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A multi-objective two stage stochastic programming model is proposed to deal with a multi-period multi-product multi-site production-distribution planning problem for a midterm planning horizon. The presented model involves majority of supply chain cost parameters such as transportation cost, inventory holding cost, shortage cost, production cost. Moreover some respects as lead time, outsourcing, employment, dismissal, workers productivity and training are considered. Due to the uncertain nature of the supply chain, it is assumed that cost parameters and demand fluctuations are random variables and follow from a pre-defined probability distribution. To develop a robust stochastic model, an additional objective functions is added to the traditional production-distribution-planning problem. So, our multi-objective model includes (i) the minimization of the expected total cost of supply chain, (ii) the minimization of the variance of the total cost of supply chain and (iii) the maximization of the workers productivity through training courses that could be held during the planning horizon. Then, the proposed model is solved applying a hybrid algorithm that is a combination of Monte Carlo sampling method, modified

One of the problems that could be addressed in the scope of supply chain management is production-distribution planning which is an operational activity that does a plan for the production process, to give an idea to management as to what quantity of materials and other resources are to be procured and when, so that the total cost of operations of the organization is kept to the minimum over that period. Production-distribution planning has attracted the attention of many researchers from several years ago [

In spite of the fact that the concepts of variance has been considered in other areas, but to the best of our knowledge, it is the first time workers productivity is considered in a multiobjective scheme to model robust production-distribution planning under uncertainty. Moreover, the idea of involving the human-related issues such as workers' skill level and workers' training is also incorporated into the model. Using this idea, we have the option of training the workers instead of firing them and then hiring new full-skilled ones. Since the expected total cost, the variance of the total cost and the workers productivity are in conflict with each other, it is proposed to model a multiobjective production-distribution problem whose solution will be a set of Pareto-optimal possible plan alternatives representing the trade-off among different objectives rather than a unique solution. Some approaches to deal with solving a multiobjective production-distribution planning under uncertainty are developed such as Possibilistic linear programming method [

According to Masud and Hwang [

The

We formulate the proposed model as a multiobjective robust stochastic mixed-integer nonlinear programming problem, after linearization, it is solved by using a hybrid algorithm that is a combination of the extended Monte Carlo sampling method, modified

The rest of the paper is organized as follows: in Section

The proposed multiobjective multiproduct multisite production-distribution problem can be described as follows.

There are

The present work formulates the production-distribution problem as a robust multiobjective nonlinear programming and tries to minimize the expected total cost of supply chain, the variability of the total cost of supply chain and the workers productivity, simultaneously, and take decisions for each period as follows:

the quantity of product

the number of

the quantity of product

the amount of demand in each customer’s zone is not met.

In our proposed model the scenario-based approach is used to represent the uncertain parameters. Due to the multiperiod multisite multiproduct nature of the model, the problem includes a large number of uncertain parameters; a resulting challenge is that a large number of scenarios are required. To reduce the model size and the number of scenarios, we use an extended Monte Carlo sampling method to generate the scenarios. Each scenario is then associated with the same probability with the summation of the probabilities for all the scenarios equal to 1. The extended Monte Carlo sampling method is an extension of the conventional Monte Carlo sampling method in which interaction between uncertainties is analyzed. Therefore value assignment for dependent uncertain parameters is controlled regarding the type and the level of possible dependencies.

In this paper, a novel multiobjective stochastic robust optimization approach is presented in which uncertainty is represented by a set of discrete scenarios (

In order to overcome the complexity of multiobjective stochastic programming problems, we combine two techniques; the modified

In this paper, we applied a modified version of

Select one of the objective functions (

Determine the grid points

The modified

Note that, at each iteration of the internal loop of modified

As mentioned before our proposed model is a multiobjective stochastic robust optimization model in which uncertainty is represented by a set of discrete scenarios, we use an extended Monte Carlo sampling approach to descretize the continuous distribution functions and generate the scenarios [

The idea behind of the L-shaped method is to first solve the master problem (the model with those constraints that do not include the second stage variables) to obtain a lower bound of the objective value. We then fix all the first stage decisions and solve each scenario sub-problem (inner-model that include second stage decisions) to get an upper bound. If the lower bound and the upper bound fall into a pre specified tolerance, then the algorithm stops. Otherwise, we add a cut by using of the duals of the scenario sub-problems and return to the master problem. We use this method whenever needed in the inner loops of modified

Flowchart of the proposed method.

Consider the supply chain network problem depicted in Figure

Cost items distribution functions.

Cost item | Probability distribution |
---|---|

Product inventory holding cost ( | Uniform |

Hiring cost (10 | Normal |

Firing cost (10 | Normal |

Salary cost (10 | Uniform |

Training cost (10 | Normal |

Production cost ( | Uniform |

Transportation cost ( | Uniform |

Shortage cost ( | Normal |

Supply chain network.

We solve the example with a sampling size of 100 scenarios. The mathematical model has 1920 integer variables, 19680 continuous decision variables and 44148 constraints. Using L-shaped method the ideal and the nadir vales for each objective functions are obtained and reported in Table

The payoff table for modified

The optimal solution for | |||
---|---|---|---|

1,733,212 | 92,705.3 | 0.25 | |

10,594,461 | 0 | 0.5 | |

4,414,184 | 281,715.4 | 0.95 | |

Ideal value (Id) | 1,733,212 | 0 | 0.95 |

Nadir value (Nd) | 10,594,461 | 281,715.4 | 0.25 |

Figure

Convergence of the L-shaped method.

Pareto curve for the expected and variance of total cost.

The behavior of

The size of the resulting stochastic programming problem is very large and increases exponentially as the number of scenarios increases. For the stochastic programming model with 100 scenario case, mathematical programming solvers could not solve the problem in reasonable amount of time due to its huge size, that is, the problem cannot be solved directly, although the deterministic model can be solved to optimality within five minutes. By using the proposed method, we can obtain the optimal solution for the 100 scenario case in around 1 h with 0.01% optimality tolerance. In Table

State of the staff upgrading versus the average workers productivity.

Upgraded level | Factory | Period | ||||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |||

1 | 1 | |||||||||||||

2 | 4 | 9 | ||||||||||||

2 | 5 | |||||||||||||

1 | 3 | 3 | 1 | |||||||||||

3 | 1 | 1 | ||||||||||||

4 | 6 | |||||||||||||

1 | 4 | 3 | ||||||||||||

0.75 | 2 | 2 | 8 | |||||||||||

1 | 10 | |||||||||||||

1 | 8 | |||||||||||||

0.5 | 1 | 8 | ||||||||||||

2 | 1 | 2 | ||||||||||||

0.25 | 1 | 6 | 3 | |||||||||||

1 | 4 |

In this section, five large scaled test problems are generated to evaluate the efficiency of the proposed algorithm. As we described earlier, for these problems, standard mathematical programming solvers cannot solve the problem in reasonable amount of time. Therefore, each test problem is solved four times with 50, 100, 500 and 1000 scenarios and compared with lower bounds of linear programming solvers obtained after one and half an hour.

To evaluate the efficiency of the algorithm, the usual relative gap (RG) between the average of best values of first objective function in Pareto set solutions (AB) (obtained from the proposed method) and the average of the lower bounds (AL) of the first objective function in Pareto set solutions (obtained by standard linear programming solver) is used and reported in Table

Comparison of the performance of the proposed algorithm with different scenario numbers.

Prob. No. | Problem info. No. of | ||||

50 scenarios | 100 scenarios | ||||

CPU Time (min) | RG% | CPU Time (min) | RG% | ||

1 | 12/20 | 17 | 1.253 | 17 | 1.358 |

2 | 20/25 | 18 | 1.668 | 18 | 1.681 |

3 | 25/30 | 20 | 2.446 | 23 | 2.588 |

4 | 25/35 | 21 | 2.477 | 24 | 2.356 |

5 | 30/40 | 25 | 2.302 | 25 | 3.01 |

Prob. No. | Problem info. No. of | ||||

500 scenarios | 1000 scenarios | ||||

CPU Time (min) | RG% | CPU Time (min) | RG% | ||

1 | 12/20 | 18 | 1.392 | 20 | 1.501 |

2 | 20/25 | 20 | 1.382 | 21 | 1.661 |

3 | 25/30 | 24 | 2.508 | 27 | 2.446 |

4 | 25/35 | 27 | 2.210 | 28 | 2.477 |

5 | 30/40 | 29 | 2.788 | 32 | 2.302 |

In this paper a multiobjective two-stage stochastic programming model is developed to deal with production-distribution planning in an uncertain supply chain considering workers productivity. In addition to the traditional production planning problem in which the total cost is considered as the main objective function we added two extra objective functions that are variability and workers productivity. Risk is described in the form of absolute deviation and indicates the variability of the total cost of supply chain and productivity is described in the form of average workers productivity among all factories in all periods. It is assumed that all of the parameters are subject to uncertainty. The proposed model is solved with a novel hybrid algorithm composed of modified