Newsvendor Model for Multi-Inputs and -Outputs with Random Yield:

Applications to Agricultural Processing Industries

Kannapha Amaruchkul

Graduate School of Applied Statitistics, National Institute of Development Administration (NIDA), Bangkok, Thailand

Keywords:

Agriculture Supply Chain, Applied Operations Research, Stochastic Model Applications, Newsvendor

Models.

Abstract:

Consider a newsvendor model, which we extend to include both multiple inputs and outputs. Different input

types possess different levels of quality, and are purchased at different prices by a processing ﬁrm. Each type

of input is processed into multiple outputs, which are sold at different prices. The yield for each output type

is random and depends on the input type. We need to determine the purchase quantities of different types

of input, before demands of different types of output are known. In our analytical results, we show that the

expected total proﬁt is jointly concave in the purchasing quantities and derive the optimality condition. Our

multi-input and -output newsvendor model is suitable for processing industries in agriculture supply chain. In

our numerical example, we apply our model to the rice milling industry, whose primary output is head rice

and byproducts are broken rice, bran and husk. Our model can help the rice mill to decide which paddy types

to procure and how much, in order to maximize the total expected proﬁt from all outputs. We also show that

the expected proﬁt can be signiﬁcantly better than using the standard newsvendor model.

1 INTRODUCTION

In a newsvendor (single-period inventory) model,

there is a single opportunity to place an order be-

fore a random demand is known. Leftovers cannot

be kept from one selling season to the next, due to

obsolescence or perishability. The applications of the

newsvendor model include a newspaper, a style good,

a Christmas tree, bakery, and fresh produce. In a clas-

sical newsvendor model, there is only one product and

no resource constraint. In a multi-item newsvendor

model, the decision maker needs to decide the order

quantities of multiple items before their random de-

mands materialize, subject to several resource con-

straints. For instance, a news vendor, who manages

a “newsstand,” offering many different newspaper ti-

tles, needs to determine how many of each title to

buy, in order to maximize the total expected proﬁt

from all titles, subject to a budget constraint and a

limited newsstand size. We modify this multi-item

model to include both multiple outputs and inputs:

Each type of input has a different quality level and

is processed into several types of outputs at different

rates. The purchase price for the high-quality input

is typically greater than the low-quality output, and

the high-quality inputs gives a better yield. After pro-

cessing, multiple outputs (e.g., a primary product and

byproducts) are obtained and sold at different prices.

Our model is suitable for agricultural processing

industries. The processing company needs to deter-

mine the purchase quantity for each type of input, be-

fore the random demands for all outputs are known,

in order to maximize the total expected proﬁt. In an

agriculture supply chain, yield and crop quality are in-

ﬂuenced by many factors including seeding time, har-

vest time, weather conditions, presence of nutrients

within the soil at the farm location, and use of fertil-

izers and pesticides. An input type can be speciﬁed

by yield and quality. For instance, in a rice process-

ing industry, outputs from milling a paddy include

head rice, broken rice, rice bran and husk (Wilasi-

nee et al., 2010). A high-quality paddy gives a high

yield of head rice. An input type can also specify

other factors such as the percent impurity, the per-

cent moisture content, the farm location, and the rice

species. In a sugarcane milling process, the yield of

raw sugar depends on the cane quality and the juice

extraction efﬁciency. The supplier of sugarcane with

the greater commercial cane sugar (CCS) receives the

higher purchasing price (Higgins et al., 1998). In a

vineyard harvesting problem, grape qualities depend

on, e.g., the time of harvest, a wine block (a group

of adjacent vineyard’s ﬁelds growing the same vari-

Amaruchkul, K.

Newsvendor Model for Multi-Inputs and -Outputs with Random Yield: Applications to Agricultural Processing Industries.

DOI: 10.5220/0007346900720081

In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 72-81

ISBN: 978-989-758-352-0

ety of grapes), sugar and acidity levels determined by

a winery’s oenologist. The end-products can be clas-

siﬁed into four different types, e.g., Vins de Prestiges

and Vins de Charme (Arnaout and Maatouk, 2010).

In many fresh fruit supply chains (e.g., apple, pear,

banana, coconut), fruits are categorized into different

grades, which usually depend on their sizes, among

other factors. The low-quality farm with many ag-

ing trees and poor soil quality produces smaller sizes,

which are bought at lower prices. Most agricultural

raw materials need to be harvested at certain times of

year. They are prone to deterioration after harvesting

and processing. They are processed into different out-

puts before the demands for the outputs are known.

The yield of the primary (most valuable) output de-

pends on the crop quality. Thus, our multi-input and

-output newsvendor model is applicable.

A literature review of the newsvendor (single-

period inventory) model includes (Qin et al., 2011)

and (Choi, 2012b). Multi-period inventory models are

covered extensively in textbooks in inventory man-

agement, e.g., (Nahmias, 2009), (Silver et al., 1998),

(Zipkin, 2000) and (Porteus, 2002). The multi-item

newsvendor model was ﬁrst formulated in (Hadley

and Whitin, 1963): The order quantity of each item

needs to be made in order to maximize the total ex-

pected proﬁt subject to a single resource constraint.

The budget constraint is a special case of the resource

constraint. When the number of products is small, the

exact solution can be obtained using a dynamic pro-

gramming approach. When the number of products

is large, several heuristics are proposed in, e.g., (Nah-

mias and Schmidt, 1984). (Moon and Silver, 2000)

extends the multi-item newsvendor problem with a

single resource constraint to include ﬁxed ordering

costs. (Lau and Lau, 1995) extends the multi-item

newsvendor problem with a single resource constraint

to multiple resource constraints. To contrast with the

newsvendor problem, they refer to their multi-item

multi-constraint model as the “newsstand problem.”

In the newsstand problem, there is no random yields;

for instance, if we place an order of 50 units of the

New York Times and 50 units of the Chicago Tri-

bune, then all 50 units of the New York Times and

50 units of the Chicago Tribune are received, and

they are put on the newsstand as long as the space

constraint is satisﬁed. In our model, if the rice mill

purchases 50 tons of paddy from a high-quality sup-

plier and 50 tons of paddy from a low-quality sup-

plier, then after the milling process, the total volume

of head rice would be less than 100 tons, and a large

fraction of the total head rice comes from the high-

quality paddy. The multi-product multi-constraint

newsvendor model is a convex optimization problem.

Heuristic solutions based on a quadratic programming

approach are presented in, e.g., (Abdel-Malek and

Areeratchakul, 2007) and (Chernonog and Goldberg,

2018). Various extensions of the multi-production

multi-constraint newsvendor model include an out-

sourcing strategy of shortages (Zhang and Du, 2010),

an allocation decision on the raw material (Xie et al.,

2018), a reservation policy to meet marketing needs

(Chen and Chen, 2010), and carbon cap and trade

mechanism (Zhang and Xu, 2013). The multi-product

newsvendor problem is reviewed in (Turken et al.,

2012) and (Choi, 2012a). In Section 1.5.2 in (Turken

et al., 2012), “nearly all the models in this chap-

ter assume single supplier.... It would be interesting

to incorporate multiple suppliers into MPNP [multi-

product newsvendor problem].” Different input types

can be interpreted as different suppliers, who pro-

vide crops with different quality levels. To the best

of our knowledge, our article is the ﬁrst study of the

newsvendor model with multiple inputs and outputs

whose yields are random.

Operations research (OR) models applied to the

agriculture supply chain are reviewed in (Kusumastuti

et al., 2016), (Soto-Silva et al., 2016) and (Ahumada

and Villalobos, 2009). The quality issue of the agri-

cultural product is one of the most distinctive char-

acteristics of the agriculture supply chain. This fea-

ture is explicitly incorporated in our multi-input and

-output newsvendor model. The demand uncertainty

and the random yield are other important character-

istics of the agriculture OR models, and they are pre-

sented in (Tan and Comden, 2012) and (Kazaz, 2004).

(Tan and Comden, 2012) studies a planning problem,

in which the production quantity at each farm is ran-

dom, depending on the planted area and the crop yield

in that farm. In our model, different input types can

be interpreted as different suppliers or farmers who

provide different crop qualities. (Kazaz, 2004) for-

mulates a two-stage stochastic programming problem

for the production planning in the olive oil industry;

the oil producer can buy olives from farmers at a unit

cost varying with the yield. In our model, the unit cost

also depends on the input type. The high-quality input

type has a high purchasing cost, and the low-quality

input type incurs a high pre-processing cost. This fea-

ture is also presented in these two papers. Neverthe-

less, they consider multiple farms (inputs) but a single

product (output), whereas in ours there are both mul-

tiple inputs and outputs.

In this article, we assume that there are multiple

types of perishable inputs and outputs. In contrast

to the standard multi-item newsvendor, the number

of inputs need not be equal to the number of outputs

in our model. In the standard multi-item newsvendor

Newsvendor Model for Multi-Inputs and -Outputs with Random Yield: Applications to Agricultural Processing Industries

model, there is a one-to-one correspondence between

each input type and each output type. In the agricul-

tural supply chain, the type of input usually represents

the quality of the crop. The better type of input results

in the larger yields of the high-value outputs. Further-

more, we allows the yield to be random. In most agri-

cultural products, the yield tends to ﬂuctuate. We de-

velop the newsvendor for multiple inputs and outputs

with random yields, and derive an optimal purchase

quantity of each type of input in order to maximize

the expected total proﬁt from all outputs.

The rest of the paper is organized as follows: Sec-

tion 2 presents the newsvendor model for multiple in-

puts and outputs with random yields, and the problem

is analyzed in Section 3. We illustrate the application

of our model to the rice processing industry and pro-

vide some managerial insights in Section 4. Finally,

the conclusion is provided in Section 5.

2 FORMULATION

Consider a newsvendor model that produces n differ-

ent outputs from m inputs. Let D

be the random de-

mand of output j. Let x

be the purchase quantity of a

type-i input. We need to decide x = (x

,...,x

)

before knowing the demands (D

,. ..,D

). For

each i = 1,2, .. ., n and j = 1, 2,. .. ,m, let U

i j

) be

the random volume of a type- j output after process-

ing x

units of a type-i input. The volume of the type- j

output depends not only the volume of the input but

also the input type. For a concrete example, in the

rice processing industry, the different moisture con-

tent paddies are categorized into different input types.

The moist paddies with the high percentage of mois-

ture content (%MC) would take longer to dry and re-

sult in a higher weight loss. For instance, a ton of

paddy with 21% MC would incur a weight loss of

100 kilograms, whereas a ton of paddy with 30%MC

would incur a weight loss of 200 kilograms (Thailand

Deparment of Internal Trade, 2018).

Note that the decision x is taken before knowing

output volume U

i j

) and consequently demand D

The probability distributions of volume outputs and

demands are described as follows. After processing,

the realization of output volume U

i j

,ω) becomes

known. Let Ω be the set of all scenarios associated

with output volume. The triplet (Ω, A ,P) is a prob-

ability space, where A is an event and P is a proba-

bility. The expected volume of the type- j output after

processing x

units of a type-i input is given as

E[U

i j

)] =

Ω

i j

,ω)P(dω)

where the above integral is a Lebesque integral. We

assume that Ω is a ﬁnite set. Then, the random vari-

able U

i j

) is discrete, and its expectation can be

written as

E[U

i j

)] =

∑

∈Ω

i j

,ω

)P(U

i j

) = U

i j

,ω

)).

Finally, the demand D

becomes known. Assume

that the demand D

is a real-valued random variable

and independent of the output vector (U

1 j

,. ..,U

m j

)

for each j. Let F

denote the distribution of de-

mand D

and F

−1

be the corresponding quantile func-

tion. Note that the quantile function is strictly increas-

ing on [0,1]. These assumptions are made to avoid

technical difﬁculties, and they are not restrictive in

practice. In strategic models like ours, the different

scenarios Ω of the volume output might be obtained

through experts’ judgments, and in many situations

there are only ﬁnitely possible outcomes (Birge and

Louveaux, 1997).

Let (y)

= max{y, 0} denote the positive part of

a real number y. Assume that the volume of a type- j

output given the input x is deﬁned as

(x) =

∑

i=1

i j

). (1)

The type- j sales are S

(x) = min(Y

(x),D

), the left-

overs are W

(x) = (Y

(x) − D

)

, and the shortages

are T

(x) = (D

−Y

(x))

. The cost parameters are

denoted by:

= per-unit cost of a type-i input

= per-unit selling price of a type- j output

= per-unit salvage value of a type- j output

= per-unit penalty cost for a type- j shortage.

The unit cost c

includes both the purchasing cost and

the pre-processing cost associated with the type-i in-

put. The high-quality input typically has a large pur-

chasing cost but a small pre-processing cost. Among

all types of outputs, the primary output usually has the

largest selling price. The penalty cost includes both a

direct cost of the shortages and a loss of goodwill.

The expected total proﬁt is deﬁned as follows:

π(x) =

∑

j=1

(x) + h

(x) − g

(x)

−

∑

i=1

(2)

∑

j=1

E[(p

− h

+ g

)min(D

∑

i=1

i j

))]

∑

j=1

∑

i=1

E[U

i j

)] −

∑

j=1

E[D

]

−

∑

i=1

. (3)

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

In the ﬁrst summation in (2), the ﬁrst term is the ex-

pected revenue, and the second term is the expected

salvage value from the leftovers from n output types.

The last term is the total cost of all m input types.

Equation (3) follows from the identity

min(a,b) = a − (a − b)

(4)

for any real numbers a and b. The ﬁrst expectation

in (3) is with respect to distributions of D

and U

i j

Assume that we are risk neutral, and our objective is

to maximize the expected total proﬁt:

max{π(x) : x ≥ 0}. (5)

Note that we allow the demand vector (D

,. ..,D

) to

be dependent. Speciﬁcally, the expression (3) for the

total expected proﬁt remains valid, since the expec-

tation of the sum of random variations is the sum of

their expected values.

Our formulation subsumes the standard multi-

product newsvendor model (Turken et al., 2012).

Suppose that the number of inputs is equal to the num-

ber of outputs (i.e., m = n). There is a one-to-one cor-

respondence between each input type and output type,

and the yield rate is 100% for i = j (i.e., U

) = x

and U

i j

) = 0 for i 6= j). In other words, the type-i

input is processed into only the type-i output. Then,

the expected total proﬁt becomes

∑

i=1

E[(p

− h

+ g

)min(D

) − (c

− h

− g

(6)

Let c

= c

− h

be the overage cost, i.e., the expected

per-unit cost of positive inventory remaining at the

end of the period. Let c

= p

− c

+ g

be the under-

age cost, i.e., the per-unit cost of unsatisﬁed demand.

Deﬁne the expected total cost as the sum of the ex-

pected overage and underage costs:

∑

i=1

E[c

− D

)

+ c

− x

)

]. (7)

Using the identity (4) again, (7) becomes

∑

i=1

E[c

− (c

+ c

)min(D

) + c

]. (8)

Comparing (6) and (8), we see that minimizing the to-

tal expected cost is equivalent to maximizing the to-

tal expected proﬁt. Our model reduces to the multi-

product newsvendor model. The (unconstrained) op-

timal purchase quantity is given as

∗

= F

−1

/(c

+ c

)) (9)

= F

−1



− c

+ g

− h

+ g



The term c

/(c

+ c

) is often referred to as the criti-

cal ratio. The optimal purchase quantity is the quan-

tile, evaluated at the critical ratio.

3 ANALYSIS

Assume that p

− h

+ g

≥ 0 for each j = 1,2,...,n.

This assumption is not restrictive; in most practical

cases, the per-unit selling price p

exceeds the per-

unit salvage value h

, and the assumption holds. Fur-

ther assume that

i j

) = A

i j

(10)

where a one-unit type-i input yields a random A

i j

-unit

type- j output. The assumption (10) states that the

volume of a type- j output is linear in the volume of

an input. The multiplicative form is extensively used

in many applied operations research model (e.g., the

linear programming model). For each input type i,

each realization (A

(ω),A

(ω),...,A

(ω)) satisﬁes

∑

j=1

i j

(ω) = 1 and A

i j

(ω) ≥ 0. An example of such

distribution is a Dirichlet distribution, which is a mul-

tivariate generalization of the beta distribution. Note

that a one-unit type-i input on average yields E[A

i j

]

units of a type- j output:

E[A

i j

] =

Ω

i j

(ω)P(dω), (11)

where the integral in (11) is a Lebesque integral. For

a discrete distribution on a sample space Ω, the ex-

pected yield (11) becomes

E[A

i j

] =

∑

∈Ω

i j

(ω



i j

= A

i j

(ω

)



In Theorem 1, we show that the multi-input and

-output newsvendor problem {π(x) : x

≥ 0} is a con-

vex programming problem.

Theorem 1. The expected total proﬁt π(x) is jointly

concave in x.

Proof. We will use the following result on convex-

ity (page 529 in (Heyman and Sobel, 1984)). Let

g : R

→ R be a concave function, and let f : R →

R be a nondecreasing concave function. Then, the

composite function h : R

→ R deﬁned as h(x) =

f (g(x)) is a concave function. For each realiza-

tion ω, f (y;ω) = min(D(ω),y) is concave and non-

decreasing, and g(x;ω) =

∑

i=1

i j

(ω)x

is linear (i.e.,

both concave and convex). From the assumption that

− h

+ g

≥ 0, we have that

− h

+ g

)min(D

(ω),

∑

i=1

i j

(ω)x

)

is concave in x for each realization ω. The ﬁrst term

in (3) is concave, since the expectation of the concave

function is also concave. The second term is linear

under the multiplicative form (10). Thus, the expected

total proﬁt π(x) is concave.

Newsvendor Model for Multi-Inputs and -Outputs with Random Yield: Applications to Agricultural Processing Industries

Theorem 1 shows that the objective function, the

expected total proﬁt, is concave. The constraint

functions are linear. Thus, the multi-input and -

output newsvendor problem (5) is convex program-

ming. Algorithms to solve a convex programming

problem are active research, and they can be classi-

ﬁed into three groups, namely gradient algorithms,

sequential unconstrained algorithms, and sequential-

approximation algorithms (Hillier and Lieberman,

2005).

Concavity of the expected proﬁt function is desir-

able, since we can guarantee that the ﬁrst-order con-

ditions provide the globally optimal solution. The op-

timality conditions are derived in Theorem 2.

Deﬁne the expected per-unit salvage of the type-i

input as:

Ω

∑

j=1

i j

(ω)

P(dω)

∑

j=1

E[A

i j

The expected per-unit price p

and penalty cost g

are

deﬁned similarly. Deﬁne the expected overage cost

as c

= c

− h

. Deﬁne the expected underage cost as

= p

− c

+ g

. The expected critical ratio associ-

ated with the type-i input is

+ c

Let i

∗

= argmax{ξ

} be the input type with the largest

critical ratio.

For short-hand, denote c

= p

− h

+ g

Theorem 2. An optimal purchase quantity that max-

imizes π(x) is x

∗

= 0 for k 6= i

∗

and x

∗

> 0 which

satisﬁes

Ω

∑

j=1

∗

(ω)

∗

(ω)x

∗

)

P(dω) = c

∗

. (12)

Proof. We want to maximize {π(x) : x ≥ 0}. Re-

call the Kuhn-Tucker optimality conditions, there ex-

ist µ

≥ 0 and

∂π(x)

∂x

+ µ

= 0 for each k = 1, 2,. .. ,n.

Then,

∂π

∂x

= 0 for x

> 0

∂π

∂x

≤ 0 for x

= 0.

The condition is also sufﬁcient since π(x) is concave

(see Theorem 1).

Let A

= (A

1 j

2 j

,. ..,A

m j

). Note that

E[S

(x)|A

= A

(ω)] = E[min(D

∑

i=1

i j

(ω)x

)]

∑

i=1

i j

(ω)x

(t)dt

where the last equation follows from the tail-sum for-

mula for expectation. Recall that

E[S

(x)] = E[E[S

(x)|A

]].

Then,

E[S

(x)] =

Ω

E[min(D

∑

i=1

i j

(ω)x

)]P(dω).

Taking the partial derivative with respect to x

and us-

ing the Leibniz’s rule, we have

∂E[S

(x)]

∂x

∂

∂x

Ω

E[min(D

∑

k=1

k j

(ω)x

)]P(dω)

Ω

∂

∂x

E[min(D

∑

k=1

k j

(ω)x

)]P(dω)

Ω

∂

∂x

∑

k=1

k j

(ω)x

(t)dtP(dω)

Ω

i j

(ω)

(

∑

k=1

k j

(ω)x

)P(dω)

where the last equation follows from the fundamental

theorem of calculus. From (3), the partial derivative

of the expected proﬁt ∂π(x)/∂x

becomes

∂π(x)

∂x

∑

j=1

Ω

k j

(ω)

(

∑

`=1

` j

(ω)x

)P(dω) − c

∑

j=1

Ω

k j

(ω)

∗

(ω)x

∗

)P(dω) − c

by construction of x

∗

. Substituting the expression for

∂π(x)/∂x

, the optimality equations become

∑

j=1

Ω

k j

(ω)

∗

(ω)x

∗

)P(dω) − c

+ µ

= 0

(13)

For k = i

∗

, we have x

∗

> 0, µ

= 0, and (13) be-

comes (12). For k 6= i

∗

, (13) becomes

= c

−

∑

j=1

Ω

k j

(ω)

∗

(ω)x

∗

)P(dω).

It follows from the deﬁnition of i

∗

= argmax{ξ

} that

≥ 0 for k 6= i

∗

. The Kuhn-Tucker conditions hold.

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

Theorem 2 allows us to reduce the problem size from

m decision variables to a single decision variable, x

∗

Several efﬁcient one-dimensional search algorithms

are available. We should purchase the “best” input

type i

∗

, and the purchase quantity is given by (12).

Suppose that the number of inputs is identical to

the number of outputs and the yield is 100%. Then,

the optimality condition (12) reduces to

− h

+ g

)

) = c

− h

which is equivalent to the optimality condition in the

multi-product newsvendor (9). Again, we see that our

multi-input and -ouput newsvendor model becomes

the standard multi-product newsvendor model.

Theorem 3 provides a sensitivity analysis: How

would the optimal order quantity change, when some

of the parameters change?

Theorem 3. Assume that the “best” input type re-

mains the same. The optimal order quantity x

∗

larger if one of the following conditions holds (ceteris

paribus):

1. The per-unit selling price p

increases.

2. The per-unit salvage value h

decreases.

3. The per-unit penalty g

increases.

4. The demand D

is stochastically larger (in the

usual stochastic order sense).

5. The per-unit cost c

decreases.

Proof. Note that the right-hand side (RHS) of the op-

timality equation (12)

Ω

∑

j=1

− h

+ g

∗

(ω)

∗

(ω)x

∗

)

P(dω)

is decreasing in x

∗

. Furthermore, for a ﬁxed x

∗

, the

RHS increases, when the per-unit selling price p

in-

creases. Thus, the optimal order quantity becomes

larger. The proof for the other cases is similar.

The directional change in Theorem 3 makes economic

sense. To determine which factors signiﬁcantly affect

the optimal order quantity and the optimal expected

proﬁt, we can create various sensitivity graphs, e.g.,

a tornado diagram and a spider graph (Hillier et al.,

2000). The method to calculate the expected proﬁt is

illustrated in the next section.

4 NUMERICAL ILLUSTRATION

We apply our model to the rice processing industry

in Thailand. Consider a rice mill which purchases

paddy from middleman and farmers. The purchase

price for paddy depends on the percent moisture con-

tent (%MC) and the head yield (HY). The paddy is

classiﬁed into four groups of moisture content:

1. Group 15.0: %MC ≤ 15.0%

2. Group 19.9: %MC between 15.1–19.9%

3. Group 24.9: %MC between 20.0–24.9%

4. Group 29.9: %MC greater than 25.0%

Paddy with the %MC greater than 15% needs to

be dried. The head yield can be divided into three

groups, namely 40%, 45% and 50%. Table 1 shows

the drying cost and the purchase price in Thai Baht

(THB) per ton. For the same head yield, the purchase

cost is higher for the paddy with the lower %MC. For

the same %MC, the purchase cost is higher for the

paddy with the larger head yield. In addition to the

%MC and the head yield, the purchasing price de-

pends on the rice varieties. The purchase prices in

Table 1 are similar to the purchase prices of Thai aro-

matic rice paddy (KDML105); see (Thailand Depar-

ment of Internal Trade, 2018).

Table 1: Purchase prices for paddy.

HY Drying %

%MC 40 45 50 Cost Loss

15.0 19600 20600 21600 0.00 0.0

19.9 18277 19210 20142 59.00 6.9

24.9 16807 17665 18522 62.10 14.4

29.9 15337 16120 16902 65.21 21.9

A type-i input is speciﬁed by the %MC and the

HY. Table 2 shows m = (4)(3) = 12 input types. The

per-unit cost of the type-i input, c

, includes both the

purchasing and drying costs. For instance, the type-

2 input with %MC=19.9 and HY=40, and the drying

cost is 59 THB/ton, so the per-unit cost c

= 18277 +

59 = 18336 THB/ton. In our example, we assume that

a farmer brings paddies to the rice mill. In some other

cases, the rice mill may travel to buy from the farmers

whose produce paddies with high quality. Then, the

input type would specify the %MC and the head yield

as well as the location of the paddy ﬁeld; the cost c

may include the (round-trip) transportation from the

rice mill to the ﬁeld.

After the paddy is dried, a weight loss occurs,

and the percent weight loss depends on the percent

moisture content (see Table 1). The dried paddy is

stored in a silo and waits for milling. The milling pro-

cess consists of husk removing from paddy, whitening

and polishing for bran removal, and ﬁnally separat-

ing head rice (the primary product) and broken rice

into different grades. The milled rice is distributed to

domestic and export markets. The rice bran can be

sold to the rice bran oil industry. The husk is used

Newsvendor Model for Multi-Inputs and -Outputs with Random Yield: Applications to Agricultural Processing Industries

Table 2: Description of 12 input types.

i %MC HY c

1 15.0 40 19600

2 19.9 40 18336

3 24.9 40 16869

4 29.9 40 15402

5 15.0 45 20600

6 19.9 45 19268

7 24.9 45 17727

8 29.9 45 16185

9 15.0 50 21600

10 19.9 50 20201

11 24.9 50 18584

12 29.9 50 16967

in paper production and biomass power generation.

The average selling prices p

, salvage values h

and

the penalty costs g

(in THB/ton) for these n = 4 out-

puts (head rice ( j = 1), broken rice ( j = 2), rice bran

( j = 3) and husk ( j = 4)) are shown in Table 3. In

Thailand, the selling price of head rice and broken

rice are published online by the government (Thailand

Deparment of Internal Trade, 2018). If the shortages

of the head rice occur, then the rice mill tries to pur-

chase the rice from other rice mills. The penalty cost

includes the average purchasing cost, the trans-

portation cost, and the loss of goodwill if the customer

does not receive the entire rice volume in time. In our

example, we assume that there are no penalty costs

for the other byproducts, i.e., g

= 0 for j > 1.

Assume that the demand for the type- j output, D

is independent and normally distributed with mean µ

and standard deviation σ

(in ton), as shown in Ta-

ble 3. The coefﬁcient of variation of demand is suf-

ﬁciently small so that the probability of negative de-

mand is negligible. (Our formulation in Section 2 and

the analytical results in Section 3 hold without nor-

mality assumption.) Let φ denote the density function

of the standard normal distribution and Φ denote the

corresponding cumulative distribution function. The

expected sales can be calculated easily:

E[min(D

)] = µ

− σ



− µ



where the standard loss function is

L(z) = φ(z) −z(1 − Φ(z)).

Table 3: Selling prices, salvage values and penalty costs.

Output type Head Broken Bran Husk

( j) 1 2 3 4

36800 12340 8999 1500

29872 5000 1000 500

1000 0 0 0

94.00 64.00 53.00 24.00

14.10 9.60 7.95 3.60

If the demand does not follow the normal distribution

but other well-known distribution, then the expected

sales can be calculated using the closed-form formula

for the limited expected value (see, e.g., Table 12.2

in (Cunningham et al., 2008)). The demands for these

four outputs are likely to be independent, since they

are different products used in different industries, and

they cannot be substituted for one another. In practice,

we may have more output types. For instance, after

being milled, the broken rice can be further graded

into ﬁve levels, and their demands may be dependent.

Our model can still be applied to the dependent out-

puts. The rice mill needs to determine the purchase

quantities of different types of paddies, before the de-

mands of all outputs are known.

Example 1. In our model, two sources of uncertainty

are the demand and the random yield. For an illustra-

tive purpose, we assume that there are 3 scenarios for

the random yield, Ω = {ω

,ω

}. The actual yields

depend on the rice variety, the operations designed for

rice kernel cracking and breakage, and the mechani-

cal and physical properties (Correa et al., 2007). In

stochastic programming approximations, we can ﬁnd

some relatively low cardinality discrete set of real-

izations that represents a good approximation of the

true underlying distribution of the yield; details can be

found in Chapter 9, (Birge and Louveaux, 1997). In

scenario ω

, one unit (ton) of the type-i paddy results

in A

(ω

) units of head rice, A

(ω

) units of broken

rice, A

(ω

) units of rice bran and A

(ω

) units of

rice husk (see Table 4). Assume that each scenario is

equally likely to occur; i.e.,

P(A

i j

= A

i j

(ω

)) = 1/3

for each ` = 1,2,3. For each input type and for the

type-1 output (head rice), the yield in scenario ω

(resp., ω

) is approximately 10–20% higher (resp.,

lower) than the yield in scenario ω

. The higher the

head yield, the higher yield for head rice. The paddy

with the higher moisture content incurs the larger per-

cent weight loss.

Note that for the paddy with the %MC less than

15% (i.e., i ∈ {1,5, 9}), there is no weight loss, and

∑

j=1

i j

(ω

) = 1. For the other types,

∑

j=1

i j

(ω

) <

1. Suppose that we deﬁne the type-5 output to be the

processing loss. Let A

= 1−

∑

j=1

i j

. Then, the out-

put vector U

i j

) = A

i j

now has a valid probability

distribution, i.e.,

∑

j=1

i j

= 1 and A

i j

≥ 0.

To use the optimality condition (12), we ﬁrst ﬁnd

the “best” type of input by ranking the expected crit-

ical ratio. From Table 5, we ﬁnd that the most prof-

itable input type is i

∗

= 10 (%MC=19.9 and HY=50).

The ranking in Table 5 also helps the rice mill select

a paddy supplier. Note that the paddy types with the

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

low head yield (i.e., i ∈ {1, 2,3, 4}) are ranked among

the lowest. It is more important to focus on the head

yield, compared to the moisture content, since it is rel-

atively cheap to dry the paddy. This ﬁnding is consis-

tent with the result found in the agricultural academic

journal (Wilasinee et al., 2010).

Upon solving the optimality equation (12), we

ﬁnd that the optimal purchase quantity is x

∗

= 187

ton. The total purchasing and drying costs is 3781345

THB. The expected revenue is 3820335 THB, and the

expected proﬁt is 125331 THB. The expected rev-

enues and proﬁts for all scenarios are shown in Ta-

ble 6.

We see that the proﬁt is very sensitive to the out-

put yield of the milling process. In scenario 2, the

head rice yield is 10–20% higher than the yield in sce-

nario 1, but the expected proﬁt in scenario 2 is double.

Our model can also be used to quantify the beneﬁt

from milling process improvement.

Example 2. We want to highlight the beneﬁt of our

Table 4: Yields of the four outputs A

i j

(ω

` i A

(ω

) A

(ω

) A

(ω

) A

(ω

)

1 1 0.4000 0.1976 0.2541 0.1483

1 2 0.3724 0.1840 0.2366 0.1380

1 3 0.3424 0.1692 0.2175 0.1269

1 4 0.3124 0.1544 0.1985 0.1158

1 5 0.4500 0.1812 0.2329 0.1359

1 6 0.4190 0.1687 0.2169 0.1265

1 7 0.3852 0.1551 0.1994 0.1163

1 8 0.3515 0.1415 0.1819 0.1061

1 9 0.5000 0.1647 0.2118 0.1235

1 10 0.4655 0.1533 0.1972 0.1150

1 11 0.4280 0.1410 0.1813 0.1057

1 12 0.3905 0.1286 0.1654 0.0965

2 2 0.4096 0.1468 0.2366 0.1380

2 3 0.3664 0.1452 0.2175 0.1269

2 4 0.3218 0.1450 0.1985 0.1158

2 5 0.4590 0.1722 0.2329 0.1359

2 6 0.4358 0.1519 0.2169 0.1265

2 7 0.3929 0.1474 0.1994 0.1163

2 8 0.3796 0.1134 0.1819 0.1061

2 9 0.5100 0.1547 0.2118 0.1235

2 10 0.4934 0.1254 0.1972 0.1150

2 11 0.4323 0.1367 0.1813 0.1057

2 12 0.4022 0.1169 0.1654 0.0965

3 1 0.3920 0.2056 0.2541 0.1483

3 2 0.3650 0.1914 0.2366 0.1380

3 3 0.3424 0.1692 0.2175 0.1269

3 4 0.2968 0.1700 0.1985 0.1158

3 5 0.4230 0.2082 0.2329 0.1359

3 6 0.3813 0.2064 0.2169 0.1265

3 7 0.3852 0.1551 0.1994 0.1163

3 8 0.3515 0.1415 0.1819 0.1061

3 9 0.4600 0.2047 0.2118 0.1235

3 10 0.4469 0.1719 0.1972 0.1150

3 11 0.4237 0.1453 0.1813 0.1057

3 12 0.3749 0.1442 0.1654 0.0965

Table 5: Expected overage and underage costs and the rank-

ing based on the expected critical raio.

i %MC HY c

Rank

1 15.0 40 6269 535 0.079 11

2 19.9 40 5739 600 0.095 9

3 24.9 40 5315 513 0.088 10

4 29.9 40 5093 219 0.041 12

5 15.0 45 6100 793 0.115 7

6 19.9 45 5802 616 0.096 8

7 24.9 45 5123 782 0.132 6

8 29.9 45 4509 883 0.164 5

9 15.0 50 5816 1169 0.167 4

10 19.9 50 5197 1313 0.202 1

11 24.9 50 4860 1124 0.188 2

12 29.9 50 4478 981 0.180 3

Table 6: Expected proﬁt.

Scenario 1 2 3

Head rice sales 84.29 87.52 81.75

Revenue 3819035 3873344 3768627

Proﬁt 113336 230482 32176

model which includes multiple outputs (i.e., the head

rice as the primary product and all byproducts). In

particular, we will identify the set of parameters such

that our model signiﬁcantly outperforms the standard

newsvendor model. For simplicity, assume that Ω =

{ω

} and that there are no penalty costs g

= 0 for

all j.

Suppose that the decision maker does not consider

multiple outputs and focuses only on the single pri-

mary output, which is the head rice (i.e., n = 1 and

j = 1). Then,

= p

E[A

] = p

(ω

)

= p

E[A

] = h

(ω

When the decision maker considers only one input

(n = 1), let i

∗

(n = 1) be the best input type and ξ

∗

(n=1)

the corresponding critical ratio. The optimality con-

dition (9) reduces to

∗

(n=1)

= A

(ω

−1

(ξ

∗

(n=1)

) (14)

= A

(ω

)[µ

+ σ

−1

(ξ

∗

(n=1)

)]

if the critical ratio ξ

∗

(n=1)

> 0; otherwise the purchase

quantity is zero.

We will vary the selling price of the primary input

from 400000 to 70000 THB. Suppose that the sell-

ing price of the head rice is large, say p

= 65000.

Then, the type-9 input has the largest critical ratio.

Based on (14), the decision maker purchases 197 tons

of the type-9 paddy and obtains the expected proﬁt of

2667857. When we consider all output types (i.e., n =

4), an optimal purchase quantity is 215 tons, and the

optimal expected proﬁt is 2702104; the percent proﬁt

loss is (2702104 − 2667857)/2702104 = 1.27%. On

Newsvendor Model for Multi-Inputs and -Outputs with Random Yield: Applications to Agricultural Processing Industries

Table 7: Proﬁt loss from ignoring byproducts.

Purchase quantity

n = 1 n = 4 Opt. proﬁt %∆ proﬁt

40000 0 186 415232 100.00

45000 155 198 862410 11.35

50000 176 205 1317887 4.32

55000 186 209 1777279 2.52

60000 192 213 2238948 1.71

65000 197 215 2702104 1.27

70000 200 218 3166306 0.99

the other hand, suppose that the selling price of head

rice is small as in the previous example, p

= 36800.

Then,

= p

9,1

(ω

) = 18400 < 21600 = c

and consequently ξ

< 0, the decision maker that ig-

nores all byproducts would not buy any paddies. Ta-

ble 7 shows the purchase quantity when the decision

maker focuses only on the head rice (n = 1), the op-

timal purchase quantity when all output types (n = 4)

are considered, the corresponding optimal expected

proﬁts, and the percent proﬁt loss, as the selling price

of the head rice varies. The purchase quantity when

the decision maker ignores all byproducts is less than

the optimal quantity. The difference between the two

quantities becomes smaller when the selling price of

the head rice becomes larger. In other words, our

model outperforms the standard newsvendor model

when the processing ﬁrm has a signiﬁcant revenue

potential from the other products (other than the pri-

mary product). We can perform an ABC classiﬁcation

on all outputs to determine their importance. If the

volumes of some byproducts are signiﬁcant or their

selling prices are not negligible, the processing ﬁrm

could beneﬁt from using our model with multiple out-

puts.

5 CONCLUSION

In summary, we formulate the multi-input and -output

newsvendor model with the random yield. The opti-

mization problem of determining the purchase quan-

tity of each input type is shown to be a convex pro-

gramming problem. In the numerical example, we use

the optimality condition to ﬁnd the optimal purchase

quantities of different types of paddy for the rice mill,

who wants to maximize the total expected proﬁt from

the head rice and its byproducts, namely the broken

rice, bran and husk.

Some future research directions are identiﬁed

below. In the standard multi-product newsvendor

model, the resource constraints are considered. We

could extend our model to include the resource con-

straints. Obtaining some structural results and de-

riving the optimality condition for the multi-input

and -output newsvendor model with resource con-

straints are interesting from a theoretical viewpoint.

From a practical viewpoint, we could consider price-

dependent demands. In many agricultural products,

the selling price itself is random, depending on var-

ious factors such as the world economic output, the

global production and consumption. In our current

model, the selling price is deterministic, and the de-

mand distribution is given exogenously. A stochas-

tic programming framework can be used to capture

the price-dependent demand. Another extension is to

consider contract farming, one of the most common

arrangements between a farmer and a private agricul-

tural processing company, wherein the farmer agrees

to produce at a pre-agreed market price for procure-

ment by the other party. A mechanism design frame-

work can be applied in order to improve efﬁciency of

the agriculture supply chain through an optimal con-

tract. Finally, when the variability of yield is signiﬁ-

cant, and the decision maker is no longer risk neutral,

the objective of maximizing the expected proﬁt would

not be suitable. Instead, one could maximize the ex-

pected utility, in which risk is explicitly taken into ac-

count. We hope to pursue these or related issues in

the future.

ACKNOWLEDGEMENTS

The problem was materialized after some discussions

with Mr. Chatbodin Sritrakul, our part-time master

student who owns a rice mill in the Northeast of Thai-

land. His independent project, a part of requirement

for a master degree in logistics management at the

school, was related to our model.

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