A modified variable neighborhood search algorithm for manufacturer selection and order acceptance in distributed virtual manufacturing network

Abstract

In this paper, we investigate a manufacturer selection and composition problem for Distributed Virtual Manufacturing Network (DVMN) with order acceptance and scheduling of deteriorating jobs, where potential manufacturers include proprietary plants and outsourced co-manufacturers. In such a problem, at the beginning of planning horizon, a manufacturing company (MC) receives a series of order requests from its customers. Due to the limited production capacity, the MC is not able to fulfill all the orders requested by the customers on time. To satisfy the order demand as much as possible, one of the MC’s choices is to establish a collaborative manufacturing platform (i.e., DVMN) that leverages the advantages of a social manufacturing network to expand their production capabilities. However, the increase in production capacity does not mean that all orders from the customers are able to be accepted and produced. In fact, standing at the profit-maximizing point of view, the order requests that do not bring profits to the company need to be rejected at the beginning of planning horizon. For the accepted orders, the MC will produce a part of them by its plants in the DVMN, and the remaining orders will be outsourced to other co-manufacturers in the DVMN. Based on some structural properties, we develop a novel Reduced Variable Neighborhood Search (RVNS) algorithm incorporated multiple random mutations neighborhood structures (NSs-MRM) to determine the composition of the DVMN, and further determine which orders should be rejected, which accepted orders should be processed by the company-owned plants in the DVMN, and which accepted orders should be outsourced to the co-manufacturers in the DVMN. In particular, the processing sequence of the orders produced in the company-owned plants are also determined. Finally, a reasonable composition of the DVMN and corresponding order acceptance as well as order scheduling on each member of the DVMN are given. To evaluate the proposed algorithm, a series of computational experiments are conducted, and the experimental results show that the proposed algorithm outperforms other existing baseline algorithms in solving both small-scale and large-scale problem instances.

Appendix

The following lemmas are reported without detailed proof, since it can be derived from the basic mathematical method and has been frequently presented in the literature on the scheduling of deteriorating jobs [29,30,31].

Lemma 1

For any order assigned into any company-owned plant, such as $o_{ijk} \left( {CP} \right)$ , the jobs in this order should be processed by the Shortest Normal Processing Time(SNPT) rule.

Proof

This Lemma can be proven by exchanging adjacent jobs, and we omit it. □

Without loss of generality, we assume that the indices of the jobs in any order satisfy the SNPT rule. That is, $p_{ijk1} \left( {CP} \right) \le \cdots p_{ijkg} \left( {CP} \right) \cdots \le p_{{ijkn_{ijk} \left( {CP} \right)}} \left( {CP} \right)$. In Lemma 2, we give the completion time of $o_{ijk} \left( {CP} \right)$ if it starts at time $T$.

Lemma 2

For any order assigned into any company-owned plant, such as $o_{ijk} \left( {CP} \right)$ , the completion time of $o_{ijk} \left( {CP} \right)$ is

$$ C(o_{ijk} \left( {CP} \right)) = \left( {1 + d} \right)^{{n_{ijk} \left( {CP} \right)}} \times {\text{T}} + \mathop \sum \limits_{g = 1}^{{n_{ijk} \left( {CP} \right)}} p_{ijkg} \left( {CP} \right)\left( {1 + d} \right)^{{n_{ijk} \left( {CP} \right) - g}} $$

Proof

This Lemma can be proven easily by mathematical induction method, and we omit it. □

Lemma 3

For any company-owned plant, such as $CP_{ij}$ , all the orders assigned into this plant should be processed by the non-increasing order of $\rho (o_{ijk} \left( {CP} \right))$ , where the $\rho (o_{ijk} \left( {CP} \right))$ is

$$ \frac{{\left( {1 + d} \right)^{{n_{ijk} \left( {CP} \right)}} - 1}}{{\mathop \sum \nolimits_{g = 1}^{{n_{ijk} \left( {CP} \right)}} p_{ijkg} \left( {CP} \right)\left( {1 + d} \right)^{{n_{ijk} \left( {CP} \right) - g}} }} $$

Proof

This proof is simple, and we omit it. □

Without loss of generality, we assume that the indices of orders in any plant satisfy the property displayed in Lemma 3. Then, the $C_{ij} \left( {CP} \right)$ can be calculated in Lemma 4.

Lemma 4

For any company-owned plant, such as $CP_{ij}$, the completion time the last order of $CP_{ij}$ is

$$ C_{ij} \left( {CP} \right) = \mathop \sum \limits_{e = 1}^{{n_{ij} \left( {CP} \right)}} \left( {1 + d} \right)^{{\mathop \sum \limits_{k = e + 1}^{{n_{ij} \left( {CP} \right)}} n_{ijk} \left( {CP} \right)}} \mathop \sum \limits_{g = 1}^{{n_{ije} \left( {CP} \right)}} p_{ijeg} \left( {CP} \right)\left( {1 + d} \right)^{{n_{ije} \left( {CP} \right) - g}} $$

where $\mathop \sum \nolimits_{{k = n_{ij} \left( {CP} \right) + 1}}^{{n_{ij} \left( {CP} \right)}} n_{ijk} \left( {CP} \right) = 0$.

Proof

This Lemma can be proven easily by mathematical induction method, and we omit it. □