Tackling Demand Stochasticity by Redistribution among Retailers in a

Two-stage Distribution System

Benedikt De Vos and Birger Raa

Department of Industrial Systems Engineering and Product Design, Ghent University,

Technologiepark 903, 9000 Ghent, Belgium

Keywords:

Supply Chain Collaboration, Lateral Transshipments, Inventory Control, Inventory-routing.

Abstract:

This paper considers a two-stage distribution system consisting of a supplier supplying several retailers with

stochastic demand rates. Replenishments by the supplier happen cyclically, based on average retailer demand

rates. Hence, the considered distribution problem between the suppliers and its retailers is a Cyclic Inventory

Routing Problem (CIRP), which we solve with a state-of-the-art heuristic solution method. To cope with the

demand stochasticity, lateral transshipments can happen between retailers in each time period. We propose

a redistribution policy that determines the quantities being redistributed through lateral transshipments based

on the desired level of customer service the retailers want to reach (using the desired ﬁll rate) and their actual

inventory levels. The cost of the daily redistribution is determined by solving a Travelling Salesman Problem

(TSP) covering the participating retailers. The potential beneﬁts of the collaboration among retailers are

analyzed by comparing the distribution costs, the redistribution costs, the inventory holding costs and the

costs of lost sales at the retailers in different scenarios.

1 INTRODUCTION

Routing and inventory control often represent a large

part of a company’s operational costs. Traditionally,

operations management focused on optimizing the in-

ternal operations of a company. However, researchers

and companies realized that operational performance

does not only depend on their own decisions, but also

on the decisions made by other players in the sup-

ply chain. Hence, focus shifted from single-ﬁrm op-

timization to a more global ’supply chain manage-

ment’ perspective to create higher operational efﬁ-

ciency. (Mentzer et al. (2001); Power (2005))

In this paper, a combination of vertical (i.e.,

among players on subsequent levels of the supply

chain) and horizontal (i.e., among players on the same

level of the supply chain) collaboration is introduced

in a distribution network. We consider a two-echelon

supply chain consisting of a supplier and his retailers.

Vertical collaboration between the supplier and his

retailers is established through Vendor-Managed In-

ventory (VMI). The retailers share demand data with

the supplier and hand over the responsibility for the

replenishment timing and quantity to the supplier.

This way, the supplier can coordinate retailer replen-

ishments better and design more efﬁcient routes. The

supplier is then faced with an integrated inventory and

vehicle routing problem known as the Cyclic Inven-

tory Routing Problem (CIRP). VMI has been exten-

sively studied and comprehensive overviews of the

existing literature are given by Moin and Salhi (2007),

Andersson (2010) and Coelho (2013).

Horizontal collaboration is established among the

retailers through lateral transshipments, which en-

ables them to cope with demand uncertainty. Lat-

eral transshipments were deﬁned by Tagaras (1999) as

”the redistribution of stock from retailers with stock

on hand to retailers that cannot meet customer de-

mands or retailers that expect signiﬁcant losses due to

high risk”. Lateral transshipments can lead to service

improvement by preventing stockouts and to reduced

inventory holding costs. Previous research made a

distinction in redistribution based on the timing of the

lateral transshipments. Redistribution can take place

at predetermined times before all demand is realized

(i.e., proactive transshipments), or at any time to re-

spond to (potential) stockouts (i.e., reactive transship-

ments). Paterson (2011) gives a review on inventory

models with lateral transshipments.

A large part of the literature on lateral transship-

ments focuses solely on inventory control, assumes a

periodic inventory review policy, only considers one

232

Vos, B. and Raa, B.

Tackling Demand Stochasticity by Redistribution among Retailers in a Two-stage Distribution System.

DOI: 10.5220/0006623202320238

In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems (ICORES 2018), pages 232-238

ISBN: 978-989-758-285-1

moment for redistribution within the order cycle or

lets lateral transshipments take place between only a

limited number of inventory points (Paterson et al.

(2011)). This paper contributes to this literature by

combining routing and inventory control at the retail-

ers through lateral transshipments. We develop an op-

erational model to determine the costs of redistribu-

tion and to analyze the beneﬁts.

2 PROBLEM DESCRIPTION

We consider a two-echelon distribution network con-

sisting of a single supplier S who delivers a single

product to a retailer set I. The supplier cyclically re-

plenishes the retailers under a VMI system, based on

the retailers’ mean demand rates d

. The actual de-

mand rates at the retailers a

i,t

(i.e., the demand rate at

retailer i in time period t) are assumed to be stochastic

with a known probability distribution.

To cope with demand uncertainty, lateral trans-

shipments among the retailers are possible. The quan-

tities that are redistributed are determined by the re-

distribution policy. This policy is aimed at achiev-

ing a target service level at all retailers. Hence, the

redistributed quantities are based on the retailers’ in-

ventory levels and their ﬁll rates. Since redistribution

will only involve relatively small quantities of goods

(compared to the supplier deliveries), we assume that

redistribution can be performed by a single vehicle.

We develop an operational solution method to de-

sign a cyclic distribution plan for the supplier to re-

plenish the retailers and a redistribution policy among

the retailers that realizes the potential beneﬁts result-

ing from lateral transshipments. The solution method

determines the distribution cost, the cost of lateral

transshipments, the inventory holding cost at the re-

tailers and the cost of lost sales. These costs are com-

pared between the baseline system in which no lat-

eral transshipments occur and the collaborative sys-

tem with lateral transshipments. Given that redistri-

bution brings along additional costs, these have to be

balanced with the potential beneﬁts of lower inven-

tory holding costs and lower lost sales.

3 DISTRIBUTION SOLUTION

ALGORITHM FOR THE CIRP

The distribution routes of the supplier are designed

using the two-phase route and ﬂeet design CIRP

heuristic (Raa and Dullaert (2017)).

In the ﬁrst phase of the CIRP heuristic, the

vehicle routes are designed using a construct-and-

improvement heuristic. An initial solution is con-

structed using a savings-based heuristic based on

the Clarke-and-Wright heuristic (Clarke and Wright

(1964)), which was adapted for the CIRP. The initial

solution is improved by using local search operators

(2-opt, relocate and exchange operators). Each local

search operator is reiterated until no more improve-

ment is found before moving on the the next operator.

A best-accept strategy is used per operator.

The route cycle times are chosen such that the dis-

tribution and inventory holding costs are minimized.

Every time a route r is made, it incurs a distribution

cost F

consisting of a vehicle loading and dispatch

cost, the delivery costs at the subset I

(i.e., the set

of retailers served by route r), and the transportation

costs to drive the route (i.e., the TSP tour visiting the

depot and the retailers in the route). To reduce dis-

tribution costs, the time between iterations of route r,

i.e. the cycle time T

, should be as long as possible.

Contrary to the distribution costs, the inventory

holding costs at the retailers that are covered by a

route increases with the route’s cycle time. The de-

livery quantity of retailer i is d

. Hence, the average

inventory level is

and the inventory holding cost

rate is

, with η

the holding cost per item per pe-

riod. The total cost rate TC

of route r thus varies with

the cycle time T

+ T

∑

i∈I

(1)

The cycle time is however restricted by the vehi-

cle capacity κ and the storage capacity of the retailers

, resulting in a maximum cycle time T

max,r

(cfr. for-

mula 2). The retailers in route r are visited each T

days, so they receive a delivery of d

items. This

delivery quantity should be lower than the storage ca-

pacity of the retailers κ

. Hence, the cycle time must

be less than or equal to min

i∈r

. Further, when mak-

ing the route r, the vehicle is loaded with

∑

i∈r

items. This load should be lower than the vehicle ca-

pacity κ, so the cycle time T

cannot be higher than

∑

i∈r

max,r



min



∑

i∈r

, min

i∈r



(2)

There is an optimal cycle time T

∗

that balances the

holding cost and the route cost and hence minimizes

the cost rate.

∗

= min

∑

i∈r

, T

max,r

) (3)

Tackling Demand Stochasticity by Redistribution among Retailers in a Two-stage Distribution System

233

The total distribution cost rate equals the sum of

all the routes r ∈ R that the supplier has to perform to

replenish all his retailers I:

∑

r∈R

(4)

The ﬁrst phase of the CIRP algorithm thus divides

the set of retailers into subsets that are each covered

by a separate route and for which the route cycle time

is chosen to minimize the cost rate.

The second phase of the CIRP algorithm assigns

the routes to vehicles and to speciﬁc periods using a

construct-and-improvement heuristic. The sequence

of the routes stays the same, but their cycle times can

be adjusted to minimize the required number of vehi-

cles and thus the ﬂeet costs.

The construction heuristic is a best-ﬁt insertion

heuristic that inserts the routes into the schedule in

such a way that the cumulative remaining time of the

vehicles to which the routes are assigned is minimal.

The routes are inserted in the schedule using two cy-

cle time selection rules. Firstly, the routes are inserted

with cycle times as close as possible to their optimal

cycle time (resulting in minimal route cost rates). Or

secondly, they are inserted with cycle times as close

as possible to their maximal cycle times (resulting in

minimal ﬂeet cost rate). This results in two initial

schedules that are passed on to the improvement step.

In the improvement step, two local search operators

are applied, namely removing any single route from

the schedule and reinserting it in the cheapest possi-

ble spot, or removing all routes made by the vehicle

with the lowest utilization from the schedule en rein-

serting them in the cheapest possible way. To escape

a potential local optimum, the schedule that results

from the improvement step is scrambled by shufﬂing

route allocations. Local search is then repeated on the

scrambled solution. A predeﬁned number of scram-

bles is set as a stopping criterion for the improvement

step.

The two-phase route and ﬂeet design approach is

then reiterated within a metaheuristic framework (Raa

and Dullaert (2017)).

4 REDISTRIBUTION STRATEGY

AMONG RETAILERS

Once the routes and their allocation to vehicles are

known, the supplier can execute his distribution plan.

Then, the actual daily demand values a

i,t

are observed

for all retailers in each period. These demand rates

are generated using their known probability distribu-

tion. Since these demand values deviate from the av-

erage demand values, the actual inventory levels and

the resulting inventory holding costs will differ from

the inventory holding costs resulting from the CIRP

solution and have to be recalculated. Hence, in order

to know the actually incurred distribution cost rate,

the inventory holding costs (T

∑

i∈I

) have to be

subtracted from the routes’ cost rates TC

Further, inventory levels may be lower than antici-

pated (after periods with higher than average demand)

and the risk of stockout may occur. This is when re-

distribution through lateral shipments is activated.

Given that the cycle times of the routes can differ,

the time periods in which the retailers are replenished

will also differ from one retailer to another. Hence,

we allow redistribution in each time period across a

certain planning horizon.

At the start of period t, every retailer i has an in-

ventory level SI

i,t

. The ending inventory of retailer i

in that period EI

i,t

is calculated by subtracting the de-

mand in that period from the start inventory (= SI

i,t

). Redistribution is assumed to happen ’overnight’

between the periods, so the starting inventory level

of the next period also includes any lateral transship-

ments.

The redistribution quantities are thus determined

based on the ending inventory level of the retailers.

The retailers are divided into three different groups

in each time period. The retailers whose ending in-

ventory is too low to reach a certain desired service

level until their next delivery from the supplier, are

called receivers (1). Retailers whose ending inventory

is more than necessary to reach their desired service

level are called contributors (2). Finally, there are the

retailers that do not want to participate to the redis-

tribution in a speciﬁc period (3). The group to which

a retailer belongs can differ from one period to the

next, since his ending inventory will change accord-

ing to his incurred demand rate, redistributed quanti-

ties from the previous period and possible deliveries

from the supplier.

The desired service level is expressed by the ﬁll

rate (i.e., the fraction of demand ﬁlled from items

available in inventory). For each time period, three

inventory levels are determined for each retailer, the

critical inventory level CI

i,t

, the threshold inventory

level TI

i,t

and the optimal inventory level OI

i,t

. The

critical inventory level CI

i,t

of a retailer is the inven-

tory level below which he wants to receive goods from

other retailers. So, a retailer is a receiver in time pe-

riod t if EI

i,t

<CI

i,t

. The threshold inventory level TI

i,t

is the inventory level above which he is willing to

share goods with other retailers. Hence, a retailer is

a contributor in time period t if EI

i,t

>TI

i,t

. The opti-

mal inventory level OI

i,t

of a retailer is the inventory

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

234

level up to which he wants to receive items from the

other retailers when his inventory has fallen below his

critical inventory level. If the inventory of retailer i in

time period t lies in the interval [CI

i,t

;TI

i,t

], he will not

participate in the redistribution in time period t. The

ﬁll rates determining the critical inventory level and

the threshold inventory level of a retailer can change

over the time periods, depending on how many time

periods there are left before he receives his next order

from the supplier (i.e., the lead time L).

Based on the critical inventory level, the optimal

inventory level and the threshold inventory level of a

retailer, the number of goods he wants to contribute

i,t

) or receive (r

i,t

) in a time period t can be deter-

mined.

i,t

= EI

i,t

− T I

i,t

(5)

i,t

= OI

i,t

− EI

i,t

(6)

When the contributor amount (

∑

i,t

) equals the

receiver amount (

∑

i,t

), they can be optimally redis-

tributed. However, these amounts will often be dif-

ferent. When the contributor amount is smaller than

the receiver amount (

∑

i,t

∑

i,t

), the receivers will

not all be able to reach their optimal inventory level,

so an appropriate decision rule is required to divide

the contributed items over the receivers. Conversely,

when

∑

i,t

∑

i,t

, we need to decide what contribu-

tors will contribute which quantities.

These decisions are also taken based on the ﬁll

rates of the receivers and the contributors. When

the contributors want to contribute more items than

receivers want to receive (

∑

i,t

∑

i,t

), the con-

tributed items are collected from the contributors in

such a way that they end up with a new inventory

level that results in the same ﬁll rate for the contribu-

tors (which is still above the required ﬁll rate). When

the receivers want to receive more items than the con-

tributors want to contribute, the contributed items are

allocated over the receivers in such a way they all end

up with a new inventory level with the same ﬁll rate

(which is still below the required ﬁll rate). The num-

ber of items a retailer i actually receives or contributes

in period t are denoted as R

i,t

and C

i,t

When the received and contributed number of

items for all retailers are calculated at the end of time

period t, the start inventory level of the retailers in the

next period SI

i,t+1

can be calculated. This is done by

adding the quantity delivered by the supplier in that

time period D

i,t+1

(which will be zero when no deliv-

ery occurs in that time period), subtracting the number

of contributed items C

i,t

and adding the number of re-

ceived items R

i,t

from the end inventory level EI

i,t

period t. The end inventory level EI

i,t

in all time peri-

ods is determined by subtracting the incurred demand

i,t

from the start inventory level SI

i,t

. Note that this

end inventory level can become negative. When it is

negative, it means that retailer i could not meet the de-

mand and lost sales occur (LS

i,t

). In periods in which

the end inventory of a retailer is higher than or equal

to zero, no lost sales occur (i.e. LS

i,t

= 0). When the

end inventory is lower than zero, lost sales amount to

-EI

i,t

. Note that the formula for the start inventory SI

i,t

takes into account lost sales from the previous period.

i,t

(

0 if there is no delivery in t

if there is a delivery in t

(7)

i,t

= max(0, EI

i,t−1

) + D

i,t

+ R

i,t−1

−C

i,t−1

(8)

i,t

= SI

i,t

− a

i,t

(9)

i,t

= −min(0, EI

i,t

) (10)

Besides the cost of distribution and redistribution,

the cost of lost sales and the inventory holding cost at

the retailers are also considered to assess the collab-

orative supply chain with lateral transshipments. The

lost sales cost rate of a retailer LSC

is calculated by

taking his average number of lost sales per time pe-

riod and multiplying it with the cost of one item that

could not be sold (λ

) (cfr. formula 11). To calculate

the inventory holding cost rate of a retailer H

, his av-

erage inventory in each time period is summed over

the planning horizon and multiplied with his inven-

tory holding cost per item per time period (η

) divided

by the number of time periods in the planning horizon

p (cfr. formula 12).

LSC

∑

t∈P

i,t

(11)

∑

t∈P

i,t

+ EI

i,t

(12)

Note that this new inventory holding cost at the re-

tailers replaces the inventory holding cost calculated

in the CIRP solution algorithm (T

∑

i∈I

, cfr. for-

mula 1), since it is the inventory holding cost that the

retailers will actually incur.

To determine the daily redistribution cost, a TSP

is solved each day. The nodes in the TSP on a spe-

ciﬁc day are the receivers and the contributors. We

assume there is no limitation on the capacity of the

third party courier, since the redistributed volumes

are small. The model takes into account a travel cost

per km (θ). The redistribution cost rate is calculated

Tackling Demand Stochasticity by Redistribution among Retailers in a Two-stage Distribution System

235

Table 1: CIRP routes supplier illustrative example.

r TC

∑

i∈r

Sequence

1 139 64 75 4 S → 8 → 4 → 5 → 2 → S

2 172.66 99.16 73.5 5 S → 10 → 1 → 11 → 7 → 13 → 12 → S

3 224 149.3 74.7 3 S → 6 → 14 → 3 → 15 → 9 → S

Total 535.66 312.46 223.2

by multiplying the distances resulting from the TSPs

with the travel cost per km and taking the average over

the planning horizon P.

RC =

∑

t∈P

T SP

.θ

(13)

The total cost comprises the distribution cost rate

of the supplier TC

(minus the inventory holding

cost at the retailers incorporated in this cost rate

(

∑

i∈I

)), the redistribution cost rate RC, the

sum of the inventory holding cost over all retailers

(

∑

i∈I

) and the sum of the lost sales cost over all

retailers (

∑

i∈I

LSPC

). This total cost is computed

for the baseline system without lateral transshipments

and for the cooperative system with lateral transship-

ments. Both cases are compared to identify the bene-

ﬁts of the lateral transshipments.

5 ILLUSTRATIVE EXAMPLE

The suggested solution approach is illustrated with an

illustrative example. The example comprises one sup-

plier replenishing 15 retailers with one type of prod-

uct. One time period is assumed to be one day. The

average daily demand rates of the retailers d

are gen-

erated randomly between 0.2 and 10 items per day

and their maximum storage capacity between 10 and

100 items. The inventory holding cost and the lost

sales cost of the retailers are assumed to be equal

for all retailers and are respectively 1.5/item/day and

100/item. The retailers’ locations are generated ran-

domly in such a way that they are uniformly dis-

tributed in a circle around the supplier

The routes resulting from the CIRP algorithm are

shown in Table 1. The total transportation cost rate

of the supplier TC

equals 535.66 (= 139 + 172.66 +

224). The inventory holding costs of the retailers have

to be subtracted from the routes’ cost rates. These

inventory holding cost rates amount to 75, 73.5 and

74.7. So, the transportation cost rates that are taken

into account for the comparison of the baseline sys-

tem without lateral transshipments and the collabora-

tive system with lateral transshipments are 64 (= 139

- 75), 99.16 (=172.66 - 73.5) and 149.3 (= 224 - 74.7),

summing to a total transportation cost rate of 312.46.

The daily demand rates of the retailers are as-

sumed to be normally distributed with mean d

and

standard deviation 0.1 d

. Given that the cycle times

of the routes of the suppliers are 4, 5 and 3, actual

daily demand rates are generated for all retailers over

a planning horizon of 60 days (= P). Table 2 shows the

actual daily demand rates for retailer 2 over 12 days.

These demand rates are generated based on his aver-

age demand rate of 4.6 and standard deviation 0.46.

The supplier receives a delivery of 18.4 items (= 4 .

4.6) on days 1, 5 and 9 since the cycle time of his

route is 4. Based on the actual demand rates, the start

and end inventory levels of the retailers in the base-

line system without lateral transshipments can be cal-

culated using formulas 8 and 9. The negative end in-

ventory levels on days 4, 8 and 12 (resp. -0.3, -0.9 and

-1.0 items) indicate that a stockout (and consequently

lost sales) occurs on these days due to the higher than

average demand over the previous days. Note that the

lost sales cannot be backlogged, so the start inventory

on day 5 equals the delivery retailer 2 receives from

the supplier and does not take into account the lost

sales of 0.3 from day 4.

The cost of lost sales and the inventory holding

cost are calculated for all retailers (cfr. formulas 11

and 12). Table 3 shows the overall lost sales cost rate

and inventory holding cost rate in the baseline system

without lateral transshipments over all retailers. The

total cost rate in the baseline system without redistri-

bution equals 618.11 (= 312.46 + 254.15 + 51.5).

When redistribution is introduced, the threshold

inventory level, the optimal inventory level and the

critical inventory level are calculated for all retailers

on all days. In our example, we assume a ﬁll rate of

99.5% for the threshold inventory level, 99% for the

optimal inventory level and 98.5% for the critical in-

ventory level. Table 4 shows the three inventory levels

for retailer 2 over 12 days. Note that on day 1 the in-

ventory levels are calculated using a lead time of 3

since it is the inventory level at the end of the day that

is compared to the threshold inventory level, the opti-

mal inventory level and the critical inventory level. At

the end of day 1, retailer 2 expects his next delivery in

3 days. Hence, the lead time of 3 days.

Based on these inventory levels and the end inven-

tory level of the retailers, the receiver and contributor

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

236

Table 2: Retailer 2 days 1 to 12 illustrative example baseline system.

t 1 2 3 4 5 6 7 8 9 10 11 12

2,t

18.4 0 0 0 18.4 0 0 0 18.4 0 0 0

2,t

4.3 4.2 4.9 5.3 5.4 4.6 5 4.3 4.9 4.9 4.4 5.2

2,t

18.4 14.1 9.9 5.0 18.4 13.0 8.4 3.4 18.4 13.5 8.6 4.2

2,t

14.1 9.9 5.0 -0.3 13.0 8.4 3.4 -0.9 13.5 8.6 4.2 -1.0

2,t

0 0 0 0.3 0 0 0 0.9 0 0 0 1

Table 3: Lost sales, inventory holding, distribution and re-

distribution cost rates.

Baseline Collaborative

∑

LSC

51.50 23.50

∑

254.15 244.00

312.46 312.46

RC 0.00 31.84

Total cost rate 618.11 611.80

amounts are determined. On day 2 for example, the

end inventory of retailer 2 is higher than his threshold

inventory (9.9>9.7), so he is willing to hand over 0.2

items to the other retailers. Table 4 also shows the to-

tal receiver and contributor amount (resp.

∑

i,t

and

∑

i,t

) over all retailers over the 12 days. On day 2

there are more receiver items requested than there are

items contributed (1.2<3.1). Hence, supplier 2 will

contribute the total 0.2 items. The receiver and con-

tributor quantities are redistributed ’overnight’. So,

the start inventory of retailer 2 on day 3 is updated for

these 0.2 contributed items (9.9 - 0.2 = 9.7). On day

5 on the other hand, the end inventory of retailer 2

falls below his critical inventory (13.2<13.9). So, re-

tailer 2 is a receiver on that day. The total contributor

amount is now higher than the total receiver amount

(4.4>0.8), so retailer 2 will receive his full number

of requested items (0.7). Again, the start inventory of

retailer 2 on day 6 takes into account this number of

received items (13.2 + 0.7 = 13.9).

The new lost sales of retailer 2 in the 12-day pe-

riod are also included in table 4. These are determined

in the same way as in the baseline system (i.e., as the

negative end inventory when the end inventory falls

below zero). Note that despite the redistribution of in-

ventory among the retailers, retailer 2 incurs lost sales

on day 4. Overall, his lost sales are lower than in the

baseline sytem though (0.3 + 0.9 + 1).

The number of lost sales, the average inventory

level and the corresponding lost sales cost rate and in-

ventory holding cost rate is calculated over all retail-

ers in the planning horizon in the collaborative sys-

tem with lateral transshipments. The results are also

shown in Table 3. The lost sales cost rate has dropped

(-54.3%) compared to the baseline system, as has the

inventory holding cost rate (-3.9%). However, the

cost of redistribution has to be taken into account. A

TSP is solved for each day within the 60-days plan-

ning horizon. On day 7 for example, retailers 2 and 13

want to receive items (resp. 0.1 and 0.9 items). They

receive these items from retailers 4, 9 and 10. So, on

day 7 a TSP is solved among retailers 2, 4, 9, 10 and

13. The total redistribution cost rate resulting from

the TSPs over the planning horizon equals 31.84.

The collaborative system with lateral transship-

ments among the retailers results in our example

only in a small decrease in the total cost rate (-

1,02%). However, lost sales have decreased consider-

ably (from 30.9 items to 14.1 items over the 60-days

planning horizon), so the customer service level has

improved too.

6 CONCLUSIONS

We studied the beneﬁts of lateral transshipments of in-

ventory among retailers in a two-stage supply chain.

The supplier replenishes the retailers in a VMI setting

and his/her distribution routes and distribution cost

are determined using a CIRP solution heuristic. Re-

plenishments by the supplier are periodic. However,

since the replenishment frequencies are determined in

the CIRP heuristic, cycle times of the retailers can

differ from one to another. Redistribution of inven-

tory is possible in all time periods. The redistributed

quantities are determined based on the inventory level

of the retailers and their desired ﬁll rate. The bene-

ﬁts of the lateral transshipments are savings created

through reductions in lost sales and inventory hold-

ing costs and increased customers service levels. The

cost reductions have to be balanced with the increase

in transportation cost caused by the redistribution of

inventory. The cost of redistribution is determined by

solving TSP problems among the retailers that want

to participate in the redistribution.

Preliminary results show that savings in inventory

costs and lost sales can occur through lateral trans-

shipments in the VMI setting. Furthermore, the cus-

tomer service levels increase. However, the savings

are highly dependent on the problem parameters, like

the cost of lost sales, the inventory holding cost at the

Tackling Demand Stochasticity by Redistribution among Retailers in a Two-stage Distribution System

237

Table 4: Retailer 2 days 1 to 12 illustrative example collaborative system.

t 1 2 3 4 5 6 7 8 9 10 11 12

2,t

18.4 0 0 0 18.4 0 0 0 18.4 0 0 0

L 3 2 1 4 3 2 1 4 3 2 1 4

T I

2,t

14.5 9.7 4.8 19.2 14.5 9.7 4.8 19.2 14.5 9.7 4.8 19.2

2,t

14.1 9.4 4.6 18.9 14.1 9.4 4.6 18.9 14.1 9.4 4.6 18.9

2,t

13.9 9.2 4.4 18.6 13.9 9.2 4.4 18.6 13.9 9.2 4.4 18.6

2,t

4.3 4.2 4.9 4.3 5.4 4.6 5 4.3 4.9 4.9 4.4 5.2

2,t

18.4 14.1 9.7 4.8 18.6 13.9 9.3 4.4 18.6 13.9 9.2 4.8

2,t

14.1 9.9 4.8 -0.5 13.2 9.3 4.3 0.1 13.7 9 4.8 -0.4

2,t

0 0.2 0 0 0 0 0 0 0 0 0 0

2,t

0 0 0 0.2 0.7 0 0.1 0.1 0.2 0.2 0 0.2

∑

i,t

0.1 1.2 0.6 4.7 4.4 3.1 7.2 7.1 9.5 10.6 11.3 10.5

∑

i,t

3.3 3.1 2.3 0.6 0.8 1.5 1 0.2 0.5 0.2 0.5 0.7

2,t

0 0 0 0.5 0 0 0 0 0 0 0 0.4

retailers and the location of the retailers (and conse-

quently the cost of distribution and redistribution).

More in-depth research into larger datasets is nec-

essary. A design of experiments should be performed

to calibrate the redistribution policy parameters across

a wide range of instances. The chosen levels of the

ﬁll rate to determine the threshold, critical and op-

timal inventory levels appear to have a large impact

on the quantities all retailers want to contribute or re-

ceive to the redistribution, and consequently on the

redistribution costs. Also the way in which we deter-

mine which retailers actually contribute items when

the contributor amount is higher than the receiver

amount (or vice versa, when the amount of receiver

items is higher than the contributor amount) must be

investigated more in detail. The preliminary analy-

sis showed for example that it can be beneﬁcial to

exclude some retailers from the redistribution in cer-

tain time periods to prevent unnecessary movements

of small quantities and to keep the redistribution cost

resulting from the TSP under control. These redistri-

bution policy parameters become certainly important

in more realistic datasets with a high number of retail-

ers.

Furthermore, when determining which retailers

should contribute and receive what quantities in the

redistribution, we must keep in mind that all retailers

want to receive beneﬁts when entering the collabora-

tion. Hence, the beneﬁts should be shared over all

participants in a fair way, otherwise some participants

will be inclined to leave the collaboration. Further

research can be performed on how all retailers can re-

ceive a fair share of the redistribution beneﬁts.

Additionally, taking into account demand variabil-

ity into the CIRP to reduce the need for redistribution

should also be investigated.

ACKNOWLEDGEMENTS

This research was supported by the Agency for In-

novation by Science and Technology in Flanders

(VLAIO).

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