MULTI OBJECTIVE OPTIMIZATION OF MULTI -ECHELON SUPPLY CHAIN NETWORK ARCHITECTURES

A single objective mathematical programming models are commonly used in many managerial and operational decision making processes .Although single objective decision models are sufficient for some decision making processes, but there are many situations, where the decisions depend upon multiple objectives. An important issue in real world supply chain management problem is how to measure the performance of a supply chain for a given set of decision variables, when involving several incommensurable and competing objectives. No matter how appropriate the methodology, if the performance measure is poor, the results could be misleading or false.


INTRODUCTION
A single objective mathematical programming models are commonly used in many managerial and operational decision making processes .Although single objective decision models are sufficient for some decision making processes, but there are many situations, where the decisions depend upon multiple objectives. An important issue in real world supply chain management problem is how to measure the performance of a supply chain for a given set of decision variables, when involving several incommensurable and competing objectives. No matter how appropriate the methodology, if the performance measure is poor, the results could be misleading or false.
It is comprehensible that there are many independent entities in a supply chain, each of them try to maximize their own inherent objective functions (or interests) in business transactions. Many of their interests will be conflicting. Thus, a specific scenario giving an optimal design configuration using traditional approaches could actually be a non-optimal design of the supply chain, when we look at the design from a systems optimization perspective. When conflicting interests occur in a problem, modeling the system using traditional optimization techniques does not commensurate intuitively with a robust formulation. The results could also be misleading in the very likely situation of a dynamic environment.
As reported in the literature, the genetic algorithm and evolutionary strategies are applied to two objectives minimization problems without constraint or with few constraints (see: Horn et al 1994, Murata et al 1996and Zitzler and Thiele 1999. Most researchers did not attempt to solve real world supply chain management problems consisting of many constraints and mix of both minimization and maximization objective functions.
In this chapter, we have considered three stage and four stage multi echelon supply chain network problems for the study. Each supply chain network problem is attempted with different sets of conflicting objectives with outstanding new intelligent NLIW-PSO algorithm as solution methodology. It is noted that NLIW-PSO is out performed with respect to other PSO algorithm used for solving three echelon supply chain network problems in the chapter 3.

THREE STAGE MULTI ECHELON SUPPLY CHAIN NETWORK
This section, specifically deals with the modeling of multi objective optimization of a three-stage supply chain network using the Non-linear inertia weight particle swarm optimization (NLIW-PSO) algorithm with weighted sum approach. The same model assumptions and parameters used in the three echelon SCN are considered in the mathematical formulation of three stage multi-objective SCN.

The mathematical formulation of multi-objective three stage multi echelon supply chain architecture
This study considers the same assumptions, model parameters and the mathematical model of the three stage multi echelon supply chain network architecture (equations 3.1 to 3.4 and constraints equations 3.7 to 3.10 of chapter 3) to quantify the relationship among all the decision variables involved in supply chain network. Two sets of conflicting objectives, the total supply chain operating Cost (TSCC) and ratio of Total Manufacturing Cost (TMC) to Total Supply Chain operating Cost (TSCC) are considered as the performance indicators. The problem of optimizing the supply chain configuration can be summarized in the following mathematical model.

Objective Functions
Set 1: A justification for using these objective functions is as follows.
Minimizing the total operating cost is an important performance metric in supply chain management problems. The second objective function denotes minimizing the ratio of manufacturing costs to total operating cost.
As can clearly be seen from Set 1, the two objective functions are conflicting because total supply chain operating cost features in the denominator in the second objective. The analysis will highlight the trade-off between the objectives. We would like to ensure that our Manufacturing costs fall within a certain permissible bound as a percentage of the total operating cost. The decision maker based on his/her knowledge and expertise would be able to make an intelligible decision in choosing a solution.

Fitness mapping of multi-response objective functions.
In our multi-objective supply chain network analysis following set of objectives have been considered for the analysis. Step 2 : The over all Overall objective function is to be Minimized and can be written as: Min Over All Obj = (w 1 ×fit 1 +w 2 ×fit 2 )+R m ×(Constraints) (5.5) where Weights ,w 1 and w 2 has to be selected such way that the w 1 +w 2 =1.
The values of w 1 and w 2 give importance to a particular objective.
(For examples, w 1 = w 2 = 0.5 gives equal importance to both objectives) and R m is the penalty parameter used in analysis (detailed explanation is given in chapter 1).

NETWORK
This section briefly describes the objective of the research problem, the model assumptions, the mathematical formulation. The problem description of the four stage multi echelon supply chain network model is same as discussed in the chapter 4.In this model, we are considering some additional assumptions and additional modeling parameters for the multi objective analysis so as to make the supply chain network complex to match with the real world applications.

Objective of study
This chapter specifically deals with the modeling and multi objective optimization of a four-stage supply chain using the Non-linear

Chain network
Supply chain cost components All the decision variables should be integers and non negative.

Objective Functions
Set 1:

Fitness mapping of multi-response objective functions
The two objectives considered are,  TSCC -which is to be minimized  GMROI -which is to be maximized But both objective functions are in different units i.e. one in Rupees and another is in Ratio. Hence the mapping of objectives is done to bring the objectives it to the same units as follows.
Step where, TSCC Min and GMROI Max have to be selected based on the problem in hand.
Step 2 : The multi -response weighted objective function of the defined problem can be expressed as Min Over All Obj = (w 1 ×fit 1 +w 2 ×fit 2 )+R m ×(Constraints) (5.29) where Weights ,w 1 and w 2 has to be selected such way that the w 1 +w 2 =1.
The values of w 1 and w 2 give importance to a particular objective (For examples, w 1 = w 2 = 0.5 gives equal importance to both objectives)and R m is the penalty parameter used in analysis (detailed explanation is given in chapter 1).

Introduction
This section discusses particle representation, velocity calculation of all PSO algorithms, general structure of optimization, experimental design and results and discussions of three stage and four stage multi objective multi echelon supply chain network optimization.

PSO algorithm
One solution in a three echelon supply chain network configuration is represented by a particle i.e., one string of integers (decision variables).
Three stage multi echelon supply chain network configuration considered in this study is represented by a particle which consists of 30 segments (see Velocity ,

Multi objective PSO algorithm for multi-echelon SCN problem
General procedural steps involved in multi objective analysis using PSO algorithm with weighted sum approach is given below.
Step 1: Initializing the particle position {X kd , d = 1,2,…,D} Where 'k' denotes the number of particles, 'D' denotes maximum number of dimensions within the minimum and maximum limits for each dimension.
Step 2: Initialize the particle velocity {v kd , d=1,2,…,D} Where 'k' denotes the number of particles, 'D' denotes maximum number of dimensions within the minimum and maximum limits for each dimension.
Step 3: Calculate the maximum velocity of the particles v max =0.5 × Maximum limit for each dimension, D Step 4: Step 5: If v kd >v max, then Set v kd = v max for all 'k' and 'd'.
Choose the weights w 1 and w 2 for objectives 1 and objective 2, such that w1+w2=1.
Set new kd v = v max for all 'k' and 'd'.
where LL d and UL d denotes the minimum and maximum limits on value of X kd with respect to dimension space d , and U (0,1) denotes the uniformly distributed random number in the range (0,1).
where LL d and UL d denotes the minimum and maximum limits on value of X kd with respect to dimension space d , and U (0,1) denotes the uniformly distributed random number in the range (0,1).

Experimental design
In this research an attempt has been made to apply best performing PSO variant i.e. NLIW-PSO algorithm to study its effectiveness for real world multi objective supply chain network management problems with the two sets of objectives. These two sets objectives are used as the performance indicators for SCN architectures.

The various parameters used in the algorithm
Pilot studies have been conducted in this research work to arrive at the best values of parameters in the proposed applications of PSO variants to multi echelon supply chain networks such as swarm size , learning parameters (c 1 and c 2 ), inertia weights and penalty increment and initial penalty parameters.
From the studies, following parameters are set and worked well for the proposed NLIW-PSO multi-objective analysis of three and four stage multi echelon supply chain network optimization. Penalty multiplication factor f 10 Penalty multiplication factor f 10

Input data for the supply chain model
Input data related to vendors, manufactures ware houses and distribution centers for analysis of three and four stage echelon supply chain network architectures are exhibited in Table 3.7 to 3.14 (for three stage SCN) and Tables 4.3 to 4.13 (for four stage SCN), the same data sets are considered for the multi objective performance analysis SCN architectures.
For four stage multi echelon SCN analysis, we require additional information regarding average monthly demand at each distribution centers and average monthly demand (i.e. one period ) occurring at all the warehouses from the various distribution centers for one year and are tabulated in Table 5.3(a) and (b). The average lead time required to replenish the goods from the different plants to the warehouses considered is given in the Table 5.3(c) and also the value of 'Z' corresponding to the service level is provided in the Table 5.4.  The weighted objective is optimized for different values of weight factors, w 1 and w 2 (varying from 0.1 to 0.9 in steps of 0.1), the results obtained by the multi-objective NLIW-PSO of performance evaluation of three stage multi echelon and four stage multi echelon supply chain network for average demand of supply chain setting-I are exhibited in the Tables 5.5 to 5.15. The corresponding the different weight combinations the decision variables are tabulated in Table 5.6. The near optimal trade off solutions obtained for all combination of weights for three echelon is shown in  Table 5.5 and would be beneficial to the system from a systems optimization perspective.   Table 5.11(a) and Table 5.11(b). With the above multi-objective performance analysis, it is concluded that as service levels increases, the TSCC of SCN also increases and as the service level increases, the GMROI decreases, because all the warehouses need to maintain safety stock depending on the service level required. GMROI varies from 2.8 to 3.6 and TSSC of SCN varies from Rs.12,37,000 to Rs.19,78,000 depending on the service level. From the Figure 5.3 to 5.9, the decision maker can, make the intelligible decision based on his expertise to satisfy his /her requirements (service levels to meet the customer demand) for the stated objectives for four stage multi echelon SCN architecture.  34 116 260 105 118 68 185 36 68 82 168 77 46 44 52 54 42 67 37 54 3 3 47 25 2 0.8 0. 2 189 10 66 234 115 184 37 99 90 307 46 180 258 132 13 137 162 88 7 27 6 1 55 76 76 71 49 56