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Hybrid algorithm for a vendor managed inventory system in a two-echelon supply chain
Accepted Manuscript Hybrid Algorithm for a Vendor Managed Inventory System in a Two-Echelon Supply Chain Ali Diabat PII: DOI: Reference:
S0377-2217(14)00213-6 http://dx.doi.org/10.1016/j.ejor.2014.02.061 EOR 12201
To appear in:
European Journal of Operational Research
Received Date: Accepted Date:
25 March 2013 28 February 2014
Please cite this article as: Diabat, A., Hybrid Algorithm for a Vendor Managed Inventory System in a Two-Echelon Supply Chain, European Journal of Operational Research (2014), doi: http://dx.doi.org/10.1016/j.ejor.2014.02.061
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Hybrid Algorithm for a Vendor Managed Inventory System in a Two-Echelon Supply Chain Ali Diabat, Engineering Systems and Management, Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emirates *Corresponding author: Email : [email protected] Abstract In this paper we address the issue of vendor managed inventory (VMI) by considering a twoechelon single vendor/multiple buyer supply chain network. We try to find the optimal sales quantity by maximizing profit, given as a nonlinear and non-convex objective function. In such complicated combinatorial optimizations, exact algorithms and optimization solvers such as CPLEX and LINGO are inefficient, especially on practical-size problems. In this paper we develop a hybrid genetic/simulated annealing algorithm to deal with this nonlinear problem. Our results demonstrate that the proposed hybrid algorithm outperforms previous methodologies and achieves more robust solutions. Keywords: Supply chain management; VMI; Hybrid algorithm Highlights • • •
Proposed a hybrid algorithm using GA- SA for the VMI model Results show better performances for hybrid algorithm Sensitivity analysis was done to demonstrate the performance of proposed algorithm
Hybrid Algorithm for a Vendor Managed Inventory System in a Two-Echelon Supply Chain Abstract In this paper we address the issue of vendor managed inventory (VMI) by considering a twoechelon single vendor/multiple buyer supply chain network. We try to find the optimal sales quantity by maximizing profit, given as a nonlinear and non-convex objective function. In such complicated combinatorial optimizations, exact algorithms and optimization solvers such as CPLEX and LINGO are inefficient, especially on practical-size problems. In this paper we develop a hybrid genetic/simulated annealing algorithm to deal with this nonlinear problem. Our results demonstrate that the proposed hybrid algorithm outperforms previous methodologies and achieves more robust solutions. Keywords: Supply chain management; VMI; Hybrid algorithm 1. Introduction Vendor managed inventory (VMI) has been studied by many researchers because of its impact on the cost efficiency of supply chain networks. Under a VMI system, the vendor determines the timing and quantity of replenishment and has access to the retailer's inventory and demand data (Darwish and Odah, 2010). A well-designed VMI system can decrease inventory levels and enhance supply chain integration, thereby reducing costs (Schenck and McInerney, 1998; Achabal et al., 2000; Cetinkaya and Lee, 2000; Angulo et al., 2004). These benefits have been realized by many successful retailers and suppliers, including Wal-Mart and Procter and Gamble (Yao et al., 2007). Due to the growing interest in enhancing supply chain integration, we consider the integrated VMI system model developed by Nachiappan and Jawahar (2007).
In order to establish the optimal parameters of a VMI system, a complicated combinatorial optimization problem with a nonlinear non-convex objective function must be solved. Finding the optimal solution to such a problem may be infeasible using traditional algorithms, particularly for larger problem sizes, so robust new methodologies for solving such problems are required. Costa and Oliveira (2001) examined meta-heuristic algorithms such as genetic
algorithms (GA), simulated annealing algorithms (SAA), and evolution strategies (ES), and concluded that these meta-heuristic algorithms are well-suited to solving such problems.
In this paper we consider two supply chain models that use VMI: a single vendor/single buyer model, and a single vendor/multiple buyer model. We try to find the sales quantity, sales price, and contract price between the vendor and the buyer for each model. A hybrid genetic/simulated annealing algorithm is developed to determine the optimal parameters for each VMI model, and our results are compared with those obtained by Nachiappan and Jawahar (2007) using a genetic algorithm on the same model and the same data.
The remainder of the paper is organized as follows. In Section 2, we provide a comprehensive literature review. The model is presented in Section 3, and the proposed hybrid algorithm is explained in Section 4. The results of a complete computational analysis are presented in Section 5, and a sensitivity analysis is provided in Section 6. Finally, Section 7 presents our conclusions and identifies future research directions. 2. Literature review Our two-echelon single vendor/multiple buyer supply chain model under VMI focuses on efficient methodologies to solve this nonlinear problem. There is substantial research on model development and formulation (Cetinkaya and Lee, 2000; Shah and Goh, 2006; Jaruphongsa et al., 2004; Zhang et al., 2007), but the primary objective of such NP-hard problems is achieving better optimal solutions. Consequently, we see growing interest in exploiting meta-heuristic algorithms, especially genetic algorithms. These algorithms perform very well on large-scale problems like VMI. The capability of achieving near optimal solutions is the most important part of the applicability of a theoretical model, so developing more robust algorithms in order to address integrated VMI systems is essential.
Many approaches to solving VMI models, both by exact and approximate algorithms, are inefficient when they are applied to real-world problems; they cannot find optimal or nearoptimal solutions in a reasonable time. For instance, the excellent work of Archetti et al. (2007) presents a mixed-integer linear programming model. They implement a branch-and-cut
algorithm to solve the model optimally, and they compare the optimal solution of the problem to two problems obtained by implementing various methods of relaxation. Because their model is linear, they could exploit the CPLEX solver to solve the problem to optimality. Their approach works particularly well on small problem sizes.
Bertazzi et al. (2005) attempt to solve complex VMI problems. They propose a heuristic algorithm, and they decompose the problem into two sub-problems (production and distribution), which are then solved individually. Siajadi et al. (2006) consider a two-echelon single vendor/multiple buyer problem. They propose an analytical solution approach, and their work indicates that more efficient solution approaches are required.
Cardenas-Barron et al. (2012) propose a heuristic algorithm to solve a nonlinear integer programming (NLIP) vendor management inventory system with multiple products and multiple constraints. Their algorithm outperforms the traditional GA in terms of the total cost, the number of evaluations of the total cost function, and the computational time required. Sadeghi et al. (2013) solve a multi-vendor, multi-retailer, single-warehouse (MV-MR-SW) problem using a hybrid meta-heuristic-based particle swarm optimization (PSO) method, and compare the performance of their algorithm with the traditional GA.
In Table 1 we summarize the various heuristic and meta-heuristic methods that have been applied to solve VMI models. In the following section we present our hybrid algorithm for solving VMI models, which is intended to address some of the shortcomings of the previous approaches.
Table 1: New solution methodology developments on VMI Row
Paper
1.
Nachiappan and Jawahar 2007
vendor/multiple-buyers
A genetic based heuristic algorithm
Sue-Ann et al.
Two-echelon single-
PSO in comparison
2012
vendor/multiple-buyers
with a hybrid GA-AIS
2.
Model Two-echelon single-
Methodology
Developments and Comments Finding better near optimal solutions and correcting the presented solutions.
PSO performs better than GA and the hybrids. Therefore, the capability of this new methodology is proved.
They have mentioned that “other metaPasandideh et al.
3.
2011
Two-echelon, one supplier, one-retailer
heuristic search algorithms such as Genetic Algorithm
and multi-product
simulated annealing may also be employed and a comparison may be made among the algorithms.”
Two-echelon, inventory 4.
Zhao et al. 2007
control and vehicle routing schedules for a
Computational results reveal the Tabu Search
effectiveness as well as the robustness of the policy and the algorithm.
distribution system
Three-echelon (a 5.
Zhao et al. 2008
supplier, a central warehouse and a group of retailers)
Sadeghi et al.,
6.
2013
Variable Large Neighborhood Search
Tabu Search algorithm.
(VLNS) algorithm
Multi-vendor multiretailer single-
It performs better than previously proposed
Hybrid PSO
Proposed algorithm performs better than
warehouse model
the traditional GA solutions.
3. Model Description This section describes the two-echelon VMI model used in this paper. The following notation is used: aj
intercept value of the cost-demand curve of the jth buyer
bj
negative slope of the cost-demand curve of the jth buyer
c, c′, c″, c‴
chromosome
C
capacity of the vendor
cp(c)
cumulative probability of survival
EOQj
economic order quantity of jth buyer
fit(c)
fitness function
Hbj
annual unit holding cost of the jth buyer in independent mode
Hs
annual unit holding cost of the vendor in independent mode
HjVMI
annual unit holding cost of the vendor in VMI mode
j
buyer identifier (j = 1 to n)
n
number of the buyers
newfit(c)
new fitness function
OSMj
sum of order and inventory holding cost to the vendor for the jth buyer
p_cross
probability of crossover
Pbj
profit of the jth buyer
Pc
channel profit
Pc opt
optimal channel profit
Ps
vendor profit
Psj
profit obtained by vendor when supplying products to the jth buyer
PDj
production distribution cost of the jth buyer
p(c)
probability of survival of chromosome ‘c’
p_mut
probability of mutation
pop_size
population size
PRj
revenue share ratio between vendor and the jth buyer
P(y)
sales price
P(yj)
sales price of the jth buyer corresponding to sales quantity ‘yj’
P(yjopt)
optimal sales price of the jth buyer
Qj
replenishment quantity to the jth buyer
R
random number
Sbj
setup cost of the jth buyer per order in independent mode
Ss
setup cost of the vendor per order in independent mode
SjVMI
setup cost of the vendor per order in VMI mode of operation
W
contract price
Wj
contract price between vendor and the jth buyer
Wjopt
optimal contract price between vendor and the jth buyer
y
aggregate annual sales quantity of the vendor
yj
annual sales quantity of the jth buyer
yjmin
minimum expected sales quantity of the jth buyer
yjmax
maximum expected sales quantity of the jth buyer
yjopt
optimal annual sales quantity of the jth buyer
θj
flow cost per unit from vendor to the jth buyer
υj
transportation resource cost per unit from vendor to the jth buyer
δ
production cost per unit The network structure is illustrated in Figure 1:
Figure 1: Two-echelon single vendor/multiple buyers supply chain model Figure 1 illustrates that a single vendor has a special contract price with any buyer. Buyers also have a special sales price. The relationship between P(y) and y is assumed to behave linearly and is given as:
(1)
s.t.
(2)
where a and -b are the intercept of P(y) axis and the slope of the sales curve, respectively, in the sales price vs. sales quantity graph. Based on the above notation and the illustration of the network in Figure 1, we can summarize the complete mathematical model as:
Maximize
∑ { 0.5 [2 + ! " + "! ]/ }
(3)
Subject to:
(4)
% &
(5)
≥ 0
(6)
The objective is maximization of total profit. The vendor costs are due to production and distribution, orders, and stock maintenance. The production cost is derived from the amount spent for producing/acquiring a single unit, ,
and the aggregate demand, , and is expressed as . Distribution cost is derived from the
multiplication of flow cost ( ) and transportation resource cost () ). It should be mentioned
that ) includes indirect costs such as mode of transport, human router cost, and administrative costs per unit (assigned as 0.5), so that the distribution cost is expressed as 0.5 . The order cost is the cost involved to replenish the batches of demand of the *+, buyer and is
expressed as " + "! /- . The total holding inventory cost is based on the sum of holding costs at the vendor and buyer locations and is expressed as + ! - /2.
Finally, if we replace the order quantity - with the optimum order quantity under the EOQ model of ./- [
01 2034 54 / ] 61 2634
then the total order and stock maintenance cost becomes
[2 + ! " + "! ]/ , which is the last term of the objective function given in Equation (3).
Equations (4) and (5) are constraints on the buyer sales quantity and on the vendor capacity, respectively. Equation (6) constrains the solution to be non-negative. The solution to the above problem gives the optimal sales quantity 78+ (for all buyers). Once the above problem has been solved, the optimal sales price for each buyer can be calculated as: 78+ 78+
(7)
Furthermore, the acceptable contract price can be calculated as (see (Nachiappan and Jawahar, 2007) for details):
978+
78+ : 78+ : + 78+ + 0.5 78+ + [2 + ! " + "! 78+ ]/
(8)
(1 + : )78+
4. Proposed hybrid Algorithm
Usually, solving a nonlinear non-concave optimization model is not an easy task, so in this paper we use a hybrid GA-SA methodology to solve the above problem. These approaches will be compared with each other in terms of their solution quality in a later section. The chromosome representation used in this paper is shown in Figure 2. The length of the strings is equal to the number of buyers involved in the problem. 1
2
3
Sales Sales Sales quantity quantity quantity of buyer of buyer of buyer 1 2 3
N
…
Sales quantity of buyer j
Figure 2. Chromosome structure 4.1 Hybrid GA-SA algorithm
← Gene index
Recent studies (Sadeghi et al., 2013; Chen et al., 2013; Sue-Ann et al., 2012) have demonstrated that hybrid metaheuristics work better than individual metaheuristics for solving nonlinear models. This paper utilizes a hybrid algorithm to solve the two-echelon singlevendor/multiple-buyer VMI model. Usually, hybridization refers to the combination of two search algorithms to solve a given problem (Chen at al., 2013). There are several ways to use hybrid metaheuristics, one of which is combining the traditional GA with any of the local search heuristics. This paper proposes a hybrid algorithm based on GA and SA which presents several advantages as follows. GA has strong global search ability, but limited premature and weak local search ability. Conversely, SA has strong local search ability with no premature issues, but weak global search ability (Li et al., 2013; Chen et al., 2013). In order to best utilize the advantages and to eliminate the disadvantages of both methods, this paper proposes a hybrid GA-SA method. The structure of the proposed hybrid GA-SA is shown in Figure 3.
Figure 3. The structure of the Hybrid GA-SA (Chen et al., 2013)
In the proposed method, SA is used a local search algorithm to prevent the algorithm from being trapped in local optima. The input for the SA comes from the output of the GA mutation process. The output from the SA is used in the elitist selection process. The algorithm runs until a termination condition (i.e. on the maximum number of iterations) is met. It is evident from the literature that the parameters used in GA and SA have a clear effect on both solution quality and solution time (Yang et al., 2009; Kao and Han, 2008). The GA and SA parameters used were based on a pilot study. The SA parameters were set as follows: the initial temperature To was set to 100, the temperature cooling coefficient (α) was set to 0.9, and after each iteration is carried out for each value of T, the new temperature (Tnew) was obtained by multiplying the initial temperature by the cooling coefficient. The algorithm iterates until the temperature falls below the freezing temperature, where the freezing temperature was set to 1. The GA parameters for this study were determined using a 1-vendor 1-buyer problem. Based on this analysis, the GA parameters were fixed as follows: population size was set to twice the number of buyers, the crossover probability p_cross was set to 0.8, the mutation probability p_mut was set to 0.05, and the number of iterations was set to 300.
5.
Computational Analysis of the hybrid algorithm In this section, we evaluate the performance of the proposed methodology. The hybrid GA-
SA algorithm was coded using the C++ programming language and the computational experiments were run on a 2.5 GHz Intel(R) core (TM) i5 computer equipped with 8 GB of RAM. We solve the model in LINGO software package. We compare the performance of our algorithm to that of the proposed genetic algorithm of Nachiappan and Jawahar (2007) on the same problem and on the same data. In order to compare the performance of our proposed algorithm, two problem cases are introduced (Case 1: single vendor/single buyer; Case 2: single vendor/ three buyers). Case 1: Single Vendor/Single Buyer For this case we evaluate the performance of the proposed algorithm and compare to the results available in the literature. The parameters that describe the problem are presented in Table 2, and the results are summarized in Table 3. Table 2: Parameters for the single vendor/single buyer problem
!
"!
:
9
300
80
0.01
1000
2000
1
0.005
9
"
C
150
6150
40
Table 3: Hybrid algorithm compared with Dong and Xu, LINGO, and GA Methodology
78+
(78+ )
78+
Dong and Xu
1535
64.65
26960.49
LINGO
1535
64.65
26960.49
1532
64.68
26960.42
1535
64.65
26960.49
GA (Nachiappan and Jawahar, 2007) Hybrid Algorithm
In Table 3, our results are compared with the exact methods of Dong and Xu (2002), and it is observed that our results are in line with their findings. In order to measure the sensitivity of the hybrid algorithm parameters, we test the Case 1 problem with different values for parameters such as the crossover probability, the mutation probability, and the number of iterations. From the literature it is evident that different parameter values are used. To find the best parameter values for the GA, we conducted three experiments. The first experiment considers a fixed crossover probability (80%), a fixed mutation probability (10%), and a varying number of iterations. The best solution obtained by varying the number of iterations is shown in Figure 4, which shows that when the number of iterations increases, the algorithm yields better solutions.
Total profit vs. number of iterations 27500 27000 26500 Total profit
26000 25500 25000 24500 24000 23500 23000 Iteration numbers
Figure 4: Total profit vs. number of iterations The second experiment considers a fixed mutation probability (10%), a fixed number of iterations (300), and varying crossover probabilities (50-90%). The best solution obtained by varying the crossover probability values is shown in Figure 5, which shows that when the crossover probability increases, the total profit also increases.
Total Profit vs. Crossover probability
Total Profit
26500 25500 24500 23500 22500 21500 50% 52% 54% 56% 58% 60% 62% 64% 66% 68% 70% 72% 74% 76% 78% 80% 82% 84% 86% 88% 90% Crossover probability
Figure 5: Total profit vs. crossover probability
The third experiment considers a fixed crossover probability (80%), a fixed number of iterations (300), and varying mutation probabilities (2-25%). The best solution obtained by varying the mutation probability values is shown in Figure 6, which shows that when the mutation probability increases, the total profit generally decreases.
Total Profit vs. Mutation probability
Total Profit
26500 25500 24500 23500 22500
25%
24%
23%
22%
21%
20%
19%
18%
17%
16%
15%
14%
13%
12%
11%
10%
9%
8%
7%
6%
5%
4%
3%
2%
21500 Mutation probability
Figure 6: Total profit vs. mutation probability From Figures 4, 5, and 6, we can infer that the best performance on the Case 1 problem is obtained when the number of iterations is set to 300, the crossover probability is set to 80%, and the mutation probability is set to 5%. We used these parameter values in the Case 2 study and in the sensitivity analysis. Case 2: Single Vendor/Three Buyers The case from Nachiappan and Jawahar (2007) is used to evaluate the performance of the proposed algorithm. Table 4 and Table 5 give the input data for the study. Table 4 gives the data for three buyers, while Table 5 gives the data for the single vendor. Table 4: Buyer-related data
1
2
3
!
7
8
9
10
20
30
"!
Table 4: Buyer-related data
1
2
3
20
19
18
0.003
0.005
0.008
2000
500
500
4000
3000
1500
0.004
0.006
0.008
1
1.2
1.2
:
Table 5: Vendor-related data 9
"
C
15
5750
7
Table 6 summarizes the results of computational analysis for LINGO, GA (corrected results of Nachiappan and Jawahar (2007)), and our proposed hybrid algorithm. Table 6: Comparison of results for LINGO, GA, and hybrid algorithm 1
78+
(78+ )
LINGO
GA
2000
2002
14
14
78+
2 Hybrid
LINGO
GA
2001
710
673
14
15.5
15.60
algorithm
9903.11 9905.509
3 Hybrid
Hybrid
LINGO
GA
675
500
500
500
15.6
14
14
14
algorithm
algorithm
9908.487
Before analyzing the results of Table 6, we should point out that the results of LINGO and GA in (Nachiappan and Jawahar, 2007) are corrected here (highlighted cells). In that paper, the results for j=2 are y@ABC 657 and Py@ABC 15.7. Even if we accept these results, the objective function would be 9851.65 (wrongly calculated as 5982.1 in (Nachiappan and Jawahar, 2007)). 6. Sensitivity Analysis In this section a complete sensitivity analysis is performed on the following parameters: the upper and lower sales quantity limits, the (negative) slope of the cost-demand curves, the
holding costs, and the order costs. Included here are the data for five buyers (Table 7) and for one vendor (Table 8). We consider the sensitivity to parameter values that are 20% higher and 20% lower than normal, so three levels will be considered: Normal, 0.8*Normal, and 1.2*Normal. Table 7: Buyer-related data
1
2
3
4
5
!
7
8
9
7
9
10
20
30
15
25
20
19
18
21
18
0.003
0.005
0.008
0.003
0.006
2000
500
500
1700
500
4000
3000
1500
3500
2500
0.004
0.006
0.008
0.005
0.007
1
1.2
1.2
1.3
1.4
"!
:
Table 8: Vendor-related data 9
"
C
15
9850
7
Effects of changes in sales quantity limits. We first consider the effects of changes in the sales quantity limits. Before presenting the results of this analysis, it should be mentioned that the corrected results of Nachiappan and Jawahar (2007) relating to the effect of changes in the lower sales quantity limits ( ) and the upper sales quantity limits ( ) at the three levels on the optimal sales quantity (78+ ) and the optimal channel profit ( 78+ ) are presented in Tables 9 and 10, respectively (the corrected results are highlighted).
Table 9: Effect of lower limit on optimal sales quantity and channel profit (corrected) n
2
3
4
5
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
1600
2000
2400
10416.03
8290.972
4560.133
2
658
670
656
1
1602
2000
2400
12068.58
9885.799
5848.97
2
638
665
650
3
415
501
600
1
1600
2000
2401
19808.07
16493.6
10030.19
2
655
659
637
3
406
501
615
4
1365
1703
2040
1
1625
2126
2400
21997.6
17602.15
12320.07
2
659
653
647
3
423
567
604
4
1361
1702
2043
5
487
522
604
Table 10: Effect of upper limit on optimal sales quantity and channel profit (corrected) n
2
3
4
5
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
2000
2000
2000
8276.516
8290.972
8266.662
2
651
670
641
1
2000
2000
2000
9892.74
9325.279
9563.101
2
673
665
501
3
500
501
502
1
2001
2000
2002
16486.4
16493.6
15053.27
2
653
659
642
3
501
501
501
4
1702
1703
1926
1
2001
2126
2394
18796.42
17602.15
14510.71
2
688
653
754
3
503
567
666
4
1700
1702
1700
5
511
522
557
The results of the sensitivity analysis for the proposed hybrid algorithm are given in Tables 11 and 12. Table 11 shows the effect of changes in the lower sales quantity limits ( ) at the three levels on the optimal channel profit ( 78+ ), while Table 12 shows the effect of changes in the upper sales quantity limits ( ) at the three levels on the optimal channel profit ( 78+ ). Table 11: Effect of lower limit on optimal channel profit n
Pcopt Level 1
2 3 4 5
Level 2
10428.1
8297.025
4569.199
12080.67
9896.891
5859.01
19821.16
16507.69
10043.27
22007.69
17614.22
12328.15
Table 12: Effect of upper limit on optimal channel profit n
Level 3
Pcopt
Level 1 2 3 4 5
Level 2
Level 3
8278.542
8292.998
8270.715
9897.806
9327.305
9569.18
16489.44
16497.65
15059.35
18798.45
17608.23
14514.76
Effects of changes in the (negative) cost slope. We consider the effects of changes in the (negative) slope of the cost-demand curves. Before presenting the results of this analysis, it should be mentioned that the corrected results of Nachiappan and Jawahar (2007) relating to the effect of changes in the (negative) slope of the cost-demand curves ( ) at the three levels on the optimal sales quantity (78+ ) and the optimal channel profit ( 78+ ) are presented in Table 13 (the corrected results are highlighted).
Table 13: Effect of (negative) cost slope on optimal sales quantity and channel profit (corrected) n
2
3
4
5
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
2000
2000
2000
7953.018
8290.972
5334.177
2
571
670
747
1
2015
2000
2002
13164.26
9885.799
4361.515
2
740
665
1137
3
511
501
500
1
2001
2000
2001
21590.95
16493.6
11440.08
2
732
659
582
3
502
501
500
4
1700
1703
1700
1
2000
2126
2001
24253.32
17602.15
13495.93
2
776
653
587
3
521
567
500
4
1700
1702
1700
5
535
522
510
The results of the sensitivity analysis for the proposed hybrid algorithm are given in Table 14, which shows the effect of changes in the (negative) slope of the cost-demand curves ( ) at the three levels on the optimal channel profit ( 78+ ). Table 14: Effect of (negative) cost slope on optimal channel profit n
2 3 4 5
Pcopt
Level 1
Level 2
Level 3
7956.058
8295.025
5339.243
13170.34
9889.852
4366.581
21596.02
16500.69
11445.15
24255.35
17604.18
13503.02
Effects of changes in holding costs. The cost of inventory is an important factor in real situations and it accounts for most of the holding cost. We consider the effects of changes in the holding costs. Before presenting the results of this analysis, it should be mentioned that the corrected results of Nachiappan and Jawahar (2007) relating to the effect of changes in the holding costs at the three levels on the optimal sales quantity (78+ ) and the optimal channel profit ( 78+ ) are presented in Table 15 (the corrected results are highlighted).
Table 15: Effect of holding costs on optimal sales quantity and channel profit (corrected) n
2
3
4
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
2000
2000
2000
8519.457
8290.972
8066.441
2
671
670
645
1
2000
2000
2037
10202.65
9885.799
9202.98
2
658
665
647
3
501
501
541
1
2067
2000
2000
16458.79
16493.6
16088.36
2
680
659
656
3
507
501
501
5
4
1700
1703
1701
1
2002
2126
2003
2
674
653
674
3
506
567
501
4
1700
1702
1701
5
508
522
522
19324.34
17602.15
18281.22
The results of the sensitivity analysis for the proposed hybrid algorithm are given in Table 16, which shows the effect of changes in the holding costs at the three levels on the optimal channel profit ( 78+ ). Table 16: Effect of holding costs on optimal channel profit n
Pcopt Level 1
2 3 4 5
Level 2
Level 3
8528.576
8294.012
8074.546
10211.77
9891.878
9209.059
16461.83
16498.67
16095.45
19333.46
17609.24
18290.34
Effects of changes in setup costs. Setup costs include costs related to purchasing (such as invoice processing costs, receipt processing costs, and so on). We consider the effects of changes in the setup costs. Before presenting the results of this analysis, it should be mentioned that the corrected results of Nachiappan and Jawahar (2007) relating to the effect of changes in the setup costs at the three levels on the optimal sales quantity (78+ ) and the optimal channel profit ( 78+ ) are presented in Table 17 (the corrected results are highlighted). Table 17: Effect of setup costs on optimal sales quantity and channel profit (corrected) n
2
3
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
2001
2000
2000
8519.07
8290.972
8059.511
2
683
670
638
1
2086
2000
2000
9535.44
9885.799
9555.487
4
5
2
695
665
624
3
514
501
501
1
2026
2000
2002
2
691
659
662
3
546
501
503
4
1717
1703
1764
1
2001
2126
2002
2
665
653
678
3
500
567
606
4
1700
1702
1704
5
500
522
521
16599.59
16493.6
15729.59
19331.32
17602.15
17936.51
The results of the sensitivity analysis for the proposed hybrid algorithm are given in Table 18, which shows the effect of changes in the setup costs at the three levels on the optimal channel profit ( 78+ ). Table 18: Effect of setup costs on optimal channel profit n
2 3 4 5
7.
Pcopt Level 1
Level 2
Level 3
8524.136
8296.038
8065.59
9537.466
9893.904
9561.566
16604.66
16499.68
15733.64
19341.45
17606.2
17943.6
Conclusions and Future Research
In this paper we developed a hybrid genetic/simulated annealing algorithm to solve the nonlinear, non-convex optimization problem associated with VMI models. For the two-echelon single vendor/multiple buyer supply chain network considered in this paper, we demonstrated that our hybrid algorithm outperforms existing algorithms, and also performed a complete sensitivity analysis on our algorithm. Among the promising research directions that future researchers may wish to explore are to extende this problem to three echelons, elevating the problem from a single vendor to multiple vendors, and utilizing other metaheuristics. Other possible extensions could be to combine the
inventory problem studied in this paper with a facility location problem and a vehicle routing problem, resulting in different optimization problems, and solving these problems using different solution methodologies. Diabat et al. (2013) and Le et al. (2013) present two such integrated optimization problems. References Achabal, D.D., McIntyre, S.H., Smith, S.A., Kalyanam, K., 2000. A decision support system for vendor managed inventory. Journal of Retailing 76 (4), 430–454. Angulo, A., Nachtmann, H., Waller, M.A., 2004. Supply chain information sharing in a vendor managed inventory partnership. Journal of Business Logistics 25, 101–125. Archetti, C., Bertazzi, L., Laporte, G., Speranza, M.G., 2007. A Branch-and-Cut Algorithm for a Vendor-Managed Inventory-Routing Problem. Transportation Science 41 (3), 382-391. Bertazzi, L., Paletta, G., Speranza, M.G., 2005. Minimizing the Total Cost in an Integrated Vendor-Managed Inventory System. Journal of Heuristics 11, 393–419. Cárdenas-Barrón, L. E., Treviño-Garza, G., & Wee, H. M. (2012). A simple and better algorithm to solve the vendor managed inventory control system of multi-product multiconstraint economic order quantity model. Expert Systems with Applications, 39(3), 3888-3895. Cetinkaya, S., Lee, C.Y., 2000. Stock replenishment and shipment scheduling for vendormanaged inventory systems. Management Science 46 (2), 217–232. Costa, L., Oliveira, P., 2001. Evolutionary algorithms approach to the solution of mixed integer nonlinear programming problems. Computers and Chemical Engineering 25, 257–266. Darwish, M.A., Odah, O.M., 2010. Vendor managed inventory model for single-vendor multi-retailer supply chains. European Journal of Operational Research 204, 473–484. Diabat, A., Richard, J. P., & Codrington, C. W., 2013. A Lagrangian relaxation approach to simultaneous strategic and tactical planning in supply chain design. Annals of Operations Research, 203(1), 55-80.
Dong, Y., Xu, K., 2002. A supply chain model of vendor managed inventory. Transportation Research Part E: Logistics and Transportation Review 38 (2), 75–95. Chen, G., Govindan, K., Yang, Z., (2013) A method to reduce truck queueing at terminal gates: managing truck arrivals with vessel-dependent time windows, International Journal of Production Economics, Vol. 141, No. 1, pp. 179-188 Jaruphongsa, W., Çetinkaya, S., Lee, C., 2004. Space capacity and delivery time window considerations in dynamic lot-sizing for a simple supply chain. International Journal of Production Economics 92, 169–180. Kao, Y. T., & Zahara, E. (2008). A hybrid genetic algorithm and particle swarm optimization for multimodal functions. Applied Soft Computing, 8(2), 849-857. Ko, H., Evans, G., 2007. A genetic algorithm-based heuristic for the dynamic integrated forward/reverse logistics network for 3pls. Computers and Operations Research, 34, 346–366. Le, T., Diabat, A., Richard, J. P., & Yih, Y., 2013. A column generation-based heuristic algorithm for an inventory routing problem with perishable goods. Optimization Letters, 7(7), 1481-1502. Li, L., Guo, H., Wang, L. and Fu, J. (2013) “A Hybrid Genetic-Simulated Annealing Algorithm for the Location-Inventory-Routing Problem Considering Returns under E-Supply Chain Environment,” The Scientific World Journal, vol. 2013, Article ID 125893, 10 pages, 2013. doi:10.1155/2013/125893 Nachiappan, S.P., Jawahar, N., 2007. A genetic algorithm for optimal operating parameters of VMI system in a two-echelon supply chain. European Journal of Operational Research 182, 1433–1452. Pasandideh, S.H.R., Akhavan Niaki, S.T., Roozbeh Nia, A., 2011. A genetic algorithm for vendor managed inventory control system of multi-product multi-constraint economic order quantity model. Expert Systems with Applications 38, 2708–2716.
Sadeghi, J., Mousavi, S. M., Niaki, S. T. A., & Sadeghi, S. (2013). Optimizing a multi-vendor multi-retailer
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algorithms. Knowledge-Based Systems, 50, 159-170. Schenck, J., McInerney, J., 1998. Applying vendor-management inventory to the apparel industry. Automation 14, 36–38. Shah, J., Goh, M., 2006. Setting operating policies for supply hubs. International Journal of Production Economics 100, 239–252. Siajadi, H., Ibrahim, R., Lochert, P., 2006. A single-vendor multiple-buyer inventory model with a multiple-shipment policy. International Journal of Advanced Manufacturing Technology 27, 1030–1037. Sue-Ann, G., Ponnambalam, S.G., Jawahar, N., 2012. Evolutionary algorithms for optimal operating parameters of vendor managed inventory systems in a two-echelon supply chain. Advances in Engineering Software 52, 47–54. Yang, J., Xu, M., & Gao, Z. (2009). Sensitivity analysis of simulated annealing for continuous network design problems. Journal of Transportation Systems Engineering and Information Technology, 9(3), 64-70. Yao, Y., Evers, P. T., Dresner, M. E., 2007. Supply chain integration in vendor-managed inventory. Decision Support Systems 43, 663– 674. Zhang, T., Liang, L., Yu, Y., Yu, Y., 2007. \An integrated vendor-managed inventory model for a two-echelon system with order cost reduction", International Journal of Production Economics 109 (1-2), 241-253. Zhao, Q.H., Chen, S., Zang, C.X., 2008. Model and algorithm for inventory/routing decision in a three-echelon logistics system. European Journal of Operational Research 191, 623–635. Zhao, Q.H., Wang, S.Y., Lai, K.K., 2007. A partition approach to the inventory/routing problem. European Journal of Operational Research 177, 786–802.
S0377-2217(14)00213-6 http://dx.doi.org/10.1016/j.ejor.2014.02.061 EOR 12201
To appear in:
European Journal of Operational Research
Received Date: Accepted Date:
25 March 2013 28 February 2014
Please cite this article as: Diabat, A., Hybrid Algorithm for a Vendor Managed Inventory System in a Two-Echelon Supply Chain, European Journal of Operational Research (2014), doi: http://dx.doi.org/10.1016/j.ejor.2014.02.061
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Hybrid Algorithm for a Vendor Managed Inventory System in a Two-Echelon Supply Chain Ali Diabat, Engineering Systems and Management, Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emirates *Corresponding author: Email : [email protected] Abstract In this paper we address the issue of vendor managed inventory (VMI) by considering a twoechelon single vendor/multiple buyer supply chain network. We try to find the optimal sales quantity by maximizing profit, given as a nonlinear and non-convex objective function. In such complicated combinatorial optimizations, exact algorithms and optimization solvers such as CPLEX and LINGO are inefficient, especially on practical-size problems. In this paper we develop a hybrid genetic/simulated annealing algorithm to deal with this nonlinear problem. Our results demonstrate that the proposed hybrid algorithm outperforms previous methodologies and achieves more robust solutions. Keywords: Supply chain management; VMI; Hybrid algorithm Highlights • • •
Proposed a hybrid algorithm using GA- SA for the VMI model Results show better performances for hybrid algorithm Sensitivity analysis was done to demonstrate the performance of proposed algorithm
Hybrid Algorithm for a Vendor Managed Inventory System in a Two-Echelon Supply Chain Abstract In this paper we address the issue of vendor managed inventory (VMI) by considering a twoechelon single vendor/multiple buyer supply chain network. We try to find the optimal sales quantity by maximizing profit, given as a nonlinear and non-convex objective function. In such complicated combinatorial optimizations, exact algorithms and optimization solvers such as CPLEX and LINGO are inefficient, especially on practical-size problems. In this paper we develop a hybrid genetic/simulated annealing algorithm to deal with this nonlinear problem. Our results demonstrate that the proposed hybrid algorithm outperforms previous methodologies and achieves more robust solutions. Keywords: Supply chain management; VMI; Hybrid algorithm 1. Introduction Vendor managed inventory (VMI) has been studied by many researchers because of its impact on the cost efficiency of supply chain networks. Under a VMI system, the vendor determines the timing and quantity of replenishment and has access to the retailer's inventory and demand data (Darwish and Odah, 2010). A well-designed VMI system can decrease inventory levels and enhance supply chain integration, thereby reducing costs (Schenck and McInerney, 1998; Achabal et al., 2000; Cetinkaya and Lee, 2000; Angulo et al., 2004). These benefits have been realized by many successful retailers and suppliers, including Wal-Mart and Procter and Gamble (Yao et al., 2007). Due to the growing interest in enhancing supply chain integration, we consider the integrated VMI system model developed by Nachiappan and Jawahar (2007).
In order to establish the optimal parameters of a VMI system, a complicated combinatorial optimization problem with a nonlinear non-convex objective function must be solved. Finding the optimal solution to such a problem may be infeasible using traditional algorithms, particularly for larger problem sizes, so robust new methodologies for solving such problems are required. Costa and Oliveira (2001) examined meta-heuristic algorithms such as genetic
algorithms (GA), simulated annealing algorithms (SAA), and evolution strategies (ES), and concluded that these meta-heuristic algorithms are well-suited to solving such problems.
In this paper we consider two supply chain models that use VMI: a single vendor/single buyer model, and a single vendor/multiple buyer model. We try to find the sales quantity, sales price, and contract price between the vendor and the buyer for each model. A hybrid genetic/simulated annealing algorithm is developed to determine the optimal parameters for each VMI model, and our results are compared with those obtained by Nachiappan and Jawahar (2007) using a genetic algorithm on the same model and the same data.
The remainder of the paper is organized as follows. In Section 2, we provide a comprehensive literature review. The model is presented in Section 3, and the proposed hybrid algorithm is explained in Section 4. The results of a complete computational analysis are presented in Section 5, and a sensitivity analysis is provided in Section 6. Finally, Section 7 presents our conclusions and identifies future research directions. 2. Literature review Our two-echelon single vendor/multiple buyer supply chain model under VMI focuses on efficient methodologies to solve this nonlinear problem. There is substantial research on model development and formulation (Cetinkaya and Lee, 2000; Shah and Goh, 2006; Jaruphongsa et al., 2004; Zhang et al., 2007), but the primary objective of such NP-hard problems is achieving better optimal solutions. Consequently, we see growing interest in exploiting meta-heuristic algorithms, especially genetic algorithms. These algorithms perform very well on large-scale problems like VMI. The capability of achieving near optimal solutions is the most important part of the applicability of a theoretical model, so developing more robust algorithms in order to address integrated VMI systems is essential.
Many approaches to solving VMI models, both by exact and approximate algorithms, are inefficient when they are applied to real-world problems; they cannot find optimal or nearoptimal solutions in a reasonable time. For instance, the excellent work of Archetti et al. (2007) presents a mixed-integer linear programming model. They implement a branch-and-cut
algorithm to solve the model optimally, and they compare the optimal solution of the problem to two problems obtained by implementing various methods of relaxation. Because their model is linear, they could exploit the CPLEX solver to solve the problem to optimality. Their approach works particularly well on small problem sizes.
Bertazzi et al. (2005) attempt to solve complex VMI problems. They propose a heuristic algorithm, and they decompose the problem into two sub-problems (production and distribution), which are then solved individually. Siajadi et al. (2006) consider a two-echelon single vendor/multiple buyer problem. They propose an analytical solution approach, and their work indicates that more efficient solution approaches are required.
Cardenas-Barron et al. (2012) propose a heuristic algorithm to solve a nonlinear integer programming (NLIP) vendor management inventory system with multiple products and multiple constraints. Their algorithm outperforms the traditional GA in terms of the total cost, the number of evaluations of the total cost function, and the computational time required. Sadeghi et al. (2013) solve a multi-vendor, multi-retailer, single-warehouse (MV-MR-SW) problem using a hybrid meta-heuristic-based particle swarm optimization (PSO) method, and compare the performance of their algorithm with the traditional GA.
In Table 1 we summarize the various heuristic and meta-heuristic methods that have been applied to solve VMI models. In the following section we present our hybrid algorithm for solving VMI models, which is intended to address some of the shortcomings of the previous approaches.
Table 1: New solution methodology developments on VMI Row
Paper
1.
Nachiappan and Jawahar 2007
vendor/multiple-buyers
A genetic based heuristic algorithm
Sue-Ann et al.
Two-echelon single-
PSO in comparison
2012
vendor/multiple-buyers
with a hybrid GA-AIS
2.
Model Two-echelon single-
Methodology
Developments and Comments Finding better near optimal solutions and correcting the presented solutions.
PSO performs better than GA and the hybrids. Therefore, the capability of this new methodology is proved.
They have mentioned that “other metaPasandideh et al.
3.
2011
Two-echelon, one supplier, one-retailer
heuristic search algorithms such as Genetic Algorithm
and multi-product
simulated annealing may also be employed and a comparison may be made among the algorithms.”
Two-echelon, inventory 4.
Zhao et al. 2007
control and vehicle routing schedules for a
Computational results reveal the Tabu Search
effectiveness as well as the robustness of the policy and the algorithm.
distribution system
Three-echelon (a 5.
Zhao et al. 2008
supplier, a central warehouse and a group of retailers)
Sadeghi et al.,
6.
2013
Variable Large Neighborhood Search
Tabu Search algorithm.
(VLNS) algorithm
Multi-vendor multiretailer single-
It performs better than previously proposed
Hybrid PSO
Proposed algorithm performs better than
warehouse model
the traditional GA solutions.
3. Model Description This section describes the two-echelon VMI model used in this paper. The following notation is used: aj
intercept value of the cost-demand curve of the jth buyer
bj
negative slope of the cost-demand curve of the jth buyer
c, c′, c″, c‴
chromosome
C
capacity of the vendor
cp(c)
cumulative probability of survival
EOQj
economic order quantity of jth buyer
fit(c)
fitness function
Hbj
annual unit holding cost of the jth buyer in independent mode
Hs
annual unit holding cost of the vendor in independent mode
HjVMI
annual unit holding cost of the vendor in VMI mode
j
buyer identifier (j = 1 to n)
n
number of the buyers
newfit(c)
new fitness function
OSMj
sum of order and inventory holding cost to the vendor for the jth buyer
p_cross
probability of crossover
Pbj
profit of the jth buyer
Pc
channel profit
Pc opt
optimal channel profit
Ps
vendor profit
Psj
profit obtained by vendor when supplying products to the jth buyer
PDj
production distribution cost of the jth buyer
p(c)
probability of survival of chromosome ‘c’
p_mut
probability of mutation
pop_size
population size
PRj
revenue share ratio between vendor and the jth buyer
P(y)
sales price
P(yj)
sales price of the jth buyer corresponding to sales quantity ‘yj’
P(yjopt)
optimal sales price of the jth buyer
Qj
replenishment quantity to the jth buyer
R
random number
Sbj
setup cost of the jth buyer per order in independent mode
Ss
setup cost of the vendor per order in independent mode
SjVMI
setup cost of the vendor per order in VMI mode of operation
W
contract price
Wj
contract price between vendor and the jth buyer
Wjopt
optimal contract price between vendor and the jth buyer
y
aggregate annual sales quantity of the vendor
yj
annual sales quantity of the jth buyer
yjmin
minimum expected sales quantity of the jth buyer
yjmax
maximum expected sales quantity of the jth buyer
yjopt
optimal annual sales quantity of the jth buyer
θj
flow cost per unit from vendor to the jth buyer
υj
transportation resource cost per unit from vendor to the jth buyer
δ
production cost per unit The network structure is illustrated in Figure 1:
Figure 1: Two-echelon single vendor/multiple buyers supply chain model Figure 1 illustrates that a single vendor has a special contract price with any buyer. Buyers also have a special sales price. The relationship between P(y) and y is assumed to behave linearly and is given as:
(1)
s.t.
(2)
where a and -b are the intercept of P(y) axis and the slope of the sales curve, respectively, in the sales price vs. sales quantity graph. Based on the above notation and the illustration of the network in Figure 1, we can summarize the complete mathematical model as:
Maximize
∑ { 0.5 [2 + ! " + "! ]/ }
(3)
Subject to:
(4)
% &
(5)
≥ 0
(6)
The objective is maximization of total profit. The vendor costs are due to production and distribution, orders, and stock maintenance. The production cost is derived from the amount spent for producing/acquiring a single unit, ,
and the aggregate demand, , and is expressed as . Distribution cost is derived from the
multiplication of flow cost ( ) and transportation resource cost () ). It should be mentioned
that ) includes indirect costs such as mode of transport, human router cost, and administrative costs per unit (assigned as 0.5), so that the distribution cost is expressed as 0.5 . The order cost is the cost involved to replenish the batches of demand of the *+, buyer and is
expressed as " + "! /- . The total holding inventory cost is based on the sum of holding costs at the vendor and buyer locations and is expressed as + ! - /2.
Finally, if we replace the order quantity - with the optimum order quantity under the EOQ model of ./- [
01 2034 54 / ] 61 2634
then the total order and stock maintenance cost becomes
[2 + ! " + "! ]/ , which is the last term of the objective function given in Equation (3).
Equations (4) and (5) are constraints on the buyer sales quantity and on the vendor capacity, respectively. Equation (6) constrains the solution to be non-negative. The solution to the above problem gives the optimal sales quantity 78+ (for all buyers). Once the above problem has been solved, the optimal sales price for each buyer can be calculated as: 78+ 78+
(7)
Furthermore, the acceptable contract price can be calculated as (see (Nachiappan and Jawahar, 2007) for details):
978+
78+ : 78+ : + 78+ + 0.5 78+ + [2 + ! " + "! 78+ ]/
(8)
(1 + : )78+
4. Proposed hybrid Algorithm
Usually, solving a nonlinear non-concave optimization model is not an easy task, so in this paper we use a hybrid GA-SA methodology to solve the above problem. These approaches will be compared with each other in terms of their solution quality in a later section. The chromosome representation used in this paper is shown in Figure 2. The length of the strings is equal to the number of buyers involved in the problem. 1
2
3
Sales Sales Sales quantity quantity quantity of buyer of buyer of buyer 1 2 3
N
…
Sales quantity of buyer j
Figure 2. Chromosome structure 4.1 Hybrid GA-SA algorithm
← Gene index
Recent studies (Sadeghi et al., 2013; Chen et al., 2013; Sue-Ann et al., 2012) have demonstrated that hybrid metaheuristics work better than individual metaheuristics for solving nonlinear models. This paper utilizes a hybrid algorithm to solve the two-echelon singlevendor/multiple-buyer VMI model. Usually, hybridization refers to the combination of two search algorithms to solve a given problem (Chen at al., 2013). There are several ways to use hybrid metaheuristics, one of which is combining the traditional GA with any of the local search heuristics. This paper proposes a hybrid algorithm based on GA and SA which presents several advantages as follows. GA has strong global search ability, but limited premature and weak local search ability. Conversely, SA has strong local search ability with no premature issues, but weak global search ability (Li et al., 2013; Chen et al., 2013). In order to best utilize the advantages and to eliminate the disadvantages of both methods, this paper proposes a hybrid GA-SA method. The structure of the proposed hybrid GA-SA is shown in Figure 3.
Figure 3. The structure of the Hybrid GA-SA (Chen et al., 2013)
In the proposed method, SA is used a local search algorithm to prevent the algorithm from being trapped in local optima. The input for the SA comes from the output of the GA mutation process. The output from the SA is used in the elitist selection process. The algorithm runs until a termination condition (i.e. on the maximum number of iterations) is met. It is evident from the literature that the parameters used in GA and SA have a clear effect on both solution quality and solution time (Yang et al., 2009; Kao and Han, 2008). The GA and SA parameters used were based on a pilot study. The SA parameters were set as follows: the initial temperature To was set to 100, the temperature cooling coefficient (α) was set to 0.9, and after each iteration is carried out for each value of T, the new temperature (Tnew) was obtained by multiplying the initial temperature by the cooling coefficient. The algorithm iterates until the temperature falls below the freezing temperature, where the freezing temperature was set to 1. The GA parameters for this study were determined using a 1-vendor 1-buyer problem. Based on this analysis, the GA parameters were fixed as follows: population size was set to twice the number of buyers, the crossover probability p_cross was set to 0.8, the mutation probability p_mut was set to 0.05, and the number of iterations was set to 300.
5.
Computational Analysis of the hybrid algorithm In this section, we evaluate the performance of the proposed methodology. The hybrid GA-
SA algorithm was coded using the C++ programming language and the computational experiments were run on a 2.5 GHz Intel(R) core (TM) i5 computer equipped with 8 GB of RAM. We solve the model in LINGO software package. We compare the performance of our algorithm to that of the proposed genetic algorithm of Nachiappan and Jawahar (2007) on the same problem and on the same data. In order to compare the performance of our proposed algorithm, two problem cases are introduced (Case 1: single vendor/single buyer; Case 2: single vendor/ three buyers). Case 1: Single Vendor/Single Buyer For this case we evaluate the performance of the proposed algorithm and compare to the results available in the literature. The parameters that describe the problem are presented in Table 2, and the results are summarized in Table 3. Table 2: Parameters for the single vendor/single buyer problem
!
"!
:
9
300
80
0.01
1000
2000
1
0.005
9
"
C
150
6150
40
Table 3: Hybrid algorithm compared with Dong and Xu, LINGO, and GA Methodology
78+
(78+ )
78+
Dong and Xu
1535
64.65
26960.49
LINGO
1535
64.65
26960.49
1532
64.68
26960.42
1535
64.65
26960.49
GA (Nachiappan and Jawahar, 2007) Hybrid Algorithm
In Table 3, our results are compared with the exact methods of Dong and Xu (2002), and it is observed that our results are in line with their findings. In order to measure the sensitivity of the hybrid algorithm parameters, we test the Case 1 problem with different values for parameters such as the crossover probability, the mutation probability, and the number of iterations. From the literature it is evident that different parameter values are used. To find the best parameter values for the GA, we conducted three experiments. The first experiment considers a fixed crossover probability (80%), a fixed mutation probability (10%), and a varying number of iterations. The best solution obtained by varying the number of iterations is shown in Figure 4, which shows that when the number of iterations increases, the algorithm yields better solutions.
Total profit vs. number of iterations 27500 27000 26500 Total profit
26000 25500 25000 24500 24000 23500 23000 Iteration numbers
Figure 4: Total profit vs. number of iterations The second experiment considers a fixed mutation probability (10%), a fixed number of iterations (300), and varying crossover probabilities (50-90%). The best solution obtained by varying the crossover probability values is shown in Figure 5, which shows that when the crossover probability increases, the total profit also increases.
Total Profit vs. Crossover probability
Total Profit
26500 25500 24500 23500 22500 21500 50% 52% 54% 56% 58% 60% 62% 64% 66% 68% 70% 72% 74% 76% 78% 80% 82% 84% 86% 88% 90% Crossover probability
Figure 5: Total profit vs. crossover probability
The third experiment considers a fixed crossover probability (80%), a fixed number of iterations (300), and varying mutation probabilities (2-25%). The best solution obtained by varying the mutation probability values is shown in Figure 6, which shows that when the mutation probability increases, the total profit generally decreases.
Total Profit vs. Mutation probability
Total Profit
26500 25500 24500 23500 22500
25%
24%
23%
22%
21%
20%
19%
18%
17%
16%
15%
14%
13%
12%
11%
10%
9%
8%
7%
6%
5%
4%
3%
2%
21500 Mutation probability
Figure 6: Total profit vs. mutation probability From Figures 4, 5, and 6, we can infer that the best performance on the Case 1 problem is obtained when the number of iterations is set to 300, the crossover probability is set to 80%, and the mutation probability is set to 5%. We used these parameter values in the Case 2 study and in the sensitivity analysis. Case 2: Single Vendor/Three Buyers The case from Nachiappan and Jawahar (2007) is used to evaluate the performance of the proposed algorithm. Table 4 and Table 5 give the input data for the study. Table 4 gives the data for three buyers, while Table 5 gives the data for the single vendor. Table 4: Buyer-related data
1
2
3
!
7
8
9
10
20
30
"!
Table 4: Buyer-related data
1
2
3
20
19
18
0.003
0.005
0.008
2000
500
500
4000
3000
1500
0.004
0.006
0.008
1
1.2
1.2
:
Table 5: Vendor-related data 9
"
C
15
5750
7
Table 6 summarizes the results of computational analysis for LINGO, GA (corrected results of Nachiappan and Jawahar (2007)), and our proposed hybrid algorithm. Table 6: Comparison of results for LINGO, GA, and hybrid algorithm 1
78+
(78+ )
LINGO
GA
2000
2002
14
14
78+
2 Hybrid
LINGO
GA
2001
710
673
14
15.5
15.60
algorithm
9903.11 9905.509
3 Hybrid
Hybrid
LINGO
GA
675
500
500
500
15.6
14
14
14
algorithm
algorithm
9908.487
Before analyzing the results of Table 6, we should point out that the results of LINGO and GA in (Nachiappan and Jawahar, 2007) are corrected here (highlighted cells). In that paper, the results for j=2 are y@ABC 657 and Py@ABC 15.7. Even if we accept these results, the objective function would be 9851.65 (wrongly calculated as 5982.1 in (Nachiappan and Jawahar, 2007)). 6. Sensitivity Analysis In this section a complete sensitivity analysis is performed on the following parameters: the upper and lower sales quantity limits, the (negative) slope of the cost-demand curves, the
holding costs, and the order costs. Included here are the data for five buyers (Table 7) and for one vendor (Table 8). We consider the sensitivity to parameter values that are 20% higher and 20% lower than normal, so three levels will be considered: Normal, 0.8*Normal, and 1.2*Normal. Table 7: Buyer-related data
1
2
3
4
5
!
7
8
9
7
9
10
20
30
15
25
20
19
18
21
18
0.003
0.005
0.008
0.003
0.006
2000
500
500
1700
500
4000
3000
1500
3500
2500
0.004
0.006
0.008
0.005
0.007
1
1.2
1.2
1.3
1.4
"!
:
Table 8: Vendor-related data 9
"
C
15
9850
7
Effects of changes in sales quantity limits. We first consider the effects of changes in the sales quantity limits. Before presenting the results of this analysis, it should be mentioned that the corrected results of Nachiappan and Jawahar (2007) relating to the effect of changes in the lower sales quantity limits ( ) and the upper sales quantity limits ( ) at the three levels on the optimal sales quantity (78+ ) and the optimal channel profit ( 78+ ) are presented in Tables 9 and 10, respectively (the corrected results are highlighted).
Table 9: Effect of lower limit on optimal sales quantity and channel profit (corrected) n
2
3
4
5
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
1600
2000
2400
10416.03
8290.972
4560.133
2
658
670
656
1
1602
2000
2400
12068.58
9885.799
5848.97
2
638
665
650
3
415
501
600
1
1600
2000
2401
19808.07
16493.6
10030.19
2
655
659
637
3
406
501
615
4
1365
1703
2040
1
1625
2126
2400
21997.6
17602.15
12320.07
2
659
653
647
3
423
567
604
4
1361
1702
2043
5
487
522
604
Table 10: Effect of upper limit on optimal sales quantity and channel profit (corrected) n
2
3
4
5
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
2000
2000
2000
8276.516
8290.972
8266.662
2
651
670
641
1
2000
2000
2000
9892.74
9325.279
9563.101
2
673
665
501
3
500
501
502
1
2001
2000
2002
16486.4
16493.6
15053.27
2
653
659
642
3
501
501
501
4
1702
1703
1926
1
2001
2126
2394
18796.42
17602.15
14510.71
2
688
653
754
3
503
567
666
4
1700
1702
1700
5
511
522
557
The results of the sensitivity analysis for the proposed hybrid algorithm are given in Tables 11 and 12. Table 11 shows the effect of changes in the lower sales quantity limits ( ) at the three levels on the optimal channel profit ( 78+ ), while Table 12 shows the effect of changes in the upper sales quantity limits ( ) at the three levels on the optimal channel profit ( 78+ ). Table 11: Effect of lower limit on optimal channel profit n
Pcopt Level 1
2 3 4 5
Level 2
10428.1
8297.025
4569.199
12080.67
9896.891
5859.01
19821.16
16507.69
10043.27
22007.69
17614.22
12328.15
Table 12: Effect of upper limit on optimal channel profit n
Level 3
Pcopt
Level 1 2 3 4 5
Level 2
Level 3
8278.542
8292.998
8270.715
9897.806
9327.305
9569.18
16489.44
16497.65
15059.35
18798.45
17608.23
14514.76
Effects of changes in the (negative) cost slope. We consider the effects of changes in the (negative) slope of the cost-demand curves. Before presenting the results of this analysis, it should be mentioned that the corrected results of Nachiappan and Jawahar (2007) relating to the effect of changes in the (negative) slope of the cost-demand curves ( ) at the three levels on the optimal sales quantity (78+ ) and the optimal channel profit ( 78+ ) are presented in Table 13 (the corrected results are highlighted).
Table 13: Effect of (negative) cost slope on optimal sales quantity and channel profit (corrected) n
2
3
4
5
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
2000
2000
2000
7953.018
8290.972
5334.177
2
571
670
747
1
2015
2000
2002
13164.26
9885.799
4361.515
2
740
665
1137
3
511
501
500
1
2001
2000
2001
21590.95
16493.6
11440.08
2
732
659
582
3
502
501
500
4
1700
1703
1700
1
2000
2126
2001
24253.32
17602.15
13495.93
2
776
653
587
3
521
567
500
4
1700
1702
1700
5
535
522
510
The results of the sensitivity analysis for the proposed hybrid algorithm are given in Table 14, which shows the effect of changes in the (negative) slope of the cost-demand curves ( ) at the three levels on the optimal channel profit ( 78+ ). Table 14: Effect of (negative) cost slope on optimal channel profit n
2 3 4 5
Pcopt
Level 1
Level 2
Level 3
7956.058
8295.025
5339.243
13170.34
9889.852
4366.581
21596.02
16500.69
11445.15
24255.35
17604.18
13503.02
Effects of changes in holding costs. The cost of inventory is an important factor in real situations and it accounts for most of the holding cost. We consider the effects of changes in the holding costs. Before presenting the results of this analysis, it should be mentioned that the corrected results of Nachiappan and Jawahar (2007) relating to the effect of changes in the holding costs at the three levels on the optimal sales quantity (78+ ) and the optimal channel profit ( 78+ ) are presented in Table 15 (the corrected results are highlighted).
Table 15: Effect of holding costs on optimal sales quantity and channel profit (corrected) n
2
3
4
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
2000
2000
2000
8519.457
8290.972
8066.441
2
671
670
645
1
2000
2000
2037
10202.65
9885.799
9202.98
2
658
665
647
3
501
501
541
1
2067
2000
2000
16458.79
16493.6
16088.36
2
680
659
656
3
507
501
501
5
4
1700
1703
1701
1
2002
2126
2003
2
674
653
674
3
506
567
501
4
1700
1702
1701
5
508
522
522
19324.34
17602.15
18281.22
The results of the sensitivity analysis for the proposed hybrid algorithm are given in Table 16, which shows the effect of changes in the holding costs at the three levels on the optimal channel profit ( 78+ ). Table 16: Effect of holding costs on optimal channel profit n
Pcopt Level 1
2 3 4 5
Level 2
Level 3
8528.576
8294.012
8074.546
10211.77
9891.878
9209.059
16461.83
16498.67
16095.45
19333.46
17609.24
18290.34
Effects of changes in setup costs. Setup costs include costs related to purchasing (such as invoice processing costs, receipt processing costs, and so on). We consider the effects of changes in the setup costs. Before presenting the results of this analysis, it should be mentioned that the corrected results of Nachiappan and Jawahar (2007) relating to the effect of changes in the setup costs at the three levels on the optimal sales quantity (78+ ) and the optimal channel profit ( 78+ ) are presented in Table 17 (the corrected results are highlighted). Table 17: Effect of setup costs on optimal sales quantity and channel profit (corrected) n
2
3
j
Yjopt
Pcopt
Level 1
Level 2
Level 3
Level 1
Level 2
Level 3
1
2001
2000
2000
8519.07
8290.972
8059.511
2
683
670
638
1
2086
2000
2000
9535.44
9885.799
9555.487
4
5
2
695
665
624
3
514
501
501
1
2026
2000
2002
2
691
659
662
3
546
501
503
4
1717
1703
1764
1
2001
2126
2002
2
665
653
678
3
500
567
606
4
1700
1702
1704
5
500
522
521
16599.59
16493.6
15729.59
19331.32
17602.15
17936.51
The results of the sensitivity analysis for the proposed hybrid algorithm are given in Table 18, which shows the effect of changes in the setup costs at the three levels on the optimal channel profit ( 78+ ). Table 18: Effect of setup costs on optimal channel profit n
2 3 4 5
7.
Pcopt Level 1
Level 2
Level 3
8524.136
8296.038
8065.59
9537.466
9893.904
9561.566
16604.66
16499.68
15733.64
19341.45
17606.2
17943.6
Conclusions and Future Research
In this paper we developed a hybrid genetic/simulated annealing algorithm to solve the nonlinear, non-convex optimization problem associated with VMI models. For the two-echelon single vendor/multiple buyer supply chain network considered in this paper, we demonstrated that our hybrid algorithm outperforms existing algorithms, and also performed a complete sensitivity analysis on our algorithm. Among the promising research directions that future researchers may wish to explore are to extende this problem to three echelons, elevating the problem from a single vendor to multiple vendors, and utilizing other metaheuristics. Other possible extensions could be to combine the
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