16th Iranian International Industrial Engineering Conference

(1)

16

^th

Iranian International Industrial Engineering Conference 22-23 January2020

Alzahra University of Tehran

Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

Seyed Meysam Mousavib

, Mohammad Hasan Esmailia

a MSc Student, Department of Industrial Engineering, Faculty of Engineering, Shahed University, Tehran, Iran E-mail: [email protected]

b Associate Professor, Department of Industrial Engineering, Faculty of Engineering, Shahed University, Tehran, Iran E-mail:sm.mousavi @ shahed.ac.ir

* Corresponding author: Mohammad Hasan Esmaili

Abstract

In the recent decade, to demonstrate the importance of customer satisfaction can mention numbers of the service provider that attempt to differentiate themselves by satisfied their customers have witnessed high growth.

In this paper, some factors that increase retailers and customers satisfaction such as driver consistent services and delivering fresh products are considered in a perishable inventory routing problem (PIRP) under fuzzy uncertainty.

The proposed model is formulated as mixed-integer programming. Two types of consistent driver services are regarded for different types of customers, including particular and typical customers. To investigate the validity of the model, the problem is solved for two values of possibility and necessity that finally demonstrates the validity of the proposed model.

Keywords: Perishable products- driver consistent-freshness-inventory routing problem

1. Introduction andLiterature review

The continuity of a business life depends on many reasons, such as investigate business performance, update strategies according to the new situation, and customer satisfaction is some of them.

The importance of the customer's satisfaction role as one of the above main reasons can be business incentives to grow and be well known in the competitive market to increase its portion more and more.

It is undeniable that a series of essential activities need to be implemented to achieve these goals since these efforts can be better with new ideas and creativity. In the recent decade, the number of related activities to customer satisfaction growth high. This fact can be a witness to the role of customers importance in today's business world. Among practical intentions that may impact customer satisfaction is providing consistent services.

The consistent services introduced by Groër et al. [1] in a vehicle routing problem for the first time, in which the principal is implemented to minimize the difference between the start times of services in different periods. Although some applications of consistent services are home meal delivery, homecare, and post-delivery services [2], the potential of issues is enough to include more topics. Furthermore,

(2)

2

16

^th Iranian International Industrial Engineering Conference

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

businesses that perform consistent services being aware of their short-term costs, more advantages will be received for longer-term and in vast aspects.

There are types of consistent services, including drivers consistency, time consistency, and quantity consistency, that studied in the vehicle routing problem (VRP). Driver consistency is defined as assigning one (consistent) or a limited number of drivers (generalized consistent) to each customer during the planning horizon [1-3]. By satisfying customers' demand in this way, the better relationship between the service provider and customers can be formed.

In Figure 1, the difference between visiting customers with driver consistent (hard) and driver generalized consistent (soft) illustrated in a spectrum that its define on an interval.

Arrival time consistency as the other type guarantees that starting times of services in different periods is almost the same. This type of consistency is studied by Campelo et al. [4], Spliet and Gabor [5], Spliet and Desaulniers [6], and Kovacs et al. [3]. Third ones, quantity consistency, is studied by Coelho et al. [7] and state that the amounts of delivered products to retailers in different periods are consistent.

A broad piece of the family’s budget devoted to food; thus wasted food is wasted money. Fruits, vegetables, dairy products, meat, and seafood are perishable foods that they consumed fresh or store to eat at another time. There is some limitation to store these types of products, for example, physical restrictions are store out in the sun and heat, under the control moisture and temperature zone, and time restriction or perhaps the main one is the finite consuming time. Neglecting these factories causes generally decreases their quality, freshness, and occasion to fall their ultimate values.

Above mentioned statements, high financial turnover of food industries, the potential of the issue to improve more by applying optimization activities during their operation process from product or packing to delivering them to final customers, and decreasing amounts of spoilage products are enough reasons to pay more attention with more detail.

Synopsize, this article demonstrated the distribution network of perishable products in a Perishable Inventory Routing Problem (PIRP). In the PIRP context, integrating operational decisions such as replenishment scheduling, inventory control, and vehicle routing is defined as the main characteristic.

In this problem, a Distribution Center (DC) regulates the number of products based on the trade-off between the pros and cons of how many should receive and how many should be store to fulfill demand on other periods and have fewer costs. According to this context, DC response to retailers that not face with shortage and satisfy their demands in each period. Incurring an overall reduction in costs that regard to make decisions in this manner can observe in [8], reported as a win-win situation.

The researches related to the PIRP investigated different facets of the problem. Many pieces of research work on environmental activities like Soysal et al. [9] that study horizontal collaboration belonged to perishable products and greenhouse gas emission in the inventory routing problem with a case study and multi-products under uncertainty on demands. In this research, various essential

Driver consistent (hard) Driver generalized consistent (soft)

Figure 1 – The difference between visiting customers with driver consistent and driver generalized consistent

(3)

3

16

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

performance indicators such as emissions cost as fuel and spoilage, and driving time has investigated.

Dai et al. [10] studied a location inventory routing problem (LIRP) for perishable products that greenhouse gas emission and capacitated that consider in the fuzzy method. To minimize the total costs, they nominate two metaheuristics algorithms, such as hybrid genetic (HGA) and hybrid harmony search (HHS). Groups of researchers investigate a different approach to delivering products (e.g., [11]), which consider last-mile delivery with patterns in a two-echelon inventory-routing problem for perishable products. Because of chosen this distributing technique, there is no holding costs belong to the customer, and just depot has it.

A group of studies has been conducted on different ways to decrease the deterioration of perishable products and deliver them on a high level of quality and freshness. For example, Coelho and Laporte [12] envisaged three different selling policies that concluded fresh first (FF), old first (OF), and optimized priority (OP), that there are differences between them to FIFO, and LIFO politics in inventory management. Amorim et al. [13] impose a make to order strategy for decreasing the number of spoilage products in a perishable production and routing problem (PPRP). Their model includes two types of perishable products: the first one with fixed shelf-life and the second one with loose shelf-life. The summery of related articles regarded the PIRP is addressed in Table 1. Despite the advantages of considering driver consistency in increasing the retailer and customer satisfaction, it did not address in the studies related to PIRP.

Table 1. A summary of recent PIRP studies

Authors

Routing Inventory Perishability Uncertainty

Consistent services

Depot Retailer Shelf

life Freshness Fuzzy Stochastic Robust Driver Time Quantity

Dai et al. [10]    

Hiassat et al. [14]  

Soysal et al. [15]  

Coelho et al. [16]  

Rohmer et al. [11]  

Crama et al. [17]    

Coelho and Laporte

[12]   

Alkaabneh et al. [8]   

This study     

(4)

4

16

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

The main aim of this paper is to increase retailer and customer satisfaction in a perishable inventory routing problem (PIRP). For this reason, driver consistency is considered and retailers are categorized into two groups, particular and typical ones. Particular retailers must be visited with a driver in the planning period (consistent driver services) and typical ones with limited number of drivers (generalized consistent services). The special situation of some retailers is the main reason to apply separation in this way. It was the first practical step to receive our aim.

The other innovation of this paper formed based on a critical issue about perishable products that they should be delivered with a high level of quality and freshness to achieve a high level of satisfaction.

This idea is implemented by considering a given percentage of delivered fresh products to retailers that ensure the specific percentage of retailers sells devote to fresh products. Determining the exact value which ensures retailers' satisfaction is not possible because given amounts influenced by different factors and ideas, so this part of the problem faced with uncertain parameters. Fuzzy possibility and necessity approach used to cope with the uncertainty. In results, implementing the above methods cause to decrease cost satisfy retailers and customers, simultaneously. The problem has been represented in Figure 2.

Figure 2 – An illustrative example of the proposed Model.

Distribution

Center Vehicle Route

DC

Period 1

DC

Period 2

DC

Period 3

Typical Retailer

Particular Retailer

(5)

5

16

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

Figure 3. Triangular fuzzy number 1

2. Problem description and mathematical formulations

In this section, a mathematical model proposed for the PIRP by considering driver consistency.

Two different types of consistency are considered for different types of retailers including typical and particular ones. Particular retailers must be visited by only one driver in the planning horizon. However, generalized driver consistency is applied for typical retailers. Therefore, the objective function contains routing, holding inventory and consistency costs. In this paper, customer satisfaction is addressed by meeting customer demands by fresh products. The predefined amount of customers’ demands at each retailer must be met by fresh product to achieve an special level of customer satisfaction. However, the exact amount of demands that must be met by fresh products to achieve this level can not be determined.

Therefore, the proportion of demand in each retailer (

L 

_i ) is defined as fuzzy parameter. The percentage of delivered fresh products to ith customer

L 

_imodeled with triangular fuzzy numbers

( , , )

i i i i

L   L L L 

_where



i i i

L   L L

for each customer i, as shown in Figure 2.

The applied method causes to increase retailer satisfaction. The results of implementing this method effect on retailers which customers are beneficiary by receiving top fresh products from their retailers.

Driver consistent services as another way to increase retailer satisfaction mentioned in this model.

Because of the different characteristics of retailers, they divided into two groups, the particular and the typical. The primary assumptions of the proposed model are as follows:

 All kinds of customers should be served by at most one vehicle in each period.

 All demands have to be fulfilled.

 There is a limited inventory level at the distribution center to satisfy customer demands

 The delivered products from the distribution center to the customers are fresh.

 Products delivered to the customers before the spoilage time.

(6)

6

16

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

Table 2. Summary of notations

Sets

N Set of nodes including depot and customers

A Set of arcs

R Set of customers

Rparticular Set of customers with hard driver consistency

R^typical Set of customers with a soft consistency (generalized)

S Set of products age

F Set of fresh age

K Set of vehicles

T Set of time periods

Parameters

cij Routing cost on arc (i,j) ϵ A

hi Holding cost per unit of product for each customer iϵR

δ Consistency cost

rt Amount of product available at the distribution center in time period tϵT

dit The demand for customer iϵR in period tϵT

Ci Inventory capacity for customer iϵR

_i

L Triangular fuzzy percentage of delivering fresh product per each customer iϵR, i ( , , )

i i i

L  L L L





α Amount of service possibility

β Amount of service necessity Q The capacity of a vehicle

(7)

7

16

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

Decision variables

gt

I

i Amount of inventory with age g at customer iϵR in period tϵT

gt

D

i The amount of i^th customer demand met by-products with age g on period t

kt

q

i The amount of fresh product which transports from DC to i^th customer by vehicle k in period t

kt

y

i 1, if customer iϵN is served by vehicle kϵK in period tϵT, otherwise 0

x

ijkt 1, if vehicle kϵK traversed the route between customers i, jϵN in period tϵT, otherwise 0

k

Z

i 1, if customer iϵR^typical is assigned to vehicle k, otherwise 0

Z

max The maximum number of different drivers assigned to a particular customer

G

ikt The sequence number of customer iϵN in the route corresponding to vehicle k in period tϵT

The proposed mathematical model is formulated as follows:

max ( , )

gt

i i ij ijkt

i N g S t T i j A k K t T

Min h I c x Z

     

  

  

(1)

Subject to:

1, 1

gt g t gt

i i i

I  I

^{ }

 D

^g^^S ^{\ {0}}

,

i R ,tT (2)

0t kt 0t

i i i

k K

I q D



  

ⁱ^^{R t}^, ^^T ⁽³⁾

kt

i t

i R k K

q r

 

  

^{t T}^ ⁽⁴⁾

gt

it i

g S

d D



 

ⁱ^^{R t}^, ^^T ⁽⁵⁾

gt

i i

g S

I C



 

ⁱ^^{R t}^, ^^T ⁽⁶⁾

(8)

8

16

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

, 1

kt g t

i i i

k K g S

q C I

^

 

 

 

ⁱ^^{R t}^, ^^T ⁽⁷⁾

kt kt

i i i

q  C y

ⁱ^^{R k K t T}^, ^ ^, ^ ⁽⁸⁾



^gt

it i

t T g F t T

d L

i

D

  

  

i R ⁽⁹⁾

0

kt kt

i i R

q Q y



 

^k^^{K t T}^, ^ (10)

kt

1

i k K

y



 

ⁱ^^{R t}^, ^^T ⁽¹¹⁾

k kt

i i

Z  y

ⁱ^^{R k K t T}^, ^ ^, ^ ⁽¹²⁾

k

1

i k K

Z



 

_i__R^particular ⁽¹³⁾

max k

i k K

Z Z



 

_i__R^typical (14)

 

\ \

ijkt jikt

j N i j N i

x x

 

  

ⁱ^^{N k}^, ^^{K t}^, ^^T (15)

 

\

ijkt ikt j N i

x v



  iR k, K t, T (16)

jkt ikt ijkt

G  G  Nx  R

ⁱ^^{R j N k}^, ^ ^, ^^{K t T}^, ^ (17)

0

1





^ikt



i R

x

_k__{K t T}_, _ ₍₁₈₎

, , 0

gt gt kt

i

D

i

q

i

I 

ⁱ^^{R g S k}^, ^ ^, ^^{K t T}^, ^ ⁽¹⁹⁾

{0,1}

kt

y

i



ⁱ^^{N k}^, ^^{K t}^, ^^T ⁽²⁰⁾

{0,1}

k

Z

i



ⁱ^^{R k}^, ^^{K t T}^, ^ ⁽²¹⁾

max

0

Z 

⁽²²⁾

{0,1}

x

ijkt



^{( , )}^{i j} ^^{A k}^, ^^{K t T}^,^ ⁽²³⁾

The objective function (1) consists of three terms, which are inventory, routing, and consistency costs, respectively. Constraints (2) and (3) guarantee the inventory flow balance at the customers for products of different ages. Constraint (4) ensures that each customer receives fresh products from DC.

(9)

9

16

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

Constraint (5) states that the demand of each customer is met with products of different ages at each period. Constraint (6) guarantees the inventory capacity of each customer at each period. Constraint (7) ensures that the amount of delivered product to ith customer is at most equal to empty space. Constraint (8) states the relationship between the decision variables q and y. Constraint (9) guarantees that the special amount of each customer’s demand should be met with fresh products. Constraint (10) ensures that vehicle capacity should be respected. Constraint (11) guarantees that each customer should be visited by at most one vehicle in each period. Vehicle assignment to a customer in each period regulated by constraint (13). Constraints (14) and (15) relate to the consistent and generalized types of driver consistency for their specific customers, respectively. Flow control constraints represented with (16) and (17). Constraint (18) is related to sub-tour elimination. Constraint (19) demonstrates that at most, one visit in each period can perform by a vehicle. Finally, domain decision variables detected by constraints (20) – (23).

To cope with uncertainty, possibility theory is applied, which is a mathematical theory for relating to certain forms. Two concepts utilized by possibility theory, the possibility, and the necessity. The amount of possibility α and necessity β defined with a fuzzy triangular function membership. Jamison and Lodwick [18], Werners [19], and Werners and Kondratenko [20] investigate the fuzzy number as a method of illustrating the uncertainty in a given value quantity by defining a possibility for the quantity.

To model delivering fresh products and based on requests which each customer order, two fuzzy approaches are applied. The decision-maker set a certain degree α ϵ [0,1] in advance and define the below:

_it ⁱ _i^gt

t T g F t T

Pos d L d

  

  

 

 

    

^{i R}^ ^; ^^[0,1] (24)

The other approach is to consider the firm condition, which is a specific degree of necessity β defined to:

gt

it i i

t T g F t T

Nec d L d

  

  

 

 

    

i R ; [0,1] (25) The preconditions (24) and (25) for mentioned constraint can be changed in this way:

gt 0

i it i

g F t T t T

Pos d d L

  

  

  

 

    

^{i R}^ ^; ^^[0,1] (26)

0

gt

i it i

g F t T t T

Nec d d L

  

 

 

 

 

    

^{i R}^ ^; ^^[0,1] (27)

The possibility Pos (serve

L 

i) and necessity Nec(serve

L 

i) which ensure each retailer receive a percentage of the fresh product during their demands period represented by a mathematical model in below:

(10)

10

16

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

(



_i)

Nec Serve

L

 _i ^gt _i

it i it

t T f F t T t T

d L d d L

   

 

   

 (  )

gt i

i it

f F t T t T

i i

it t T

d d L

d L L

  





 



1, ^gt _i

i it

f F t T t T

d d L

  







( )

gt

i

i it

f F t T t T

i i

it t T

d d L

d L L

  





 



^gt _i

i

it i it

t T f F t T t T

d L d d L

   

 

  

(28)

0, _i^gt _it _i

f F t T t T

d d L

  

 





1, i^gt it i

f F t T t T

d d L

  







(29)

0, ^gt _i

i it

f F t T t T

d d L

  







Werners and Drawe [21] stated the following relation between possibility and necessity:

 

( _i) 1 ( _i) 0

Pos serve

L

 Nes serve

L

 (30)

For α,β >0 mentioned constraints are modeled by as crisp equivalents for the fuzzy constraint (9), respectively:

(



_i)

Pos serve

L

   i i it ^gti

t T g F t T

( L (1 ) L ) d d



  

    

 

ⁱ^^R^^[0,1] (31) (



_i)

Pos Serve

L



(11)

11

16

Title: Perishable inventory routing problem with consistent driver services and deliver fresh products under fuzzy uncertainty

Figure 4. Represent delivering products with α=1, β=1 in a). first period b). second period c). third period

b c

: :

a

(



_i)

Nec serve

L

   _i _i _it ^gt_i

t T g F t T

( L (1 )L ) d ˆ d

  

    

 

ⁱ^^R^^[0,1] (32) Now the problem is solved with mentioned objective function and constraints (2-8), (10-23) and (31) and (32).

3. Statistics and Data Analysis

The represented mathematical model is solved by Julia Software and accomplished on an Intel Core i7 with 1.8 GHz CPU and 8 GB of RAM. In order to evaluate the proposed model, ten customers have considered which six of them are particular, and the others are typical. Four vehicles with the same capacity and three-time periods are considered. The values of parameters generated from Coelho and Laporte [12].

The obtained routes in different periods under different values of possibility and necessity are illustrated in Figures 3 and 4. Figure 3 represents the problem with α=0.7, β=0 in three periods. It is clear that based on the problem assumptions in the first period, all of the retailers received fresh products and satisfied their demands. In the second period, the difference between the two given values of possibility and necessity will be shown. The results reported 11867 units for routing costs and 105.25 unit for inventory costs. Represented results in figure 4 which solved for α=0.7, β=0 reported 9514 unit for routing and 704.2 unit to inventory costs.

In accordance, the existing difference of 598.95 units in inventory costs showed the efficiency of the proposed model, which happens according to the different dedicated degree for α, β.

The results such as routing and inventory costs for different vehicle capacity and different values of possibility and necessity reported in Table 3. According to them, increasing α, β values cause to increase routing costs because the insurance to deliver fresh product and model avoid to store more products. By increasing vehicle capacities, this growth trend is slower, that enough or maybe residual capacity can be the reason.