Aggregation methodologies for perishability management in

548

European J. Industrial Engineering, Vol. 11, No. 4, 2017

Aggregation methodologies for perishability management in production and distribution system Minh Ho Ngoc and Narameth Nananukul* Industrial Engineering, School of Manufacturing Systems and Mechanical Engineering, Sirindhorn International Institute of Technology, Thammasat University, Pathum Thani, 12120, Thailand Email: [email protected] Email: [email protected] *Corresponding author Abstract: An existing integrated model such as the production-inventoryrouting-problem (PIDRP) can be used to generate operational decision in the planning horizon. However, for certain products, such as food, freshness of the product are also an important factor to be considered. Typically, determining the solutions for large instances of PIDRP is not possible. The contribution of this research is to propose two aggregation techniques that can manage large instances of PIDRP that consider perishability of products. The first technique is based on a clustering algorithm that considers constraints from delivery capacity. The second technique is a tailored aggregation for producing subsets of products that can be ordered jointly. Computational results show that the proposed techniques can be used to reduce problem size while maintaining good quality solution within acceptable run time. The gaps of the solutions for problem sets up to 30 retailers, 20 products and eight periods are less than 5%. [Received 8 December 2016; Revised 27 March 2017; Accepted 21 April 2017] Keywords: aggregation methodology; integrated production and distribution.

perishability

management;

the

Reference to this paper should be made as follows: Ngoc, M.H. and Nananukul, N. (2017) ‘Aggregation methodologies for perishability management in production and distribution system’, European J. Industrial Engineering, Vol. 11, No. 4, pp.548–568. Biographical notes: Minh Ngoc Ho earned his Master degree in Logistics and Supply Chain Systems Engineering from Sirindhorn International Institute of Technology, Thammasat University. Currently, he is a Logistics Inventory Controller at Unilode Aviation Solutions, Thailand. He also had experience with Vinamilk, Vietnam, as a Technical and Procurement Engineer. Narameth Nananukul earned his PhD and Master degrees in Operations Research and Industrial Engineering from the University of Texas at Austin and Texas A&M University, USA. He has over ten years experience in applying operations research and statistics to real-world problems for many companies in USA. His research interests include developing decision support and business intelligent systems for production planning, transportation,

Copyright © 2017 Inderscience Enterprises Ltd.

Aggregation methodologies for perishability management

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integrated production and distribution and inference systems. His teaching interests include classes in engineering management and energy technology such as production planning, logistics and supply chain management, quality management, statistics, optimisation and renewable energy. This paper is a revised and expanded version of a paper entitled ‘Integrated production and distribution with perishability management in logistics system’ presented at International Conference on Trends in Multidisciplinary Business and Economic Research (TMBER), Bangkok, Thailand, 25–26 March 2016.

1

Introduction

Many companies try to maintain a good service level offered to their customers, in order to be more competitive than their competitors. Decisions related to production and distribution are considered the most important factors in any supply chain. However, traditionally, they were decided separately. In order to increase the efficiency of the supply chain operation, an integrated framework that integrates production and distribution planning was introduced. It was shown by Chandra and Fisher (1994) that considering the production scheduling and vehicle routing decision in an integrated framework can result in cost saving up to 20%. Various models have been proposed in order to manage a supply chain in an integrated framework. For example, a powerful system called vendor managed inventory (VMI) was developed and applied to manage inventory and distribution in a supply chain. In VMI, vendors can monitor customer inventory levels continuously. As a result, they can better consolidate the delivery to their customers, which leads to lower operational cost. Another integrated model is the production-inventory-routing-problem (PIDRP) (Lei et al., 2006; Bard and Nananukul, 2008). The model coordinates the decisions for production, level of inventory, and distribution routing in the planning horizon. Yang and Lin (2012) considered cooperative inventory models that collaborate supply chain partners in order to improve the performance of the inventory control. They proposed a single-vendor multiple-buyer integrated inventory model with uncertain demand based on normal distribution. Liao et al. (2015) proposed a two-stage supply chain model that coordinates production and transportation schedules. The authors developed an electromagnetism-like mechanism algorithm for searching for heuristic solution. Renna (2015) studied three main issues of supply chain network design that includes architecture, coordination mechanism and the local plant decisions. The focus in on providing useful information for choosing coordination approaches to be used in production planning. For certain industries such as agriculture and food industries, the products are perishable, in another word, the quality of products changes over time. As a result, the consideration of perishability in various processes in the supply chain (purchasing, production, inventory management and distribution) is crucial to satisfy customer requirements. Without considering the perishability aspects, decision makers are unable to make production plans as well as distribution decision. They cannot control the spoilage cost of delivering products to customers.

550

M.H. Ngoc and N. Nananukul

Therefore, the coordination of the PIDRP and perishability management is a practical way to help decision makers in logistics management manages the quality of perishable products. A lot-sizing decision determines the batch for each type of product to be produced in each period while considering the loss value of a product. The allocation between manufacturing plant and retailer may have an effect on the shelf life of a perishable product. As a result, distribution planning and vehicle routing are integrated with lot-sizing decision in order to better coordinate different decisions in the supply chain while considering the perishability of products. Typically, a PIDRP that considers multiple products is very complex, and the size of a practical PIDRP is very large (Ngoc and Nananukul, 2016). To determine the solutions for large instances of PIDRP is not possible. This research focuses on developing aggregation techniques that can be used to manage large instances of a PIDRP that considers the perishability of products. The first technique is based on a clustering algorithm that considers constraints from delivery capacity. The second technique is a tailored aggregation for producing subsets of products that can be ordered jointly. The contents of this paper are arranged as follows: Section 2 provides literature reviews in the areas of lot sizing and the coordination of production and distribution planning, In Section 3, the problem statement for this research is introduced, follows by Section 4 where aggregation techniques are presented. Section 5 reports computational results of the PIDRP with perishability consideration. Managerial insights that describe practical application of the proposed methodologies are provided in Section 6. Section 7 provides conclusion of the research. Finally, the appendix includes detailed description of the PIDRP considered in this article.

2

Literature review

In the area of lot sizing, a single item lot size problem was proposed by Gutiérrez et al. (2003). Gutiérrez et al. (2007) extended the basic lot-sizing model by considering backlogging. Lot sizing with multi-items was developed by Pochet and Wolsey (1991) as a MIP model. A capacitated lot-sizing problem that considers multiple items and unrelated parallel machines was studied by Fiorotto et al. (2015). Lot sizing with the consideration of job shop scheduling was formulated by Karimi-Nasab and Modarres (2015). A lot-sizing model that considers uncertainty of processing times was studied by Koca et al. (2015). The problem was formulated as a nonlinear mixed integer problem. With perishability consideration, Hsu (2003) proposed an economic lot size (ELS) model with perishability consideration. Both perishable inventory and backorder costs are considered in the objective function. A lot size model for a perishable product that considers infinite planning horizon was formulated by Abad (2000). Finite production and partial backordering were taken into account. An exponential function was used to represent the deterioration. A lot-sizing model that allows rework was studied by Teunter and Flapper (2003). Both rework time and cost were considered in the model. An integrated production and distribution framework that focuses on perishable food production was proposed by Wang et al. (2009). Several costs, such as production setup cost, raw material cost, holding cost, and perishable cost were considered in the proposed framework. A mixed-integer linear programming model that considers the shelf life of yoghurt production was proposed by Lütke entrup et al. (2005). The objective is to balance shelf life issues and common costs in food industries.

Aggregation methodologies for perishability management

551

When considering perishability, dairy product delivery problem was analysed by Tarantilis and Kiranoudis (2001) using a fixed fleet vehicle routing model. Tarantilis and Kiranoudis (2002) considered a vehicle routing problem with multiple depots for fresh meat delivery. Hsu et al. (2007) considered a stochastic VRPTW model for food delivery with the objective function to minimise inventory costs from the deterioration of perishable food, and energy costs. A vehicle routing problem that considers time windows (VRPTW) for the distribution of fresh vegetables was developed by Osvald and Stirn (2008). The objective is to minimise inventory cost for the deterioration of food, and energy cost. In the area of coordinating production and distribution planning, Chandra and Fisher (1994) investigated an integrated model with multiple plants and multiple products. Ozdamar and Yazgac (1999) also presented a production-distribution model with the objective to minimise the overall cost from factory, inventory at warehouses, and transportation cost. An integrated model with multi-echelon environment was studied by Jayaraman and Pirkul (2001) using a mixed integer programming model. Lei et al. (2006) proposed a two-phase solution for a PIDRP problem in order to optimise production, inventory, and transportation operations. An integrated optimisation model for production and distribution that considers a single plant was proposed by Fumero and Vercellis (1999). Also, Bard and Nananukul (2008) presented an integrated production-inventory-distribution-routing problem with the objective of satisfying customer demand with minimum cost. The problem was extended by Armentano et al. (2011) to consider a multi-item problem with deterministic demand. In the area of perishability management, Amorim and Almada-Lobo (2014) proposed a multi-objective optimisation model that takes into account both the distribution cost and the freshness state of the delivered products. Also, Amorim et al. (2012) presented a multi-objective model that trade-offs between the cost and the freshness of a product in an integrated framework. A mathematical programming model with heuristic algorithms was proposed by Sel et al. (2015) to consider a yoghurt production problem. The model minimises the loss of product value, production cost, transportation cost, and unmet demand cost. Bilgen and Çelebi (2013) presented an integrated model for dairy product production and distribution problem in order to consider both the effect of shelf life and total cost. Ngoc and Nananukul (2016) proposed a mixed integer linear programming model with a loss function for the PIDRP. The model determines production schedule, inventory level, and distribution routing for multi-products. Computation results that compare the solutions from short and long shelf life were provided. The literature review is summarised in Table 1, the research scopes are provided and the research gap is identified by the model that considers production plans, inventory, VRP and perishability management. Therefore, this research aims to propose the methodologies to solve multiple perishable products problem with the consideration of production planning, inventory and vehicle routing problem.

3

Problem statement

The focus of this research is on developing methodologies for solving the PIDRP proposed by Ngoc and Nananukul (2016) that can be used to handle practical cases with

552

M.H. Ngoc and N. Nananukul

large data instances. We address a production planning and distribution scheduling and consider perishability in a multi-product dairy plant. A set of retailers has a non-negative demand for each type of product over the planning horizon that must be satisfied. The model considers a production plant as well as retailers. The plant can produce multiple products with limited capacity. The plant and retailers are allowed to carry inventories with specified holding costs. The products will be delivered from the plant to retailers by a fleet of vehicles. The model will generate optimal routes for each period in the planning horizon. Literature review summary

x

Amorim and Almada-Lobo (2014)

x

2

Bard and Nananukul (2008)

x

x

x

x

3

Fumero and Vercellis (1999)

x

x

x

x

4

Lei et al. (2006)

x

x

x

x

5

Wang et al. (2009)

x

x

x

6

Gutiérrez et al. (2003)

x

x

x

7

Gutiérrez et al. (2007)

x

x

x

8

Koca et al. (2015)

x

x

x

9

Abad (2000)

x

x

x

10

Teunter and Flapper (2003)

x

x

x

11

Osvald and Stirn (2008)

x

x

x

12

Tarantilis and Kiranoudis (2001)

x

x

x

13

Tarantilis and Kiranoudis (2002)

x

x

x

14

Çetinkaya et al. (2006)

x

x

15

Armentano et al. (2011)

x

x

x

x

16

Chandra and Fisher (1994)

x

x

x

x

17

Ozdamar and Yazgac (1999)

x

x

x

x

18

Amorim et al. (2012)

x

x

x

x

19

Bilgen and Çelebi (2013)

x

x

x

x

20

Sel et al. (2015)

x

x

x

x

21

Pochet and Wolsey (1991)

x

x

22

Lütke entrup et al. (2005)

x

The gap

x

Inventory

1

Multi

Authors

Single

x

Order

Perishability

Research area Production

Item type

VRP

Table 1

x

x x

x

x

x

Aggregation methodologies for perishability management

553

In addition, the model also considers the shelf life of each product, to make sure that the freshness requirement is achieved. This is represented by the duration between the ready time of the product until the time when the product is purchased. In general, the manufacture and retailers define the shelf life of each product. A shelf life dependent loss function is used to represent the loss of product value. It is assumed that the loss value has a linear relationship with every additional lost day of shelf life. The detailed description of the model can be found in the Appendix.

4

Aggregation methodologies

4.1 Clustering technique for handling large data instances Due to the complexity of the PIDRP model presented in the previous section, it is unlikely to be able to generate a solution for large datasets. As a result, a grouping techniques based on K-mean clustering algorithm was proposed. We proposed an enhanced K-mean clustering algorithm that performs clustering of a given number of retailers while maintaining clusters’ size, such that the delivery capacity is feasible. Without loss of generality, we assumed that all retailers in the same cluster will be delivered by the same vehicle. Parameter ‘k’ is used to represent the number of clusters. The objective is to classify a given number of retailers into k clusters. Among the retailers, the Euclidean distances are defined by using the coordinates (x,y), and the key element is to determine how many ‘k’ centroids and the coordinates for these centroids. The result at the end of the process is to assign retailers to these k clusters. A reasonable value for parameter ‘k’ is obtained when the locations of retailers are tightly surrounding the centroids, and the delivery capacity within each cluster is not violated. Otherwise, a new value of k needs to be determined again. Also, the distance among retailers and these new centroids need to be recalculated. A possible approach is to iterate the value of k and choose the one that minimises the sum of square root of the distance between the retailers to the centroids. In general, we may notice a good number of “k” when the total distance cannot be reduced. The clusters are constructed based on an enhanced K-mean algorithm to form k clusters of customers. Figure 2 summarises the enhanced K-mean algorithm. The algorithm starts with the initialisation step, where the number of clusters, α, is set equal to the number of transportation resources, θ. Also, the deviation from centres, ψ, is set to 0, initially. The proposed algorithm selects a retailer randomly and assigns it to each cluster. Then, other retailers are assigned to the closest clusters. Every time a cluster includes a new member, the cluster centroid is recalculated. Note that in this step the feasibility of transportation capacity needs to be checked, if the transportation capacity of any cluster is violated, the retailer will not be included to the cluster. When all retailers are assigned, the percentage change of total deviation of distance from all core clusters compared to the deviation from the previous iteration, dev, is then calculated, based on the following formula: Δψ ψ

× 100

554 Figure 1

M.H. Ngoc and N. Nananukul Network flow representation of production-distribution problem for multiple products

If dev is less than or equal to the minimum required percentage, devmin (5% is a practical value), then the number of clusters should not be increased, the number of generated clusters is acceptable, and the algorithm terminates. On the other hand, if dev is more than devmin or there is transportation capacity violation, then increasing the number of clusters is necessary and will decrease the total distance. As a result, α is incremented before the algorithm is repeated until the centroids of all clusters are fixed and the stop condition aforementioned is satisfied. In general, the algorithms based on K-mean clustering algorithm are used to process large number of data points. The average runtime for the proposed enhanced K-mean algorithm is acceptable. For the test datasets used for testing, the average runtime is less than 1 minute.

4.2 Tailored aggregation for producing selected subsets of products With the result after clustering the retailers, the demand of each cluster needs to be aggregated. To determine the frequency of delivery, tailored aggregation is applied to determine products that will be delivered jointly. The tailored aggregation procedure considers common order costs, S, and a product-specific order cost, sk, for each product k. Based on Chopra and Meindl (2013), there are four steps that can be used to determine the subsets of products that will be delivered jointly. In the first step, the most frequently ordered product is determined by using equations (1) and (2). Note that the number of products is l, h represents holding cost, and Dk represents demand rate of product k.

Aggregation methodologies for perishability management Figure 2

555

Procedure of enhanced K-mean clustering

n = max k∈{1,..,l} {nk } , nk =

hCk Dk , k ∈ {1, … , l} 2( S + sk )

(1)

and k ∗ = arg max k∈{1,..,l} {nk }

(2)

In the second step, the ordering frequencies of other products are evaluated by using equation (3).

556

M.H. Ngoc and N. Nananukul nk =

hCk Dk , k ∈ {1, … , l}, k ≠ k ∗ 2 sk

(3)

Then, the relative frequency of product k is calculated relative to nk by using equation (4). mk = ⎡⎢ n nk ⎤⎥

(4)

In the third step, equation (5) is used to calculate the ordering frequency of the most frequently ordered product k*. n=

∑

l k =1

hCk mk Dk

( ∑

2 S+

l k =1

sk mk

(5)

)

Finally, the order frequency of each product is calculated by using equation (6). nk =

n , k ∈ {1, … , l} mk

(6)

When considering perishability of the products, the cycle time based on the aforementioned procedure needs to be adjusted. In order to satisfy the perishability requirement of product k of customer i, the cycle time cannot exceed SLk ∗ MinRateik .

Thus, if the cycle time is more than SLk ∗ MinRateik , it will be set to SLk ∗ MinRateik . A procedure for tailored aggregation with perishability management is described in Figure 3.

4.3 Numerical illustration In this section, a small test case is used to illustrate how the proposed procedures in Sections 4.1 and 4.2 work. The planning time horizon is assumed to be four days, and there are five retailers. The production plant can produce two different types of products. The locations of retailers, as well as the demand of each product during the planning horizon, are summarised in Table 2. Table 2

Locations of retailers and the demand of each product

Location

X

Y

Demand of product 1

Demand of product 2

Production plant

30

42

0

0

Retailer 1

62

89

100

50

Retailer 2

54

76

150

20

Retailer 3

12

25

150

150

Retailer 4

15

13

50

150

Retailer 5

51

72

50

30

Aggregation methodologies for perishability management Figure 3

557

Tailored aggregation procedure with perishability management Start

Step 1: Identify the most frequently ordered product k*

Step 2: Calculate ordering frequencies of all products

Step 3: Recalculate ordering frequencies of product k*

Step 4: Recalculate ordering frequencies of all products

Step 5: Determine cycle times of all products

If the cycle time ≤ SLk * MinRateik

No

Step 6: Set the cycle time to SLk * MinRateik

Yes

Step 7: Calculate order sizes of all products

Step 8: Generate order plan for retailer

End

To apply the clustering methodology in Section 4.1, the total distance is calculated with the different number of clusters, K. The total distance and percentage reduction of distance for different values of K is summarised in Table 3.

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M.H. Ngoc and N. Nananukul

Table 3

Distance and percentage of distance decrease

Number of cluster

Total of distance

Percentage of distance decrease

1

192.0577

100.00%

2

57.39198

70.12%

3

40.44431

29.53%

4

32.63373

19.31%

5

32.63379

0.00%

From Table 3, the total distance decreases dramatically from 1 cluster to 2 clusters, and the percentage of reduction is 70.12%. Then, it gradually decreases from 3 to 5 clusters. As a result, two clusters should be chosen. Figure 4 shows a scatter plot of the clusters. Next, the centers of all clusters were determined and the distance matrix was calculated, as shown in Table 4. Figure 4

Table 4 From/to

Scatter plot of the clusters (see online version for colours)

Distance between clusters and the production plant Plant

Cluster 1

Cluster2

Plant

0

38

58

Cluster 1

38

0

70

Cluster 2

58

70

0

Once the clusters are defined, the tailored aggregation procedure mentioned in section 4.2 is then applied, to determine order frequency and order size of each product within each cluster. For each type of product, the common order cost S, product-specific order cost sk, and the holding costs, are assumed to be 500$, 100$, and 10$, respectively. Also, the shelf life of each product is set to 3 days, and the retailer requires a minimum residual shelf life of 60%. Then, the tailored aggregation procedure with perishability management from Figure 3 was applied, as shown in Tables 5 to 13. Table 5 shows Step 1 of the procedure. The most frequently ordered product in each cluster is determined. In cluster 1, product 1 is the most frequently ordered product and product 2 is the most frequently ordered product in cluster 2.

Aggregation methodologies for perishability management Table 5

559

Step 1 of tailored aggregation procedure with perishability management Step 1: ni =

hCi Di 2( S + si )

Prod. 1

Prod. 2

Cluster 1

1.291

1.581*

Cluster 2

1.581*

0.91

In Step 2, the ordering frequencies of all products are calculated and then the ratios of ordering frequencies to the highest order frequency in each cluster are calculated as shown in Table 6. Table 7 shows Step 3 of the procedure which recalculates the ordering frequencies of product with the highest order frequency in each cluster. Table 6

Step 2 of tailored aggregation procedure with perishability management Step 2: ni =

hCi Di 2si

mi = ⎡⎢ n ni ⎤⎥

Prod. 1

Prod. 2

Prod. 1

Prod. 2

Cluster 1

3.2

-

1

-

Cluster 2

-

2.2

-

1

Table 7

Step 3 of tailored aggregation procedure with perishability management

Step 3: n =

∑

l i =1

2( S +

hCi mi Di

∑

l i =1

si mi )

Prod. 1

Prod. 2

Cluster 1

-

1.89

Cluster 2

1.69

-

Then, the highest order frequency of product in each cluster is updated in Step 4 as shown in Table 8. Table 9 summarises cycle time of products in each cluster. Supposing that the shelf life of each product is 3 days and the retailer requires a minimum residual shelf life of 60% ( MinRateik = 0.6) for selling products, the remaining shelf life that the retailer needs are calculated as shown in Table 10. Table 8

Step 4 of tailored aggregation procedure with perishability management

Step 4: ni =

n mi

Prod. 1

Prod. 2

Cluster 1

1.89

-

Cluster 2

-

1.69

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M.H. Ngoc and N. Nananukul

Table 9

Step 5 of tailored aggregation procedure with perishability management

Step 5: Define the cycle time (planning horizon/ni) Prod. 1

Prod. 2

Cluster 1

2.12 ≈ 3

2.12 ≈ 3

Cluster 2

2.37 ≈ 3

2.37 ≈ 3

Table 10

The remaining shelf life calculation

Product

The remaining shelf life

1

1.8 ≈ 2

2

1.8 ≈ 2

In Step 6, the resulted cycle time from Step 5 is then adjusted in order to satisfy the remaining shelf life from Table 10. This is shown in Table 11. Table 12 reports order sizes of products in each cluster. Finally, order plan for products in each cluster is generated as shown in Table 13. The quantity of products within each cluster can then be aggregated and used as an input to the PIDRP model. The size of the resultant model is much smaller than the size of the original PIDRP model, which makes it more practical for implementation. The data used in the numerical illustration is from a dataset for PIDRP used by Bard and Nananukul (2008). Table 11

Step 6 of tailored aggregation procedure with perishability management

Step 6: The adjusted cycle time (days) Prod. 1

Prod. 2

Cluster 1

2

2

Cluster 2

2

2

Table 12

Step 7 of tailored aggregation procedure with perishability management

Step 7: Order size Prod. 1

Prod. 2

Cluster 1

100

150

Cluster 2

150

50

Table 13

Step 8 of tailored aggregation procedure with perishability management

Step 8: Order plan

Product

Days 1 100

2

3

Cluster 1

1

100

2

150

150

Cluster 2

1

150

150

2

50

50

4

Aggregation methodologies for perishability management

5

561

Computational results

In this section, several datasets were used to test the effectiveness of the methodologies presented in Section 4. The models were solved by using the IBM ILOG CPLEX optimisation studio version 12.6.0.0, using a computer with an Intel Core I5-3337U CPU @ 1.80 GHz and 4.0 GB memory. Datasets are based on standard datasets for the PIDRP model used by Bard and Nananukul (2008). Datasets consist of 12 instances containing 10, 20, and 30 retailers, 10 and 20 types of product, and four and eight periods for planning horizon. The number of available vehicles is 5, with the vehicle capacity of 5,000 units, for all of instances. The initial inventory levels are set to zero. The maximum inventory levels at retailer sites are assumed to be 600 for each type of product, and the maximum inventory level at the production plant is unlimited. For perishability consideration, the loss value of each product is assumed to be 0.5, and the minimum requirement shelf life at retailers is set to 60%. The computational time is limited to 2 hours and the test instances are grouped into two categories, short and long shelf life. First, the computational results based on the PIDRP with perishability management proposed by Ngoc and Nananukul (2016) are presented. The results are displayed in Tables 14 and 15, respectively.

No.

R

τ

K

No. of total variables/binaries

Gap (%)

Solution

Heuristic gap (%)

Heuristic solution

Results from solving the PIDRP with perishability management for short shelf life products No. of constraints

Table 14

1

10

4

10

36,150

34,451/2,460

0.10

0.10

4.5

256,755.61

2

10

4

20

69,380

66,481/2,500

0.13 480,648.8

3

497,632.96

3

10

8

10

74,530

64,111/4,920

0.81 288,439.4

4.4

301,214.73

4

10

8

20

143,220

123,381/5,000

0.18 288,439.4

2.34

306,262.41

5

20

4

10

124,216

122,351/8,860

0.18 366,033.0

0.3

365,518.00

6

20

4

20

238,020

235,881/8,900

2.68 789,837.1

–1.4

778,722.48

7

20

8

10

252,110

224,811/17,720

2.68 867,381.8

–0.56

863,562.76

8

20

8

20

484,700

431,981/17,800

4.69 823,317.4

–2.14

805,382.71

9

30

4

10

263,630

264,251/19,260

4.69

*

*

234,067.19

10

30

4

20

506,660

509,281/19,300

4.69

*

*

456,527.59

11

30

8

10

533,690

483,511/38,520

4.69

*

*

526,423.23

12

30

8

20

1,026,180

928,581/38,600

0.48

*

*

994,194.5

Note: *No solution.

M.H. Ngoc and N. Nananukul

562

13

10

4

10

36,150

34,451/2,460

0.11 50,293.71

6.1

53,308.13

14

10

4

20

69,380

66,481/2,500

0.14 96,927.58

2.62

99,389.80

15

10

8

10

74,530

64,111/4,920

0.78 63,522.19

4.1

66,435.21

16

10

8

20

143,220

123,381/5,000

0.22 69,984.49

–0.13

69,629.12

17

20

4

10

124,216

122,351/8,860

1.3

47,550.43

–0.7

46,647.32

18

20

4

20

238,020

235,881/8,900

3.2

101,126.8

0.12

99,910.82

Heuristic solution

K

Heuristic gap (%)

τ

Solution

R

Gap (%)

No.

No. of total variables/binaries

Results from solving the PIDRP with perishability management for long shelf life products No. of constraints

Table 15

19

20

8

10

252,110

224,811/17,720

1.5

116,140.8

–0.31

116,844.09

20

20

8

20

484,700

431,981/17,800

4.69 116,732.8

–1.5

112,539.33

21

30

4

10

263,630

264,251/19,260

0.14

*

*

59,222.18

22

30

4

20

506,660

509,281/19,300

0.17

*

*

114,788.1

23

30

8

10

533,690

483,511/38,520

1.39

*

*

132,346.1

24

30

8

20

1,026,180

928,581/38,600

1.97

*

*

249,662

Note: *No solution.

For the short shelf life case, the shelf life is assumed to be 3 days. For the long shelf life case, the shelf life is 14 days. Tables 14 and 15 report computational results for both cases. Notice that for problems with similar size (defined by the number of retailers, number of periods, number of products), the average total cost for the long shelf life case is less than the one from the short shelf life case. This is because the supplier is required to deliver products more frequently for short shelf life products. Notice that the solutions for problem numbers 9 to 12 and problem numbers 21 to 24 could not be obtained within 2 hours by using Cplex because of the large number of constraints and variables. The PIDRP model with perishability consideration can be used to handle the problems of multiple products and multiple retailers from 4 to 8 periods within 2 hours. However, for the cases in which the number of retailers is 30, Cplex failed to achieve the solution within a practical run time, 2 hours. As a result, the application of an enhanced K-mean clustering and tailored aggregation methods were used to reduce the problem size so that it is possible to generate a good quality solution with acceptable run time. In Tables 14 and 15, columns 1–4 give description of each test problem which includes number of retailers, number of periods and number of products. Columns 5 and 6 represent number of constraints and variables for the test problem. Column 7 reports the gap from LP solution when the runtime reaches the time limit. Column 8 represents the solution from Cplex. Column 9 reports the gap between heuristic solution (column 10) and the solution from Cplex (column 8).

Aggregation methodologies for perishability management

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For problems that Cplex can provide solutions, problems 1–8 from Table 14 and problems 13–20 from Table 15, on overage the gap from LP solution is 1.46%. Although, Cplex cannot provide the solution within time limit, the LP gap is within acceptable range (less than 5%). For problems that Cplex cannot provide any solution, problems 9–12 from Table 14 and problems 21–24 from Table 15, on average the LP gap of heuristic solutions is 0.58% which is also acceptable. To determine the quality of heuristic solution, the heuristic gap is provided in column 9. For small test problem instances, problems 1–4 from Table 14 and problems 13–16 from Table 15, the average heuristic gap is 3.37%. This shows that for small test problems, the solutions obtained from Cplex are better that heuristic solutions. However, for medium test problems (problems 5–8 from Table 14 and problems 17–20 from Table 15), the average heuristic gap is –0.77%. This shows that for larger problems, heuristic solutions are slightly better that the solutions from Cplex.

6

Managerial insights

Typically, the PIDRP can be applied to the retail industry where a few central manufacturers are used to supply products to customer sites. The PIDRP with perishability consideration can be directly applied to manage products such as food or diary products. Based on the computational results from Section 5, results can be categorised into two groups, the first group contains small and medium data instances where Cplex can provide solutions within time limit and the second group contains large data instances where Cplex cannot provide any solution within time limit. Data instances in the first group have number of retailers up to 20 with number of periods up to 8 and number of products up to 20. These are problem numbers 1 to 8 from Table 14 and problem numbers 13 to 20 from Table 15. If the focus is on the quality of the solution, the analyst should focus on the solution with less LP gap value. This will generally lead to more saving opportunity. For small data instance (problem numbers 1 to 4 from Table 14 and problem numbers 13 to 16 from Table 15), this corresponds to solutions provided by Cplex which have the average LP gap less than 1 percent. Notice that the quality of the heuristic solution is worse that Cplex solution for small data instance, this is shown by positive heuristic gaps. For medium data instance (problem numbers 5 to 8 from Table 14 and problem numbers 17 to 20 from Table 15), the analyst has an option to choose between Cplex solution and heuristic solution. In some cases (those with negative heuristic gap), heuristic solution is better than Cplex solution. As a result, for medium data instances, if the solution quality has the highest priority, determine and compare both Cplex and heuristic solutions. For large data instances (problem numbers 9 to 12 from Table 14 and problem numbers 21 to 24 from Table 15), if the expected solution time is limit (less than 2 hours), heuristic solution is the only option. However, if there is no constraint on solution time, the analyst can generate and compare both Cplex and heuristic solutions.

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Conclusions

In this research, we proposed two aggregation methodologies for the PIDRP model with perishability consideration in order to reduce the size of the model for large instances. The first is based on a customised clustering technique based on K-mean algorithm. The second is a tailored aggregation technique that groups products based on ordering frequency. The key limitation of the original PIDRP model is the impractical computational time when the model is tested with large instances. This is due to a significant increase in the number of variables and constraints in the production and distribution model. As shown in the computational section, after applying the proposed aggregation methodologies, the size of problems can be reduced and the results can be obtained within time limit. As a future research direction, a possible extension of this research is to propose other heuristic algorithms which can improve the solving time and the efficiency of the model for dealing with large instances. The other possible extension is to examine different shelf life dependent loss functions for other types of perishable products such as fresh food (e.g., meat, fish, vegetables or bakery). To be able to apply the methodologies proposed in this research practically, future scope of the research includes implementing an effective disruption management which can be used to take into consideration of uncertainty of factors from the operation. This could involve developing stochastic models for the PIDRP model with perishability consideration and disruption management methodologies.

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Appendix This section provides description of the indices, parameters of the PIDRP (Ngoc and Nananukul 2016).

Indices and sets N

set of retailers; N = {1, …, R}

N0

set of retailers and production plant; N0 = N ∪ {0}

i, j

indices for retailers, where 0 represents the production plant, i, j ∈ N0

T

set of periods or days in planning horizon; T0 = T ∪ {0} and | T | = τ

t

index for periods or days, t ∈ T0

s

index for periods of delivery; S ∈ T

K

set of products, k = {1, …, K}

V

set of vehicles, v = 1{1, …, V}.

Parameters dik,t

demand of product k of retailer i on day t (units)

Q

capacity of vehicles (units)

k

C

production capacity of product k at production plant (units)

cij

cost of travelling from retailer i to retailer j ($)

k Ipmax

maximum inventory for product k at production plant (units)

Irik,max

maximum inventory for product k at retailer i (units)

fk

set up cost for product k when a production takes place ($)

cp

k

production cost for product k ($ per product)

k

holding cost for product k at inventory production plant ($ per unit per day)

hp

hrik

holding cost for product k at inventory retailer i ($ per unit per day)

MinRateik

minimum requirement shelf life of product k at retailer i (% of shelf life)

SLk

shelf life of product k (days) k

lost

maximum loss value of product k ($/ 1 unit of product per day).

Aggregation methodologies for perishability management

567

Decision variables xijvt

1 if vehicle v visits from node i to node j on day t; 0 otherwise (binary)

k qivt

quantity of product k is delivered to retailer i by vehicle v on day t (units)

k wijvt

quantity of product k on vehicle v on the route from i to j on day t (units)

ptk.s

quantity for product k produced on day t for the delivery on day s (units)

ztk

1 if a production for product k takes place on day t; 0 otherwise (binary)

Iptk

inventory level of product k at production plant on day t (units)

Iritk

inventory level of product k at retailer i on day t (units).

Objective function Min Z =

∑ ∑ ∑ ∑c

∗ xijvt +

∑ ∑ hp

+

ij

t∈T i∈N 0 j∈N 0 v∈V

+

k

∑ ∑ ∑∑

k

∗ ztk + cp k ∗ ptk, s )

t∈T0 \{τ } k∈K s∈T

∗ Iptk

t∈T0 \{ τ } k∈K

+

∑ ∑∑ ( f

∑ ∑∑ hr

i

k

∗ Iritk

(1)

t∈T0 \{ τ } i∈N k∈K

ptk, s *

t∈T0 s∈T , s > t i∈N k∈K

lost k ∗ ( s − t ) (1 − MinRateik ) ∗ SLk

Constraints Iptk = Iptk−1 +

∑ p − ∑∑ q k t .s

s∈T

Iritk = Irik,t −1 +

k ivt ,

∀t ∈ T , k ∈ K

(2)

i∈N v∈V

∑q

k ivt

− ditk , ∀i ∈ N , t ∈ T , k ∈ K

(3)

v∈V

∑

ptk.s ≤ C k ztk , ∀t ∈ T0 \ {τ}, k ∈ K

(4)

s∈T , s > t

ptk.s = 0, ∀t ∈ T0 , s ∈ T , k ∈ K : t > s : ( s − t ) > (1 − MinRateik ) ∗ SLk

(5)

∑∑ q

(6)

k ivt

≤ Iptk−1 , ∀t ∈ T , k ∈ K

i∈N v∈V k p0,1 ≥

∑(d

k i ,1

− Irik,0 ), ∀k ∈ K

(7)

∑

(8)

i∈N

∑∑ q

k ivs

i∈N v∈V

≤

t∈T0 , s > t

ptk, s , ∀k ∈ K , s ∈ T

568

M.H. Ngoc and N. Nananukul

∑x

=

∑x

≤ 1, ∀t ∈ T , v ∈ V

ijvt

i∈N 0 i≠ j

∑x

jivt ,

∀j ∈ N 0 , t ∈ T , v ∈ V

(9)

i∈N 0 i≠ j

0 jvt

(10)

j∈N

∑∑x

≤ 1, ∀j ∈ N , t ∈ T

(11)

xiivt = 0, ∀i ∈ N 0 , t ∈ T , v ∈ V

(12)

∑w

−

(13)

∑w

≤ Q ∗ xijvt , ∀i ∈ N , j ∈ N , t ∈ T , v ∈ V

ijvt

i∈N 0 v∈V

k ijvt

i∈N 0 i≠ j

k ijvt

∑w

k jivt

= q kjvt , ∀j ∈ N , t ∈ T , k ∈ K , v ∈ V

i∈N 0 i≠ j

(14)

k∈K

k 0 ≤ Iptk ≤ Ipmax , 0 ≤ Iritk ≤ Irik,max , ∀i ∈ N , k ∈ K , t ∈ T \{τ}

(15)

Ipτk = Iriτk = 0, ∀i ∈ N , k ∈ K

(16)

pτk, τ = 0, ∀k ∈ K

(17)

k wijv 0 = 0, ∀i ∈ N , j ∈ N , t ∈ T , v ∈ V , k ∈ K

(18)

wik0vt = 0, ∀i ∈ N , t ∈ T , v ∈ V , k ∈ K

(19)

k k xijvt ∈ {0, 1}, ztk ∈ {0, 1}, ptk, s , qivt , wijvt ≥ 0, ∀i ≠ j ∈ N 0 , t ∈ T0 , s ∈ T , v ∈ V

(20)

Equation (1) represents the total operating cost which includes the cost from the loss value of products. Constraint (2) determines the inventory level for product k on day t at the production plant. Constraint (3) shows the inventory level at each retailer. Constraint (4) represents production capacity at plant. Constraint (5) restricts the delivery days of products in order to satisfy freshness constraints. Constraint (6) ensures that only the inventory at the factory on day t – 1 can be used for delivery on day t. Constraint (7) initialises the production on day 0 in order to ensure the feasibility of demand of products on the day 1. Constraint (8) limits the delivered quantity of product k on day s by the inventory of product k produced on earlier days. Constraints (9)–(14) represent the constraints of the routing problem. Constraints (15) and (16) represent the bounds of the inventory at both the production plant and retailers’ sites. Constraint (17) forces the amount of production to be zero at the end of the planning horizon. Constraint (18) omits the delivery to retailers on day 0 and constraint (19) ensures that there is no product returned to the production plant. Finally, constraint (20) defines the types of decision variables.