Effects of variable prepayment installments on pricing and inventory decisions with power demand pattern and non-linear holding cost under carbon cap-and-price regulation

Khan, Md. Al-Amin e al.
A icle
E ec s o a iable p epaymen ins allmen s on p icing and in en o y
decisions wi h powe demand pa e n and non-linea holding cos
unde ca bon cap-and-p ice egula ion
Ope a ions Resea ch Pe spec i es
P o ided in Coope a ion wi h:
Else ie
Sugges ed Ci a ion: Khan, Md. Al-Amin e al. (2024) : E ec s o a iable p epaymen ins allmen s on
p icing and in en o y decisions wi h powe demand pa e n and non-linea holding cos unde
ca bon cap-and-p ice egula ion, Ope a ions Resea ch Pe spec i es, ISSN 2214-7160, Else ie ,
Ams e dam, Vol. 12, pp. 1-24,
h ps://doi.o g/10.1016/j.o p.2023.100289
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E ec s o a iable p epaymen ins allmen s on p icing and in en o y
decisions wi h powe demand pa e n and non-linea holding cos unde
ca bon cap-and-p ice egula ion
Md. Al-Amin Khan
a
,
b
, Leopoldo Edua do C´
a denas-Ba ´
on
a
,
c
,
*
, Ge a do T e i˜
no-Ga za
d
,
A mando C´
espedes-Mo a
a
, Imelda de Jesús Loe a-He n´
andez
a
, Neale R. Smi h
a
a
Tecnol´
ogico de Mon e ey, School o Enginee ing and Sciences, E. Ga za Sada 2501 Su , C.P. 64849, Mon e ey, Nue o Le´
on, Mexico
b
Depa men o Ma hema ics, Jahangi naga Uni e si y, Sa a , Dhaka 1342, Bangladesh
c
Yonsei F on ie Lab, Yonsei Uni e si y, Seoul 03722, Sou h Ko ea
d
Ing am School o Enginee ing, Texas S a e Uni e si y, 601 Uni e si y D i e San Ma cos, TX 78666, USA
ARTICLE INFO
Keywo ds:
Va iable p epaymen ins allmen
Powe demand
Non-linea holding cos
Ca bon emissions
Ca bon a i ules
ABSTRACT
Regula o s’ inc easingly s ingen ca bon ules o p o ec he en i onmen a e encou aging p ac i ione s o
modi y hei ope a ional ac i i ies ha a e accoun able o eleasing emissions in o he a mosphe e. The eby,
p ac i ione s dealing wi h p oduc in en o y planning a e seeking p ope managemen s a egies no only o
inc ease p o i s bu also o educe eleased ca bons om ope a ions. In addi ion, inc easing unce ain y in supply
ope a ions has mo i a ed supplie s o impose p epaymen mechanisms in ecen decades. This s udy examines
he bes p epaymen ins allmen policy o a p ac i ione o he i s ime, whe e he consump ion beha io o
consume s changes as a esul o he combined e ec s o uni selling p ice and s o age ime. Mo eo e , o make
he p esen in en o y planning mo e ealis ic, he uni holding cos unc ion is adop ed as a powe unc ion o he
in en o y uni ’s s o age pe iod. The goal o his s udy is o p o ide he bes combined ins allmen o ad ance
paymen , p ice, and eplenishmen s a egies o a p ac i ione unde cap-and-p ice, cap-and- ade, and ca bon
ax en i onmen al guidelines by ensu ing maximum p o i . Fo his pu pose, an algo i hm is c ea ed by
combining all de i ed heo e ical esul s om he analy ical s udy, whe eas he e icacy o he algo i hm is
assessed h ough he examina ion o i e illus a i e nume ical ins ances. A ple ho a o no ewo hy managemen
insigh s o he p ac i ione a e ob ained by in es iga ing he dynamic shi s in op imal s a egies esul ing om
luc ua ions in sys em pa ame e s. The esul s e eal ha i he demand is low in he nascen phases o he
business cycle, hen he p uden app oach o he p ac i ione en ails p ocu ing a compa a i ely smalle lo -size
using a modes numbe o paymen equencies and hen se ing a ela i ely small uni selling p ice o inc ease
p o i s.
1. In oduc ion
To achie e he goals o sus ainable de elopmen , one o he g ea es
global challenges is con olling he emission o g eenhouse gasses, and
he e o e, many coun ies a ound he wo ld a e execu ing en i on-
men al egula ions o lessen he nega i e impac o emissions on clima e
change. I is impo an o no e ha , due o ac i i ies like wa ming,
eezing, ligh ing, and p oduc handling, wa ehousing in he business
indus y is one o he majo sou ces o ca bon emissions. Fu he mo e,
main aining a company’s cash low by selec ing a sui able way o
collec ing paymen s om cus ome s is one o he mos di icul jobs. An
e ec i e way o a business o sus ain cash low and lowe o de can-
cella ions o pos ponemen s is h ough p epaymen o ad ance paymen
a angemen s. The i ems’ demand a e du ing he s o age p ocess is
ano he c ucial ac o ha has a subs an ial impac on any company’s
in en o y managemen p ac ices. Th ough he du a ion o s o age,
cus ome p e e ences o i em consump ion may a y. Fo ins ance, due
o hei eshness, ce ain commodi ies like packaged oods, ui s, and
mea wi h expi a ion da es a e in highe demand ea ly in he s o age
pe iod. On he o he hand, since hey end o d y ou wi h ime, allowing
* Co esponding au ho .
E-mail add esses: [email p o ec ed] (L.E. C´
a denas-Ba ´
on), [email p o ec ed] (G. T e i˜
no-Ga za).
Con en s lis s a ailable a ScienceDi ec
Ope a ions Resea ch Pe spec i es
jou nal homepage: www.else ie .com/loca e/o p
h ps://doi.o g/10.1016/j.o p.2023.100289
Recei ed 20 June 2023; Recei ed in e ised o m 3 No embe 2023; Accep ed 8 No embe 2023
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
2
cus ome s o u ilize hem o longe pe iods, goods like po a oes, ga lic,
and da es may expe ience a ise in demand owa ds he end o he in-
en o y cycle. As opposed o his, he demand o goods like u ni u e,
elec ical appliances, iles, and d ape ies o en emains s eady
h oughou he cycle. To accoun o hese a ying cha ac e is ics, he
demand pa e n is assumed o be a mix o wo ac o s: he impac o
s o age ime, which is modelled as a powe pa e n, and he impac o
p icing, which is ep esen ed in a linea o m.
Taking hese ac o s in o accoun simul aneously, he cu en s udy
digs in o he op imiza ion o ins allmen plans o p ac i ione s by
ho oughly examining he e ec s o a iable p epaymen ins allmen s
on p o i abili y. This s udy ocuses pa icula ly on a case in which
cus ome demand is iewed as a composi e unc ion ha combines he
impac o s o age ime ollowing a powe pa e n and p icing s uc u e
in a linea o ma . All o hese ac o s a e aken in o accoun in he
con ex o s ic ax laws ha egula e ca bon emissions. This esea ch
add esses he en i onmen al impac om he pe spec i e o ca bon
emissions es ic ions, in addi ion o o e ing p ac i ione s use ul in-
sigh s in o inancial decision-making. I is a ho ough and o wa d-
looking examina ion since i akes in o accoun he dual objec i es o
maximizing p o i and upholding ecologically app op ia e p ac ices.
1.1. Mo i a ion
The En i onmen al P o ec ion Agency (EPA) o he US e ealed on
i s websi e ha indus y, he hi d la ges emission sou ce, accoun ed o
nea ly one-qua e o he g eenhouse gasses eleased in he yea 2020
(h ps://www.epa.go /ghgemissions/sou ces-g eenhouse-gas-emission
s). Fu he mo e, he eu opean commission (EC) has e ealed ha he
bigges con ibu o o global wa ming is ca bon emissions om human
ac i i ies (h ps://clima e.ec.eu opa.eu/clima e-change/causes-c
lima e-change_en). To cu b he inc easing end o global wa ming,
go e nmen s in de eloped coun ies ha e aken se e al ini ia i es, one
o which is he en o cemen o s ingen ca bon ules on businesses o
educe ca bon emissions. Finnish au ho i ies implemen ed a ca bon ax
policy o he i s ime in he wo ld in 1990, and a yea la e , Swedish
au ho i ies also imposed a ax egula ion on ca bon emissions. The
Wo ld Bank epo s ha 68 ca bon p icing measu es ha e been adop ed
o a e expec ed o be employed by June 2022 in 46 di e en na ions
(h ps://www.we o um.o g/agenda/2022/07/ca bon- ax-emissions-co
un ies/). Fig. 1 highligh s di e en coun ies’ ax a es in US dolla s
agains each me ic on o CO
2e
(ca bon dioxide equi alen ). A he
beginning o he yea 2022, U uguayan au ho i ies es ablished a ax
egula ion agains ca bon emissions o he i s ime in hei coun y,
and he ax a e is he highes in he wo ld a $137 pe me ic on o CO
2e
as o Ap il 2022. No icing he posi i e consequences in e ms o educing
ca bon emissions, such as in U uguay, many coun ies a e going o enac
mo e s ic ax egula ions on companies. As a esul , companies in
di e en coun ies a e awa e o hei ope a ional ac i i ies du ing he
cou se o business, which a e sou ces o ca bon emissions, and hey a e
seeking app op ia e amendmen s o hei bes business policies unde
a ious ca bon egula ions.
Wa ehousing in he business sec o is one o he signi ican sou ces o
ca bon emissions because o wa ming, eezing, ligh ing, and p oduc
handling ope a ions [30]. Acco ding o he Wo ld Economic Fo um
[46], 13% o eleased emissions om supply chain ac ions a e p oduced
om s o age ac i i ies in logis ics acili ies, including p oduc handling
ope a ions. Fig. 2 highligh s he amoun o eleased CO
2e
(ca bon di-
oxide equi alen ) annually in Sweden om indus ial p ocessing and
p oduc consump ion om 2011 o 2020. The e o e, companies a e now
planning hei in en o y by aking in o accoun he amoun o ax on he
eleased emissions unde he imposed ca bon ax egula ions. In he
mean ime, wi h ambi ions o signi ican ly educe CO
2e
, many coun ies
a e e ising hei egula ions by imposing highe ax a es on each uni
o emissions. As an example, he ax a e in Sweden was $123 pe on o
CO
2e
in 2019, $126 pe on o CO
2e
in 2020, and $130 pe on o CO
2e
in
2022 (which is he second highes a e in he wo ld).
1,2
Due o an in-
c ease in he ax a e, Sweden’s o al CO
2e
emissions om indus ial
p ocessing and p oduc consump ion dec eased by 16.43% (1.3 million
ons) in 2020 in con as o he p e ious yea . Fu he mo e, he Po u-
guese go e nmen nea ly doubled he ax a e in 2020 o
€
24 pe on o
CO
2e
emissions, while he No wegian go e nmen inc eased he ax a e
o 8.16% in 2019 [1]. Unde hese s ic ax egula ions, companies
ough o me iculously p epa e hei bes in en o y and business plans
wi hou comp omising hei p o i abili y.
One o he mos challenging asks o any company is o main ain i s
cash low by choosing an app op ia e me hod o collec ing paymen s
om cus ome s. In addi ion, cancela ion o pos ponemen o o de s can
become a nigh ma e o any manu ac u e o supplie , and hence, con-
olling clien s’ o de cancella ions o pos ponemen s is ano he big
challenge o he manu ac u e s o supplie s o a ious p oduc s. A
p epaymen o ad ance paymen a angemen is an e ec i e one o a
company, no only o main ain cash low bu also o educe o de can-
cella ions o pos ponemen s. As a esul , p epaymen mechanisms ha e
become a common s a egy o many companies (supplie s) o a ious
p oduc s in he p esen business sec o . Fig. 3 illus a es he di e en
exis ing paymen me hods o companies in Eu ope in he yea 2019. As
seen om Fig. 3, hal o he o al companies in Eu ope o e ed p epay-
men me hods (pa ially o ully) in 2019. Howe e , as i is no possible
o ea n any e enue om he in es ed p epaymen amoun be o e
ecei ing p oduc s, he e is an oppo uni y expense o capi al cha ge
agains he paid amoun in ad ance. To educe his oppo uni y o
capi al cos o a p ac i ione , companies o e he mul iple-ins allmen
acili y o accomplishing he p epaymen [16]. Mos impo an ly, hey
did no conside any ansac ion cha ge o ins allmen s, hough he e is
always a ansac ion cha ge o all p epaymen ins allmen s. Fo
ins ance, when ins allmen s a e comple ed by exchanging money
in e na ionally o one bank accoun o ano he accoun , a ansac ion
cos is incu ed acco ding o he exis ing ules; when he ins allmen s a e
comple ed by he p ac i ione i sel o by sending an agen , he ans-
ac ion cha ge comp ises he anspo a ion cos , and ime and labo
cos s. Consequen ly, he esea ch issue ha a ises in his case is: Wha is
he ideal p epaymen ins allmen me hod o minimize capi al cos s o
inc ease p o i s?
1.2. Resea ch ques ions and goals
The clien s’ beha io ega ding i ems’ consump ion may a y o e
he s o age pe iod. Fo ins ance, o he eason o hei eshness, ce ain
i ems (such as packaged oods, ui s, mea wi h an expi a ion da e, e c.)
a e in mo e demand ea ly on in he s o age pe iod. Again, some p od-
uc s, like po a oes, ga lic, da es, e c., may be in mo e demand owa ds
he end o he in en o y cycle because o hei d ying condi ions, which
encou age consume s o consume o longe pe iods o ime, whe eas
o he p oduc s, like u ni u e, elec ical appliances, iles, and cu ains,
ha e basically consis en demand h oughou he cycle. To adop hese
cha ac e is ics, he clien s’ demand pa e n is assumed o be an addi i e
combina ion o he e ec s o s o age ime in powe pa e n and p ice in
linea o m. Now, he ollowing esea ch que ies appea :
1
h ps:// ax ounda ion.o g/sweden-ca bon- ax- e enue-g eenhouse-gas-e
missions/
2
h ps://www.ca bonp icingleade ship.o g/blogs/2019/10/18/should-
e e y-coun y-on-ea h-copy-swedens-ca bon- ax
3
h ps://www.s a is a.com/s a is ics/483590/p ices-o -implemen ed-ca bon
-p icing-ins umen s-wo ldwide-by-selec -coun y/
4
h ps://www.s a is a.com/s a is ics/412202/annual-g eenhouse-gas
-emissions- om-indus ial-p ocessing-in-sweden/
5
h ps://www.s a is a.com/s a is ics/931239/sha e-o -paymen -me hods
-o e ed-by-co po a ions-in-eu ope/
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
3
Fig. 1. Coun y-wise ax a es o ca bon emissions in 2022 (S a is a, ID: 483,590
3
).
Fig. 2. Annual CO
2e
emissions in Sweden om indus ial p ocessing and p oduc use om 2011 o 2020 (S a is a, ID: 412,202
4
).
Fig. 3. Paymen app oaches o businesses in Eu ope in 2019 (S a is a, ID: 931,239
5
).
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
4
(i) How do en i onmen al guidelines a ec a p ac i ione ’s bes in-
en o y planning when demand ollows a powe pa e n in he
s o age phase?
(ii) Wha is he ideal p epaymen ins allmen me hod o minimize
capi al cos s o inc ease p o i s unde en i onmen al guidelines?
(iii) The p ac i ione ’s p o i pe uni o ime is impac ed by ca bon
laws o wha ex en ?
The esea ch goals o his s udy a e o add ess he esea ch ques ions
aised abo e app op ia ely, which a e succinc ly and sys ema ically
summa ized as ollows:
•To examine he e ec s o a a iable p epaymen ins allmen s a egy
on he p o i abili y o a p ac i ione .
•To unde s and how cus ome demand, modeled as a composi e
unc ion o s o age ime and p icing s uc u e, in luences
p o i abili y.
•To p o ide insigh s in o inancial decision-making o p ac i ione s
by conside ing he impac o s ic ax laws go e ning ca bon
emissions.
•To balance he dual objec i es o maximizing p o i and adhe ing o
en i onmen ally esponsible p ac ices.
2. Li e a u e e iew
The pu pose o his wo k is o in eg a e he p epaymen ins allmen
policy in o an in en o y planning p oblem o a p ac i ione ha
simul aneously includes he p icing and eplenishmen s a egies unde
ca bon a i egula ions such as cap-and-p ice (C&P), cap-and- ade
(C&T), and ca bon ax (CT) guidelines. In he li e a u e, hough he
p icing and eplenishmen s a egies unde ca bon a i egula ions a e
in es iga ed, hese s a egies a e no in es iga ed unde ca bon a i
egula ions a e inco po a ing he simul aneous e ec s o s o age ime
and uni selling p ice on he clien s’ consump ion beha io , while no
exis ing wo k is a ailable dealing wi h he p epaymen ins allmen
policy o a p ac i ione . This sec ion p o ides an ou line o he exis ing
li e a u e ela ed o his esea ch wo k.
Due o he na u e o oday’s compe i i e business, p ac i ione s mus
c ea e e icien in en o y managemen models o maximize p o i s a e
mee ing cus ome demands a a ai p ice wi hin a easonable ime
ame in o de o p e en losing ma ke sha e o i als. Clien s’ con-
sump ion beha io o any p oduc is no always he same; a he , i
luc ua es o e he s o age pe iod due o a ious ealis ic cha ac e is ics
like age o in en o y, eshness o in en o y, selling p ice, a ailabili y o
i ems, and o he s. When demand is in e connec ed wi h in en o y ages
and i s s uc u e is cha ac e ized by he quo ien o p oduc s’ s o age
du a ion and cycle leng h, hen he demand is e e ed o as “powe
demand,” which was in en ed in in en o y analysis by Naddo [24].
Relaxing a ixed business cycle in powe demand, Sicilia e al. [38]
explo ed he bes in en o y planning o a company whe e i s business
cycle is adop ed as a a iable. A e wa ds, Sicilia e al. [39] analyzed he
bes eo de poin o a p ac i ione unde powe demand in a
manu ac u ing sys em. Taking in o conside a ion he impe ec p o-
duc ion, Kesha a z a d e al. [13] u he ex ended he manu ac u ing
sys em wi h a powe demand o Sicilia e al. [39]. These s udies did no
in es iga e he consequences o selling p ice along wi h he conse-
quences o p oduc s’ age on he consump ion beha io . A ianada e al.
[3] desc ibed wo in en o y p oblems in which one akes in o accoun
he mul iplica i e impac s o p ice and p oduc s ock age on demand,
while he o he akes in o accoun he addi i e e ec s o p ice and
p oduc s ock age. Desc ibing he impac s o p ice as a logi unc ion and
o he p oduc ’s s ock age as powe demand, he ideal p ice and s ock
plans o a company a e de i ed by San-Jos´
e e al. [33]. Subsequen ly,
San-Jos´
e e al. [34] u he in es iga ed demand as a composi e unc ion
comp ising wo dis inc and sepa able componen s: one associa ed wi h
he ma ke p ice and he o he wi h he du a ion o s o age, while he
e ec s o p ice and s o age ime a e desc ibed as linea and powe
pa e ns, espec i ely. Then, hey explo ed he op imal p ice and s ock
plans o a p ac i ione unde he consequences o non-linea holding
cos s. La e , C´
a denas-Ba ´
on e al. [4] u he expanded he s udy o
San-Jos´
e e al. [34] in a s ockou en i onmen . In his esea ch di ec ion,
some no ewo hy wo ks a e San-Jos´
e e al. [35–37]. Howe e , all wo k
published ela ed o powe demand has a common ea u e ha paymen
ollows a cash-on-deli e y p ocess ha is incompa ible wi h many cu -
en business en i onmen s.
As supply chains become mo e agile, supplie s a e e y conce ned
abou cancelling o pos poning o de s om hei clien s. As a esul ,
supplie s ask o a comple e o ac ional ad ance paymen om hei
cus ome s, and his paymen me hod is known as “p epaymen ” o
“ad ance paymen ” in in en o y managemen li e a u e. This p epay-
men mechanism is widely popula in China’s apidly ising au omo i e
indus y sec o [48]. Acco ding o Zia and Taleizadeh [49], and Dua y
e al. [7], he e a e wo key bene i s o supplie s om he p epaymen
mechanism: (i) hey can con ol o de cancela ion ma kedly, and (ii)
hey ha e he oppo uni y o acc ue in e es on he p epaymen sum. On
he o he hand, as i is no possible o ea n any e enue om he in es ed
p epaymen amoun be o e ecei ing p oduc s, he e is an oppo uni y
expense o capi al cha ge o he p ac i ione agains he p epaymen
sum [29]. To educe his oppo uni y o capi al cos o a p ac i ione ,
supplie s o e a mul iple-ins allmen acili y o accomplishing he
p epaymen [42]. Taleizadeh e al. [42] explo ed he consequences o
using many paymen s as p epaymen s a he han a single paymen .
La e , Taleizadeh [43] ex ended he p epaymen ag eemen o decay
i ems. Acco ding o he a iable p epaymen legisla ion Khan e al. [17]
sugges ed, me chan s would be compelled o p epay a iny po ion o he
acquisi ion cos o g ea e quan i ies acqui ed. Howe e , all he
abo e-s a ed published wo ks ela ed o p epaymen did no conside
any ansac ion cha ge o ins allmen s, hough he e is always a
ansac ion cha ge o e e y ins allmen o p epaymen , such as money
exchanging cha ges (na ionally o in e na ionally) om one bank o
ano he , anspo a ion cha ges, and ime and labo cos s.
Du ing he holding cos compu a ion o any in en o y p ocedu e, i is
gene ally p esumed ha he uni ca ying expense is linea in he
p oduc ’s s ock du a ion, hough his assump ion may be incompa ible
o many p oduc s in p ac ice. Adop ing he expense o ca ying each
uni as a non-linea unc ion o he p oduc ’s s ock du a ion, Naddo
[24] desc ibed an in en o y p ocedu e o a p ac i ione . Weiss [45]
showed ha when he p oduc s’ alue alls non-linea ly wi h espec o
hei s ock du a ion, he in en o y models wi h he expense o ca ying
each uni as a non-linea unc ion o he p oduc ’s s ock du a ion can be
sui able. The model ha Weiss [45] p e iously looked a was eex-
amined in Fe guson e al.’s wo k [9], which cla i ied he model’s
unc ion as an app oxima ion o he bes o de ing s a egy o pe ishable
commodi ies. Taking in o accoun he non-linea i y wi h espec o a
p oduc ’s s ock du a ion, no only in holding cos bu also in demand,
Pando e al. [26] examined he bes in en o y planning o a p ac i ione
o maximize p o i . San-Jos´
e e al. [31] included bo h a a iable uni
ca ying expendi u e, which was modeled as a non-linea powe ela-
ionship wi h s o ing ime, and a cons an uni ca ying expense, which
e lec ed he expenses o accommoda ing p oduc s in he wa ehouse.
Al a es and Ghai han [2] ound ha he expense o holding each uni is
di ec ly dependen on he uni acquisi ion cos and also exhibi s a lin-
ea ly inc easing ela ionship wi h he p oduc ’s s ock du a ion. La e ,
Shaikh e al. [40] conduc ed ano he analysis inco po a ing a linea ly
inc easing uni holding expense o decay i ems, while Khan e al. [15]
explo ed he consequences o a linea ly inc easing uni holding cos
unc ion on he p o i o a p ac i ione dealing wi h p oduc s ha ing a
maximum li e ime. Some o he in es iga ed in en o y models unde a
non-linea uni holding cos a e Khalilpou aza i and Pasandideh [14],
Paknejad e al. [25], Pando e al. [27], Edala pou and Mi zapou
Al-e-Hashem [8] and Pando e al. [28].
The wa ming, eezing, ligh ing, and p oduc handling p ocedu es
Md.A.-A. Khan e al.

Ope a ions Resea ch Pe spec i es 12 (2024) 100289
5
in ol ed in wa ehousing a e among he majo causes o ca bon emis-
sions wi hin a supply chain [30]. Acco ding o he Wo ld Economic
Fo um [46], 13% o g eenhouse gas emissions om supply chain busi-
ness ope a ions a e p oduced om s o age ac i i ies in logis ics acil-
i ies, including p oduc handling ope a ions. Au ho i ies in a ious
coun ies (see Fig. 1) impose a numbe o egula ions on companies wi h
he aim o seeking he bes in e es s o ou socie y a he han indi idual
in e es s ( he in e es s o he companies) [12]. Conside ing he o de ing
and holding ope a ions as ca bon emission sou ces in an in en o y
p ac ice o a p ac i ione , Hua e al. [11] in es iga ed he impac s o
C&T egula ion on he in en o y planning. The ideal o de ing s a egy
unde he ca bon cap (CC) legisla ion was u he explo ed by Chen e al.
[5] by aking p ocu ed p oduc s in o accoun as an addi ional cause o
emissions in a p ac i ione ’s s ock sys em. La e , analyzing he in es -
men s a egy o a p ac i ione o sh ink eleased ca bons om se e al
ope a ions in an in en o y sys em, Top al e al. [44] in es iga ed he
op imal in en o y decision unde h ee ca bon a i egula ions (C&T,
CT, and CC). Ho elaque and Bi onneau [10] in eg a ed no only he
ca bon a i s bu also he impac s on cus ome s’ demand o he eleased
ca bon amoun s by a p ac i ione in an in en o y sys em. Chen e al. [6]
analyzed an adjus men s a egy o a e aile unde he C&T egula ion
and a c edi pe iod oppo uni y. The ideal eo de s a egies o se e al
me chan s ope a ing ou o a single wa ehouse whe e a ca bon a i is
applied o he emissions om pu chasing and s o ing ope a ions we e
co e ed by Li and Hai [22]. A e wa d, he s ock managemen plan o a
p ac i ione wi h ime-sensi i e clien demand unde ca bon a i ules
adop ing a pa icula business cycle was examined by Xu e al. [47].
They did no , howe e , ake in o accoun how p ices a ec cus ome s’
pu chasing decisions. Lee [21] con as ed he bes p ac ice wi h C&T,
CC, CT, and CO (ca bon o se ) ules a e u he examining he bes
pu chasing s a egy o a co po a ion unde he C&P egula ion.
Recen ly, Khan e al. [18] discussed he signi icances o ca bon a i on
he bes ci cula indexing s a egy policy o a manu ac u e unde he
CT egula ion. Fu he mo e, a compa ison o simila s udies is unde -
aken o highligh he esea ch’s uniqueness, as shown in Table 1.
The discussion pe o med abo e e eals ha he e is only one p e-
ious pape [47] ha has s udied he in en o y s a egy o a p ac i-
ione wi h ime-sensi i e clien demand unde ca bon a i egula ions
adop ing a speci ic business cycle. They did no , howe e , ake in o
accoun how p ices a ec cus ome s’ pu chasing decisions. This wo k
a emp s o in es iga e he bes in en o y planning o a p ac i ione
unde ca bon a i egula ions by inco po a ing he simul aneous e -
ec s o s o age ime and uni selling p ice on he clien s’ consump ion
beha io . In addi ion, he exis ing p epaymen li e a u e on in en o y
con ol ocuses pa icula ly on accomplishing he p epaymen by a
single ins allmen o ixed mul iple ins allmen s o e ed by he supplie .
Khan e al. [19] examined he e ec s o a iable ad ance paymen in-
s allmen s on he op imum s ocking selec ion and ook in o accoun
ansac ion expenses ela ed o he paymen ins allmen s in ad ance
paymen a angemen s. Using a i-exponen ial unc ion o p ice and a
powe unc ion o ime o model he demand s uc u e, which was
assumed o e lec consume beha io , hey examined he ins allmen
policy o p epay unde a discoun depending on he amoun o pu chase.
I is impo an o no e ha hey ound all he heo e ical esul s in he
absence o a ca bon egula o y policy. Wi h he demand s uc u e
conside ed o be an addi i e combina ion o he in luence o s o age
ime in powe pa e n and p ice in linea o m unde a ious en i on-
men al guidelines, he cu en s udy makes he i s a emp o de e -
mine he op imal s ock managemen and ma ke p ice plan as well.
Mo eo e , Khan e al. [20] in es iga ed p icing, in en o y, and ins all-
men decisions o g owing i ems only, a speci ic kind o i em ha in-
c eases in en o y le els du ing i s eeding ime. I is no ewo hy ha
hei de eloped model is no sui able o many gene al p oduc s like
cosme ic p oduc s, elec ical equipmen , compu e s, clo hes, u ni u e,
Table 1
Majo li e a u e s udy o e he p epaymen ins allmen mechanism and powe demand pa e n.
Au ho (s) Demand in ol es P epaymen
ins allmen
Holding
cos
P oduc s ype En i onmen al
egula ions
Objec i e
unc ion
Time
impac s as
P ice impac s
as
O he s Va iable Fixed G owing Gene al (non-
g owing)
CM PM
Taleizadeh e al.
[42]
Con. √ Con. √ √
Taleizadeh [43] Con. √ Con. √ √
Top al e al. [44] Con. Con. √ C&T, CT, CC √
Zia and Taleizadeh
[49]
Con. √ Con. √ √
Ho elaque and
Bi onneau [10]
Linea o m Emissions
le el
Con. √ CT √
Zhang e al. [48] Random
a iable
√ √ √
San-Jos´
e e al. [32] PDP Con. √ √
Chen e al. [6] Con. Con. √ C&T √
San-Jos´
e e al. [33] PDP Logi o m Con. √ √
San-Jos´
e e al. [34] PDP Linea o m Non-linea √ √
San-Jos´
e e al. [35] PDP Exponen ial
o m
Con. √ √
Khan e al. [16] Linea o m √ Linea √ √
Xu e al. [47] Gene al
o m
Con. √ CC, C&T √
Lee [21] Con. Con. √ C&T, CC, CT, CO √
Khan e al. [17] Con. √ Con. √ √
C´
a denas-Ba ´
on
e al. [4]
PDP Linea o m Non-linea √ √
San-Jos´
e e al. [36] PDP Powe o m Non-linea √ √
Manna e al. [23] In e al-
alued
√ In e al-
alued
√ √
Khan e al. [19] PDP Exponen ial
o m
√ Con. √ √
Khan e al. [20] PDP Powe o m √ Non-linea √ C&P, C&T, CT √
This wo k PDP Linea o m √ Non-linea √ C&P, C&T, CT √
PDP: powe demand pa e n; Con.: cons an ; CM: cos minimiza ion; PM: p o i maximiza ion.
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
6
and many o he s. In en o y planning o non-g owing i ems di e s
signi ican ly om ha o g owing i ems. Non-g owing i ems a e ypi-
cally managed wi h a ocus on demand and s o age conside a ions. In
con as , g owing i ems in ol e unique challenges ela ed o b eeding,
eeding, and ca e o e ime. Fo non-g owing i ems, he emphasis is on
main aining op imal s ock le els o mee cus ome demand e icien ly,
while o g owing i ems, i in ol es managing he li ecycles and heal h
o he li es ock o ensu e a s eady supply. Thus, i is c ucial o conduc
dis inc in es iga ions in o in en o y planning o g owing i ems and
gene al (non-g owing) i ems. As a esul , one o he big limi a ions o
Khan e al. [20] is ha i is no sui able o many gene al (non-g owing)
p oduc s. The p esen s udy ies o co e his limi a ion by de e mining
he op imal p icing, in en o y, and ins allmen decisions o gene al
p oduc s. The e o e, his esea ch makes he i s a emp o inspec he
bes p epaymen ins allmen s a egy o a p ac i ione by examining he
e ec s o a iable p epaymen ins allmen s on p o i wi h demand as an
addi i e combina ion o he e ec s o s o age ime in powe pa e n and
p ice in linea o m unde s ic ax egula ions on eleased ca bons.
The ollowing is a lis o he s udy’s emaining sec ions: Sec ion 3
cla i ies he in en o y p oblem de ini ion, including he equi ed no a-
ion and assump ions. The co esponding mixed-in ege op imiza ion
p oblem is o mula ed in Sec ion 4, while all he necessa y analy ical
indings o a i e a he op imal s ocking and ins allmen plans a e
de i ed in Sec ion 5. Sec ion 6 deals wi h he cla i ica ion o he ope -
a ional e iciency o he es ablished algo i hm by demons a ing i e
nume ical examples. The consequences o luc ua ions in sys em pa-
ame e s a e examined in Sec ion 7, and se e al managemen insigh s
a e ex ac ed in Sec ion 8. Sec ion 9 highligh s he conclusion and
salien indings o he wo k.
3. P oblem de ini ion
Suppose a p ac i ione c ea es an o de o a speci ic kind o i em
om an ou side supplie by p epaying a pa icula p opo ion o he cos
o he pu chase o e he cou se o a a iable numbe o ins allmen s
(single o mul iple iden ical ins allmen s) om he ime he o de is
placed o he ime he o de is deli e ed. The es o he cos o he
pu chase is paid a he momen he p ac i ione ecei es he o de , and
hen he in en o y p ocess commences a he p ac i ione ’s s o age, and
he s o ed i ems a e consumed h ough he consump ion pe iod. The
clien s’ beha io ega ding i ems’ consump ion may a y o e he
s o age pe iod such ha in en o y may all apidly a he ini ial s ages o
a he inal s ages o he s o age pe iod. To adop his cha ac e is ic, he
clien s’ demand pa e n is assumed o be an addi i e combina ion o he
e ec s o s o age ime in powe pa e n and p ice in linea o m. In
addi ion, he p ac i ione holds i ems in s ock bea ing he cos , which is
a powe o m o he s o ing pe iod. I he p epaymen is comple ed
wi hin a single ins allmen , hen he p ac i ione has a highe oppo u-
ni y o capi al cos . This capi al cos dec eases when he p epaymen
ins allmen numbe also inc eases. On he o he hand, each paymen
ins allmen leads o ansac ion expenses o he p ac i ione (ei he by
money ansi ioning om one bank accoun o ano he bank accoun o
by he expense o anspo a ion, labo , o ime). Consequen ly, he e is
a ade-o be ween he ansac ion cha ge and capi al expense o he
p ac i ione wi h espec o he ins allmen equency. Fu he mo e, he
p ac i ione incu s addi ional cos s o he amoun o ca bon emissions
in acco dance wi h he en i onmen al egula ions (C&P, C&T, and CT)
imposed by he au ho i y / go e nmen because he p ac i ione ’s
a ious business ac i i ies elease signi ican amoun s o emissions in o
he su oundings. The majo p oblem o his s udy is o add ess he
ollowing ques ion: wha a e he mos e ec i e p epaymen ins allmen ,
p ice, and s ock s a egies o p ac i ione s so ha hey may make he
g ea es p o i s possible in he sho es amoun o ime?
To cla i y he abo e in en o y p oblem o he p ac i ione ma he-
ma ically, he ollowing no a ion and hypo hesis a e adop ed:
3.1. No a ion
Pa ame e s:
ac Uni acquisi ion cos
θ P epaymen p opo ion o he o al cos o pu chasing
Lp Allowed lead ime by he supplie
ic Ra e o in e es
u Maximum clien demand a e, excluding he impac s o p ice and ime
P ice sensi i i y ac o in demand (>0)
yh The g ea es selling p ice pe uni
α
The ime-dependen demand’s scale ac o wi h
α
∈ ( − ∞,∞)
ξ Index o powe demand pa e n wi h ξ>0
O Fixed o de ing expense o e e y o de , which is independen o
pu chased quan i y uni s
g0 Fixed s o ing cos o e e y uni
g Time- a ying holding cos pa ame e o e e y uni
γ Index o ime- a ying cumula i e holding expense
c Expense o ansac ions o each ins allmen o paymen s

O Quan i y o ca bon eleased du ing cycle se up p ocedu es
g0 Fixed quan i y o ca bon emissions pe uni p oduced by lodging
g Time-sensi i e emission ac o om holding ope a ions
ac The quan i y o ca bon emissions pe uni o pu chased goods
 c The quan i y o ca bon emissions eleased du ing e e y single paymen
ins allmen
π
Limi o he p ac i ione ’s eleased ca bons pe uni o ime
p Ca bon a i pe uni o emissions when he o al amoun o emissions
exceeds
π
Rewa d a e pe uni o emissions when he o al emi ed ca bon is below
π
Decision a iables:
y P ice o e e y uni (y≥ac)
τ
Du a ion o each in en o y cycle
I Ins allmen numbe o accomplishing p epaymen (I ∈ {1,2,3, ...})
Dependen -decision a iables:
Q Pu chased quan i y uni s o e e y cycle
Func ions:
q( )A ailable s ock amoun in he s o age a ∈ [0,
τ
]
ω
( )Cumula i e holding expenses o keep each uni a any ime
ρ
(y, )Demand s uc u e
δ(y)Auxilia y unc ion
λ(y,
τ
,I )To al ca bon oo p in s pe uni o ime h oughou one in en o y cycle
Y(y,
τ
,
I )
P ac i ione ’s p o i pe uni o ime
3.2. Hypo hesis
(i) This s udy makes e e ence o a single p ac i ione ’s in en o y
p ocess o an inde ini e planning ho izon in which he p ac i-
ione sells a single i em.
(ii) The lead ime is ixed and based on he quan i y o de ed. Du ing
he lead ime pe iod Lp uni s o ime, he p ac i ione has o
p epay he θ p opo ion o he o al cos o pu chasing using I
iden ical mul iple ins allmen s. Finally, he es (1−θ)p opo -
ion o he o al cos o pu chasing is paid cash-on-deli e y o
ecei e all he goods.
(iii) Owing o money ans e o shipping o manpowe o ime
expendi u e in p epaymen ins allmen s, he p ac i ione bea s
he ansac ion cha ge o e e y ins allmen .
(i ) As he demand is de e minis ic, he s ockou si ua ion is no
encoun e ed by he p ac i ione du ing he in en o y p ocess.
( ) To hold each uni in s ock, he cumula i e holding expense con-
sis s o wo pa s: (a) a ixed accommoda ing cos independen
om s o age ime, and (b) a a iable cos as a powe unc ion o
he s o age ime in s ock [31]. This cos s uc u e is assumed as
ω
( ) = g0+g γ, whe e g0≥0, g>0 and γ≥1.
( i) Ca bon emissions om all ope a ions a e included in he in-
en o y coo dina ion. As a esul , he p ac i ione incu s an
addi ional cos o he amoun o ca bon emi ed in acco dance
wi h he ca bon egula ions (C&P, C&T, and CT) imposed by he
au ho i y / go e nmen .
( ii) Clien s’ demand
ρ
(y, ) luc ua es o e he in en o y cycle due o
he esul an e ec o uni selling p ice and s ock age. Simila o
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
7
San-Jos´
e e al. [34], clien s’ demand s uc u e is assumed as
ρ
(y, )
=
ρ
1(y) +
ρ
2( ), whe e he i s e m, ha is, he consequence o
uni selling p ice, is gi en by
ρ
1(y) = u− y,wi h u>0, >0 and ac≤y≤u
,
and he second e m, ha is, he e ec o in en o y age, is de ined
by
ρ
2( ) = (
α
ξ)(
τ
)1
ξ−1
,wi h −∞<
α
<∞ and ξ>0.
4. Model o mula ion
Wi hou limi ing he gene ali y, le us assume he p ac i ione o de s
Q uni s o a single i em p io o Lp uni s o ime om ecei ing p oduc s
unde a p epaymen egula ion imposed by he supplie . In acco dance
wi h his egula ion, he p ac i ione p epays θacQ amoun o he o al
acquisi ion p ice acQusing I (≥ 1)equal ins allmen s and ecei es he
o de by cash-on-deli e y o he esidual (1−θ)acQ amoun o he o al
cos o pu chasing. Once he p oduc s a e ecei ed, he main ac i i ies o
he in en o y p ocedu e (s o age and sale) begin (see Fig. 4). The con-
sump ion o in en o ies now s a s wi h a beginning quan i y o Q uni s,
which dec eases o e ime
τ
wi h he cus ome demand a e
ρ
( ,y)as
ollows:
The ime-sensi i e componen in demand, i.e., (
α
ξ)(
τ
)1
ξ−1, ep esen s
he consume s’ pu chase decisions based on he in en o y age. Cus-
ome s’ pu chasing pa e ns a e ca ego ized in one o h ee ways,
depending on he s o age du a ion o he in en o ies: (i) because o hei
cu en condi ion, some i ems (such as p ocessed meals, ui s, mea
wi h an expi y da e, e c.) a e in highe demand in he beginning han in
he end, and his pe cep ion is ep esen ed by ξ>1; (ii) some i ems,
such as po a oes, da es, and ga lic, migh be in g ea e demand la e in
he s o age cycle because o hei d ying condi ions, which allow con-
sume s o consume hem o longe , and his si ua ion is e lec ed o 0
<ξ<1; and (iii) he e a e some goods (such as household i ems,
elec ical de ices, iles, cu ains, e c.) whose demand is p ac ically
consis en ly s eady h oughou he cycle, and his ea u e is s a ed o
ξ=1. Again, he p ice-sensi i e componen in demand, i.e., u− y is a
declining and a ine unc ion o y such ha y≤yh whe e yh=u
is he
g ea es selling p ice pe uni . In addi ion, he pa ame e u a icula es
he maximum clien demand, excluding he impac s o p ice and ime,
and is p ice sensi i i y. I is no ewo hy ha he p ice-dependen
po ion o he demand unc ion becomes cons an , ha is, independen
o he e ec o p ice, when =0 and yh ends o in ini y.
Le q( ,y) = {q( |y):0≤ ≤
τ
}be he i em s ocking p ocess du ing
an in en o y pe iod
τ
, whe e q deno es he amoun o a ailable s ock a
ime ∈ [0,
τ
]. As all he s o ed i ems a e consumed a he demand a e
ρ
( ,y)du ing he in en o y pe iod
τ
, he pu chased quan i y amoun o
e e y cycle is p ecisely iden ical o he o al demand du ing he pe iod
(0,
τ
]. The e o e,
Q=∫
τ
0
ρ
( ,y)d =∫
τ
0{u− y +(
α
ξ)(
τ
)1
ξ−1}d = (u− y +
α
)
τ
(2)
In addi ion, he a ailable s ock amoun a ∈ [0,
τ
]is desc ibed as
ollows:
q
′
( |y) = − {u− y +(
α
ξ)(
τ
)1
ξ−1},0< <
τ
(3)
wi h he auxilia y condi ions q(0|y) = Q and q(
τ
|y) = 0.
F om Eq. (3), he a ailable s ock amoun a ime ∈ [0,
τ
]is:
q( |y) = (u− y)(
τ
− ) +
ατ
{1−(
τ
)1
ξ},0≤ ≤
τ
.(4)
Since he p ac i ione pu chases Q uni s a he uni acquisi ion p ice
ac and hen sells hem all a he uni selling p ice y, he p ac i ione ’s
sales e enue pe uni o ime (SR) and acquisi ion p ice pe uni o ime
(AC) a e y(u− y +
α
)and ac(u− y +
α
), espec i ely.
The p ac i ione p epays θacQ amoun o he o al cos o pu chasing
acQ by using I (≥ 1)iden ical mul iple ins allmen s wi hin he p o ided
ime du a ion LP uni s o ime and ecei es he o de by paying ins an ly
he esidual (1−θ)acQ amoun o he o al cos o pu chasing. Conse-
Fig. 4. G aphic p esen a ion o mul iple p epaymen ins allmen s in an in en o y p ocedu e wi h a powe demand.
ρ
( ,y) = u− y +(
α
ξ)(
τ
)1
ξ−1
,wi h u>0, >0,−∞<
α
<∞,ξ>0,and s∈[ac,u
](1)
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
8
quen ly, he p epaymen amoun incu s an addi ional oppo uni y o
capi al cos o he p ac i ione , and his capi al cos (CC) pe uni ime is
calcula ed om Fig. 4(a) as ollows:
ic
τ
[(θacQ
I )Lp
I (1+2+3+... +I )]=(I +1
2I )θicacLp(u− y +
α
)(5)
E e y p epaid ins allmen esul s in a ansac ion expense c o he
p ac i ione (due o money p oceeding om one bank accoun o
ano he bank accoun , anspo , pe sonnel, o ime cos ); he e o e, he
o al ansac ional expense pe uni o ime (TC) is cI
τ
.
The expense o holding ope a ions pe uni o ime (HC) o he
p ac i ione is
1
τ
∫
τ
0
(g0+g γ)
ρ
( ,y)d =g0(u− y +
α
) + g(u− y
γ+1+
α
γξ +1)
τ
γ(6)
Now, he p ac i ione ’s o al eleased emissions amoun in a uni
ime is
λ(y,
τ
,I )=
O
τ
+ac(u− y +
α
) + cI
τ
+g0(u− y +
α
)
+g(u− y
γ+1+
α
γξ +1)
τ
γ(7)
The i s and second e ms in he exp ession o λ(y,
τ
,I )indica e he
eleased emissions amoun in a uni ime om se up and pu chasing
ope a ions, espec i ely. When he p epaymen ins allmen p ocess in-
ol es any ehicle, hen he eleased emissions amoun in a uni ime
om ins allmen ac i i y is gi en by he hi d e m o λ(y,
τ
,I ); o he -
wise,  c is ze o. Addi ionally, he ou h and i h componen s show he
amoun o eleased emissions om holding ac ions pe uni o ime.
Depending on he en i onmen al egula ions es ablished by he
au ho i y o go e nmen , he p ac i ione may be ined o ewa ded o
hei ca bon oo p in . Due o he ac ha C&T and CT ules a e speci ic
examples o he C&P egula ion, he p ac i ione ’s p o i is calcula ed
using he C&P egula ion in his s udy and exp essed as ollows:
In Eq. (8), he e m p(λ(y,
τ
,I ) −
π
)+=max{0,p(λ(y,
τ
,I ) −
π
)} in-
dica es he a i amoun , whe eas he e m (
π
−λ(y,
τ
,I ))+=
max{0, (
π
−λ(y,
τ
,I ))} deno es he ewa d amoun o he p ac i ione .
Fo he index c=p, , le us de ine
No e ha
Y(y,
τ
,I )={Yc(y,
τ
,I ),i λ(y,
τ
,I )≥
π
Y (y,
τ
,I ),i λ(y,
τ
,I )≤
π
(10)
Finally, he desc ibed in en o y p ac ice gene a es he ollowing
mixed-in ege maximiza ion p oblem
max
(y,
τ
,I )∈ΓYc(y,
τ
,I ),(11)
whe e Γ≡{(y,
τ
,I ):ac≤y≤yh=u
,
τ
>0 and I ∈ {1,2,3,...}}
5. Solu ion echnique
The e a e h ee decision a iables y,
τ
, and I in he objec i e unc ion
Yc(y,
τ
,I ), whe e wo o hem (y and
τ
) a e con inuous and he emaining
one (I )is discon inuous o e he posi i e in ege se . E iden ly, because
he ins alla ion numbe mus be a posi i e in ege , he di e en ial
op imiza ion app oach canno be used o de e mine he bes I∗
. In o de
o add ess he desc ibed mixed-in ege op imiza ion issue in Eq. (11), a
Y(y,
τ
,I )=[y−ac{1+(I +1
2I )θicLp}−g0](u− y +
α
) − O+ cI
τ
−g(u− y
γ+1+
α
γξ +1)
τ
γ
−p(λ(y,
τ
,I )−
π
)++ (
π
−λ(y,
τ
,I ))+
(8)
Yc(y,
τ
,I )=[y−ac{1+(I +1
2I )θicLp}−g0](u− y +
α
) − O+ cI
τ
−g(u− y
γ+1+
α
γξ +1)
τ
γ
−c(λ(y,
τ
,I )−
π
)
=[y−ac{1+(I +1
2I )θicLp}−g0−c(ac+g0)](u− y +
α
)
−(O+c
O)
τ
−( c+c c)I
τ
− (g+cg)(u− y
γ+1+
α
γξ +1)
τ
γ+c
π
(9)
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
15
I∗
=⌈1
2{−1+
1+2acθicLp
τ
∗
( c+c c)(u− y∗+
α
)
√}⌉, he bes business cycle is
τ
∗
=[{O+c
O+ ( c+c c)I∗
}/{(g+cg)γ(u− y∗
γ+1+
α
γξ+1)}]1
γ+1, he ideal num-
be o p oduc s o p ocu e isQ∗= (u− y∗+
α
)[{O+c
O+ ( c
+c c)I∗
}/{(g+cg)γ(u− y∗
γ+1+
α
γξ+1)}]1
γ+1, and he p ac i ione ’s maximum
p o i is
Y∗
c=[y∗−ac{1+(I∗
+1
2I∗
)θicLp}−g0−c(ac+g0)](u− y∗+
α
)
− (γ+1){O+c
O+( c+c c)I∗
γ}γ
γ+1{(g+cg)(u− y∗
γ+1+
α
γξ+1)}1
γ+1
+c
π
.
P oo . The p oo is simila o ha o
P oposi ion 3
. □
Co olla y 4. (a) Acco ding o he C&T egula ion, when ca bon
emissions a e p iced a p pe uni , le (y∗,
τ
∗,I∗
)
ep esen he op imal solu ion o he p ac i ione
de e mined h ough he es ablished algo i hm. When
y∗<yh and c<acθicLp
τ
∗
α
2−p c, hen he bes
paymen ins allmen numbe is I∗
=⌈1
2{−1+
1+2acθicLp
τ
∗
( c+p c)(u− y∗+
α
)
√}⌉, he
bes business cycle is
τ
∗=[{O+c
O+ ( c+p c)
I∗
}/{(g+pg)γ(u− y∗
γ+1+
α
γξ+1)}]1
γ+1, he ideal numbe
o p oduc s o p ocu e is Q∗= (u− y∗+
α
)
[{O+p
O+ ( c+p c)I∗
}/{(g+pg)γ(u− y∗
γ+1+
α
γξ+1)}]1
γ+1, and he p ac i ione ’s maximum p o i is
(b) Acco ding o he CT egula ion, when ca bon emissions a e p iced a p
pe uni , le (y∗,
τ
∗,I∗
) ep esen he op imal solu ion o he p ac i-
ione de e mined h ough he es ablished algo i hm. When y∗<yh
and c<acθicLp
τ
∗
α
2−p c, hen he op imal s a egies o he p ac i ione
a e iden ical o wha he C&T ule equi es. Howe e , he p ac i-
ione ’s maximum p o i is Y∗
c=[y∗−ac{1+(I∗
+1
2I∗
)θicLp}−g0−
p(ac+g0)](u− y∗+
α
) − (γ+1){O+p
O+( c+p c)I∗
γ}γ
γ+1
{(g+pg)(u− y∗
γ+1+
α
γξ+1)}1
γ+1.
P oo . The p oo is simila o ha o Co olla y 1. □
5.2. Some pa icula cases
The necessa y condi ions o inco po a ing he in en o y model
sugges ed in his wo k in o p e iously examined models by o he s a e
ou lined in his subsec ion.
(i) The in en o y model o San-Jos´
e e al. [34] is de i ed when he
p epaymen echnique does no exis , any emission con ol
guideline is no en o ced, and no ixed lodging expense is
included o holding ope a ions pu poses (i.e., θ=0, c=0, p=
=0, and g0=0).
(ii) I I is known, c=0, p= =0, =0,
α
=0, ξ=1, g0=0, γ=
1, and y is ixed, hen i is in e ed ha he model o Taleizadeh
e al. [42] ope a es in he absence o sho ages.
(iii) The ma hema ical amewo k o Sicilia e al. [38] in a s ockou
si ua ion is deduced i θ=0, c=0, p= =0, u=0, =0, g0=
0, γ=1, and y is cons an in he p esen s udy.
Fig. 7. A isual ep esen a ion o he p ac i ione ’s p o i is shown in com-
pa ison o I unde Example 1.
Table 2
Bes business ac ics and he p ac i ione ’s g ea es p o i unde a ious en i onmen al ules o Example 1.
Regula ion
π
p y∗
τ
∗I∗
Q∗λ(y∗,
τ
∗,I∗
)Y(y∗,
τ
∗,I∗
)
C&P 70 3 6 $89.9768/uni 2.795 mon hs 4 111.8631 uni s 67.1913 ons $1823.506
C&T 70 3 3 $88.1719/uni 2.758 mon hs 5 115.3629 uni s 69.5350 ons $1818.282
CT 0 3 NA $88.1719/uni 2.758 mon hs 5 115.3629 uni s 69.5350 ons $1608.282
Y∗
c=[y∗−ac{1+(I∗
+1
2I∗
)θicLp}−g0−p(ac+g0)](u− y∗+
α
)
− (γ+1){O+p
O+ ( c+p c)I∗
γ}γ
γ+1{(g+pg)(u− y∗
γ+1+
α
γξ +1)}1
γ+1
+p
π
.
Md.A.-A. Khan e al.

Ope a ions Resea ch Pe spec i es 12 (2024) 100289
16
(i ) The cu en in en o y app oach becomes he Smi h e al. [41]
model when a linea consump ion s uc u e is used by adop ing θ
=0, c=0, p= =0,
α
=0, ξ=1, and g0 =0.
( ) The in en o y amewo k examined by Fe guson e al. [9] is
ob ained by conside ing θ=0, c=0, p= =0, u =0, =0,
and g0=0 in he p esen s udy.
6. Nume ical indings and ema ks
This sec ion deals wi h he cla i ica ion o he unc ional e icacy o
he es ablished algo i hm by in es iga ing se e al nume ical examples.
Since he algo i hm o e s i e po en ial scena ios o he bes -selling
p ice by conside ing a ious pa ame e alues, i e nume ical illus a-
ions a e adop ed o elucida e he wo king e icacy o he implemen ed
algo i hm.
Example 1. The pa ame e s used in he cu en illus a ion a e aken
om Example 1 o he wo k by San-Jos´
e e al. [34], accompanied by he
inclusion o all o he essen ial da a equi ed o alignmen wi h he
a o emen ioned model, which a e summa ized as ollows: O =
$200/o de , u=120uni s/mon h, =1,
α
=10uni s/mon h, ξ =0.5,
ac=$40/uni , γ=1.5, g0=$1/uni , g=$0.6/uni /mon h, Lp =
0.5mon h, θ=0.6, ic=$0.1/mon h, c=$1.6/ins allmen , 
O =
50 ons/o de , ac=0.4 on/uni ,  c=0.6 on/ins allmen , g0 =
0.5 on/uni , g=0.15 on/uni /mon h, p=$3/ on, =$6/ on, and
π
=
70 ons/mon h. As a esul , he g ea es uni selling p ice o he p ac-
i ione is yh=u
=$120. Now, he policies o he p ac i ione a e
a ained by employing he algo i hm in he ollowing way:
Se j=1 and I(j)
=1.
I e a ion 1. Since 10 =
α
<u− [ac{1+(I +1
2I )θicLp}+g0+ (ac+
g0)]=72.4, compu e he poin y0=u+
α
2 +1
2[ac{1+(I +1
2I )θicLp}+
g0+ (ac+g0)]=88.8. F om Eq. (16), ind he alue 
Y
′
(yh|I ) = −
59.9707. As 
Y
′
(yh|I )<0, de e mine y(1)=a g
y∈(88.8,120)
{
Y
′
(y|I ) = 0} =
90.1803. Exploi ing Eqs. (14) and (15), one inds
τ
(1)=2.7663 and
Y(1)=1811.152, espec i ely. Since acθicLp
τ
(1)
2(u− y(1)+
α
) −  c=
Table 3
Nume ical esul s o Example 2.
I e a ion I(j)
y0 y1 y(j)
τ
(j)I(j+1)
Condi ions
j =1 1 111.2250 118.6076 120 8.1423 ⌈6.5408⌉.=
7
α
<u− [ac{1+(I +1
2I )θicLp}
+g0+ (ac+g0)]=77.55, 
Y
′
(yh|I ) = 15.5984 >0, 
Y
′
(y1|I ) =
11.79188 >0, acθicLp
τ
(1)
2(u− y(1)+
α
) −  c=252.8818 > c and I(1)
∕= I(2)
.
j =2 7 111.0000 118.5176 120 8.1865 ⌈6.5598⌉.=
7
α
<u− [ac{1+(I +1
2I )θicLp}
+g0+ (ac+g0)]=78, 
Y
′
(yh|I ) = 15.5093 >0, 
Y
′
(y1|I ) = 11.4887 >0,
acθicLp
τ
(2)
2(u− y(2)+
α
) −  c=254.2743 > c and I(2)
=I(3)
=7.
Table 4
Bes business ac ics and he p ac i ione ’s g ea es p o i unde a ious en i onmen al ules o Example 2.
Regula ion
π
p y∗
τ
∗I∗
Q∗λ(y∗,
τ
∗,I∗
)Y(y∗,
τ
∗,I∗
)
C&P 74 3 6 $120/uni 8.1865 mon hs 7 491.1892 uni s 72.4475 ons $4769.196
C&T 74 3 3 $120/uni 8.9658 mon hs 9 537.9475 uni s 74.3647 ons $4767.101
CT 0 3 NA $120/uni 8.9658 mon hs 9 537.9475 uni s 74.3647 ons $4545.101
Table 5
Nume ical esul s o Example 3.
I e a ion I(j)
y0 y1 y(j)
τ
(j)I(j+1)
Condi ions
j =1 1 103.7272 114.4085 109.3755 4.7532 ⌈0.5777⌉.=
1
α
<u− [ac{1+(I +1
2I )θicLp}
+g0+ (ac+g0)]=58.5456, 
Y
′
(yh|I ) = 12.4399 >0, 
Y
′
(y1|I ) = −
5.229<0, y=a g
y∈(103.7272,114.4085)
{
Y
′
(y|I ) = 0} = 109.3755,
1787.194 =
Y (y|I )>
Y (yh|I ) = 1759.781,
acθicLp
τ
(1)
2(u− y(1)+
α
) −  c=1.1394 < c and I(1)
=I(2)
=1.
Fig. 8. A isual ep esen a ion o he p ac i ione ’s p o i is shown in com-
pa ison o y and
τ
unde Example 3.
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
17
62.4908 >1.6= c, se I(2)
=⌈1
2{−1+

1+2acθicLp
τ
(1)
( c+  c)(u− y(1)+
α
)
√}⌉=⌈3.1⌉=4. As I(2)
∕= I(1)
, se j =2
and p oceed o S ep 3.
I e a ion 2. As 10 =
α
<u− [ac{1+(I +1
2I )θicLp}+g0+ (ac+g0)
]=72.85, he poin y0=u+
α
2 +1
2[ac{1+(I +1
2I )θicLp}+g0+ (ac+
g0)]=88.575. Since 
Y
′
(yh|I ) = − 57.3207 <0, ind he uni selling
p ice y(2)=a g
y∈(88.575,120)
{
Y
′
(y|I ) = 0} = 89.9768. Mo eo e , om Eqs.
(14) and (15), one has
τ
(2)=2.795 and Y(2)=1823.506, espec i ely.
As acθicLp
τ
(2)
2(u− y(2)+
α
) −  c=63.5179 >1.6= c, se I(3)
=⌈1
2{−
1+
1+2acθicLp
τ
(2)
( c+  c)(u− y(2)+
α
)
√}⌉=⌈3.1273⌉=4. Clea ly, I(3)
=
I(2)
, j>1 and Y(2)>Y(1). The e o e, (y∗,
τ
∗,I∗
,Y∗
c) = (89.9768,2.795,4,
1823.506). The op imal pu chased quan i y o he p ac i ione , ac-
co ding o Eq. (2), is Q∗=111.8631uni s. Wi hin Fig. 6, he p ac i-
ione ’s op imal p o i poin is shown as a dis inc g een do , si ua ed
upon he p o i unc ion’s su ace, which is con ingen upon he
con inuous a iables y and
τ
. I should be no ed ha he paymen e-
quency is se a 4, which co esponds o he ideal alue de e mined o
Example 1. The p ac i ione ’s p o i is also plo ed in Fig. 7 when he
paymen ins allmen numbe anges om 1 o 15. Mo eo e , Fig. 7
demons a es he alidi y o he deduced ma hema ical indings in
Theo em 3 and p o es ha he p ac i ione ’s p o i is s ic ly conca e
e sus he paymen ins allmen numbe I o Example 1.
Now he de i ed ideal decisions and highes p o i o he p ac i-
ione in Example 1 a e compa ed wi h hose ound in Example 1 o San-
Jos´
e e al. [34]. The op imal policies o Example 1 in San-Jos´
e e al. [34]
a e y∗=$85.6472/uni and
τ
∗=2.11779mon hs whe e he p ac i-
ione ’s maximum p o i is Y∗=$1867.18. Bo h he bes uni p ice and
business pe iod o he p ac i ione in his s udy a e highe han hose
om San-Jos´
e e al. [34]. Since he p ac i ione in his s udy bea s
se e al addi ional cos s, such as capi al o oppo uni y cos s on he
p epaymen amoun , ansac ion cos s due o p epaymen ins allmen s,
and ca bon emissions cos s, he op imal uni selling p ice (y∗=
$89.9768)is highe han he op imal selling p ice in Example 1 o
San-Jos´
e e al. [34] in o de o encoun e hese cos s. As he uni selling
p ice is highe , he clien ’s demand in his s udy is lowe han he de-
mand a e in Example 1 o San-Jos´
e e al. [34] and hence, he op imal
business pe iod o his example is also highe han ha o Example 1 o
San-Jos´
e e al. [34]. Al hough bo h y∗and
τ
∗a e highe , he p ac i-
ione ’s highes p o i o his example is lowe han ha o Example 1 o
San-Jos´
e e al. [34] due o se e al addi ional cos s in his s udy. How-
e e , his s udy helps p ac i ione s igu e ou no only how o accom-
plish he p epaymen in o de o maximize hei p o i bu also how o
conduc hei business h ough se e al ca bon emission egula ions ha
ha e no been s udied by San-Jos´
e e al. [34]. Table 2 de ails how
a ious en i onmen al ules (C&P, C&T, and CT) a ec he bes business
Fig. 9. A isual ep esen a ion o he p ac i ione ’s p o i is shown in com-
pa ison o I unde Example 3.
Fig. 10. G aphic illus a ion o he unc ion 
Y
′
(y|I )when I =1 unde
Example 3.
Table 6
Bes business ac ics and he p ac i ione ’s g ea es p o i unde a ious en i onmen al ules o Example 3.
Regula ion
π
p y∗
τ
∗I∗
Q∗λ(y∗,
τ
∗,I∗
)Y(y∗,
τ
∗,I∗
)
C&P 74 3 6 $109.3755/uni 4.7532 mon hs 1 174.0839 uni s 57.3372 ons $1787.194
C&T 74 3 3 $106.3614/uni 4.7715 mon hs 1 189.1339 uni s 63.5458 ons $1746.479
CT 0 3 NA $106.3614/uni 4.7715 mon hs 1 189.1339 uni s 63.5458 ons $1524.479
Table 7
Nume ical ou comes o Example 4.
I e a ion I(j)
y0 y1 y(j)
τ
(j)I(j+1)
Condi ions
j =1 1 119.3673 148.5469 120 3.7386 ⌈0.5693⌉.=
1
α
<u− [ac{1+(I +1
2I )θicLp}
+g0+p(ac+g0)]=81.2654, 
Y
′
p(yh|I ) = 7.2575 >0, yh=120 ≤148.5469 =
y1, acθicLp
τ
(1)
2(u− y(1)+
α
) − p c=3.3817 < c and I(1)
=I(2)
=1.
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
18
ac ics and he p ac i ione ’s g ea es p o i o Example 1.
The bes business p ac ices o he p ac i ione unde C&P, C&T, and
CT ules a e compa ed in Table 2. Because he C&P policy pe mi s he
p ac i ione o ea n mo e when he o e all quan i y o emissions is lowe
han he C&T o CT guidelines, i gene a es he p ac i ione ’s highes
p o i among he h ee. Acco ding o he C&P ule, he ewa d o e e y
on o emissions ha is eleased in o he a mosphe e is o en g ea e han
he penal y. The e o e, as is seen in Table 2, his policy encou ages
p ac i ione s o p oduce ewe ca bon emissions. Despi e he ac ha
he C&T policy pe mi s a ewa d o each uni o ca bon ha is eleased
when he o e all emission is less han he cap, he o al ca bon emission
in Example 1 is highe han he C&T since bo h he a i and incen i e
o each uni o ca bon eleased a e he same. Wi h he aim o p o ec ing
he en i onmen , he au ho i y o go e nmen migh app o e he C&P
p og am wi h a bigge ewa d han he a i o each uni o emissions.
The ideal p ice and business pe iod unde he C&P legisla ion a e lowe
han hose unde he C&T and CT laws, bu he ideal paymen ins all-
men numbe and p ocu ed amoun a e la ge unde he C&P p og am.
The p ac i ione ’s p o i unde he C&T scheme is lowe han ha o he
CT egula ion, despi e he ac ha he p ac i ione ’s combined bes
business ac ics unde he C&T and CT laws a e iden ical. These indings
suppo he conclusions d awn om he analysis in P oposi ion 4 and
Co olla y 4. Acco ding o he nume ical indings in Example 1, he
en i onmen al guideline C&P is he op imal p og am when he ewa d is
g ea e han he a i o uni emissions, bo h o he p ac i ione (in
ega d o p o i ) and o he go e nmen (in ega d o emissions
amoun ).
Example 2. Wi h he ollowing excep ions, his ins ance uses he
iden ical alues o he pa ame e s as Example 1: O =$1600/o de ,
α
=
60uni s/mon h, ξ=25, ac=$35/uni , γ=2, and
π
=74 ons/mon h.
The e o e, he maximum p ice o he p ac i ione is yh =u
=$120.
Now, Table 3 p esen s he nume ical esul s o he algo i hm-based so-
lu ion app oach. The e o e, he ideal p ice, business pe iod, and pay-
men ins allmen numbe o he p ac i ione a e y∗=$120/uni ,
τ
∗=
8.1865mon hs, and I∗
=7 ins allmen s, espec i ely, whe e he p ac i-
ione ’s highes p o i is Y∗=$4769.196. Exploi ing Eq. (2), he ideal
numbe o p oduc s o p ocu e is Q∗=491.1892uni s. Addi ionally,
Table 4 p o ides a nume ical assessmen o he e ec s o C&P, C&T, and
CT on he bes business ac ics and he p ac i ione ’s highes p o i
wi hin Example 2. The s a is ical da a in Table 4 suppo he indings in
P oposi ion 2 and Co olla y 2.
Example 3. Wi h he ollowing excep ions, his ins ance uses he
iden ical alues o he pa ame e s as Example 2: O=$1000/o de ,
α
=
26uni s/mon h, ac=$55/uni , Lp=0.11mon h, θ=0.1, and ic=
$0.09/mon h. The compu a ional indings o he solu ion echnique
using he algo i hm a e shown in Table 5. Thus, he ideal p ice, business
pe iod, and paymen ins allmen numbe o he p ac i ione a e y∗=
$109.3755/uni ,
τ
∗=4.7532mon hs, and I∗
=1 ins allmen , espec-
i ely, whe e he p ac i ione ’s highes p o i is Y∗=$1787.194.
Exploi ing Eq. (2), he ideal numbe o p oduc s o p ocu e is Q∗=
174.0839uni s. In Fig. 8, he p ac i ione ’s highes p o i poin is shown
as a dis inc g een do , si ua ed upon he p o i unc ion’s su ace, which
is con ingen upon he con inuous a iables y and
τ
. I should be no ed
ha he paymen equency is se a 1, which co esponds o he ideal
alue de e mined o Example 3. Addi ionally, Fig. 9 is c ea ed by
showing he p ac i ione ’s p o i when he paymen ins allmen numbe
anges om 1 o 10 and ensu es, by g aphic display, he alidi y o he
deduced ma hema ical indings in Theo em 3 ha he p ac i ione ’s
p o i is s ic ly dec easing agains I o Example 3. Fig. 10 e eals he
beha io o he i s -o de de i a i e 
Y
′
(y|I =1)wi h espec o he
selling p ice pe uni . Mo e in e es ingly, in his case, he e is an addi-
ional solu ion a y=$118.7187 o he equa ion 
Y
′
(y|I =1) = 0.
Howe e , a y=$118.7187, he p ac i ione has a ela i e minimum
p o i as $1753.421 =
Y(118.7187|I =1)<
Y(yh|I =1) = $1759.781.
In addi ion, Table 6 e eals a nume ical assessmen o he e ec s o C&P,
C&T, and CT on he bes business ac ics and he p ac i ione ’s highes
p o i wi hin Example 3. The s a is ical da a in Table 6 suppo he
indings in P oposi ion 2 and Co olla y 2.
Example 4. Wi h he ollowing excep ions, his ins ance uses he
iden ical alues o he pa ame e s as Example 2:
α
=80uni s/mon h,
ac=$35/uni , ξ=2, Lp=0.11mon h, θ=0.1, ic=$0.09/mon h, and
c=$4/ins allmen . The compu a ional indings o he solu ion
Table 8
Bes business ac ics and he p ac i ione ’s g ea es p o i unde a ious en i onmen al ules o Example 4.
Regula ion
π
p y∗
τ
∗I∗
Q∗λ(y∗,
τ
∗,I∗
)Y(y∗,
τ
∗,I∗
)
C&P 74 3 6 $120/uni 3.7386 mon hs 1 299.09 uni s 119.08 ons $6018.771
C&T 74 3 3 $120/uni 3.7386 mon hs 1 299.09 uni s 119.08 ons $6018.771
CT 0 3 NA $120/uni 3.7386 mon hs 1 299.09 uni s 119.08 ons $5796.771
Table 9
Nume ical esul s o Example 5.
I e a ion I(j)
y(j)
τ
(j)I(j+1)
Condi ions
j =1 1 46.1539 4.3424 ⌈1.7936⌉.=2
α
>u− [ac{1+(I +1
2I )θicLp}
+g0+ (ac+g0)]= − 3.76, acθicLp
τ
(1)
2(u− y(1)+
α
) −  c=22.4543 > c and I(1)
∕= I(2)
.
j =2 2 46.1539 4.3602 ⌈1.798⌉.=2
α
>u− [ac{1+(I +1
2I )θicLp}
+g0+ (ac+g0)]= − 2.98, acθicLp
τ
(2)
2(u− y(2)+
α
) −  c=22.5612 > c and I(2)
=I(3)
=2.
Table 10
Bes business ac ics and he p ac i ione ’s g ea es p o i unde a ious en i onmen al ules o Example 5.
Regula ion
π
p y∗
τ
∗I∗
Q∗λ(y∗,
τ
∗,I∗
)Y(y∗,
τ
∗,I∗
)
C&P 70 3 6 $46.1539/uni 4.3602 mon hs 2 43.602 uni s 28.5465 ons $213.4404
C&T 70 3 3 $46.1539/uni 4.3744 mon hs 3 43.7442 uni s 28.6837 ons $89.3012
CT 0 3 NA $46.1539/uni 4.3744 mon hs 3 43.7442 uni s 28.6837 ons -$120.6988
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
19
Table 11
In luences o u, ,
α
and ξ on he p ac i ione ’s ideal business ac ics and highes p o i .
α
u=100 u=120 u=140 u=160
y∗
τ
∗I∗
Y∗y∗
τ
∗I∗
Y∗y∗
τ
∗I∗
Y∗y∗
τ
∗I∗
Y∗
ξ=0.5
0.5 5 130.0198 2.8518 4 3429.605 148.0938 2.6068 5 5257.858 168.0066 2.4323 5 7496.894 187.9391 2.2929 5 10,137.46
10 133.1524 2.7207 5 3838.282 153.0503 2.5204 5 5776.307 172.9733 2.3640 5 8116.104 192.9126 2.2370 5 10,857.27
15 138.1011 2.6210 5 4281.225 158.0121 2.4434 5 6320.136 177.9434 2.3020 5 8760.608 197.8886 2.1857 5 11,602.31
20 143.0566 2.5331 5 4749.612 162.9782 2.3740 5 6889.298 182.9165 2.2453 5 9430.374 202.8666 2.1382 5 12,372.56
0.75 5 95.1690 3.0450 4 1749.494 106.4910 2.7307 5 2787.379 119.7206 2.5276 5 4125.318 132.9759 2.3694 5 5731.720
10 98.3879 2.8974 4 1975.700 109.7721 2.6295 5 3092.778 123.0152 2.4497 5 4498.282 136.2791 2.3068 5 6172.034
15 101.6267 2.7726 4 2219.539 113.0603 2.5404 5 3415.295 126.3142 2.3796 5 4888.251 139.5853 2.2497 5 6629.280
20 103.1230 2.6346 4 2482.838 116.3541 2.4611 5 3754.870 129.6169 2.3161 5 5295.188 142.8940 2.1972 5 7103.433
1 5 77.8875 3.2866 3 990.5486 87.5753 2.9244 4 1636.091 95.6098 2.6381 5 2506.418 105.5183 2.4560 5 3595.181
10 80.2318 3.0929 3 1125.454 89.9768 2.7950 4 1823.506 98.0640 2.5478 5 2755.980 107.9836 2.3852 5 3895.540
15 82.6084 2.9350 3 1274.145 90.6175 2.6531 5 2029.809 100.5240 2.4676 5 3018.440 110.4526 2.3211 5 4208.703
20 85.0076 2.8027 3 1436.370 93.0707 2.5610 5 2254.305 102.9886 2.3956 5 3293.749 112.9247 2.2626 5 4534.639
1.25 5 67.6908 3.6486 3 602.2129 75.2608 3.1295 3 1013.158 81.1775 2.7687 5 1589.062 89.0678 2.5554 5 2366.669
10 69.4565 3.3707 3 680.9471 77.1310 2.9642 3 1135.362 83.1221 2.6621 5 1764.292 91.0273 2.4741 5 2582.846
15 71.2836 3.1580 3 771.5472 79.0257 2.8267 3 1268.650 85.0746 2.5687 5 1949.999 92.9915 2.4014 5 2809.376
20 73.1492 2.9875 3 873.5618 80.9209 2.7201 4 1412.856 87.0332 2.4860 5 2146.117 94.9596 2.3357 5 3046.221
ξ=1
0.5 5 130.0663 2.9127 4 3435.895 148.1161 2.6503 5 5261.693 168.0233 2.4662 5 7500.343 187.9523 2.3204 5 10,140.62
10 133.2076 2.8259 5 3846.591 153.0903 2.5998 5 5783.678 173.0039 2.4268 5 8122.776 192.9370 2.2884 5 10,863.39
15 138.1744 2.7628 5 4293.137 158.0664 2.5525 5 6330.787 177.9856 2.3894 5 8770.302 197.9165 2.2663 6 11,611.26
20 143.1440 2.7045 5 4764.842 163.0441 2.5080 5 6903.011 182.9685 2.3540 5 9442.918 202.9026 2.2371 6 12,384.22
0.75 5 95.2306 3.1229 4 1756.456 106.5180 2.7824 5 2791.498 119.7402 2.5668 5 4128.976 132.9910 2.4004 5 5735.035
10 98.4927 3.0326 4 1988.818 109.8200 2.7224 5 3100.654 123.0507 2.5213 5 4505.333 136.3068 2.3644 5 6178.458
15 101.7626 2.9511 4 2238.182 113.1246 2.6669 5 3426.627 126.3628 2.4786 5 4898.466 139.6236 2.3302 5 6638.632
20 105.0389 2.8769 4 2504.515 116.4314 2.6151 5 3769.404 129.6763 2.4384 5 5308.369 142.9414 2.2977 5 7115.553
1 5 77.9746 3.3925 3 998.3917 87.6276 2.9919 4 1642.631 95.6333 2.6838 5 2510.324 105.5358 2.4913 5 3598.682
10 80.3740 3.2701 3 1140.027 90.0670 2.9137 4 1835.897 98.1061 2.6308 5 2763.477 108.0155 2.4503 5 3902.305
15 82.7871 3.1624 3 1294.629 90.6953 2.8025 5 2041.958 100.5809 2.5812 5 3029.260 110.4964 2.4116 5 4218.525
20 85.1953 3.0783 4 1462.185 93.1629 2.7409 5 2269.814 103.0575 2.5349 5 3307.665 112.9786 2.3749 5 4547.338
1.25 5 67.8316 3.8106 3 611.4657 75.3320 3.2182 3 1020.426 81.2063 2.8234 5 1593.269 89.0885 2.5962 5 2370.389
10 69.6697 3.6241 3 697.7270 77.2341 3.1273 4 1148.983 83.1729 2.7599 5 1772.323 91.0645 2.5488 5 2590.007
15 71.5375 3.4680 3 794.7192 79.1615 3.0354 4 1287.997 85.1424 2.7013 5 1961.536 93.0422 2.5043 5 2819.741
20 73.4267 3.3345 3 902.3164 81.0971 2.9526 4 1437.354 87.1143 2.6469 5 2160.895 95.0215 2.4624 5 3059.582
ξ=2
0.5 5 130.1104 2.9699 4 3441.569 148.1368 2.6906 5 5265.129 168.0388 2.4973 5 7503.422 187.9643 2.3455 5 10,143.42
10 135.0953 2.9504 4 3859.284 153.1298 2.6769 5 5790.425 173.0335 2.4868 5 8128.839 192.9544 2.3456 6 10,868.95
15 138.2537 2.9123 5 4304.417 158.1229 2.6635 5 6340.730 178.0284 2.4764 5 8779.263 197.9504 2.3372 6 11,619.51
20 143.2440 2.8942 5 4779.568 163.1161 2.6503 5 6916.043 183.0233 2.4662 5 9454.693 202.9464 2.3290 6 12,395.07
0.75 5 95.2901 3.1973 4 1762.774 106.5434 2.8305 5 2795.200 119.7584 2.6027 5 4132.248 133.0048 2.4287 5 5737.989
10 98.6030 3.1718 4 2001.101 109.8680 2.8141 5 3107.905 123.0856 2.5905 5 4511.767 136.3335 2.4191 5 6184.282
15 101.9165 3.1470 4 2256.120 113.1929 2.7981 5 3437.288 126.4128 2.5785 5 4907.959 139.6622 2.4097 5 6647.247
20 105.2306 3.1229 4 2527.831 116.5180 2.7824 5 3783.348 129.7402 2.5667 5 5320.826 142.9853 2.4092 6 7126.926
1 5 78.0613 3.4963 3 1005.580 87.6776 3.0559 5 1648.546 95.6553 2.7262 5 2513.828 105.5519 2.5237 5 3601.809
10 80.5311 3.4604 3 1153.898 88.2910 2.9813 5 1834.647 98.1478 2.7118 5 2770.351 108.0465 2.5128 5 3908.460
15 82.9881 3.4389 4 1314.837 90.7803 2.9616 5 2053.503 100.6405 2.6977 5 3039.383 110.5411 2.5020 5 4227.618
20 85.4603 3.4054 4 1488.338 93.2699 2.9423 5 2284.871 103.1333 2.6838 5 3320.924 113.0358 2.4913 5 4559.282
1.25 5 67.9804 3.9782 3 620.1044 75.4016 3.3039 3 1027.049 83.0820 2.9332 4 1602.586 89.1077 2.6339 5 2373.719
10 69.9276 3.9191 3 714.1623 77.3626 3.2868 4 1161.891 85.0674 2.9142 4 1777.836 91.1011 2.6211 5 2596.551
15 71.8781 3.8634 3 818.2840 79.3393 3.2581 4 1306.796 85.2151 2.8400 5 1972.420 93.0947 2.6085 5 2829.391
20 73.8190 3.8248 4 932.4896 81.3168 3.2303 4 1461.730 87.2063 2.8234 5 2175.119 95.0885 2.5962 5 3072.239
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
20
echnique a e p esen ed in Table 7 using he solu ion algo i hm. As a
esul , he ideal p ice, business pe iod, and paymen ins allmen numbe
o he p ac i ione a e y∗=$120/uni ,
τ
∗=3.7386mon hs, and I∗
=1
ins allmen , espec i ely, while he p ac i ione ’s highes p o i is Y∗=
$6018.771. Exploi ing Eq. (2), he ideal numbe o p oduc s o p ocu e
is Q∗=299.09uni s. Addi ionally, Table 8 p esen s a nume ical
assessmen o he e ec s o C&P, C&T, and CT on he ideal business
ac ics and he p ac i ione ’s highes p o i o Example 4. Because he
a i o uni emissions is iden ical and he o e all emissions exceed he
cap, he bes p ac ices o he p ac i ione in his ins ance a e iden ical
unde all o he a o emen ioned en i onmen al guidelines. The s a is i-
cal da a in Table 8 suppo he indings in P oposi ion 1 and Co olla y 1.
Example 5. Wi h he excep ion o he alue o in demand, his
ins ance akes he iden ical pa ame e alues as in Example 1 in o ac-
coun . The new alue o he p ice-sensi i e pa ame e ( )is 2.6. The
compu a ional indings o he solu ion echnique, which makes use o
he algo i hm, a e shown in Table 9. The e o e, he ideal p ice, business
pe iod, and paymen ins allmen numbe o he p ac i ione a e y∗=
$46.1539/uni ,
τ
∗=4.3602mon hs, and I∗
=2 ins allmen s, espec-
i ely, while he p ac i ione ’s highes p o i is Y∗=$213.4404.
Exploi ing Eq. (2), he ideal numbe o p oduc s o p ocu e is Q∗=
43.602uni s. Table 10 epo s ha his in en o y p oblem is unp o i -
able unde CT egula ion, while bo h ca bon C&P and C&T policies help
he p ac i ione o make he sys em p o i able by allowing ewa ds om
he emaining emission amoun s o he cap.
7. Sensi i i y in es iga ion
The sec ion deals wi h cha ac e izing how he ideal paymen in-
s allmen , p ice, and s ock decisions o he p ac i ione o maximize
p o i change om luc ua ions in sys em pa ame e s unde he C&P
en i onmen al guideline.
7.1. Resul ing om he demand pa ame e s (u, ,
α
, and ξ)
An explo a ion is conduc ed in his subsec ion o in es iga e he
consequences o u, ,
α
, and ξ on he ideal business ac ics and he
highes p o i o he p ac i ione in acco dance wi h he compu a ional
esul s p esen ed in Table 11 when u∈ {100,120,140,160}, ∈ {0.5,
0.75,1,1.25},
α
∈ {5,10,15,20}and ξ∈ {0.5,1,2}in Example 1. In
addi ion, Table 11 displays he compu a ional ou comes o ano he 191
new in en o y examples, and hese ou comes help o b ing ou he
ollowing beha io al p ope ies o he demand pa ame e s on he
p ac i ione ’s p o i and ideal business decisions as well.
I he highes possible ma ke demand (u)inc eases unde ixed
alues o he p ice sensi i e pa ame e ( ), scale pa ame e (
α
), and he
powe index (ξ), he demand always inc eases, and he e o e all he
i ems a e consumed soone ; ha is, he op imal s o age du a ion (
τ
∗)
always dec eases. Because clien demand is high, he p ac i ione se s a
high p ice (y∗)when he maximum po en ial ma ke size (u)is high. Due
o highe clien demand, he in en o y planne o de s a highe lo -size
and hence, he ideal paymen ins allmen s a egy (I∗
)exhibi s an
ascending end in educing he en i e capi al expense om he p e-
paymen sum. Since a high demand helps o sell all p oduc s quickly, he
en i e holding cos sh inks signi ican ly o a high po en ial ma ke size
(u)and he e o e, he p ac i ione ’s maximum p o i (Y∗)inc eases
p ecipi ously. To inc ease he p o i , an impo an ake-home measu e-
men om he abo e indings is ha he p ac i ione should adop some
e ec i e ma ke ing s a egies (such as ad e ising h ough di e en
media, allowing discoun acili ies, among o he s) o aise he po en ial
ma ke demand.
The p ice-sensi i e coe icien ( )in demand has a nega i e way
consequence on he ideal p ice and paymen ins allmen policies when
all emaining demand pa ame e s a e held cons an . As an inc emen in
Table 12
In luences o ca bon policy pa ame e s p and on he p ac i ione ’s ideal business ac ics and highes o Example 1.
Cap-and-p ice policy Cap-and- ade policy Ca bon ax policy
p y∗
τ
∗I∗
Y∗λ(y∗,
τ
∗,I∗
)p y∗
τ
∗I∗
Y∗λ(y∗,
τ
∗,I∗
)p y∗
τ
∗I∗
Y∗λ(y∗,
τ
∗,I∗
)
3 4 88.7776 2.7636 4 1819.306 68.6054 3 3 88.1719 2.7580 5 1818.282 69.5350 3 88.1719 2.7580 5 1608.282 69.5350
3 6 89.9768 2.7950 4 1823.506 67.1913 3 3 88.1719 2.7580 5 1818.282 69.5350 3 88.1719 2.7580 5 1608.282 69.5350
3 8 91.2012 2.8160 3 1830.880 65.5296 3 3 88.1719 2.7580 5 1818.282 69.5350 3 88.1719 2.7580 5 1608.282 69.5350
4 3 88.1719 2.7580 5 1818.282 69.5350 4 4 88.7776 2.7636 4 1819.306 68.6054 4 88.7776 2.7636 4 1539.306 68.6054
6 3 88.1719 2.7580 5 1818.282 69.5350 6 6 89.9768 2.7950 4 1823.506 67.1913 6 89.9768 2.7950 4 1823.506 67.1913
8 3 88.1719 2.7580 5 1818.282 69.5350 8 8 91.2012 2.8160 3 1830.880 65.5296 8 91.2012 2.8160 3 1270.880 65.5296
2 6 89.9768 2.7950 4 1823.506 67.1913 2 2 87.5773 2.7419 5 1818.166 70.2342 2 87.5773 2.7419 5 1678.166 70.2342
4 6 89.9768 2.7950 4 1823.506 67.1913 4 4 88.7776 2.7636 4 1819.306 68.6054 4 88.7776 2.7636 4 1539.306 68.6054
8 6 89.9768 2.7950 4 1823.506 67.1913 8 8 91.2012 2.8160 3 1830.880 65.5296 8 91.2012 2.8160 3 1270.880 65.5296
6 4 88.7776 2.7636 4 1819.306 68.6054 6 6 89.9768 2.7950 4 1823.506 67.1913 6 89.9768 2.7950 4 1823.506 67.1913
6 8 91.2012 2.8160 3 1830.880 65.5296 6 6 89.9768 2.7950 4 1823.506 67.1913 6 89.9768 2.7950 4 1823.506 67.1913
Md.A.-A. Khan e al.

Ope a ions Resea ch Pe spec i es 12 (2024) 100289
21
he p ice-sensi i e coe icien pa ame e implies a p ecipi ous d op in
demand, he op imal eplenishmen pe iod (
τ
∗)inc eases signi ican ly
when he alue o inc eases. Con e sely, he bes pu chased quan i y
by he p ac i ione sh inks, so he op imal paymen ins allmen numbe
(I∗
)exhibi s a downwa d end. Consequen ly, o highe alues o , he
o e all holding expense o he p ac i ione inc eases ma kedly, and
he e o e, he maximum p o i exhibi s an upwa d end.
Simila o he po en ial ma ke demand pa ame e (u), he scale
pa ame e (
α
)o powe demand a ec s he op imal p icing and ins all-
men policies in a posi i e way when all emaining demand pa ame e s
a e held cons an . I
α
in powe demand ises, he consequences o ime
on demand a e in ensi ied, and hence, he s o ed i ems a e consumed
wi h a highe demand, and he op imal eplenishmen du a ion sh inks.
To mee a highe clien demand, he p ac i ione inc eases he pu -
chased quan i y and se s a highe uni selling p ice. Wi h he ambi ion o
lessen he capi al expense, he p ac i ione inc eases he p epaymen
ins allmen equency i he alue o
α
inc eases. Since a highe alue o
α
helps o sell all p oduc s quickly, he o al holding cos sh inks
signi ican ly, and he e o e, he p ac i ione ’s maximum p o i (Y∗)in-
c eases no iceably.
The powe index (ξ)is a g ea in luence on he beha io o he
op imal s a egies o he p ac i ione . When ξ∈ (0,1), he s o ed i ems
a e consumed wi h low demand a he beginning o he business cycle
and, he e o e, a high lo -size inc eases he whole holding expense
ma kedly. Thus, he p ac i ione pu chases a ela i ely small lo -size so
ha all p oduc s a e consumed wi hin a sho pe iod o ime. A he same
ime, o a small lo -size, he p ac i ione also uses a small numbe o
ins allmen equencies o accomplish he p epaymen condi ion. To
diminish he o al holding expense, he p ac i ione se s a ela i ely
small uni p ice. A summa y o he salien indings om he discussion
abo e is ha i he demand is low a he beginning o he business
pe iod, hen he p ac i ione should p ocu e ela i ely a small lo -size
h ough a small numbe o ins allmen equencies, and hen se a
ela i ely small uni selling p ice o inc ease p o i s. On he o he hand,
when ξ∈ (1,∞), he s o ed i ems a e consumed wi h a high demand a
he beginning o he business pe iod, and he e o e, he p ac i ione
o de s a ela i ely la ge lo -size o each cycle, and hen uses a highe
p epaymen ins allmen equency o sh ink he capi al expense agains
he ad ance paymen sum. As he demand is high a he beginning o
each business pe iod, he p ac i ione inc eases he uni selling p ice o
inc ease p o i s. F om Table 11, i is obse ed ha when u=100uni s/
mon h, =1.25 and
α
=20uni s/mon h, hen o he indices o powe
demand ξ=0.5, ξ=1, and ξ=2, he op imal p icing policies a e: y∗=
$73.1492/uni , y∗=$73.4267/uni , and y∗=$73.8190/uni , espec-
i ely; he op imal eplenishmen policies a e:
τ
∗=2.9875 mon hs,
τ
∗=
3.3345 mon hs, and
τ
∗=3.8248 mon hs, espec i ely; he ideal pay-
men ins allmen policies a e: I∗
=3 ins allmen s, I∗
=3 ins allmen s,
and I∗
=4 ins allmen s, espec i ely; he highes p o i s a e: Y∗=
$873.5618, Y∗=$902.3164, and Y∗=$932.4896, espec i ely. In e-
g a ing he abo e analysis, in o de o maximize p o i , he no able
obse a ion is ha he p ac i ione ough o p ocu e a compa a i ely
la ge lo size, adop a g ea e equency o paymen ins allmen s, and
hen es ablish a ela i ely high uni p ice i he demand s ays high a he
beginning phases o he business p ocess.
7.2. Signi icances o he a i p and he ewa d agains each uni
emission
Unde he cap-and-p ice, cap-and- ade, and ca bon ax egula ions,
Table 12 is cons uc ed o explo e he consequences o he penal y p and
he ewa d o each uni ca bon emission on he p ac i ione ’s bes
s a egies, maximum p o i , and he o al discha ged ca bons.
The p ac i ione does no elease lowe emissions using he a i p
han he amoun o ca bon emissions using he ewa d when he a i p
is no highe han he ewa d agains e e y emission uni . On he
Table 13
In luences o O, ac, g0, g, c, ic and γ on he p ac i ione ’s ideal business ac ics and highes p o i o Example 1.
Pa ame e alue γ=1 γ=1.5 γ=2 γ=2.5
y∗
τ
∗I∗
Y∗y∗
τ
∗I∗
Y∗y∗
τ
∗I∗
Y∗y∗
τ
∗I∗
Y∗
O 100 89.9232 3.5951 4 1900.117 89.8217 2.5502 4 1860.852 87.9250 1.9999 4 1841.506 87.8496 1.7693 4 1830.732
200 90.0775 4.0067 4 1873.806 89.9768 2.7950 4 1823.506 88.0944 2.2290 4 1794.256 88.0128 1.9415 4 1776.891
300 90.2098 4.3995 5 1849.971 90.1360 3.0027 4 1789.016 88.2496 2.4197 4 1751.254 88.1643 2.0830 4 1727.223
400 90.3393 4.7450 5 1828.100 90.2858 3.1919 4 1756.733 88.3945 2.5851 4 1711.306 88.3072 2.2044 4 1680.592
ac 20 77.9659 3.5368 4 2808.552 77.8930 2.5258 3 2775.308 77.8070 2.0799 3 2753.352 77.7304 1.8309 3 2737.819
30 83.0992 3.7304 5 2304.773 83.0290 2.6340 4 2270.541 82.9423 2.1529 4 2247.421 82.8773 1.8804 3 2230.912
50 95.2595 4.2919 5 1493.619 95.1796 2.9381 4 1442.253 95.0761 2.3462 3 1406.153 94.9531 2.0267 3 1379.963
60 100.4724 4.6198 5 1166.470 100.4093 3.1118 4 1114.132 100.2831 2.4663 4 1076.718 100.1828 2.1082 3 1049.129
g0 0.5 89.8231 3.9949 4 1893.831 89.7220 2.7886 4 1843.581 87.8413 2.2250 4 1815.272 87.7599 1.9386 4 1797.947
0.75 89.9503 4.0008 4 1883.803 89.8494 2.7917 4 1833.528 87.9679 2.2270 4 1804.748 87.8863 1.9400 4 1787.403
1.25 90.2047 4.0126 4 1863.841 90.1042 2.7982 4 1813.516 88.2210 2.2310 4 1783.795 88.1393 1.9430 4 1766.410
1.5 90.3319 4.0185 4 1853.908 90.2317 2.8014 4 1803.558 88.3476 2.2330 4 1773.366 88.2658 1.9445 4 1755.960
g 0.3 89.9079 4.4931 5 1901.319 89.8550 3.0525 4 1850.039 88.0012 2.4919 4 1820.225 87.9398 2.1365 4 1800.928
0.45 89.9991 4.2196 4 1887.153 89.9179 2.9137 4 1836.326 88.0504 2.3457 4 1806.513 87.9785 2.0285 4 1788.189
0.75 90.1522 3.8235 4 1861.123 90.0324 2.6918 4 1811.441 88.1345 2.1325 4 1783.120 88.0439 1.8692 4 1766.695
0.9 90.2237 3.6638 4 1849.014 90.0851 2.6009 4 1800.023 88.1715 2.0510 4 1772.880 88.0722 1.8076 4 1757.375
c 0.8 90.0640 4.0106 5 1874.705 89.9715 2.7879 4 1824.652 88.0783 2.2275 5 1795.782 88.0078 1.9365 4 1778.541
1.2 90.0669 4.0185 5 1874.206 89.9742 2.7915 4 1824.079 88.0919 2.2256 4 1794.974 88.0103 1.9390 4 1777.715
2 90.0798 4.0129 4 1873.407 89.9957 2.7868 3 1822.939 88.0970 2.2323 4 1793.538 88.0153 1.9440 4 1776.067
2.4 90.0822 4.0192 4 1873.009 89.9977 2.7895 3 1822.508 88.0996 2.2356 4 1792.822 88.0366 1.9402 3 1775.307
ic 0.05 89.8918 3.9781 3 1889.114 89.7899 2.7791 3 1839.413 87.9118 2.2191 3 1810.483 87.8304 1.9342 3 1793.372
0.075 89.9821 4.0022 4 1881.300 89.8918 2.7816 3 1831.381 87.9995 2.2274 4 1802.122 87.9316 1.9353 3 1784.949
0.125 90.1615 4.0305 5 1866.530 90.0724 2.7974 4 1816.011 88.1759 2.2372 5 1786.453 88.1077 1.9426 4 1769.027
0.15 90.2531 4.0348 5 1859.367 90.1679 2.7998 4 1808.533 88.2670 2.2386 5 1778.933 88.1852 1.9487 5 1761.315
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
22
con a y, when he ewa d is smalle han he a i p, he eleased
ca bon amoun s unde he ewa d a e no lowe han he eleased
emissions unde he a i p.
When he ewa d a e is highe han o equal o he penal y a e p,
he p ac i ione amends he ope a ions so ha he eleased ca bons a e
educed. In his si ua ion, he p ac i ione educes he p epaymen in-
s allmen equency (I∗
) o sh ink eleased emissions, hough he capi al
cos inc eases agains he p epaymen amoun . To manage his highe
capi al cos , he p ac i ione inc eases he ideal uni p ice (y∗), and
hence, he bes business pe iod (
τ
∗)also inc eases when he awa d a e
inc eases. Due o a high ewa d a e , he p ac i ione g ow hs he o al
p o i by educing he o al eleased emissions. Consequen ly, he C&P
en i onmen al guideline ensu es a highe o equal p o i o he p ac-
i ione han he C&T p og am when he ewa d a e is g ea e han o
equal o he a i a e p. Fu he mo e, an impo an ake-home mea-
su emen o he ca bon scheme egula o (au ho i y o go e nmen ) is
ha a ewa d a e allowance g ea e han he a i a e agains uni
eleased ca bon helps o educe eleased ca bons ma kedly. As a esul ,
om bo h economic and en i onmen al pe spec i es, a C&P mechanism
pe o ms be e wi h a g ea e ewa d a e han he penal y a e agains
e e y emission uni .
The p ac i ione ’s g ea es p o i acco ding o he C&T mechanism,
on he o he hand, will be g ea e han o equal o he highes p o i
acco ding o he C&P mechanism i he penal y a e p is g ea e han he
incen i e a e agains uni emission. In his ins ance, unde he C&T
egula ion, he p ac i ione educes he p epaymen ins allmen e-
quency and inc eases he uni selling p ice. As a esul , he C&T policy
p o ides a highe o equal p o i han he C&P policy when p > . Thus,
om bo h economic and en i onmen al pe spec i es, a ca bon C&T
mechanism wo ks be e when a C&P egula ion is imposed wi h p > .
As he ca bon ax mechanism does no include any cap on ca bon
elease, he p ac i ione does no ha e any oppo uni y o bene i by
emi ing a lesse amoun o ca bon om he ope a ional ac i i ies.
Mo eo e , he p ac i ione pays ax o e e y eleased emission uni , and
he e o e, he CT mechanism p o ides he lowes maximum p o i o
he p ac i ione among he conside ed egula ions. I is wo h no ing
ha he op imal s a egies o he p ac i ione acco ding o he CT and
C&P mechanisms a e iden ical, as he CT is a pa icula o m o he C&T
egula ion wi h he ze o cap.
7.3. Consequences o O, ac, g0, g, c, ic, and γ
Table 13 highligh s ha when he o de ing expense (O) o a single
cycle inc eases wi h ixed holding cos elas ici y (γ), he p ac i ione
enla ges he pu chased quan i y, and hence, he ideal paymen ins all-
men numbe (I∗
)and business cycle (
τ
∗)s a egies e eal an upwa d
end. Though he bes uni selling p ice (y∗)inc eases, he p ac i ione ’s
maximum p o i alls when O inc eases unde ixed γ. Howe e , when
he holding expense elas ici y (γ)inc eases wi h any ixed o de ing cos
(O), he p ac i ione dwindles he pu chased quan i y, and hence, he
ideal paymen ins allmen numbe (I∗
), p ice (y∗), and business cycle
(
τ
∗)s a egies e eal a downwa d end. Thus, he p ac i ione ’s highes
p o i alls when γ inc eases unde ixed O. This in es iga ion a icula es
ha he p ac i ione should pu chase a ela i ely small lo -size o a
sho business cycle o inc ease p o i when he o de ing expense (O) o
a single cycle is low.
I he uni acquisi ion cos (ac)inc eases wi h ixed holding cos
elas ici y (γ), he p ac i ione se s a highe selling p ice o each uni
(y∗), and he e o e, he clien demand alls ma kedly. This is he main
eason why he op imal business cycle (
τ
∗)sh inks when he uni
acquisi ion cos (ac)inc eases. A he same ime, a downwa d end is
iden i ied on he ideal paymen ins allmen s a egy (I∗
)and he
maximum p o i o he p ac i ione . Thus, he manu ac u e should y
o pu chase p oduc s om he supplie by nego ia ing he uni acquisi-
ion cos (ac)and hen se a lowe uni p ice o inc ease demand.
The ixed accommoda ion cos o a single uni (g0)does no play any
signi ican ole in he ideal paymen ins allmen ac ics (I∗
)o he
p ac i ione . Ne e heless, he bes uni p ice (y∗)inc eases g adually,
and he e o e, he op imal business cycle also inc eases slowly when he
ixed accommoda ing cos (g0)inc eases. Since a high ixed accommo-
da ing cos o a single uni (g0)is a cause o educe he demand, and
hence, inc eases he o al holding cos , he p ac i ione ’s maximum
p o i alls sligh ly i he ixed accommoda ing cos o a single uni (g0)
inc eases. Consequen ly, o inc ease p o i s, he p ac i ione should
educe he ixed accommoda ion cos o a single uni by amending he
ac i i ies app op ia ely.
Unde a ixed holding cos elas ici y (γ), he bes business cycle (
τ
∗)
educes as he ime-sensi i e holding cos pa ame e (g)inc eases. To
manage he inc eased holding cos due o a highe alue o g, he bes
uni p ice (y∗)inc eases sligh ly. Howe e , he maximum p o i o he
p ac i ione alls when g inc eases. Mo eo e , i he holding cos elas-
ici y (γ)unde a ixed g, he bes business cycle (
τ
∗) o he p ac i ione
sh inks signi ican ly as he pu chased lo -size dec eases o educe he
whole holding expense. A sligh dec ease is ecognized on he p icing
s a egy o he p ac i ione when he ime sensi i e holding cos
pa ame e (g)inc eases. In e es ingly, he op imal p epaymen ins all-
men numbe almos emains s able wi h changes in he ime-sensi i e
holding cos pa ame e (g).
An insigni ican consequence on he p ac i ione ’s p icing and
eplenishmen cycle s a egies is obse ed when he ansac ion cha ge
o e e y paymen ins allmen ( c)inc eases wi h ixed holding cos
elas ici y (γ). Howe e , he ideal paymen ins allmen numbe (I∗
)
shows an upwa d end when c inc eases wi h ixed γ. In addi ion, i he
holding expense elas ici y (γ)inc eases wi h s able c, hen he bes p ice
(y∗), and business pe iod (
τ
∗) all signi ican ly. Thus, he maximum
p o i o he p ac i ione dwindles when c inc eases wi h ixed γ.
Though he in e es cha ged a e (ic)does no g ea ly in luence he
p ac i ione ’s bes p icing (y∗), and business cycle (
τ
∗)decisions, i does
s imula e he p ac i ione o inc ease he equency o op imal p epay-
men ins allmen s (I∗
). Since any p epaymen amoun o e a long pe iod
a a high in e es a e incu s high capi al cos s, he p ac i ione makes an
e o o make all he paymen s in he allo ed ime in a numbe o
modes ins allmen s. An impo an inding om he esea ch is ha
when he in e es cha ged a e is high, he p ac i ione ough o aise he
paymen ins allmen numbe s o educe he capi al expendi u e om he
p epaymen .
Table 14
Impac s o emission pa ame e s 
O, ac,  c, g0, and g on he p ac i ione ’s ideal
business ac ics and highes p o i o Example 1.
Pa ame e Value y∗
τ
∗I∗
Y∗λ(y∗,
τ
∗,I∗
)

O 30 89.7855 2.4995 3 1868.773 59.4666
40 89.8758 2.6590 4 1845.507 63.6110
60 88.2189 2.8472 5 1807.578 73.0090
70 88.2646 2.9326 5 1797.197 76.3216
ac 0.2 89.3653 2.7797 4 1871.901 59.7861
0.3 89.6710 2.7873 4 1847.612 63.5195
0.5 90.2826 2.8027 4 1799.584 70.8016
0.6 90.5885 2.8105 4 1775.845 74.3502
 c 0.4 89.9689 2.7844 4 1825.227 66.9141
0.5 89.9728 2.7897 4 1824.366 67.0529
0.7 89.9967 2.7881 3 1822.723 67.0607
0.8 89.9996 2.7921 3 1822.078 67.1645
g0 0.3 89.3653 2.7797 4 1871.901 59.7861
0.4 89.6710 2.7873 4 1847.612 63.5195
0.6 88.3238 2.7617 5 1805.756 73.5238
0.7 88.4757 2.7653 5 1793.276 77.4822
g 0.11 89.8807 2.9942 4 1844.436 63.7308
0.13 89.9300 2.8885 4 1833.695 65.5270
0.17 90.0215 2.7113 4 1813.799 68.7450
0.19 88.2245 2.6326 4 1808.509 72.4551
Md.A.-A. Khan e al.
Ope a ions Resea ch Pe spec i es 12 (2024) 100289
23
7.4. Consequences o ca bon emission pa ame e s 
O, ac,  c, g0, and g
Among he ca bon emission pa ame e s, he amoun o eleased
emissions du ing cycle se up p ocedu es (
O), quan i y o ca bon emis-
sions om he ac i i y o pu chase a uni i em (ac), and ixed quan i y o
eleased emissions om pe uni accommoda ion ac i i y (g0)ha e
no able impac s on he p o i . When hese h ee ca bon amoun s (
O, ac,
and g0) a e educed, he p ac i ione ’s p o i inc eases ma kedly.
The e o e, he p ac i ione should y o educe he eleased emissions
by app op ia ely modi ying he se up, pu chasing, and accommoda ion
ac i i ies. The p ac i ione ’s op imal business cycle (
τ
∗)expe iences a
la ge luc ua ion when he quan i y o ca bon eleased du ing cycle
se up p ocedu es (
O)and ime-sensi i e emissions ac o o holding
ope a ions (g) a y. Table 14 e eals ha in o de o inc ease p o i s, he
p ac i ione should inc ease he pu chased quan i y when he quan i y
o ca bon eleased du ing cycle se up p ocedu es (
O)inc eases, while
he pu chased quan i y should be educed when he ime-sensi i e
emissions ac o o holding ope a ions (g)inc eases.
8. Manage ial insigh s
Using he key ou comes om he p e ious sec ion as a ounda ion,
he subsequen insigh s o he p ac i ione a e de eloped:
•When he demand is low a he beginning o he business pe iod, he
p ac i ione should p ocu e a ela i ely small lo -size h ough a small
numbe o ins allmen equencies and hen se a ela i ely small uni
selling p ice o inc ease p o i s. Howe e , he p ac i ione ough o
p ocu e a compa a i ely la ge lo size by adop ing a g ea e e-
quency o paymen ins allmen s and hen es ablish a ela i ely high
uni p ice i he demand s ays high in he beginning phases o he
business p ocess.
•A ewa d a e allowance g ea e han he a i a e pe uni eleased
ca bon helps o educe eleased ca bons signi ican ly. As a esul ,
om bo h economic and en i onmen al pe spec i es, a C&P mech-
anism pe o ms be e wi h a g ea e ewa d a e han he penal y
a e agains e e y emission uni .
•The p ac i ione should pu chase a ela i ely small lo -size o a sho
business cycle o inc ease p o i when he o de ing expense (O) o a
single cycle is low.
•The in en o y manage should y o pu chase p oduc s om he
supplie by nego ia ing he uni acquisi ion cos (ac)and hen se ing
a lowe uni p ice o inc ease demand.
•When he in e es cha ged a e is high, he p ac i ione ough o aise
he paymen ins allmen numbe s o educe he capi al expendi u e
agains p epaymen .
•The p ac i ione should inc ease he pu chased quan i y when he
quan i y o ca bon eleased du ing cycle se up p ocedu es (
O)in-
c eases, while he pu chased quan i y should be educed when he
ime-sensi i e emission ac o o holding ope a ions (g)inc eases.
9. Conclusions
This s udy examines he signi icance o en i onmen al egula ions
(cap-and-p ice, cap-and- ade, and ca bon ax) on he p o i abili y o a
p ac i ione dealing wi h a pa icula i em unde an ad ance paymen
mechanism. Taking in o accoun he ansac ion ee agains paymen
ins allmen (s) and he capi al expense agains p epaymen amoun , he
ideal ins allmen policy o he p ac i ione is de e mined analy ically.
The s o ed i ems’ consump ion a e is join ly connec ed wi h uni p ice
and ime, mo e p ecisely, in he addi ion o m o a linea -p ice unc ion
and a powe - ime unc ion. An algo i hm is designed o ob ain he
op imal combined paymen ins allmen , p ice, and s ock ac ics by
ensu ing he g ea es p o i o he p ac i ione h ough inco po a ing all
he de eloped analy ical indings. By s udying i e nume ical illus a-
ions, he e icacy o he algo i hm is subsequen ly es ablished.
Obse ing he al e ing pa e ns o he bes business ac ics due o a i-
a ions in he sys em pa ame e s, salien managemen insigh s o he
p ac i ione a e ex ac ed. Fo ins ance, an impo an ake-home mea-
su emen is ha i he demand is low in he beginning phases o he
business pe iod, hen he p ac i ione should p ocu e a ela i ely small
lo -size h ough a small numbe o ins allmen equencies and hen se a
ela i ely small uni selling p ice o inc ease p o i s. F om bo h eco-
nomic and en i onmen al pe spec i es, a C&P en i onmen al mecha-
nism pe o ms be e wi h a g ea e incen i e a e han he a i a e
agains e e y eleased emission uni . The p ac i ione should pu chase a
ela i ely small lo -size o a sho business cycle o inc ease p o i when
he o de ing expense o a single cycle is low. Ano he salien insigh
om he analysis is ha he p ac i ione should inc ease he p epaymen
ins allmen numbe s o lessen he capi al expense agains p epaymen
when he in e es cha ged a e is high. By app op ia ely modi ying he
se up, pu chasing, and accommoda ion ope a ions o educe he
eleased emissions, he p ac i ione ’s p o i inc eases ma kedly.
The cu en s udy possesses se e al ema kable bene i s. I helps he
p ac i ione o make he bes ins allmen decision o accomplish he
p epaymen o lessen he capi al expense and inc ease p o i s. Mo eo e ,
his wo k helps o assess he e iciency o di e en en i onmen al
mechanisms in e ms o bo h economic and en i onmen al pe spec i es.
Howe e , he disc e e ea u e o he p epaymen ins allmen equency
esul s in a ai ly complex solu ion app oach, which is one downside o
his wo k.
The inc easing unce ain y o supply some imes leads o a s ockou
si ua ion o he p ac i ione bu is no in es iga ed when modeling he
in en o y p ocedu e. Thus, allowing sho ages du ing modeling would
be wo h no ing in u u e esea ch. Explo ing in es men plans o lowe
ca bon oo p in s is ano he way o go u he in o his s udy.
Decla a ion o Compe ing In e es
a. The au ho s decla e ha he e is no con lic o in e es . b. This
manusc ip is he au ho s’ o iginal wo k and has no been published no
has i been submi ed simul aneously elsewhe e. c. All au ho s ha e
checked he manusc ip and ha e ag eed o he submission.
Da a a ailabili y
The da a is included in he pape
Re e ences
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manu ac u ing– emanu ac u ing sys ems unde ca bon ax policy. Eu J Ope Res
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[2] Al a es HK, Ghai han AM. In en o y and p icing model wi h p ice-dependen
demand, ime- a ying holding cos , and quan i y discoun s. Compu Ind Eng 2016;
94:170–7.
[3] A inada T, He bon A, Spiegel U. Op imal o de ing and p icing policy o demand
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155:406–17.
[4] C´
a denas-Ba ´
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