Journal of Applied Engineering Science

doi:10.5937/jaes16-17696 Paper number: 16(2018)4, 561, 523 - 537

DESIGN OF AN INVENTORY MANAGEMENT SYSTEM IN

AN AGRICULTURAL SUPPLY CHAIN CONSIDERING THE

DETERIORATION OF THE PRODUCT: THE CASE OF SMALL

CITRUS PRODUCERS IN A DEVELOPING COUNTRY

Cristhian Guillermo Acosta Imbachi1*, Andrés Felipe Cano Larrahondo1, Diego León Peña Orozco2,

Leonardo Rivera Cadavid1, Juan José Bravo Bastidas1

1Department of Industrial Engineering, Universidad del Valle, Colombia

2University Corporation Minuto de Dios, Department of engineering, Industrial Engineering, Colombia

Inventory management along the agrifood supply chain is a subject of great interest due to the constraints related

with the perishable condition of product. Signiﬁ cant problems like demand forecasting, inventory management and

transportation was evidenced within the supply chain studied. Additionally, the management of perishables and their

lifecycle are the most frequently issue in this kind of supply chain.An inventory management policy is deﬁ ned taking

in consideration the optimal quantity for an order and the time for ordering so as to ward off costs related with un-

derstock or overstock. This paper presents a mathematical model for inventory management in agricultural supply

chains considering perishability. The supply chain studied involves a retailer, a producer and a supplier. The advan-

tages of integrating inventory management along the supply chain are discussed and ﬁ nally some recommendations

and research opportunities set forth.

Key words: Inventory management; Perishable product; Fruit supply chain; Integration in supply chain

INTRODUCTION

Around 2,500 million people work part-time or full-time

in 500 million small farms around the world. However,

although the land is occupied by these producers, it only

represents 12% of all agricultural land covering more

than 80% of the worldwide food demand [01]. The contri-

bution by family farmers and small farmers to the millen-

nium goal related to food security, poverty reduction and

sustainable development recognized by the General As-

sembly of the United Nations. For this reason, the need

to count with new methodologies that enable greater ef-

ﬁ ciency and competitiveness in the supply chain of small

agricultural producers looms up.

Agricultural supply chains are a subject of great interest,

where an effort in the coordination of the actors, activ-

ities and resources is required in order to meet the re-

quirements of thecustomers [02], in this sense [03] afﬁ rm

that in order to improve the customer satisfaction and

reduce the cost, the cooperative relationship between

the companies must be built, especially in perishables

supply chain. However, due to the conditions in which

these chains operate, different problems come into view.

Among the most important issueslie demand forecast-

ing, inventory management and transport [04]. There

are other factors such as margins of intermediation, in-

frastructure and geographical conditions that can have

serious repercussions in terms of post-harvest losses,

which affect the economy of the members of the chain.

Literature related to the performance of fruit supply

chains conﬁ rms that inventory management is a topic

of increasing interest because of the restriction that the

perishable condition of the product implies. Considering

this constraint, it is important to determine the inventory’s

frequency of revision so that the product’s deterioration

cycle is lower than the revision cycle, with the purpose

of avoiding product losses caused by an inadequate in-

ventory policy. This policy must also consider the opti-

mal quantity of orders and the time products should be

ordered to prevent costs due to “understock” or “over-

stocking” [05, 06, 07]. In this sense [08] developed a mul-

ti-echelon inventory model for a deteriorating item from

an integrated perspective and determine the optimal de-

livery quantities for each echelon. On the other hand, au-

thors such as [09] have shown the beneﬁ ts of integration

in the supply chain thanks to the sharing of information,

which enables coordinated decisionmaking therein.

In this regard, [10] deﬁ ne four supply chain archetypes

described as the traditional supply chain, shared-infor-

mation supply chain, supplier supply chain (managed by

the supplier) and synchronized supply. The latter repre-

sents a scenario where information is transmitted in real

time among the members of the chain, such as their lev-

els of inventory, product in transit and sales, achieving

the best management amongst the archetypes present-

ed. In this case, the structure of the chain is centralized

and its performance signiﬁ cantly improved thanks to in-

formation sharing. Nevertheless, not all chains are cen-

tralized, so it is necessary to create collaboration mech-

anisms in decentralized chains to achieve some level of

integration. Authors such [11] have deﬁ ned integration as

a necessary strategy for getting into new markets. In this

respect[12]validate the importance of information in de-

cisionmaking in the chain by putting forward a classiﬁ ca

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* Cl. 13 #100-00, Cali, Valle del Cauca, Colombia, cristhian.acosta@correounivalle.edu.co

Journal of Applied Engineering Science Vol. 16, No. 4, 2018

ISSN 1451-4117

tion of integration mechanisms in decentralized chains

where they include information technologies, shared in-

formation, joint decision making and contract models.

This paper aims to approach the management of inven-

tories in the framework of an agricultural supply chain

through a mathematical model that considers the dete-

rioration of the product over time and demonstrates the

advantages of integrating of its echelons. The character-

istics of the chain under study are presented, then the

mathematical model used to determine the inventory

policy is described, subsequently the results are laid out

and a sensitivity analysis is carried out by varying some

parameters. Finally,conclusions, recommendations and

opportunities for new studies are developed.

PROBLEM STATEMENT

As described above, the study of agricultural supply

chains is of great interest mainly due to coordination

problems in the chain that affect, among other aspects,

the ﬁ nancial results that reﬂ ect in the form of low income-

for the case of small farmers. There are some determin-

ing factors for the appropriate performance of the chain

set forth by [13], such as the variability of demand and

prices; the availability of workers; the yield of the crop;

labor costs and those associated with the recollection

of products; the use of means of transportation that bal-

ance out the time it takes to reach the market and the

cost; post-harvest management of crops; the degree

of maturity of the product; the maximum time for deliv-

ery; the availability of products; transportation time and

delivery costs. Other authors such as [14], assure that

the handling of perishable products and their life cycle

are the most differentiating factors of this type of chain.

On the other hand, authors how [15], present fruits and

vegetables as products that have features of freshness,

perishability [16], timeliness, logistics performance [17].

This makes agrifood supply chains constantly-varying

complex systems that involve several echelons such as

suppliers, distributors, marketers, wholesalers and retail-

ers, among others, which makes it behave as a multi-

disciplinary system that attempts to satisfy the demands

of the ﬁ nal customer through effective coordination of

information ﬂ ows, products and ﬁ nancial resources [18].

A problem of great interest, identiﬁ ed in the literature re-

lated to the performance of agricultural supply chains,

especially fruit chains, is the management of invento-

ries along the chain, due to the restriction involved in

the product’s lifecycle. Due to the aforementioned rea-

sons, inventory management becomes a critical issue

when dealing with perishable products such as fruits,

which require compliance with strict quality requisites.

These demands cannot be satisﬁ ed only with the desire

of small producers to do it, it is also necessary to syn-

chronize from suppliers’ supply times, to producers and

intermediaries, and from intermediaries to retailers, in or-

der to achieve an adequate estimation of amounts, con-

sumption times in each echelon and eventually the best

performance in the chain. The model presented below

seeks to represent the behavior of these relationships in

a decentralized fruit supply chain, assuming a scenario

where information is shared for decision making regard-

ing the inventory along the chain.

MATERIALS AND METHODS

Development of mathematical model

a. Interaction between supply chain echelons

under study

The chain under study consists of three echelons, name-

ly: a single supplier, a producer and a retailer, as shown

in Figure 1. The supplier is responsible for starting the

ﬂ ow of the product to the other echelons, below. The

next member of the chain is the producer, who has a

warehouse for raw materials and another warehouse to

store the ﬁ nished product. Finally, facing the consumer is

the retail echelon, in charge of directly satisfying his de-

mand; the former is also the one that initiates the ﬂ ow of

information, sharing the expected demand with the other

actors in the supply chain. The interaction in the supply

chain begins at the time the retailer makes an order that

attempts to satisfy the demand during a planning period

T. This information about the demand is shared with the

other echelons and sent to the next supply chain actor

(the producer) through an order. The latter must satis-

fy this order by means of deliveries, at ﬁ xed intervals of

time, placed in retailer warehouse. Upon receipt of the

order, the producer initiates a value-adding process on

the raw material in order to generate the ﬁ nishedproduct

Figure 1: Supply Chain Inventory Management for each echelon

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for the agrifood supply chain under study.

As shown in Figure 1, there is an inversely proportion-

al relationship between the inventories held by the pro-

ducer, since the raw material inventory is reduced over

time when the value-adding process starts, while that of

ﬁ nished product increases. Since the production rate is

higher than the demand’s, the producer accumulates the

inventory of ﬁ nished product for deliveries carried out in

a period after the time of production. In this case, it is as-

sumed that the accumulated inventory is enough to meet

the number of remaining deliveries taking into consid-

eration the amount of product that suffers deterioration.

To start off his productive process, the producer requires

raw material, which is provided by the supplier. This ech-

elon must perform the supplying of inputs from an exter-

nal supplier (not considered in this case study), with the

objective of maintaining good quality raw material for the

producer. The supplier is responsible for providing the

raw material required by the producer for itsprocessing.

This is achieved by delivering equal quantities at ﬁ xed

time intervals.

b. Assumptions

• Mathematical inventory model for a single perisha-

ble product in an integrated supply chain.

• The aim is to determine the optimal number of deliv-

eries and lot sizes that minimize the total cost of the

supply chain.

• Demand and constant production over time.

• The demand rate is lower than the production rate.

• The planning period is known.

• The Lead Time is zero, shortage is not allowed.

• One item is only considered.

• The parameters that represents the deterioration

rates are constant and deterministic.

• An order is satisﬁ ed through multiple deliveries.

• The supplier delivers the same amount of raw mate-

rial to the producer.

c. Variables and parameters

SYMBOL DESCRIPTION

T Planning time

T1Production time used by the producer T1= npt+t3

T2Cycle time used by the provider T2=(np+1)t

N Number of deliveries received by the retailer from the producer for each planning time T

npNumber of deliveries received by the retailer from the producer during T1

t3Production time from the point np to the end of production

qBLot-size per delivery from the producer to retailer of ﬁ nished goods

QBTotal order quantity of ﬁ nished goods for retailer per planning time T.

qpFinished goods quantity produced in time t.

qPW Quantity of raw material per delivery received in the producer’s warehouse from the supplier.

qnPW Quantity of raw material delivery received in the producer’s warehouse from supplier in last delivery.

QSSupplier’s total order quantity of raw materials for period T2

QPW Quantity of raw material harvested by the supplier per delivery

QnPW Quantity of raw material harvested by the supplier in last delivery

ApTotal ﬁ nished goods inventory for producer in T.

IB(t) Retailer’s ﬁ nished goods inventory level at time t.

IPI(t) Producer’s ﬁ nished goods inventory level at time t.

IPW(t) Producer’s warehouse raw materials inventory level at time t.

IS(t) Supplier’s Raw materials inventory level at time t.

TCBRetailer total cost.

TCPProducer’s total cost.

TCPW Producer’s warehouse total cost.

TCSSupplier’s Total cost.

TC Global total cost(TCS + TCPW + TCP + TCB)

D Retailer’s demand rate for ﬁ nished goods.

A Finished goods’ ordering cost for retailer per order.

Table 1: Notation used in the Mathematical Model

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FBFinished goods’ receiving cost for retailer perreception.

HBFinished goods’ unitary holding cost per unit of time for retailer.

PBDeterioration cost per unit of ﬁ nished goods for retailer.

QBDeterioration rate for retailer’s ﬁ nished goods

P Producer’s production rate for ﬁ nished goods.

SPProducer’s setup cost for each setup.

FPFinished goods’ delivery cost for producer per deliver.

HPFinished goods’ unitary holding cost per unit of time for producer

PPDeterioration cost per unit of ﬁ nished goods for the producer.

QPDeterioration rateof ﬁ nished goods for the producer.

FPW Raw materials’ receiving cost per reception for the producer at warehouse.

HPW Raw materials’ unitary holding cost per unit of time for producer’s warehouse

PpW Deterioration cost per unit of ﬁ nished goods for producer’s warehouse.

QPW Deterioration rate for producer’s raw materials

S Raw materials’ ordering cost per order for the supplier.

FSRaw materials ‘ delivery cost for the supplier per delivery.

HSRaw materials’ unitary holding cost per unit of time for the supplier.

PSDeterioration cost per unit of raw materials for the retailer.

QSDeterioration rate for supplier’s raw materials

Note. Source: Adapted from [8].

d. Retailer: Finished product inventory model

According to [19], the inventory level of ﬁ nished goods at

time t’ can be expressed as follows:

(1)

Solving (1) in its extreme points by the integrating factor

method:

And by multiplying on both sides of the equation (1) by

, then:

,

(2)

The right part of equation (2) can be expressed as the

derivative of a product as follows:

The result obtained by integrating both sides of the equa-

tion and solving the respective integrals can be written

as:

(3)

Multiplying on both sides of (3) by and simplifying:

Applying the boundary condition IB(t) in (4) to obtain the

(4)

integration constant’s value, results in:

Using the constant’s value in the expression (4) and sim-

plifying:

Therefore, by setting the boundary condition IB(0) in (5) it

(5)

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is possible to obtain the initial order quantity. In this way,

IB (0)=qB and the resulting expression is:

To obtain the expression forinventory amount of standby

(6)

ﬁ nished products, the equation (6) must be integrated

between 0 and t, as follows:

The total cost of offered goods can be expressed as the

(7)

sum of the cost of the order, the cost of reception, the

cost of maintenance and the cost of deterioration. There-

fore, the total cost of the goods offered to the retailer

during a planning period T, can be expressed as:

Using the following approach:

(8)

Proposed by [8] and substituting in the retailer total cost

equation, the following is obtained:

e. Producer’s warehouse: Raw material inventory

(9)

level

The raw material inventory level of the producer’s ware-

house at time t’can be expressed as follows:

At the extreme points of equation (9), the inventory will

(10)

be expressed by the following equations:

Integrating the expressions (11) and (12) from 0 to t and

(11)

(12)

from 0 to t3 respectively, in the same way as shown in the

retailer’s model, the inventory quantities of standby raw

material at time t and t’are obtained as:

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(13)

(14)

Therefore, according to [8], the producer’s total cost for

raw material warehousing per cycle T could be repre-

sented as the sum of the maintenance cost, the receiving

cost and deterioration cost, as follows:

(15)

f. Inventory level of ﬁ nished products by the

producer

When the retailer orders a quantity of ﬁ nished product

from the producer in period T, the producer begins the

production and delivery at time t.

In the ﬁ rst period, the producer achieves a production

of ﬁ nished product equal to in period t. The producer’s

inventory level of ﬁ nished product in the ﬁ rst period can

be represented as follows:

(16)

Solving equation (16), analogous to the differential equa-

tion that represents retailer’s model bearing the integrat-

ing factor method, the inventory level for the extreme

points will be:

(17)

Thus, the inventory level of ﬁ nished products for the nth

delivery after the ﬁ rst one is obtained, according to [8]

with the following differential equation:

(18)

The inventory of ﬁ nished product before the i-delivery is .

Solving the equation (18) by as initial condition,it is pos-

sible to determine the inventory level for the i-th delivery,

like this:

(19)

According to [8], from the previous expression (19) the

lot size from the non-production period can be deter-

mined as:

(20)

In accordance with the model of [8], as soon as the

producer reaches the amount of ﬁ nished product that

the retailer needs per order cycle time T, production is

stopped. The production time is deﬁ ned in (npt+t3). How-

ever, the producer continues to deliver a constant quan-

tity of merchandise, until the entire amount of production

of ﬁ nished products has been delivered to the retailer.

This occurs at time T; when the level the producer’sin-

ventory of ﬁ nished product is equal to zero.

In [19] argue that the ﬁ nal inventory after time t consider-

ing a constant rate of deterioration, can be expressed as:

At time (npt+t3), producer stops production. The inventory

(21)

amount of ﬁ nished goods is Qnon/product ;after time (t-t3) ,

the inventory becomes Qnp1 and in compliance with the

expression of[19], the outcome will be as follows:

The ﬁ nal quantity of inventory of ﬁ nished products for

(22)

time (np+1) can be derived as:

The inventory amount of ﬁ nished product from the pro-

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ducer for the nth time is equal to the lot size per delivery.

From (23), the lot size of ﬁ nished products can be ob-

tained during the non-production period at t=t3 as:

With Qnon/product(1) and Qnon/product(2) , the values of np y t3

(24)

can be determined. To ﬁ nd the corresponding values

for np, the algorithm proposed by[8] needs be followed,

which consists on assuming Qnon/product(1), Qnon/product(2).

Thus, the equation is equal to zero and t’ is replaced

by t. Subsequently, for each value of n, every np val-

ue is obtainedalong with the parameters needed for the

equation’s solution such as the production rate of the

producer, the rate of deterioration and the lot size of ﬁ n-

ished products per delivery from the producer to the re-

tailer.

To estimate t3, a similar procedure is performed, where

t’ is replaced by t’. Then, Qnon/product(1) and Qnon/product(2 are

equated. In this way, it is possible to ﬁ nd every for its

corresponding value.In [8] an algorithm it is presentedto

ﬁ nd the value of time and , however, in this document

we propose an alternative procedure to determine the

value of by matching the expressions and which cor-

respond to the quantity of inventory at the end of the pro-

duction period and the quantity of inventory at beginning

the period of depletion once production has stopped,

respectively. The resulting equation is expressed as a

function F [t3] throughwhich we obtain the value of that

makes the function equal to zero, this point is deﬁ ned

as the point of intersection with the X axis, which corre-

sponds to value for a number of n deﬁ ned deliveries.

ThroughWolfram Mathematica, this analysis is validat-

ed by making the graph of the mathematical expression,

the value obtained with this method is observed to cor-

respond to the value obtained algebraically. In ﬁ gure 2an

arbitrary interval of (-0.003; 0.003) is obtained for the pa-

rameters associated with a number of deliveries n = 100,

a value of t3 = 0.0145134 corresponding to the point of

intersection with the X axis.

Figure 2: Graphic method for t3 calculation

Note. Source: Author’ own elaboration

The deterioration of the ﬁ nished products’ quality dur-

ing cycle T is the sum of deteriorated ﬁ nished products

quantity from period 1 to n and is written as follows:

(25)

The deterioration cost and the maintenance cost of the

ﬁ nished products per order cycle time T can be obtained

in the following way:

(

(

(

(

(

(

(2

(

(

(

(

4

)

(26)

Producer’s total ﬁ nished goods cost for planning period

T could be represented as the sum of the setup cost, the

delivering cost, the cost of maintenance and the deterio-

ration cost, as follows:

(27)

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g. Supplier Inventory Model

Considering that the supplier’s opening inventory is des-

ignated as, [19] denote the amount of raw materials per

delivery from the supplier to the producers’ warehouse,

as well as the last quantity provided by the supplier, as

follows:

Should (28) y (29)de integrated from 0 tot, the ﬁ nal in-

ventory of raw materials can be obtained, as:

(28)

(29)

(30)

The same applies for:

In this manner, the quantity of the supplier’s order for

raw material thatsatisﬁ es the demand of the producers’

warehouse from time 0 to t is also deﬁ ned as:

(31)

Eventually, supplier’s total cost according to [19], could

be represented as the sum of the maintenance cost, set-

up cost, delivery cost and deterioration cost:

(32)

(33)

Finally, the global cost of the integrated inventory model

could be represented as the sum of the supplier, produc-

er and retailer’s individual costs, as follows:

(34)

As presented in equation (34), the Total Cost is the ob-

jective function that must be minimized and it is obtained

through the sum of costs assumed per each echelon in

the supply chain. In order to ﬁ nd the minimum Total Cost,

a number of iterations must be executed, varying the

number of deliveries n. In each iteration the order quan-

tity and total quantities between echelons are derived.

Then, the values of the Total Cost from the different iter-

ations are analyzed and the number of deliveries n that

minimizes the objective function TC is deﬁ ned.

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Table 2: Parameters used in the execution of the model

TABLE OF PARAMETERS USED IN THE EXECUTION OF THE MODEL

BUYER - RETAILER

Parameter Computing Method Value Units

D Time forecast series for constant demand 3.718 [Unit/month]

A Cellphone plan, paperwork order 0,4 [ US$/Order]

FB Cost of loading and unloading per worker per truck 10,8 [US$/Reception]

HB Warehouse leasing cost, area occupied by citrus fruits, inventory

turnover

0,4 [US$/Unit-Month]

PB It is calculated as a 35% from original sales price. 0,7 [US$/Unit-Month]

THB Percentage of lemons damaged by deterioration 0,08

PRODUCER – INTERMEDIARY

P Ability to select good quality lemons and pack them in 1kg nets, with

an available time of 480 minutes per working day, during 28 days a

month

6.240 [Nets of 1kg/

month]

SP Using production capacity, the number of wages required to satisfy

the demand during the planning horizon (1 month) is reckoned and

then multiplied by the value of each one(US$ 9,03/day)

167,7 [US$/order prepa-

ration]

FP Value of retailer transportation cost plus the cost of loading 61,4 [US$/delivery ]

HP The leasing cost between the area of ﬁ nished goods and raw mate-

rial warehouse is prorated. In each part of the warehouse, 1/10 of

production capacity can be stored.

0,1 [US$/Kg*month]

Pp In this case, up to 70% of the original price can be sold. It means

that 30% of the original price fails to be received.

0,3 [US$/Kg]

ThP Percentage of lemons damaged by deterioration 0,01 -

Fpw Cost of loading and unloading a truck per worker 10,9 [US$/reception]

Hpw The leasing cost between the area of ﬁ nished goods and raw mate-

rial warehouse is prorated. In each part of the warehouse, 1/10 of

production capacity can be stored.

0,1 [US$/Kg* month]

Ppw What it is not being perceived if it is sold without packing but con-

sidering CABAZA price market. Bulk sale price: US$54,2; Cost:

US$43,4; Bulk proﬁ t: US$10,8; Deteriorating cost / kg: US$10,8 /

60Kg

0,2 [US$/kg]

Thpw Percentage of lemons damaged by deterioration 0,10 -

SUPPLIER – FARMER

Hs Calculation of monthly maintenance costs of one hectare; estimated

production per tree a month

0,2 [US$/Kg*month]

Ps Probability that a tree does not develop correctly multiplied by the

cost of the tree

0,05 [US$/Unit]

S The survey provides the estimated annual cost of reception, given

that S is for the planning horizon, it is divided into 12

90,3 [US$/order]

Fs Cost of pickup and freight is US$ 72,23 for transportation from the

supplier to the producer or broker’s warehouse

14,8 [US$/delivery

(harvest)]

ThS Percentage of lemons damaged by deterioration 0,15 -

Note.Source: Author’ own elaboration.

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Table 3: Quantities delivered by the echelons and percentage of deteriorated product along the supply

chain for different delivery frequencies

nn

pqBQBqPqPW QPW qnPW QnPW QSPercentage of

Deterioration

1 - 3.871 3.871 5.938 6.563 7.721 4.286 5.042 5.042 36%

2 1 1.897 3.793 3.043 3.199 3.470 851 923 4.393 18%

3 1 1.256 3.768 2.046 2.115 2.233 1.854 1.958 4.190 13%

4 2 939 3.755 1.541 1.580 1.645 764 795 4.086 10%

5 3 750 3.748 1.236 1.261 1.302 118 122 4.028 8%

6 3 624 3.743 1.031 1.049 1.078 735 755 3.987 7%

7 4 534 3.739 885 898 919 277 284 3.959 6%

8 4 467 3.737 775 785 801 720 735 3.939 6%

9 5 415 3.735 690 697 710 366 373 3.922 5%

10 6 373 3.733 621 627 637 84 85 3.910 5%

15 9 249 3.728 415 417 422 73 73 3.871 4%

20 12 186 3.725 311 313 315 67 68 3.852 4%

25 15 149 3.724 249 250 252 64 64 3.840 3%

30 18 124 3.723 208 208 209 61 62 3.832 3%

35 21 106 3.722 178 179 179 60 60 3.827 3%

40 24 93 3.722 156 156 157 59 59 3.823 3%

45 27 83 3.721 139 139 139 58 58 3.820 3%

50 30 74 3.721 125 125 125 57 57 3.817 3%

55 33 68 3.721 113 114 114 56 56 3.815 3%

60 36 62 3.720 104 104 104 56 56 3.813 3%

65 39 57 3.720 96 96 96 55 56 3.812 3%

70 42 53 3.720 89 89 89 55 55 3.811 2%

75 45 50 3.720 83 83 83 55 55 3.809 2%

80 48 47 3.720 78 78 78 54 55 3.808 2%

85 51 44 3.720 73 73 74 54 54 3.808 2%

90 54 41 3.720 69 69 70 54 54 3.807 2%

100 60 37 3.719 62 62 63 54 54 3.806 2%

RESULTS AND DISCUSSION

a) Delivery amounts among echelons

The mathematical model proposed in this work was

solved through Matlab using the parameters deﬁ ned in

table 2. The results obtained in Matlab were exported to

Microsoft Excel in order to display them in a clearer and

more practical way.

As a summary, table 3 shows the delivery quantities that

each echelon is supposed to

carry along the chain. It can be clearly seen that increas-

ing delivery quantities, lowers the amount of product that

deteriorates along the chain.

By increasing the frequency of delivery in the planning

time T, there will be a smaller time interval between de-

liveries. Therefore, the deterioration rate has low conse-

quences for the short time that the product lingers before

reaching the consumer.

b) Optimal independent costs and integrated supply

chain optimal cost

In Table 4 it is possible to determine the lowest overall

cost or optimum total cost of the citrus supply chain un-

der study.It is also possible to establish what the optimal

cost for each independent echelon is and the delivery

conditions for each speciﬁ c situation. The lowest total

cost for the chain is US$ 1.210 a month. This is achieved

when the number of deliveries in the planning period

(one month) is equal to 5.

Analyzing the optimal independent costs, the supplier

must make 8 deliveries during time T, in this case the

total cost of the chain becomes 8% more expensive than

Note.Source: Author’ own elaboration.

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n n_p TC_B TC_PT TC_S TC OptimalEchelon

1 - 833 709 1007 2549

2 1 428 603 529 1560

3 1 302 602 383 1288 Producer

4 2 245 637 329 1211

5 3 215 691 303 1210 Integrated SC

6 3 199 738 276 1213

7 4 191 800 272 1264

8 4 187 855 258 1301 Supplier

9 5 187 921 262 1370 Retailer

10 6 189 989 269 1447

15 9 216 1315 288 1819

25 15 303 1983 356 2642

30 18 352 2319 395 3067

35 21 403 2657 436 3496

40 24 454 2995 477 3927

45 27 506 3333 520 4359

50 30 558 3672 563 4793

55 33 611 4011 605 5227

60 36 664 4350 648 5662

65 39 717 4689 692 6098

70 42 770 5029 735 6534

75 45 824 5367 779 6970

80 48 877 5707 822 7406

85 51 931 6046 867 7844

90 54 985 6385 910 8280

100 60 1092 7065 997 9154

the lowest total cost for the global solution in the supply

chain.

From the producer’s perspective, only three deliveries

should be made, generating a greater total cost along the

chain compared with the global optimum. In this scenario

Note. Source: Author’s own elaboration

the total cost of the chain is 6% more expensive than the

integrated supply chain. The reason why the number of

optimal deliveries for the producer is so low is due to cost

of delivering, since the product must be transported to

the customer's warehouse (retailer), located in a city 90

km away. The cost of making each delivery is US$ 61.4.

That is why it is more beneﬁ cial for the producer to store

raw materials and ﬁ nished products and assume mainte-

nance and deterioration costs. The aforementioned im-

pacts on the increase of the other echelons’ costs. Final-

ly, from the retailer's approach, deliveries must be made

every 3 days.In other words, to achieve the optimum cost

of the retail echelon, 9 deliveries must be made in the

planning period of one month; in this case, the total cost

of the chain would be 13% greater than the solution ob-

tained through the mathematical model.

According with results shown in table 4 it is possible ob-

tain the value for n that minimize the total cost on the

studied supply chain. Results are presented in Figure 3

as a graphic that shows different values for n, as well as

the n where the minimum TC is obtained.

Figure 3: Graphical representation TC function and the

optimal value obtained.

the deterioration of the product: the case of small citrus producers in a developing country

533

Table 4: Local and global costs for several delivery frequencies

Journal of Applied Engineering Science Vol. 16, No. 4, 2018

ISSN 1451-4117

c) Inventory control policy for echelons in the lemon

supply chain

Taking into account the results presented above and

considering the deﬁ ned planning horizon (one month),

it is possible to formulate the inventory control policy for

each echelon belonging to the supply chain, thus re-

sponding to the three fundamental questions: Frequen-

cy of revision, moment of ordering and quantity to order.

According to [20]in the case of perishable goods, since

they have a limited lifetime, in order to be able to comply-

ing with the shelf life time, the length time in the supply

chain should not exceed the lifetime value, as each unit

of time that is exceeded in the supply chain represent

deterioration in the products condition. In this sense the

inventory level revisionshould be deﬁ ned as continuous

in each echelon.

Retailer Inventory Management Policy: Review the in-

ventory level continuously; order an amount equal to 750

units of 1 kg of selected lemon when the inventory level

in the warehouse reaches zero.

Producer Inventory Management Policy: Check the in-

ventory level continuously; order an amount equal to

1261 kilograms of virgin lemon, when the inventory of

raw material in the warehouse reaches zero.

Supplier inventory management policy:Review the in-

ventory level continuously; when the level of inventory

of virgin lemon in the warehouse reaches zero, the sup-

plier must harvest an amount equal to 1302 kilograms of

virgin lemon during the ﬁ rst 3 deliveries. Nonetheless, in

the last delivery only 122 kilograms of virgin lemon need

be sent to the producer.

Costing of the current supply chain

In order to determine the costs of the current citrus sup-

ply chain, the deliveries made in a planning period are

deﬁ ned with the information provided by the echelons

through the surveys carried out. Delivery schedule can

be seen in table 5excluding Sundays; 14 deliveries are

made per month. This will be the input used to determine

the cost of the current chain.

When the current number of deliveries is reckoned, the

cost per each component of the relevant total cost can be

computed with the purpose of costing the supply chain

in its current conditions, as presented in table 6. Sub-

sequently, the comparison with the results of the costs

obtained through the proposed model is shown.

Table 5: Current Delivery Schedule: One-month planning period

CURRENT DELIVERY SCHEDULE

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Monday Tuesday Wednesday

Note. Source: Author’s own elaboration

Once the optimal cost of the inventory control system has

been attained through the model formulated, and after

deﬁ ning each of the operational costs related to the man-

agement of inventory in the supply chain in its current

conditions. It is possible to make a comparison between

both scenarios; in the ﬁ rst scenario (model implemented)

total cost is the objective function minimizedand it is the

sum of costs assumed per each echelon in the supply

chain. Take in account that to ﬁ nd the minimum Total

Cost, a number of iterations must be executed, varying

the number of deliveries n. Then, the values of the To-

tal Cost from the different iterations are analyzed and

the number of deliveries n that minimizes the objective

function TC is deﬁ ned. A summary for the current situa-

tion and the proposed situation results are presented in

table 7. The total monthly cost related to inventory man-

agement in the current supply chain is US$1.862, while

in the proposed model the optimal cost is US$1.210 a

month. A 35% reduction is achieved with the speciﬁ ca-

tions of policies, asproposed by the mathematical model

developed herein.

It is important to highlight the realization that comes to

light with this model regarding the dominance held by the

retailer over the other actors. Currently, the frequency

of deliveries is deﬁ ned by him and the other echelons

must adapt to his orders without issuing any judgment.

From the retailer's point of view, the model suggests to

increase the frequency of receptions up to reaching nine

deliveries a month, the highest frequency suggested in

the optimal local analysis. The high frequency suggested

for the retailer by the model, validates the chain's current

operation, where a large number of deliveries is required

by the retailer.

Effect of variations in deteriorating rate on optimal

result

In order to understand deterioration rate impact in the

Supply chain, a sensitivity analysis is shown in Table 8.

Just one of the rates is modiﬁ ed at the same time remain

the rest constant. Original rate value for each MAX sce-

nario is increased arbitrarily by 75% and each rate for

each MIN scenario is reduced by the same percentage.

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534

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RETAILER

Cost of Reception 152 US$/month

Ordering cost 0,4 US $/month

Maintenance cost 34 US $/month

Buyer Monthly Cost 186,4

PRODUCER

Deliveries 860 US$/month

Preparation 168 US$/preparation of order per month

Maintenance Cost 36 US$/month

Producer Monthly Cost 1064

PRODUCER WAREHOUSE

Reception of Raw Material 152 US$/month

Raw Material Maintenance Cost 36 US$/month

Product Warehousing Monthly Cost 188

SUPPLIER

Delivery 15 US$/month

Maintenance 319 US$/month

Reception of Supplies 90 US$/month

Supplier Monthly Cost 424

Table 6: Costs of the current agricultural supply chain

Note.Source: Author’s own elaboration

Table 7: Comparison of the current cost and the optimal cost obtained through the model

Total monthly cost of the current chain US$ 1862

Total monthly cost of the proposed chain US$ 1210

Improvement 35%

Note.Source: Author’s own elaboration

In general, despite huge deterioration rate variation, the

total cost and local costs of each echelon do not vary

more than 3%. In addition,deterioration rate has a great-

er impact on the producer echelon, while in the oppo-

site extreme, the less impact is in the supplier, reaching

around 0.5% of variation in the total cost.

COST ThS=0,15 ThPW=0,1 ThP=0,1 ThB=0,08 ORIGINAL

SCENARIO

MAX MIN MAX MIN MAX MIN MAX MIN

0,2625 0,0375

TCS187 187 187 187 187 187 196 178 187

TCp602 602 611 591 640 560 604 601 602

TCb262 255 259 258 263 257 259 258 258

TC 1216 1203 1216 1203 1241 1175 1226 1189 1210

Table 8: Comparison of the current cost and the optimal cost obtained through the model

Note.Source: Author’s own elaboration

the deterioration of the product: the case of small citrus producers in a developing country

535

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ISSN 1451-4117

In general, despite huge deterioration rate variation, the

total cost and local costs of each echelon do not vary

more than 3%. In addition,deterioration rate has a great-

er impact on the producer echelon, while in the oppo-

site extreme, the less impact is in the supplier, reaching

around 0.5% of variation in the total cost.

Model limitations

In order to adjust the proposed model to the more real

situation it would be interested to consider agricultural

producer cultivate different product. In that sense this

model has some limitations because does not consider

several fruits but just lemon as a single product in supply

chain. In addition, the capacity of the lemon truck trans-

porting is not included in the analysis. Finally model only

takes in account one single member per echelon when in

this kind of supply chain frequently has a several players

in each of them.

CONCLUSIONS AND OPPORTUNITIES FOR FUTURE

RESEARCH

Efﬁ ciently managing a supply chain involves its overall

optimization. Since supply chains operate under different

conditions due to their nature, it is necessary to imple-

ment an adequate inventory control system bearingthe

integration of each of its echelons as its fundamental

principle in order to maximize the total value generated

by it. An integrated supply chain should seek real-time

information sharing by its members with the aim of gen-

erating such synergy, so that the highest possible perfor-

mance is achieved.

The pertinence of the three-level supply chain proposed

in this work, namely stated as supplier, producer and

retailer, was validated. Data on the costs related to the

management of inventories in each echelon were ob-

tained, which became an important input for the pro-

posed study.

Finally, it is evident that there are large price ﬂ uctuations

of citrus and high-rising supply costs in the chain under

study. In times of oversupply, when the prices fall sig-

niﬁ cantly, the need to minimize logistics costs along the

chain is met through an integrating model approach.

This research has shown that an optimal supply chain

works perfectly from an integration approach and not

from a particular point of view. The mathematical model

used allowed to deﬁ ne the optimal quantity for orders,

the moment of ordering and the frequency with which the

inventory of the citrus chain under study in the rural area

of Tulua- Andalucia, Valle del Cauca, Colombia, must be

revised.

The integration of the supply chain through the exchange

of information, as posed by the synchronized supply

chain archetype where information on demand and the

inventory level is shared, enables the adequate func-

tioning of the chain's overall performance, allowing the

producer to ﬁ thisyield to the information provided by the

next echelon in the supply chain, thus reducing logistics

costs.

With the goal of validating and proving the mathemati-

cal model with the data collected from the supply chain

under study, and determining ordering optimal quantity

and time, and inventory review frequency, the total cur-

rent chain cost per month was determined (taking into

account that one hectare is sufﬁ cient to satisfy the de-

mand along the supply chain under study), as the sum of

the cost incurred by each echelon from an individualis-

tic approach. To that end, the current delivery schedule,

the reception, delivery, maintenance, loading, unloading

and the ordering cost were taken into consideration. The

current total cost was compared with the optimal cost

obtained through the mathematical model and under an

integration approach, wherein the total cost of inventory

management in the citrus chain under study was found

to have reduced around 35%.

This paper can be the basis for future research consider-

ing some additional conditions in which the citrus supply

chain currently operates. It is relevant to study this model

bearing in mind a bigger supplier basis and bigger num-

ber of retailers. Furthermore, this model does not con-

sider the transport lemon truck capacity and therefore

it is highly recommended to raise this restriction for the

model to resemble reality even more. It is also proposed

to study the rate of deterioration by analyzing the prob-

lem from the chemical and biological standpoints of the

product in different scenarios in order to determine this

percentage more accurately and obtain more reliable re-

sults through this mathematical model.

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Paper submitted: 03.06.2018.

Paper accepted: 27.10.2018.

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CC BY-NC-ND 4.0 terms and conditions.