1.20.531 = 1.40 M=2 51. - 1.00.5 = 1.15 Q = 1000 . . . Even though this example problem is small, the MILP formulation is lengthy. Therefore, due to arti- cle-size limitations, the MILP formulation for the ex- ample is not included in this work. The interested reader should find, in an attempt to generate the MILP for this problem. the following numbers of constraints: 4 for Set (2): 8 for Set (3): 8 for Set (4): 60 for Sets (5) and (6), combined: 20 binary integer variable decla- rations for Set (7): 12 binary integer variable declara- tions for Set (8); Set (9), non-negativity, generally needs not be explicitly stated for most solver packages; and 4 for Set (10). The interested reader is also en- couraged to contact the authors for a copy of the MILP formulation for this example. This model was converted for input to Hyper- LINDO/PC v.6.01 (henceforth, referred to as LINDO). The solution for this model was obtained by LINDO in under 10 seconds with an Intel Pen- tium48 processor (3.00 GHz) The optimal makespan is 14.57 and, to be succinct, only the com solution output will be shown: machine. 2. Consider the example problem, which is given in Section 4.2 of Article 4 (Santos and Rou 2007). Suppose that the stages in the example problem have been swapped. That is, machines 1 and 2 of Stage 2 becomes machines 1 and 2 of Stage 1, and machines 1, 2 and 3 of Stage 1 becomes machines 1, 2 and 3 of Stage 2 in the new example problem. Then, the optimal schedule obtained by solving the MILP model given in Section 4.1 is as: Stage 1 Machine 1: Job 2 - Job 1 Machine 2: Job 4 - Job 3 Stage 2 Machine 1: Job 2 Machine 2: Job 4 - Job 3 Machine 3: Job (a) Draw the Gantt chart of the optimal schedule. (b) Using the optimal schedule, determine the values of all decision variables, and write each constraint explicitly. (c) When you compare the optimal schedules of the example problem given in the article and the new example described above, what do you observe