Modeling and Solving the Collaborative Supply Chain Planning Problems

Y. T. Chena, Z. H. Chea, Tzu-An Chiangb, C. J. Chianga and Zhen-Guo Chec,1

aDepartment of Industrial Engineering & Management – National Taipei University of Technology, Taiwan, R.O.C. bDepartment of Commerce Automation & Management – National Pingtung Institute of Commerce, Pingtung (900), Taiwan, R.O.C. cInstitute of Information Management – National Chiao Tung University, Taiwan, R.O.C.

Abstract: Improving overall performance and collaborative value is an important issue to supply chain management. By planning a collaborative supply chain, this paper intends to establish a supplier rating package, which includes production and distribution plans. The aim is to establish a supplier procurement value rating system by combining Analytic Hierarchy Process (AHP) and Rough Sets Theory, and construct a two-objective optimization mathematical model (including procurement value and costs) for planning the supply chain. Finally, a Genetic Algorithm (GA) is applied under capacity constraints to obtain the best solution, thus achieving the planning of collaborative supply chain.

Keywords: Collaborative Supply Chain, Analytic Hierarchy Process, Rough Sets Theory, Genetic Algorithm.

1 Introduction

In today’s global market of increased competition and pressure, the relationship between upstream and downstream partners in a supply chain grows more complex, therefore, the management and integration capacity of supply chain partners are crucial components to the success of a company [5]. Through a collaborative supply chain, two or more independent companies could plan and operate a collaborative supply chain, in order to obtain greater operating success [8]. As more investments and resources are available, companies could reduce risks and create more advantages through collaboration, thus, collaborative supply chains play a crucial role in complex manufacturing environments [3]. Moreover, it is important to determine the optimum partners for production, assembly, and completion of final products in a collaborative manufacturing environment [9].

1 Corresponding Author E-mail: bcc102a@hotmail.com

Y. T. Chen, Z. H. Che, T. -A. Chiang, C. J. Chiang and Z. -G. Che 566

Dogan et al. [6] considered qualitative and quantitative factors in supply partner evaluation problems, and integrated AHP and the possibilities of multi-objective linear programming, in order select appropriate partners and determine optimum production quantities. Many scholars have extensively applied AHP to solve partner selection problems [2, 7]. Xia and Wu [11] suggested that the Rough Sets Theory can reduce subjective judgments of AHP. In addition, GA is commonly applied to solve single-/multi-objective production and work management problems, given its effective search performance for global solutions [1, 12].

Therefore, this research, based on the concept of a collaborative supply chain, aimed to construct a production and distribution planning and decision-making model that were most appropriate to collaborative supply chain. First, it established a supplier procurement value rating system (Section 2) by combining AHP and the Rough Set Theory. A two-objective optimization mathematical model (Section 3) was constructed under a collaborative supply chain for supplier selection, production, and distribution planning, which considered procurement value and costs. Finally, a GA was employed to solve the optimization mathematical model (Section 4) for obtaining satisfactory decisions within a short period.

2 Procurement Value Rating Model Development

The procurement value rating model was constructed using AHP, after improved by the Rough Sets Theory, in order to reduce the degrees of subjective judgments of a traditional AHP [11]. The detailed steps are as follows, with an example of central assembly plant C1:Step 1: Determine the suppliers’ rating level and factors, of which the parameters include 3 main criteria: price (pr), quality (qu), and service (se), and 7 sub-criteria: technology level (tl), defect (de), reliability (re), on-time delivery (od), supply capacity (sc), maintenance cycle (mc), and warranty period (wp). Step 2: Calculate criteria weights. Sub-step 2.1: Resolve AHP rating errors by attributing significance concepts to the Rough Sets Theory. Sub-step 2.1.1: Design tables are based on the main criteria, with the fields of price, quality, and service; Table 1 lists definitions represented by 1-3 in the criteria, and Table 2 lists the different groups of criteria. Sub-step 2.1.2: A rating team completes the decision column in Table 2. Take the main criteria as an example, if “1” is entered into the decision column, it represents that the supplier is selected; if “0” is entered, it denotes that the supplier is not selected. For instance: group 1 indicates a moderate price, and satisfactory levels of quality and service; if the decision column is 1, it indicates that the supplier is selected.

Table 1. The meaning of value 1–3 for different main criteria and sub-criteria

Value pr qu se tl de re od sc mc (Week)

Wp (Month)

1 Low Good Good High Low High Good Great Short Long 2 Middle Middle Middle Middle Middle Middle Middle Middle Not too long Not too long 3 High Poor Poor Low High Low Poor Small Long Short

Modeling and Solving the Collaborative Supply Chain Planning Problems 567

Table 2. Decision table [11]

Main criteria Quality Service combinations pr qu se Decision tl de re Decision od sc mc wp Decision

1 2 1 1 1 1 1 2 G 1 1 2 2 G 2 3 1 1 0 1 2 1 G 1 2 2 1 G 3 1 2 2 1 1 2 2 M 1 2 3 3 M 4 2 2 2 1 2 1 2 G 2 2 2 2 M 5 3 2 1 0 2 2 1 M 2 1 1 3 M 6 1 2 3 0 2 2 2 M 3 2 1 1 P 7 1 3 1 0 1 1 3 M 3 1 2 2 M 8 1 1 3 1 3 1 2 M 1 3 2 3 M 9 2 2 1 1 3 3 2 P 3 2 2 1 P 10 2 2 3 0 2 3 2 P 3 1 1 1 M 11 3 3 1 0 2 2 1 1 M 12 3 2 2 0 2 3 2 2 P 13 2 3 1 0 2 2 2 3 M 14 2 1 3 0 2 2 3 2 P 15 1 1 3 3 M 16 2 3 1 3 P 17 1 1 1 3 G 18 1 1 3 1 G

G = Good, M = Middle, P = Poor

Sub-step 2.2: According to the decision column in Table 2, the significance of the main criteria could be obtained by the following procedures: U|IND{pr,qu,se}={{1},{2},{3},{4},{5},{6},{7},{8},{9},{10},{11},{12},{13},{14}}; U|IND{Decision}={{2,5,6,7,10,11,12,13,14},{1,3,4,8,9}}={Y1,Y2}; U|IND {qu,se}={{1,2},{3,4,12},{5,9},{6,10},{7,11,13},{8,14}}={X1,X2,X3,X4,X5,X6};p(X1)=2/14; p(X2)=3/14; p(X3)=2/14; p(X4)=2/14; p(X5)=3/14; p(X6)=2/14; p(Y1|X1)=1/2; p(Y1|X2)=1/3; p(Y1|X3)=1/2; p(Y1|X4)=1; p(Y1|X5)=1; p(Y1|X6)=1/2;p(Y2|X1)=1/2; p(Y2|X2)=2/3; p(Y2|X3)=1/2; p(Y2|X4)=0; p(Y2|X5)=0; p(Y2|X6)=1/2. The significance of price is SGF(pr,{qu,se},{Decision})=H({Decision}|{qu,se})- H({Decision}|{pr,qu,se})=-2/14[(1/2)log(1/2)]×3-3/14[(1/3)log(1/3)+(2/3)log(2/ 3)]= 0.1882 and the significance of other main criteria and sub-criteria can be obtained by the same procedures. Sub-step 2.3: Conduct pairwise comparison for relative significance according to a hierarchical structure. In AHP, the weight of every option is originated from eigenvectors placed in a pairwise comparison matrix. Take the main criteria for example, form matrix J by the significance of criteria, then turn the matrix into maximum eigenvectors, and obtain the weights of price, quality, and service: 0.4321, 0.2346, and 0.3333, respectively, and the maximum eigenvector ( max) is 3. The weights of the sub-criteria are calculated in the same way.

11.42070.7715

0.703910.5430

1.29611.84151

21

12

11

1

nnnn

n

ww

ww

ww

ww

ww

ww

J

Sub-step 2.4: Check whether the pairwise comparison matrix constructed by the Rough Sets Theory is consistent through the consistency index CI=( max-n)/(n-1),where n is the rank of the judgment matrix; if CI=0, it represents complete consistency. Step 3: Calculate the final assessment of suppliers for central assembly plant C1, as listed in Table 3.

Y. T. Chen, Z. H. Che, T. -A. Chiang, C. J. Chiang and Z. -G. Che 568

Sub-step 3.1: Multiply the weights of the main criteria and sub-criteria to obtain the global weights. Sub-step 3.2: Consider the quantitative information of suppliers, and then convert by certain proportions to obtain the suppliers’ rating. Sub-step 3.3: Combine the global weights and suppliers’ rating, then multiply the total to obtain the final rating of the suppliers.

Table 3. Final rating of central assembly plant C1

Selection factors (Global weights) pr(0.43)

tl (0.06)

de(0.12)

re(0.05)

od(0.13)

sc (0.10)

mc (0.07)

wp(0.03) Final

rating Supplier Raw data Rating Raw

data Rating Raw data Rating Raw

data Rating Raw data Rating Raw

data Rating Raw data Rating Raw

data Rating

M11 0.7 0.44 3 0.29 0.05 0.27 0.80 0.33 0.80 0.31 120 0.48 4 0.23 1 0.25 0.370 M12 1.2 0.26 3 0.29 0.05 0.27 0.80 0.33 0.85 0.33 50 0.20 3 0.31 1 0.25 0.271 M13 1 0.31 2 0.43 0.03 0.45 0.85 0.35 0.90 0.35 80 0.32 2 0.46 2 0.50 0.358

3 Assumptions and Mathematical Foundation

The assumptions are: 1) the preceding sequences of central assembly plants are known; 2) no shortages of inventory occur; 3) the suppliers face the constraint of maximum capacity; 4) the raw material/processing cost of the suppliers does not vary with the delivery quantity, but is calculated by units of raw material/processing costs; 5) the transportation costs of suppliers does not vary with the delivery quantity, and is calculated by unit transportation costs.

The notations for the mathematical model are listed below: i, l Central assembly plant index, i=1,2,…, I; l=1,2,…, LI, L Total number of central assembly plant j Supplier index, j=1,2,…, JiJi Total number of suppliers in the central assembly plant iQSij Quantity to be ordered by supplier j in the central assembly plant i

YTil otherwiseiprelationshdeliveryhaveandplantsassemblyCentral

01 li

QTil Order quantity transported from central assembly plant i to lWij Weight of supplier j in the central assembly plant iPPij Raw material/processing cost of supplier j in the central assembly plant iTPil Transportation price from central assembly plant i to lCij Max. supply capacity of supplier j in the central assembly plant iDi Total demand of central assembly plant i

The optimal mathematical model for collaborative supply chain planning is as: 21 - ZZZMax (1)

iJ

j ijjiI

iQSWZ

111 (2) ii J

j ililI

i

J

j ijjiI

iQTTPQSPPZ

11112 (3)

st ijij CQS Ii ...3,2,1 iJj ...3,2,1 (4)

Modeling and Solving the Collaborative Supply Chain Planning Problems 569

iJ

j ij DQSi

1Ii ...3,2,1 (5)

Objective function Eq. (1) is the total procurement value (Z1) minus total cost (Z2), thus, maximizing the target function (Z). Eq. (2) is the total procurement value of suppliers. Eq. (3) is the total raw material/processing costs, as well as transportation costs. Eq. (4) meets the capacity constraints of suppliers in the central assembly plant. Eq. (5) ensures that the demand of every central assembly plant is equal to the total supply of the suppliers.

4 GA Solving Model for Mathematical Model

The GA solving model proposed in this research was used to efficiently formulate supplier selection, production, and distribution planning in a collaborative supply chain. The GA computational procedures are described below: Step 1: Chromosome coding: binary coding is used for chromosomes selected by the supplier; a central assembly plant may cooperate with several suppliers, as shown in Figure 1.

M11,M12 … M21 ,M22 … Mn1,Mn2 …… …0 1 … 1 0 … … … 1 0 …

C1 C2 Cn

Gen value 0,1

Figure 1. Chromosome sturcture

Step 2: Generate the initial population: the initial supplier combination generates feasible populations under capacity constraints (Eq. (4)), and supply and demand constraints (Eq. (5)). Step 3: Calculate the fitness function: substitute individual values into the optimization mathematical model for calculation. As two objects have different units, the objects are standardized according to (Eq. (6)). Of which, min,if and

max,if are min and max values of the i -th target in existing generations.

)/()( min,max,min, iiiii fffff ni ,...,2,1 6Step 4: Reproduction: the common Roulette Wheel Selection method of GA is adopted for reproduction; the percentage of individual fitness function values, as calculated from the previous step, to the total fitness function is taken as selection probability. The individuals of larger fitness functions are easily selected. Step 5: Crossover: single-point crossover [12] is employed to randomly extract two chromosomes from the population; a tangent point is randomly generated on the chromosome, and then the gene codes, after the tangent point of two chromosomes are exchanged. Step 6: Mutation: single-point mutation of the common GA is adopted. First, randomly extract a chromosome, then randomly generate a mutation position, and randomly change the gene codes of the position within a reasonable range (under capacity constraints (Eq. (4)), and supply and demand constraints (Eq. (5)).

Y. T. Chen, Z. H. Che, T. -A. Chiang, C. J. Chiang and Z. -G. Che 570

Step 7: Generate new offspring by gene evolution modes, as shown in Steps 4-6; if the optimal fitness value of the offspring is superior to that of the current population, replace it within the population as the new value for the next-generation evolution; otherwise, retain the original population to the next generation. Step 8: Check whether the stop conditions are met: the stop conditions indicate the executed generations; the predetermined generation number is set and entered prior to operation, and it is required to stop when the evolution number reaches the generation.

5 Illustrative Example

The case of a supply chain network [4] is shown in Figure 2, where C1-C10 aredifferent central assembly plants in the supply chain network, of which every central assembly plant may cooperate with 3 suppliers. The unit procurement value, costs, and capacity of every supplier are shown in Figure 2. A single product is considered in this research, while a collaborative supply chain plan is implemented under 60 products required by the central assembly plant.

(0.370a,0.7b,60c)

(0.271,1.2,25)(0.358, 1 ,40)

C1

C6C2

C3

C4

(0.308, 1 ,35)

(0.351, 1 ,30)

(0.341,1.3,40)

(0.260,1.1,25)

(0.364,0.8,45)

(0.376,0.7,35)

(0.385,0.6,38)

(0.300,1.3,28)

(0.315,1.3,45)

(0.273,1.4,33)

(0.381,0.9,25)(0.346,1.4,40)

C5

(0.326,0.7,30)

(0.295,1.2,28)

(0.379,1.1,48)

C6

C6

(0.315,1.3,40)

(0.355,1.4,35)

(0.330,1.5,20)

C8

C6

(0.328,0.9,33)

(0.399,0.8,38)

(0.273,1.3,25)

C7

C6

(0.440,0.6,40)

(0.274,0.8,28)

(0.286,1.4,58)

C9

C6

(0.243,1.2,38)

(0.324, 1 ,45)

(0.433,0.6,18)

C10

6,4,5

4,5,2

3,3,2

2,3,2

2,2,2

4,4,5

5,2,5

4,2,2

2,3,5

3,3,2

4,3,2

3,5,2

M13

M12

M11 1d,2e,3f

5,2,26,4,4

5,3,4

1,3,4

2,5,3

5,2,5

3,1,4

3,4,4

5,2,2

2,3,6

2,2,3

2,5,5

5,4,6

6,2,3

2,5,4

4,5,52,1,3

M23

M22

M21

M33

M32

M31

M43

M42

M41

M53

M52

M51

M63

M62

M61

M73

M72

M71

M83

M82

M81

M93

M92

M91

M103

M102

M101

a: unit procurement value rating of supplier; b: unit raw material/processing cost of supplier; c: capacity limit of supplierd, e, f: unit transportation costs from upstream supplier to the 1st, 2nd, and 3rd suppliers of downstream central assembly plant

Figure 2. Collaborative supply chain network

The procurement value of every supplier is calculated by the rating system in Section 2, and the relevant data are entered into the optimization mathematical model. Next, according to the parameter settings from the GA of [10], this research proposed 16 solution models, which parameters are listed in Table 4. Every model is subjected to 15 times of calculation to ensure that all suppliers’ capacity constraints and demands are met. The calculated results are listed in Table 4. The results indicate that model 15 has better calculated result, with the mean gross procurement value as 235, and mean gross cost as 2345. The optimum production

Modeling and Solving the Collaborative Supply Chain Planning Problems 571

and distribution plan of a collaborative supply chain is shown in Figure 3. As seen, the suppliers selected by central assembly plant C6 are M61 and M63, of which M61has to process 12 units from supplier M52 of upstream central assembly plant C5,and M63 has to process 13 units from supplier M52, and 35 units from M53 of upstream central assembly plant C5. In addition, M61 has to transport 12 units to supplier M73 of downstream central assembly plant C7, and M63 has to transport 38 of 48 units to downstream supplier M72, and the remaining 10 units to downstream supplier M73.

Table 4. Parameter combinations and execution results

Model Parameter combination Result Generation Population Crossover rate Mutation rate Mean total procurement value Mean total cost

1 200 10 0.3 0.03 230 2481 2 200 10 0.3 0.05 231 2478 3 200 10 0.6 0.03 230 2500 4 200 10 0.6 0.05 230 2486 5 200 50 0.3 0.03 234 2386 6 200 50 0.3 0.05 235 2358 7 200 50 0.6 0.03 234 2384 8 200 50 0.6 0.05 235 2366 9 500 10 0.3 0.03 229 2485 10 500 10 0.3 0.05 229 2449 11 500 10 0.6 0.03 229 2492 12 500 10 0.6 0.05 230 2453 13 500 50 0.3 0.03 233 2397 14 500 50 0.3 0.05 235 2408 15 500 50 0.6 0.03 235 2345 16 500 50 0.6 0.05 235 2375

(0a,25b,35c)

C1

C2

C4

(0,25,5)

(0,0,30)

(0,25,0)(0,0,35)

(0,3,35)(0,22,0)

C5

C6

C8

C7

C9

C10

( 5, 0, 0)

(35,0,0)

(0,20,0)

(0, 0,12 )

(0,38,10)

(0,38,0)(0,4,18)

(0,40,0)

(0,2,18)

(42)

(18)

(12,0,13)

(0,0,35)(0,5,20)

(5,30,0)

M11

M22

M23

M72

M73

M81

M82

M52

M53

M61

M63

M91

M92

M102

M83

M103

C3

M32

M33

M41

M43

a, b, c :quantities transported from upstream supplier to the 1st, 2nd, and 3 rd suppliers of downstream central assembly plant

Figure 3. Production and distribution plan for collaborative supply chain

6 Conclusions

This research proposed a collaborative supply chain planning model, which allows decision-makers to select suitable suppliers, and determine the production and distribution quantities of all suppliers. According to the model, the supplier’s procurement value rating system is established by combining Rough Sets Theory and AHP. An optimal mathematical model is constructed by considering the

Y. T. Chen, Z. H. Che, T. -A. Chiang, C. J. Chiang and Z. -G. Che 572

supplier’s procurement value, cost, capacity constraints, and demand. Finally, GA is used to solve the mathematical model and the experimental results indicated that GA can equal to the production and distribution planning problems of a collaborative supply chain.

In the current form of the proposed GA approach, however, it may not be effective in dealing with more complex problems. For example, it cannot find the quality result in the problem when more events such as inventories of products are taken into consideration. For further research we thought about extending this developed approach to more complex problems such as collaborative supply chain planning problems involving different weights of considered factors, inventories, etc.

7 References

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