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Stochastic Inventory Routing Problem Under B2C E-commerce

Xie Binglei,

Traffic Engineering Research Center,

Harbin Institute of Technology Shenzhen

Graduate School, 518055, Shenzhen,

Guangdong China

E-mail: [email protected]

An Shi, Wang Jian

School of Management,

Harbin Institute of Technology, 150001

Harbin, Heilongjiang, China E-mail:[email protected]

[email protected]

Abstract

In traditional B2C, customers’ stocks are usually

controlled by themselves, while the supplier needs

only to meet requests of customers by sending right commodity on time. The operation mode results in

disadvantages of hindering saving of cost and

improvement of efficiency. Being one of the key

contents in vendor managed inventory, inventory routing problem will help supplier to lower operation

cost, improve efficiency and increase satisfaction of

customers. In the paper, a heuristic algorithm is

designed to solve stochastic inventory routing problem. In the algorithm, followed by converting

stochastic demands of customers to regular demands,

distribution period of every customer is calculated by

the modified EOQ. Finally, approximate solution of stochastic inventory routing problem can be obtained

by solving periodic vehicle routing problem.

Keywords: inventory routing problem; logistics;

vendor management inventory; B2C

1. Introduction

In traditional B2C, customers’ stocks are usually

controlled by themselves, while the supplier needs only

to meet requests of customers by sending right

commodity on time. The disadvantages of the mode is

the following:

(1)Broken balance in logistics distribution makes

reduction of distribution cost more difficult.

(2)Inventory of supplier and factory is hard to

control.

To ameliorate the situation, the method of vendor

managed inventory (VMI) emerges. Being one of the

Project 70203003 and 70271022 supported by National Natural

Science Foundation of China

key contents in VMI, inventory routing problem (IRP)

aimed at optimizing total cost of inventory and cost,

involves calculating distribution frequency and

distribution quantity, and designing routes. IRP

emphasizes coordination between inventory and

transportation, and devotes itself to enhance function

of whole logistics system. IRP will be helpful to

decrease logistics costs of enterprise, promote

operation efficiency, and improve service level.

IRP was put forward in 1970’s firstly, which a

special version was solved by Beltreami and Bodin

with a very simple method. To solve IRP, different

methods were put forward, which can be classified into

three categories.

(1) Dividing all customers to several subsets firstly

according to a certain rule, then optimizing sub-

problem in every subset, finally integrating all subsets

to optimize wholly.

(2) Decomposing IRP to inventory problem and

vehicle routing problem, then solving them separately,

at last integrating the problems together to optimize.

(3) Solving IRP with short term followed by

simplifying long scheduling term to short term. But in

the process of simplification, long term cost is possible

to mismatch with the sum of short term costs.

Along with thorough investigation in IRP, stochastic

factors were involved in the course of decision making.

Dror, Ball and Golden studied a IRP, in which every

customer owned a fuel tank with a certain capacity,

and daily consumption of fuel was assumed to a

normal independent stochastic variable[1]. The

problem was expressed as a integer programming

model with two stages, and a method based on multi-

stage decomposition was provided. Henceforth, several

works around the result appeared to solve IRP with

stochastic demand. Dror and Trudeau studied a IRP

with stochastic demands by assessing influence of

present fund value on whole system[2]. Constructing

new incremental cost, Berman with Larson brought out

Proceedings of the 2005 IEEE International Conference on e-Business Engineering (ICEBE’05) 0-7695-2430-3/05 $20.00 © 2005 IEEE

a dynamic programming method to solve stochastic

and dynamic IRP[3]. Anily and Bramel studied FPP in

IRP with stochastic demands, and thought that the

method can not guarantee customers served uniformly

in all regions[4].

In the paper, a heuristic algorithm to address IRP

with stochastic demand is proposed from tactic level.

2. Description of Stochastic IRP

The stochastic IRP in the paper can be stated as, to

find distribution period, distribution quantity, and a set

of routes at minimum total cost, assuming that a

supplier with boundless commodity is responsible for

maintaining inventory of customers, whose daily

usages are independent stochastic variables.

The problem can be defined in a complete graph

( , , )G V A D= , where

},,,{ 10 nvvvV = , a set of nodes with node 0v

denoting the depot of supplier and nodes 1v , ,,2v nv

corresponding to the customers;

},,:),{( VvvjivvA jiji ∈≠= , a set of arcs

joining the nodes;

)( ijdD = , denoting the travel distance (cost)

between node iv and jv .

Starting and ending at the depot, m identical

vehicles with known capacity Q and daily duration

H service all customers. The daily usage of customer

i with maximal inventory level iC is assumed as a

stochastic variable iξ with mean iµ and standard

deviation iσ . Because the usage which customer

consumes everyday changes stochastically, stockout is

inevitable to happen. At the moment, a extra vehicle

will be dispatched to replenish inventory of the

customer. While decreasing satisfaction of customers,

stockout would increase transportation cost. In the

paper, the maximal stockout probability of customer i

is assumed to be ip in a planning period. Since

storage cost is a linear function with storage quantity

and storage duration, unit storage fee is assumed to be

h .

3 Handling Stochastic Demand and

Obtaining Distribution Period

To simplify the problem, an approximate method is

applied to handle stochastic demand. Assuming that max

iT is the maximal distribution period which

guarantees stockout probability of customer i no more

than ip , then we have

i

T

k

iki pCP

i

≤>=

}{

max

1

ξ (1)

!n formula (1), kiξ is a stochastic variable expressing

usage of customer i in the k -th day.

According to central limit theorem, if iy denotes a

stochastic variable obeying standard normal

distribution, formula (1) can be transformed as

i

ii

iiii p

T

TCyP ≤

−> }{

2max

max

σ

µ (2)

When )(xΦ is a distribution function of standard

normal distribution, formula (2) is equivalent to

i

i

iii pT

TC−≥

−Φ 1)(

2max

max

σ

µ (3)

After maxiT is determined, the maximal allowable

usage of customer i is expressed by max/ iii TCe = (4)

In a distribution period, if usage of customer i is

equal to ie , the stock will fall to zero at the end of the

period. Through the transformation which avoids

solving complex stochastic model, iξ can be

substituted by ie .

In IRP, if total cost is sum of fixed cost (FC),

variable cost (VC), cost of storage, and cost of

stockout (OC), total average cost servicing customer i

in unit time is

iii

iiii heT

T

OCFCVCAC

2

1)(+

++= (5)

Assuming that capacity of vehicle is bigger than

total usage of one customer in distribution period, the

distribution period of customer i can be obtained by

21

)(2 ++=

i

iiiEOQi

he

OCFCVCT (6)

But the above assumption is untenable due to

stochastic demand of customer. Let *iT to be a feasible

distribution period, which can be obtained by

}/,min{*i

EOQii eQTT = (7)

If a stockout of customer i results in a loss iW , then

we have

iii WpOC = (8)

Assuming iVC associated with the visiting cost of

vehicle, the value of iVC is much more difficult to


identify than iOC . Even though the tour in which

customer i is visited has been determined, how to

assign routing cost of a whole tour is still a hard

problem.

Firstly, the saving of conveyance in retail rather than

conveyance directly is

)(2

1

0 nTSPdsaving

n

i

i −==

(9)

in which, )(nTSP is routing cost of tour visiting depot

and all n customers.

If the saving is assigned to customers according to

the weight

1

/n

i i j

j

w e e=

= (10)

the routing cost assigned to customer i is

savingwdVC iii −= 02 (11)

After all variables in formula (6) are expressed, we

can calculate the distribution period of every customer

simply. It is noted that, distribution periods of all customers are not bound to be integers. Let

][EOQ

ii TT = , in which ][⋅ denotes the maximal integer

no more than EOQ

iT . To handle non-integral

distribution period, interval between two replenishing

inventory of customer i is either iT or 1+iT . Though

the above method approximates distribution period, it

will improve generality and practicability of EOQ.

4 Solving Periodic VRP

After the distribution period of customer is determined, we can observe that the problem is similar

to periodic VRP. Periodic VRP is also a decision-

making problem with two phases, in which daily

service sets of customers in planning period must be determined firstly, daily routes are obtained by solving

general VRP. If feasible routes in a planning phase

including T days is known, a heuristic algorithm to

solve periodic VRP is designed, which details are described as the following.

Step1: Determining a sequence of customers based on

heuristic ordering rule. Let U be a sequence, based on

the rule of first arranging customers by non-decreasing

iT , then arranging customers with same iT by non-

increasing ie .

Step2: Planning dates of every customer replenished in

U successively. At the beginning, we determine

distribution dates of the first customer in U ; then draw

customer from U in order, and guaranteeing route

feasibility, distribution dates of customer is determined

according to minimal inserting cost. The operation is

repeated until all customer are drawn from U .

Let the set of customers serviced in the t -th day be

tR , distribution dates of customer i will involve

several combinations, which constitute set iS . The rule

of minimal inserting cost is to arrange customer i in

all dates of a certain combination iSr ∈ at minimal

incremental cost. The incremental cost inserting

customer i to tR is the minimal value caused by

inserting the customer to a appropriate position in a

certain route visited in the t -th day.

Step3: Obtaining daily routes of the planning period.

Based on the algorithm in [5], we can obtain daily

routes.

Step4: Improving the value of objective by modulating distribution dates of customers in the planning period.

If assigning customer i to all dates in a new

combination r′ drawn from iS can result in saving

cost, the former combination r is substituted by r′ .Though the above steps can obtain a plan to

replenish inventory and a set of routes, total logistics

cost will be decreased in Step5 and Step6 further.

Step5:Reassigning routing cost to customers by

formula (9)-(11).

Step6: Calculating new distribution period of every

customer by formula (6).

Step7: Executing Step1-Step4. If the modified

distribution period can save total logistics cost above

α %, Step5 will succeed; otherwise, the algorithm is

terminated.

5 Computational Results

Given a planning period 2 weeks, 40 customers disperse stochastically and consistently in region of

]100,0[]100,0[ × , in which the coordinate of supplier is

assumed to be (50,50). While the maximal storage iC

of customer i is chosen within the limits between 30

and 60, the average daily usage of that is chosen within

the limit between 10 and 20. Because the inventory of

customer i must be replenished immediately once

stockout happens, cost iW caused by a stockout is

approximated as the sum of activating cost F of

vehicle and routing cost ),0( iRC . Assuming that

),0( iRC is a linear function of id0 , we can substitute

),0( iRC with idr 0.2 , in which r is the cost of unit

kilometer. The acceptable maximal probability of

stockout in a distribution period is equal to 0.02. The

maximal capacity of vehicle is 200, average speed of


vehicle is 50km/h, and the maximal duration of a

vehicle is 8 hours.

Figure 1. Iterations of the Heuristic Algorithm

From Fig. 1, we can observe that, in the early iteration, the value of objective function is improved

obviously, but with development of iteration, the

improvement is decelerated gradually until the

algorithm is terminated. In the paper, the objective function of IRP consists of

four parts, which is related closely to r F and h .

The experiments show that different choice of r F

and h will affect all four costs, and the results are

demonstrated in table1.

Table 1 Results of Different Combinations Including r, F and h

r F h FC RC SC OC

1.5 160.71 243.80 334.67 34.82 50

2 167.85 249.78 426.68 34.82

1.5 314.28 239.45 352.64 48.16

1

100

2 321.42 251.75 438.64 48.16

1.5 157.14 355.48 337.87 45.57 50

2 160.71 364.18 420.16 45.57

1.5 300.00 351.03 360.15 58.90

1.5

100

2 307.14 369.52 431.25 58.90

The phenomena can be explained as the following.

Since stockout cost is only related to r and F ,

increasing h and maintaining r and F will not make

stockout cost increasing. Because increasing F and

maintaining r and h means that the curtailed

distribution period of customers may result in

increasing times of serving customers in a planning period, fixed cost and variable cost will increase.

Increasing r and maintaining F and h may result in

prolonging of distribution period of customers, that

means decreasing times of serving customers in the planning period and .increasing inventory cost.

From Table 1, we can observe the following

phenomena, increasing h and maintaining r and F ,

stockout cost remains constantly, storage cost raises

remarkably, while both fixed cost and variable cost

raise slightly; increasing F and maintaining r and h ,

fixed cost raises remarkably, while stockout cost and

inventory cost show tendency to raise; increasing r

and maintaining F and h , variable cost, storage cost

and stockout cost will increase, while fixed cost will

decrease.

6 Conclusions

Since inventory control problem and routing

problem are two important and interrelated closely

items in logistic management, integrating them will reduce further logistic costs in the process of electronic

commerce. The IRP has been proven to be NP-hard,

leading the majority of researchers to focus on

heuristic and meta-heuristic method to derive approximate solutions.

In the paper, minimizing total cost of inventory and

transportation, we propose a heuristic algorithm to

solve a stochastic IRP. The algorithm is simple, practical and effective.

Future work will be conducted to improve the

proposed algorithm. Alternative heuristic rules and

meta-heuristic features will be examined to enhance efficiency and reducing computational cost.

References [1] Dror M., Ball M., Golden B.. A computationaal

comparison of algorithms for the inventory routing

problem. Annals of Operations Research. 1985, 1: 3-23

[2] Dror M., Levy L.. A vehicle routing improvement

algorithm comparison of a greedy and a matching

implementation for inventory routing. Computers and

Operations Research. 1986, 13: 33-45

[3] Berman O., Larson R.. Deliveries in an inventory/

routing problem using stochastic dynamic programming.

Transportation Science. 2001, 35: 192-213

[4] Anily, Shoshaha. Bramel, Julien A probabilistic

analysis of a fixed partition policy for the inventory-

routing problem. Naval Research Logistics. 2004, 51:

925-948.

[5] Li J., Xie B., Guo Y.. Genetic Algorithm for Vehicle

Scheduling -Problem with Non-full Load. Systems

Engineering—Theory Methodology Application. 2000,

19(3): 235-23

Times of iteration

Total

cost


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