# International Journal of Industrial Engineering Computations

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4 332 At the same time, the measure of system is calculated taking the expectation of the variance of random idle times, i.e. E(V). By definition,. But L j, the idle time in j th workstation, is,. Thus, V,,. Hence the expectation of V can be simplified as, =,,, =,,, =,,,, =,,, Thus, E(V)=, +, 2, (4) Since the task times are random variables, the condition for completion of tasks in a workstation within the assigned cycle time can be best described in terms of chance constraints is as follows,., 1, where 0 1, j = 1,2,.,N. Equivalently it can be expressed as follows, or, 1Φ Φ or, Φ Φ Therefore,. (5) Since we have,,,, (6)

5 D. Roy and D. Khan/ International Journal of Industrial Engineering Computations 2 (2011) 333 and Var(,,, =,, 2. (7) Using Eq. (6) and Eq. (7), Eq. (5) can be rewritten as,, or,,, or,,, or,,, 2. (8) Thus, the chance constraints regarding cycle time can be reduced to the following deterministic constraints,, 2,. (9) Finally, combining Eq. (1) to Eq. (4) and Eq. (9), a deterministic problem for stochastic model formulation of the optimization problem can be written as follows, min E(B) min E(V) 1, subject to, 1 i,, i p i ( iii ) a( i, j) = 0,1 i, j +, +,. j 2, To assign equal importance to each workstation we consider = j. Further may be considered 0.05 for which = One way of dealing with dual objective is to combine them with weights or priorities. In that case the reduced objective can be written as follows, min Z 1,,, where 0, 1, 1. However, we prefer to sequentially undertake the task of minimization by generating in the first instant feasible solutions under the objective of minimization of E(B) under 1 0 and then obtaining the final solution by imposing the second objective of minimization of E(V) with 0 1. To generate the set of feasible solutions we consider a sequential approach of assigning trial cycle time and resulting in slack time, to be assigned to each workstation meeting the optimality condition for the first objective of (10). (10)

6 The Algorithm To numerically solve the proposed optimization problem we have adopted the following algorithm where the overall problem has been subdivided into sub problems based on trial cycle time. For each trial cycle time, simulation approach has been adopted to arrive at all possible feasible solutions. 1. Calculate the theoretical minimum number of workstations, N min, using the following, 1 2. Calculate the minimum cycle time, C min, using the relation, 1 3. Set the cycle time at C min 4. Make an attempt to get feasible solution following the algorithm of Roy and Khan (2010) with usual cycle time constraints replaced by (10)(iv) 5. If no feasible solution is obtained then increase N min by 1 and go to step 3 6. Within the generated set calculate E(V) for each set and save the E(V) value 7. Compare the E(V) with the previous value of E(V), If the current E(V) is lower than the previous one, then save the current value of E(V) 8. When all the feasible sets are over, the final solution to the optimization problem is achieved. In this sequential simulation method, if the set of feasible solutions is finite we can arrive at global solution. It is analytically difficult to claim global optimality in case solution set is not finite. However the Kuhn Tucker condition being both necessary and sufficient for the problem we may check for optimality. 5. Worked Out Example Consider an assembly line balancing problem from Ray Wild (2004) depicted in Fig 1. where the task number is represented by the figure within a circle Fig. 1. Precedence diagram of workstations. This problem is summarized in a tabular form in terms of work elements, immediate predecessor(s), expected task durations and their variability are given in Table 1. For this particular problem, we first get the minimum number of workstations with cycle time C = 35 with N min =5. Therefore the minimum trial cycle is 1, i.e. 29. Since the cycle time C is 35, the trial cycle time starts with 29 and goes upto 35. But among the initial trial cycle times as 29, 30, 31, 32, 33, 34, 35 we get the first feasible solution at C t = 33.Thus, our trial cycle times are 33, 34 and 35. The final configuration has been obtained from trial cycle time as 34 with slackness 1. This configuration is presented in Table 2.

7 D. Roy and D. Khan/ International Journal of Industrial Engineering Computations 2 (2011) 335 Table 1 Precedence relationships and task times of work elements Work element Immediate predecessor Expected activity Time Variance of activity time , , , ,11, , 19, Table 2 Final optimum configuration Work stations C E(V) 34 2,3,6,7,8 1,4,5,11 9,10,12,13,14 15,16,19 17,18,20, The optimal configuration consists of 5 workstations with work elements 2, 3, 6, 7, 8 assigned to workstation 1, work elements 1, 4, 5, 11 assigned to workstation 2, work elements 9, 10, 12, 13, 14 assigned to workstation 3, in workstation 4 work elements 15, 16, 19 are assigned and work elements 17, 18, 20, 21 assigned to workstation 5 for trial cycle time 34. Finally, the corresponding value of E(V) comes out as Conclusion A mathematical programming approach is presented for balancing an assembly line with twin objectives of minimization of balancing loss and system loss. As system loss arises out of variations in human behavioral, stochastic setup is needed for describing the situation, representing of the problem and arriving at the final solution of the same. Reduction of the stochastic setup into deterministic constraints has been implemented under normality assumption. A sequential approach has been installed to arrive at the final solution. We believe the approach is generic and it can be used to solve different line assembly problems in reasonable computation time. Final choice can be made based on minimum number of workstations and minimum value of expected variance of the idle times, proposed herein as a measure of system loss. References Agarwal, S. & Tiwari, M. K. (2008). A collaborative ant colony algorithm to stochastic mixed-model U-shaped disassembly line balancing and sequencing problem. International Journal of Production Research, 46(6),

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