research article research on optimal policy of single

Research ArticleResearch on Optimal Policy of Single-Period InventoryManagement with Two Suppliers

Baimei Yang Lihui Sui and Peipei Zhu

School of Business Shanghai Dianji University Shanghai 201306 China

Correspondence should be addressed to Baimei Yang ningmeirhotmailcom

Received 23 June 2014 Accepted 12 September 2014 Published 20 November 2014

Academic Editor Pu-yan Nie

Copyright copy 2014 Baimei Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study a single-period inventory control problemwith two independent suppliersWith the first supplier the buyer incurs a highvariable cost but negligible fixed cost with the second supplier the buyer incurs a lower variable cost but a positive fixed cost At thesame time the ordering quantity is limited We develop the optimal inventory control policy when the holding and shortage costfunction is convex We also conduct some numerical experiments to explore the effects of the fixed setup cost K and the orderingcapacity Q on the optimal control policy

1 Introduction

In this paper we consider a single-period inventory systemwith two suppliers and different ordering cost structures Weassume that the two suppliers are independent of each otherwhich indicates that the decision of one supplier will notinfluence the other onersquos decision And we could only orderfrom one supplier At the same time the ordering quantity islimited

Several studies have been conducted attempting to extendthe analysis of a single product systemwith two suppliers Foxet al [1] analyze a periodic-review inventory model wherethe decision maker can buy from either of the two suppliersWith the first supplier the buyer incurs a high variablecost but negligible fixed cost with the second supplier thebuyer incurs a lower variable cost but a substantial fixedcost Consequently ordering costs are piecewise linear andconcave Chen et al [2] establish a new preservation propertyof quasi-119870-concavity under certain optimization operationsand apply the result to analyze joint inventory-pricingmodelsfor single-product periodic-review inventory systems withconcave ordering costs Caliskan-Demirag et al [3] consider astochastic periodic-review inventory control system in whichthe fixed cost depends on the order quantity

This problem is also related to the optimal control ofa single-product system with finite capacity and setup cost

When the setup cost is zero Federgruen and Zipkin [4 5]have shown that the optimal strategy for the capacitatedinventory control problem is known as the modified base-stock policy Several studies have been conducted on thisproblem For instance Shaoxiang and Lambrecht [6] andChen [7] point out that the generally known result is that theoptimal policy can only be partially characterized in the formof119883-119884 bands In Gallego and Scheller-Wolf [8] the structureof the policy between the bands is further refined using twonumbers 119904 and 119904

1015840 in four possible regions Chao et al [9] havestudied a dynamic inventory and pricing optimization prob-lem in a periodic review inventory systemwith setup cost andfinite ordering capacity in each period Gavish and Graves[10] study one-product productioninventory problemwith afixed setup cost under continuous review policy De Kok [11]deals with a one-product productioninventory model withlost sales

If the decisions of the two suppliers can affect each otherthis problem is changed into a problem with capacity con-straints and game theory Some experts havemade deep stud-ies on this area Nie [12] considers commitment for storablegoods under vertically integrated structures Nie and Chen[13] andNie [14] focus on the duopoly substitutability productwith an upstream input subjected to capacity constraintsChen andWen [15] analyze co-op advertising behavior basedon a dual-brand model with a single manufacturer and a

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 417319 5 pageshttpdxdoiorg1011552014417319

2 The Scientific World Journal

single retailer And more applications of game theory toproblems in economics can be found in Nie et al [16]

In this paper we consider a single-period inventory sys-temwith two independent suppliers and investigate the struc-ture of the optimal inventory control When we order fromthe first supplier the ordering cost only includes variablecost When we order from the other supplier the orderingcosts include a fixed cost and variable cost Moreover theordering capacity is finite Then we establish the model anddevelop the optimal policy for both general convex holdingand shortage cost function and piecewise linear holding andshortage cost function We also conduct several numericalexperiments to explore the effects of the fixed ordering cost119870 and the ordering capacity119876 on the optimal control policy

The rest of this paper is organized as follows In the nextsection we present the model Themain result and the proofsare provided in Section 3 In Section 4 are the numericalresults and concluding remarks are in Section 5

2 Model

Consider a single-period inventory systemwith two indepen-dent suppliers Here the decision of one supplier will notinfluence the other onersquos decision When we order from thefirst supplier the ordering cost only includes variable cost 119888

1

When we order from the other supplier the ordering costsinclude a fixed cost119870 and variable cost 119888

2 Here 119888

1gt 1198882 And

we could only order from one supplierIn the single-period inventory system the initial inven-

tory level is zeroThere is a finite ordering capacity119876 for bothsuppliers That is the ordering quantity from each suppliercannot exceed 119876 Let 119909 be the order quantity Then there willbe 0 le 119909 le 119876

The sequence of events during a period is as follows(1) replenishment order is placed (2) replenishment orderarrives (3) random demand is realized and (4) all costs arecomputed

In the period there are 119873 nonnegative demands from1198631to 119863119873 The possibility for 119863

119899is 119901119899 119899 = 1 119873 And

sum119873

119899=1119901119899= 1 A cost 119867(119909) is incurred at the end of period

if the inventory level after demand realization is 119909 whichrepresents inventory holding cost if 119909 ge 0 and shortage costif 119909 lt 0 Assume that 119867(119909) is convex Then the expectedholding and shortage cost given that the inventory level afterreplenishment is 119909 is 119864[119867(119909 minus 119863

119899)] which is also convex

Let 119862119894denote the minimum cost when purchasing from

supplier 119894 Then

1198621= min0le119909le119876

1198881119909 + 119864 [119867 (119909 minus 119863

119899)]

1198622= min0le119909le119876

119870119868 [119909 gt 0] + 1198882119909 + 119864 [119867 (119909 minus 119863

119899)]

(1)

where 1[119860] is the indicator function taking value 1 ifstatement 119860 is true and zero otherwise The objective is to

characterize the optimal ordering strategy that minimizes theexpected cost Hence the optimal equation is

119862 = min 1198621 1198622

= min0le119909le119876

1198881119909 + 119864 [119867 (119909 minus 119863

119899)]

119870119868 [119909 gt 0] + 1198882119909 + 119864 [119867 (119909 minus 119863

119899)]

(2)

3 Analysis and Results

In this section we will analyze the two suppliers separatelyfirst Then we compare the two cost functions and obtain theresults In addition we will give one special case

31 Supplier 1 The cost function of ordering from supplier 1is simple Due to the convexity of 119864[119867(119909 minus119863

119899)] it is obvious

that 1198881119909 + 119864[119867(119909 minus 119863

119899)] is also convex Hence it is easy to

obtain the optimal policy for ordering from supplier 1

Lemma 1 Assume that

11988811199091+ 119864 [119867 (119909

1minus 119863119899)] = min 119888

1119909 + 119864 [119867 (119909 minus 119863

119899)] (3)

Then the optimal policy when ordering from supplier 1 is

(1) no order if 1199091le 0

(2) order 1199091if 0 lt 119909

1lt 119876 and

(3) order capacity 119876 if 1199091ge 119876

32 Supplier 2 The cost function of ordering from supplier 2is complex Actually the cost function could be restructuredas

1198622= min0le119909le119876

119864 [119867 (minus119863119899)] 119870 + 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)] (4)

It is obvious that 1198882119909 +119864[119867(119909 minus119863

119899)] is also convexThen

we will obtain the following lemma

Lemma 2 Assume that

11988821199092+ 119864 [119867 (119909

2minus 119863119899)] = min 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)]

1198961= 119864 [119867 (minus119863

119899)] minus 1198882min 119909

2 119876

minus 119864 [119867 (min 1199092 119876 minus 119863

119899)]

(5)



(2) order 1199092if 0 lt 119909

2lt 119876 and 119870 lt 119896

1

(3) no order if 0 lt 1199092lt 119876 and 119870 ge 119896

1

(4) order capacity 119876 if 1199092ge 119876 and 119870 lt 119896

1 and

(5) no order if 1199092ge 119876 and 119870 ge 119896

1

From Lemma 2 we could regard 1198961as a benchmarking of

119870 If 119870 ge 1198961 the optimal policy of ordering from supplier 2

is always no order

The Scientific World Journal 3

33 Combination From the previous analysis we will findthat the values of both 119909

1and 119909

2are independent of ordering

capacity 119876 and fixed ordering cost 119870 Moreover due to1198881

gt 1198882 there is always 119909

1lt 1199092 Then there would be six

possibilities 1199091lt 1199092le 0 119909

1le 0 lt 119909

2lt 119876 119909

1le 0 lt 119876 lt 119909

2

0 lt 1199091lt 1199092lt 119876 0 lt 119909

1lt 119876 lt 119909

2 and 119876 le 119909

1lt 1199092

The following theorem characterizes the structure of theoptimal inventory and pricing policy for each period

Theorem 3 Suppose that

11988811199091+ 119864 [119867 (119909

1minus 119863119899)] = min 119888

1119909 + 119864 [119867 (119909 minus 119863

119899)]

11988821199092+ 119864 [119867 (119909

2minus 119863119899)] = min 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)]

1198961= 119864 [119867 (minus119863

119899)] minus 1198882min 119909

2 119876

minus 119864 [119867 (min 1199092 119876 minus 119863

119899)]

1198962= 1198881min 119909

1 119876 + 119864 [119867 (min 119909

1 119876 minus 119863

119899)]

minus 1198882min 119909

2 119876 minus 119864 [119867 (min 119909

2 119876 minus 119863

119899)]

(6)

If 119870 ge 1198961 then the optimal ordering policy is


(2) order 1199091from supplier 1 if 0 lt 119909

1lt 119876 and

(3) order capacity 119876 from supplier 2 if 1199091ge 119876

And if 119870 lt 1198961 then the optimal ordering policy is

(1) no order if 1199091lt 1199092le 0

(2) order 1199092from supplier 2 if 119909

1le 0 lt 119909

2lt 119876

(3) order capacity 119876 from supplier 2 if 1199091le 0 lt 119876 lt 119909

2


1lt 1199092lt 119876 and 119870 le

1198962


1lt 1199092lt 119876 and 119870 ge

1198962

(6) order capacity 119876 from supplier 2 if 0 lt 1199091lt 119876 lt 119909

2

and 119870 le 1198962


1lt 119876 lt 119909

2and 119870 ge

1198962

(8) order capacity 119876 from supplier 2 if 1199092gt 1199091ge 119876 and

119870 le 1198962 and


119870 ge 1198962

Here we omit the proof of Theorem 3 From Theorem 3we will find if 119870 ge 119896

1we will not order from supplier 2 On

the contrary when 119870 lt 1198961 we will not order from supplier 1

if 1199091lt 0 Moreover when 119870 lt 119896

1and 119909

2gt 1199091gt 0 we will

order min1199092 119876 from supplier 2 if 119870 le 119896

2and min119909

1 119876

from supplier 1 if 119870 ge 1198962 If 119870 = 119896

2 ordering from either

supplier could be optimal

34 Special Case In the previous analysis we only assumethat 119867(119909) is convex The most common function of hold-ing and shortage cost is piecewise linear for instance

(119909) = ℎ119909++ 119887119909minus Here 119909

+= max119909 0 119909minus = maxminus119909 0

ℎ gt minusmin1198881 1198882 and 119887 gt max119888

1 1198882 ℎ gt minusmin119888

1 1198882

indicates that ℎ could be positive or negative When ℎ gt 0 itimplies that there is need to take some costs to control extrainventory while minusmin119888

1 1198882 lt ℎ lt 0 implies that we could

obtain some incomes when selling extra stock However theincome could notmake up the unit costThe latter is shown inthe newsboy problem 119887 gt max119888

1 1198882means that the shortage

cost is less than the unit costWhen the holding and shortage cost function is piecewise

linear the result is simplified because both 1199091and 119909

2are

positive Here we omit the proof Then the optimal policy isas follows

Theorem 4 When the holding and shortage cost function ispiecewise linear suppose that

11988811199091+ 119864 [119867 (119909

1minus 119863119899)] = min 119888

1119909 + 119864 [119867 (119909 minus 119863

119899)]

11988821199092+ 119864 [119867 (119909

2minus 119863119899)] = min 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)]

1198962= 1198881min 119909

1 119876 + 119864 [119867 (min 119909

1 119876 minus 119863

119899)]

minus 1198882min 119909

2 119876 minus 119864 [119867 (min 119909

2 119876 minus 119863

119899)]

(7)

Then the optimal ordering policy is to order min1199091 119876

from supplier 1 if 119870 ge 1198962and min119909

2 119876 from supplier 2 if

119870 le 1198962 If 119870 = 119896

2 ordering from either supplier could be

optimalHere we also omit the proof And we will find that if the

holding and shortage cost function is piecewise linear noorder will not be the optimal policy Moreover we only needto compare119870 and 119896

2to decide the ordering strategy

4 Numerical Results

In order to explore the effects of the fixed setup cost 119870 andthe ordering capacity 119876 on the optimal control policy weconduct several numerical experiments for a simple inven-tory problem with piecewise linear holding and shortage costfunction In the subsequent numerical experiments we usethe following basic settings 119888

1= 5 119870 = 20 119888

2= 3 and

119876 = 10 The demands could be 3 6 8 10 and 15 And thecorresponding possibilities are 119901

1= 01 119901

2= 02 119901

3= 03

1199014= 03 and 119901

1= 01 In the piecewise linear holding and

shortage cost function we assume that ℎ = 2 and 119887 = 7

41 Effect of119870 Firstly we consider the effect of fixed orderingcost 119870 Let 119870 = 0 1 30 The result is shown in Figure 1

FromFigure 1 it is obvious that fixed ordering cost119870doesnot affect the optimal policy when ordering from supplier 1However when ordering from supplier 2 the minimal costis increasing on 119870 If 119870 is large enough the optimal policyfor ordering from supplier 2 will change from ordering somequantity to no order As a whole the best supplier will changefrom supplier 2 to supplier 1 when 119870 is increasing

42 Effect of 119876 Then we consider the effect of orderingcapacity 119876 Let 119876 = 1 10 The result when 119870 = 10 is


0 5 10 15 20 25 3030

35

40

45

50

55

60

Fixed ordering cost K

Min

imal

cost

Supplier 1Supplier 2

Total

Figure 1 Minimal cost under different119870

1 2 3 4 5 6 7 8 9 1040

42

44

46

48

50

52

54

56

58

60

Ordering capacity C

Min

imal

cost

whe

nK

=10


Total

Figure 2 Minimal cost under different 119876 when 119870 = 10

shown in Figure 2 and the one when 119870 = 20 is shown inFigure 3

Then we will find that when ordering capacity 119862 is smallthe optimal ordering quantity is increasing and minimal costis decreasing on 119862 However when ordering capacity is largeenough the optimal policy is constant Moreover when 119862 isincreasing the choice of supplier will change from supplier 1to supplier 2 when119870 is small while the choice will be alwaysto order from supplier 1 when 119870 is large

5 Conclusion

In this paper we consider a single-period inventory systemwith two suppliers When we order from the first supplierthe ordering cost only includes variable cost When we order

1 2 3 4 5 6 7 8 9 1040

42

44

46

48

50

52

54

56

58

60

Ordering capacity C

Min

imal

cost

whe

nK

=20


Total


from the other supplier the ordering costs include a fixed costand variable cost Andwe could only order from one supplierMoreover there exists ordering capacity constraint Then weestablish the model and develop the optimal policy We alsoconduct several numerical experiments to explore the effectsof the fixed ordering cost 119870 and the ordering capacity 119876 onthe optimal control policy In the future research we willstudy the multiperiod inventory model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Project for Training andSupporting Young Teachers in Shanghai Universities (noZZSDJ13007) and Leading Academic Discipline Project ofShanghai Dianji University (no 10XKJ01) Sincere thanksare offered to the anonymous reviewers for their helpfulsuggestions

References

[1] E J Fox R Metters and J Semple ldquoOptimal inventory policywith two suppliersrdquoOperations Research vol 54 no 2 pp 389ndash393 2006

[2] X Chen Y Zhang and S X Zhou ldquoPreservation of quasi-119870-concavity and its applicationsrdquoOperations Research vol 58 no4 pp 1012ndash1016 2010

[3] O Caliskan-Demirag Y F Chen and Y Yang ldquoOrdering poli-cies for periodic-review inventory systems with quantity-dependent fixed costsrdquo Operations Research vol 60 no 4 pp785ndash796 2012


[4] A Federgruen and P Zipkin ldquoAn inventory model with limitedproduction capacity and uncertain demands I The average-cost criterionrdquo Mathematics of Operations Research vol 11 no2 pp 193ndash207 1986

[5] A Federgruen and P Zipkin ldquoAn inventory model withlimited production capacity and uncertain demands II Thediscounted-cost criterionrdquoMathematics of Operations Researchvol 11 no 2 pp 208ndash215 1986

[6] C Shaoxiang andM Lambrecht ldquoX-Y band andmodified (s S)policyrdquo Operations Research vol 44 no 6 pp 1013ndash1019 1996

[7] S Chen ldquoThe infinite horizon periodic review problem withsetup costs and capacity constraints a partial characterizationof the optimal policyrdquo Operations Research vol 52 no 3 pp409ndash421 2004

[8] G Gallego and A Scheller-Wolf ldquoCapacitated inventory prob-lems with fixed order costs some optimal policy structurerdquoEuropean Journal of Operational Research vol 126 no 3 pp603ndash613 2000

[9] X Chao B Yang and Y Xu ldquoDynamic inventory and pricingpolicy in a capacitated stochastic inventory system with fixedordering costrdquo Operations Research Letters vol 40 no 2 pp99ndash107 2012

[10] B Gavish and S C Graves ldquoA one-product productioninven-tory problem under continuous review policyrdquo OperationsResearch vol 28 no 5 pp 1228ndash1236 1980

[11] A G de Kok ldquoApproximations for a lost-sales productioninventory control model with service level constraintsrdquo Man-agement Science vol 31 no 6 pp 729ndash737 1985

[12] P-Y Nie ldquoCommitment for storable goods under verticalintegrationrdquo Economic Modelling vol 26 no 2 pp 414ndash4172009

[13] P-Y Nie and Y-H Chen ldquoDuopoly competitions with capacityconstrained inputrdquo Economic Modelling vol 29 no 5 pp 1715ndash1721 2012

[14] P-Y Nie ldquoEffects of capacity constraints on mixed duopolyrdquoJournal of Economics vol 112 pp 283ndash294 2013

[15] Y-H Chen and X-W Wen ldquoVertical cooperative advertisingwith substitute brandsrdquo Journal of Applied Mathematics vol2013 Article ID 480401 8 pages 2013

[16] P Y Nie TMatsuhisa X HenryWang and P A Zhang ldquoGametheory and applications in economicsrdquo Journal of AppliedMathematics vol 2014 Article ID 936192 2 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


single retailer And more applications of game theory toproblems in economics can be found in Nie et al [16]

In this paper we consider a single-period inventory sys-temwith two independent suppliers and investigate the struc-ture of the optimal inventory control When we order fromthe first supplier the ordering cost only includes variablecost When we order from the other supplier the orderingcosts include a fixed cost and variable cost Moreover theordering capacity is finite Then we establish the model anddevelop the optimal policy for both general convex holdingand shortage cost function and piecewise linear holding andshortage cost function We also conduct several numericalexperiments to explore the effects of the fixed ordering cost119870 and the ordering capacity119876 on the optimal control policy

The rest of this paper is organized as follows In the nextsection we present the model Themain result and the proofsare provided in Section 3 In Section 4 are the numericalresults and concluding remarks are in Section 5

2 Model

Consider a single-period inventory systemwith two indepen-dent suppliers Here the decision of one supplier will notinfluence the other onersquos decision When we order from thefirst supplier the ordering cost only includes variable cost 119888

1

When we order from the other supplier the ordering costsinclude a fixed cost119870 and variable cost 119888

2 Here 119888

1gt 1198882 And

we could only order from one supplierIn the single-period inventory system the initial inven-

tory level is zeroThere is a finite ordering capacity119876 for bothsuppliers That is the ordering quantity from each suppliercannot exceed 119876 Let 119909 be the order quantity Then there willbe 0 le 119909 le 119876

The sequence of events during a period is as follows(1) replenishment order is placed (2) replenishment orderarrives (3) random demand is realized and (4) all costs arecomputed

In the period there are 119873 nonnegative demands from1198631to 119863119873 The possibility for 119863

119899is 119901119899 119899 = 1 119873 And

sum119873

119899=1119901119899= 1 A cost 119867(119909) is incurred at the end of period

if the inventory level after demand realization is 119909 whichrepresents inventory holding cost if 119909 ge 0 and shortage costif 119909 lt 0 Assume that 119867(119909) is convex Then the expectedholding and shortage cost given that the inventory level afterreplenishment is 119909 is 119864[119867(119909 minus 119863

119899)] which is also convex

Let 119862119894denote the minimum cost when purchasing from

supplier 119894 Then

1198621= min0le119909le119876

1198881119909 + 119864 [119867 (119909 minus 119863

119899)]

1198622= min0le119909le119876

119870119868 [119909 gt 0] + 1198882119909 + 119864 [119867 (119909 minus 119863

119899)]

(1)

where 1[119860] is the indicator function taking value 1 ifstatement 119860 is true and zero otherwise The objective is to

characterize the optimal ordering strategy that minimizes theexpected cost Hence the optimal equation is

119862 = min 1198621 1198622

= min0le119909le119876

1198881119909 + 119864 [119867 (119909 minus 119863

119899)]

119870119868 [119909 gt 0] + 1198882119909 + 119864 [119867 (119909 minus 119863

119899)]

(2)

3 Analysis and Results

In this section we will analyze the two suppliers separatelyfirst Then we compare the two cost functions and obtain theresults In addition we will give one special case

31 Supplier 1 The cost function of ordering from supplier 1is simple Due to the convexity of 119864[119867(119909 minus119863

119899)] it is obvious

that 1198881119909 + 119864[119867(119909 minus 119863

119899)] is also convex Hence it is easy to

obtain the optimal policy for ordering from supplier 1

Lemma 1 Assume that

11988811199091+ 119864 [119867 (119909

1minus 119863119899)] = min 119888

1119909 + 119864 [119867 (119909 minus 119863

119899)] (3)



(2) order 1199091if 0 lt 119909

1lt 119876 and

(3) order capacity 119876 if 1199091ge 119876

32 Supplier 2 The cost function of ordering from supplier 2is complex Actually the cost function could be restructuredas

1198622= min0le119909le119876

119864 [119867 (minus119863119899)] 119870 + 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)] (4)

It is obvious that 1198882119909 +119864[119867(119909 minus119863

119899)] is also convexThen

we will obtain the following lemma

Lemma 2 Assume that

11988821199092+ 119864 [119867 (119909

2minus 119863119899)] = min 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)]

1198961= 119864 [119867 (minus119863

119899)] minus 1198882min 119909

2 119876

minus 119864 [119867 (min 1199092 119876 minus 119863

119899)]

(5)



(2) order 1199092if 0 lt 119909

2lt 119876 and 119870 lt 119896

1

(3) no order if 0 lt 1199092lt 119876 and 119870 ge 119896

1

(4) order capacity 119876 if 1199092ge 119876 and 119870 lt 119896

1 and

(5) no order if 1199092ge 119876 and 119870 ge 119896

1

From Lemma 2 we could regard 1198961as a benchmarking of

119870 If 119870 ge 1198961 the optimal policy of ordering from supplier 2

is always no order



1and 119909






1le 0 lt 119909

2lt 119876 119909

1le 0 lt 119876 lt 119909

2

0 lt 1199091lt 1199092lt 119876 0 lt 119909

1lt 119876 lt 119909

2 and 119876 le 119909

1lt 1199092



11988811199091+ 119864 [119867 (119909

1minus 119863119899)] = min 119888

1119909 + 119864 [119867 (119909 minus 119863

119899)]

11988821199092+ 119864 [119867 (119909

2minus 119863119899)] = min 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)]

1198961= 119864 [119867 (minus119863

119899)] minus 1198882min 119909

2 119876

minus 119864 [119867 (min 1199092 119876 minus 119863

119899)]

1198962= 1198881min 119909

1 119876 + 119864 [119867 (min 119909

1 119876 minus 119863

119899)]

minus 1198882min 119909

2 119876 minus 119864 [119867 (min 119909

2 119876 minus 119863

119899)]

(6)




1lt 119876 and



(1) no order if 1199091lt 1199092le 0


1le 0 lt 119909

2lt 119876


2


1lt 1199092lt 119876 and 119870 le

1198962


1lt 1199092lt 119876 and 119870 ge

1198962


2

and 119870 le 1198962


1lt 119876 lt 119909

2and 119870 ge

1198962


119870 le 1198962 and


119870 ge 1198962





1and 119909



2and min119909

1 119876





(119909) = ℎ119909++ 119887119909minus Here 119909



1 1198882 ℎ gt minusmin119888

1 1198882







2are



11988811199091+ 119864 [119867 (119909

1minus 119863119899)] = min 119888

1119909 + 119864 [119867 (119909 minus 119863

119899)]

11988821199092+ 119864 [119867 (119909

2minus 119863119899)] = min 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)]

1198962= 1198881min 119909

1 119876 + 119864 [119867 (min 119909

1 119876 minus 119863

119899)]

minus 1198882min 119909

2 119876 minus 119864 [119867 (min 119909

2 119876 minus 119863

119899)]

(7)




119870 le 1198962 If 119870 = 119896





4 Numerical Results


1= 5 119870 = 20 119888

2= 3 and


1= 01 119901

2= 02 119901

3= 03

1199014= 03 and 119901







0 5 10 15 20 25 3030

35

40

45

50

55

60


Min

imal

cost


Total


1 2 3 4 5 6 7 8 9 1040

42

44

46

48

50

52

54

56

58

60

Ordering capacity C

Min

imal

cost

whe

nK

=10


Total




5 Conclusion


1 2 3 4 5 6 7 8 9 1040

42

44

46

48

50

52

54

56

58

60

Ordering capacity C

Min

imal

cost

whe

nK

=20


Total





Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1and 119909






1le 0 lt 119909

2lt 119876 119909

1le 0 lt 119876 lt 119909

2

0 lt 1199091lt 1199092lt 119876 0 lt 119909

1lt 119876 lt 119909

2 and 119876 le 119909

1lt 1199092



11988811199091+ 119864 [119867 (119909

1minus 119863119899)] = min 119888

1119909 + 119864 [119867 (119909 minus 119863

119899)]

11988821199092+ 119864 [119867 (119909

2minus 119863119899)] = min 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)]

1198961= 119864 [119867 (minus119863

119899)] minus 1198882min 119909

2 119876

minus 119864 [119867 (min 1199092 119876 minus 119863

119899)]

1198962= 1198881min 119909

1 119876 + 119864 [119867 (min 119909

1 119876 minus 119863

119899)]

minus 1198882min 119909

2 119876 minus 119864 [119867 (min 119909

2 119876 minus 119863

119899)]

(6)




1lt 119876 and



(1) no order if 1199091lt 1199092le 0


1le 0 lt 119909

2lt 119876


2


1lt 1199092lt 119876 and 119870 le

1198962


1lt 1199092lt 119876 and 119870 ge

1198962


2

and 119870 le 1198962


1lt 119876 lt 119909

2and 119870 ge

1198962


119870 le 1198962 and


119870 ge 1198962





1and 119909



2and min119909

1 119876





(119909) = ℎ119909++ 119887119909minus Here 119909



1 1198882 ℎ gt minusmin119888

1 1198882







2are



11988811199091+ 119864 [119867 (119909

1minus 119863119899)] = min 119888

1119909 + 119864 [119867 (119909 minus 119863

119899)]

11988821199092+ 119864 [119867 (119909

2minus 119863119899)] = min 119888

2119909 + 119864 [119867 (119909 minus 119863

119899)]

1198962= 1198881min 119909

1 119876 + 119864 [119867 (min 119909

1 119876 minus 119863

119899)]

minus 1198882min 119909

2 119876 minus 119864 [119867 (min 119909

2 119876 minus 119863

119899)]

(7)




119870 le 1198962 If 119870 = 119896





4 Numerical Results


1= 5 119870 = 20 119888

2= 3 and


1= 01 119901

2= 02 119901

3= 03

1199014= 03 and 119901







0 5 10 15 20 25 3030

35

40

45

50

55

60


Min

imal

cost


Total


1 2 3 4 5 6 7 8 9 1040

42

44

46

48

50

52

54

56

58

60

Ordering capacity C

Min

imal

cost

whe

nK

=10


Total




5 Conclusion


1 2 3 4 5 6 7 8 9 1040

42

44

46

48

50

52

54

56

58

60

Ordering capacity C

Min

imal

cost

whe

nK

=20


Total





Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 3030

35

40

45

50

55

60


Min

imal

cost


Total


1 2 3 4 5 6 7 8 9 1040

42

44

46

48

50

52

54

56

58

60

Ordering capacity C

Min

imal

cost

whe

nK

=10


Total




5 Conclusion


1 2 3 4 5 6 7 8 9 1040

42

44

46

48

50

52

54

56

58

60

Ordering capacity C

Min

imal

cost

whe

nK

=20


Total





Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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