supply chain model with stochastic lead time, trade-credit...
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Research ArticleSupply Chain Model with Stochastic Lead Time, Trade-CreditFinancing, and Transportation Discounts
Sung Jun Kim and Biswajit Sarkar
Department of Industrial & Management Engineering, Hanyang University, Ansan, Gyeonggi-do 15588, Republic of Korea
Correspondence should be addressed to Biswajit Sarkar; [email protected]
Received 26 October 2016; Revised 22 January 2017; Accepted 14 February 2017; Published 18 May 2017
Academic Editor: Mohammad D. Aliyu
Copyright Β© 2017 Sung Jun Kim and Biswajit Sarkar. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.
This model extends a two-echelon supply chain model by considering the trade-credit policy, transportations discount to make acoordination mechanism between transportation discounts, trade-credit financing, number of shipments, quality improvement ofproducts, and reduced setup cost in such a way that the total cost of the whole system can be reduced, where the supplier offerstrade-credit-period to the buyer. For buyer, the backorder rate is considered as variable. There are two investments to reduce setupcost and to improve quality of products. The model assumes lead time-dependent backorder rate, where the lead time is stochasticin nature. By using the trade-credit policy, the model gives how the credit-period would be determined to achieve the win-winoutcome. An iterative algorithm is designed to obtain the global optimum results. Numerical example and sensitivity analysis aregiven to illustrate the model.
1. Introduction
Supply chain indicates that the relation among the supplychain players is forever to obtain maximum profit togetherand individual profit always. The aim of this model is toreduce the total supply chain cost.
Supply chain management (SCM) is the collaborationamong suppliers, manufacturers, retailers, and customers.Practically, the aim of the SCMmodel is tominimize the totalcost or tomaximize the total profit throughout the channel. Inthis direction, the idea of integrated vendor-buyer inventorymanagement has been successfully considered since last fewdecades. In some practical situations, lead time and setupcost can be controlled and reduced in various ways. It is atrend by shortening the lead time and reducing setup cost;the safety stock can be minimized. Thus, the target is alwaysto decrease the stockout loss and improve the service level forthe customer as to increase the competitive edge in businesswithin the SCM environment. Thus, the controllable leadtime and setup cost reduction are the key concepts to obtainsuccessful business and have attracted extensive researchattention [1]. Reduced setup in the basic inventorymodel was
investigated by Porteus [2] which is the key research idea forcost reduction policy in a supply chain.
Ouyang et al. [3] developed a continuous review inven-tory model for lead time and ordering cost reductions withpartial backorders. This model initiated single-vendor multi-buyer with ordering cost reduction. In the same direction,Woo et al. [4] and Chang et al. [5] developed several modelswith cost reduction policies. Zhang et al. [6] developed theintegrated vendor-managed inventory (VMI) model for atwo-echelon system with ordering cost reduction. Recently,Shin et al. [7] discussed a continuous review inventorymodel with transportation cost discount and a service levelconstraint, whereas Huang [8] introduced another new costreduction policy through order processing. Recently, Sarkar[9] introduced another cost reduction policy with variabledemand under imperfect production process. The above-mentioned models are several major contributions in thisfield.
In reality, transportation cost is not always constant. But,many papers used the concept of constant transportationcosts. Thus, it is too much important to consider the costas variable. By using the single-setup-multidelivery (SSMD),
HindawiMathematical Problems in EngineeringVolume 2017, Article ID 6465912, 14 pageshttps://doi.org/10.1155/2017/6465912
https://doi.org/10.1155/2017/6465912
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2 Mathematical Problems in Engineering
the number of types of transportation increases always.The primary aim of using SSMD policy is to reduce theholding cost of buyer, but as a result, the transportationcost increases. Therefore, there will be a trade-off betweenthem to control the cost of the whole system. Ganeshan [10]developed amodel formanaging supply chain inventory, withmultiretailer, single-warehouse, and multisupplier. Recently,Sarkar et al. [11, 12] developed two SCMmodels with constantand variable transportation costs.
Nowadays transportation mode selection with positivemanufacturing lead time is more effective in SCM system[13]. Ertogral et al. [14] developed a vendor-buyer supplychain model for production and shipment issues with trans-portation cost. Kang and Kim [15] discussed a coordinationof inventory holding and transportation management ina two-echelon supply chain model. Chung [16] proposedan integrated-inventory model with transportation cost andtwo-level trade-credit policy.
In real life situation, everybody prefers the best quality ofproducts with the cheapest price. As a result, all industrieshave to make good quality products with at least cheaperprice. That is why, in many cases, some investment can bedone to reduce the setup cost and improve the quality ofproducts. In this direction, Ouyang et al. [17] developed anintegrated-inventory model with quality improvement, setupcost reduction, and stochastic lead time in an imperfectproduction process. Basically, they used the concept ofPorteus [2] regarding the investment to reduce setup cost andimprove the productβs quality.They simultaneously optimizedlot size, quality improvement parameter, setup cost, safetystock, and lead time to obtain the minimum cost of theintegrated model. Sarkar and Moon [18] extended Ouyang etal.βs [17] model with variable backorder rate. Yoo et al. [19]introduced inspection process with commercial return andrework during imperfect production for quality investmentand quality cost analyses.
In this highly competitive business environment, compa-nies always desire for trade-credit policy for the entire cus-tomers.Thus, trade-credit plays an important role in modernbusiness system. Vendors offer trade-credit-period to buyersto encourage sales, promote market shares, and reduce on-hand stock. Goyal [20] proposed an economic order quantitymodel under conditions of permissible delay-in-payments.Aggarwal and Jaggi [21] discussed about ordering policiesof deteriorating items under permissible delay-in-payments.Jamal et al. [22] extended the permissible delay-in-paymentsconcepts with allowable shortage and deterioration. Teng [23]discussed an economic order quantity under the conditionsof permissible delay-in-payments. Chang [24]wrote a note onpermissible delay-in-payments for (π, π) inventory systemswith ordering cost reduction.
Jaber and Osman [25] explained about the coordinationof delay-in-payments and profit sharing in a two-echelonsupply chain model. Luo [26] examined a buyer-vendorintegrated-inventory model with credit-period incentives.Huang [8] proposed an integrated-inventorymodel under theconditions of order processing cost reduction andpermissibledelay-in-payments. Sarkar et al. [27] extended an integrated-inventory model with variable lead time, defective units,
and delay-in-payments. They assumed stochastic lead timein combination with delay-in-payments to reduce total costof the system. Recently, Sarkar [28] discussed some conceptof discount policies from vendor to buyer with variablebackorder for buyer and multi-inspection for vendor byconsidering the fixed lifetime constrains of products. Thismodel emphasized a coordination policy within the supplychain by some special discounts if the buyer agrees to buysome order quantities which are decided by vendor.
In some inventory system, such as fashionable items, thelength of the waiting time for the next replenishment woulddetermine whether the backorder is accepted or not. There-fore, backorder rate is variable and dependent on the waitingtime for the next replenishment [29]. Pan et al. [30] optimizedan inventory model with reorder point, variable lead time,and backorder discount considerations. Pan and Hsiao [31]formulated an integrated-inventory model with controllablelead time and backorder discount considerations. Lee et al.[32] developed a computational algorithmic procedure foroptimal inventory policy involving ordering cost reductionand backorder discounts when the lead time demand iscontrollable. Lo et al. [33] introduced lead time and safetyfactor in mixed inventory models with backorder discounts.Lin [34] discussed an integrated vendor-buyer inventorymodel with a backorder price-discount and an effectiveinvestment to reduce ordering cost. Huang [8] designed asimple and an efficient algorithm involving ordering costreduction and backorder price-discount on inventory systemunder variable lead time.
1.1. Problem Description. This paper illustrates a chan-nel coordination mechanism between transportation dis-counts, trade-credit financing, number of shipments, qualityimprovement of products, and reduced setup cost in a two-echelon supply chain model. Table 1 shows the distinctionbetween existing model and this model.The work is differentfrom [18, 35] for the purpose of transportation discountsand trade-credit financing. It is differing from [36] fromtransportation discount and setup cost reduction and qualityimprovement of products. It is totally differing from [37] asit is a supply chain model and [37] is a basic EPQ model. Toshow this all, we have added Table 1 for the same. The aimof this model is to minimize the total cost throughout thesupply chain network under single-supplier and single-buyerfor a single type of product and single-setup-multidelivery(SSMD) policy. The supplier offers trade-credit-period to thebuyer and the buyer uses the delay time to increase his/herprofit. A continuous review inventory model is consideredfor both supplier and buyer. For buyer, the backorder rate isconsidered as variable. An investment is used to reduce setupcost and another investment is used to improve the qualityof products. To reduce the total supply chain cost, the modelassumes lead time-dependent backorder rate, where the leadtime is stochastic in nature. By using the trade-credit policy,the model gives how the credit-period would be benefited forthe whole system.The paper is designed as follows: Section 2
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Mathematical Problems in Engineering 3
introduces the mathematical model. In Section 3, numericalexample is given. Section 4 gives the conclusions of themodel.
2. Mathematical Model
2.1. Notation. The following notation are used to develop themodel.(i) Decision Variables
π: buyerβs order quantity (units).π΄ π : supplierβs setup cost per setup ($/setup).πΏ: length of lead time in unit time (days).π: probability of the production process whichmay goto out-of-control state during producing a lot.π: safety factor for reorder point.π: number of lots delivered from the supplier to thebuyer in one production cycle, a positive integer.
(ii) Parameters
π΄π: buyerβs ordering cost per order ($/order).π·: average demand per unit time (units/year).βπ: buyerβs inventory holding cost per unit per unittime ($/unit/year).βπ : supplierβs inventory holding cost per unit per unittime ($/unit/year).π: stockout cost per unit short ($/unit short).π: reorder point (units).π: standard deviation of the lead time demand.π0: initial probability of the production process whichmay go to out-of-control state during producing a lot.πΌ: annual fractional cost of capital investment toreduce setup cost ($/year).πΌ(π΄, π): total investment for setup cost reduction fromπ΄0 to π΄ and quality improvement from π0 to π.πΏ π: length of the lead time components for π =1, 2, . . . , π.ππ: crashing cost per unit time πΏ π ($/unit time).π: lead time demand which has distribution functionπΉ (units).πΈ(π): mathematical expectation ofπ.π§+: max{π§, 0}, where π§ is any random variable.πΈ(π β π)+: expected shortage quantity at the end ofthe cycle.π: production rate (unit/unit time).ππ: buyerβs interest or opportunity cost in annualpercentage.ππ : supplierβs interest or opportunity cost in annualpercentage.π‘ππ: transportation cost for πth unit, π = 1, 2, . . . , π($/unit).
π: length of credit-period (unit time).π : rework cost per unit defective item ($/unit defectiveitem).ππ: purchasing cost per unit ($/unit).
2.2. Assumptions. The following assumptions are consideredto formulate this model. These assumptions are mainlyadopted from Sarkar and Moon [18] and Arkan and Hejazi[36].
(i) The study considers a supply chain model for a singletype of products with the single-setup-multidelivery(SSMD) policy and controllable lead time.
(ii) The lead time πΏ has π mutually independent compo-nents. The πth component has a normal duration ππand the minimum duration π‘π with crashing cost perunit time ππ with π1 β€ π2 β€ π3 β€ β β β β€ ππ. The lead timedemand π follows a normal distribution with meanπ·πΏ and standard deviation πβπΏ (Ouyang et al. [35]).
(iii) Let πΏ0 = βππ=1 ππ and πΏ π be the length of the leadtime with components 1, 2, 3, . . . , π crashed to theirminimum duration. Then, πΏ π can be considered asπΏ π = πΏ0 β βππ=1(ππ β π‘π) and the lead time crashingcost per cycle π (πΏ) can be expressed as π (πΏ) = ππ(πΏ π βπΏ) +βπβ1π=1 ππ(ππ β π‘π) for = 1, 2, . . . , π (see for referenceOuyang et al. [35]).
(iv) This model considers the variable backorder rate π½with respect to lead time (Sarkar and Moon [18]).
(v) Logarithmic expressions are assumed for both qual-ity improvement and setup cost reduction (Porteus,[37]).
(vi) The trade-credit financing is considered to make it acost-reduced supply chain.
(vii) The supplier provides a transportation cost discount,when the buyer places the order of π units.
2.3. Model Formulation. The model considers the single-setup-multidelivery (SSMD) policy in a single-suppliersingle-buyer supply chain model. If the buyer orders quantityπ when the on-hand inventory reaches the reorder point π,that is, it considers (π, π) continuous review inventorymodel,to save holding cost of the buyer, the supplier produces ππquantities, which will be delivered to the buyer π times inone production cycle. Thus, the expected cycle length forthe supplier is ππ/π· and for the buyer is π/π·, respectively.Therefore, the ordering cost per unit time for the buyer isπ΄ππ·/π.
If the inventory level reaches the reorder point π, whereπ = π·πΏ + ππβπΏ, π·πΏ = the expected demand during the leadtime, ππβπΏ = safety stock (SS), and π = safety factor, thebuyer places an order of quantityπ.Thus, before receiving anorder, the inventory is πβπ·πΏ and after receiving the order, theinventory is π + (π β π·πΏ). Hence, the average inventory overa cycle can be written asπ/2+ πβπ·πΏ. Therefore, the holdingcost per unit per unit time of the buyer is βπ(π/2 + π β π·πΏ).
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4 Mathematical Problems in Engineering
Table 1: Distinction between previous and this model.
Author(s) Supply chainmodelVariable lead
timeVariablebackorder
Setup costreduction
Qualityimprovement
Transportationdiscounts
Trade-creditfinancing
Porteus, 1986 β βOuyang et al., 1996 βMoon and Choi, 1998 βHariga and Ben-Daya, 1999 βOuyang and Chuang, 2001 β βOuyang et al., 2002 β β βLee, 2005 β βLin, 2008 β βSarkar and Majumder, 2013 β β βSarkar and Moon, 2014 β β β βSarkar et al., 2014 β β βSarkar et al., 2015 β βThis research β β β β β β β
Themodel assumes that the lead time demandπ follows anormal distribution with meanπ·πΏ, standard deviation πβπΏ,and safety factor π. Thus, the reorder point π = π·πΏ+ππβπΏ. Ifπ < π, then shortage occurs. Hence, the expected shortage atthe end of the cycle is πΈ(π β π)+, and then expected shortagecost per unit time is (ππ·/π)πΈ(π β π)+.
The concept of Ouyang et al. [35] for lead time crashingcost is used in this model. The lead time πΏ has π mutuallyindependent components. The πth component has a normalduration ππ and the minimum duration π‘π with crashing costper unit time ππ with π1 β€ π2 β€ π3 β€ β β β β€ ππ. LetπΏ0 = βππ=1 ππ and πΏ π be the length of the lead time withcomponents 1, 2, 3, . . . , π crashed to their minimum duration.Then, πΏ π can be written as πΏ π = πΏ0 β βππ=1(ππ β π‘π) andthe lead time crashing cost per cycle π (πΏ) can be expressedas π (πΏ) = ππ(πΏ π β πΏ) + βπβ1π=1 ππ(ππ β π‘π) for π = 1, 2, . . . , π.Thus, the lead time crashing cost per unit time is (π·/π)π (πΏ).Therefore, the total expected cost per unit time to the buyercan be expressed as follows:
ππΆπ (π, πΏ) = π΄ππ·π + βπ (π2 + π β π·πΏ)+ ππ·π πΈ (π β π)+ + π·ππ (πΏ) .
(1)
In reality, the fixed or constant backorder rate is veryrare and it is found only in case of life saving drugs, costlyproducts, or others. But for any low-cost products, it isgenerally variable.Thus, based on lead time of this model, weuse the concept of Sarkar and Moon [18] for the backorderrate as a function of the lead time as follows:
π½ = 11 + ππβπΏπ (π) ,π being a constant, 0 < π < β. (2)
Thus, total expected cost per unit time for the buyer,considering the partial backorder, can be expressed as
ππΆπ (π, πΏ, π)= π΄ππ·π + βπ (π2 + π β π·πΏ + (1 β π½) πΈ (π β π)+)+ [π + π0 (1 β π½)]π·π πΈ (π β π)+ + π·ππ (πΏ) .
(3)
Using the above, the expected shortage at the end of thecycle can be expressed as
πΈ (π β π)+ = β«βπ(π β π) ππΉ (π₯)
= πβπΏ {π (π) β π (1 β Ξ¦ (π))}= πβπΏπ (π) ,
(4)
where π(π) = π(π) β π(1 β Ξ¦(π)), π(π) and Ξ¦(π) are thestandard normal distribution function and the cumulativedistribution function of the normal distribution, respectively.Thus, the safety factor π can be treated as a decision variableinstead of π. Therefore, total expected cost per unit time forthe buyer considering the partial backorder can be written as
ππΆπ (π, πΏ, π) = π΄ππ·π + βπ (π2 + ππβπΏ) + πβπΏπ (π)β [βπ ππβπΏπ (π)1 + ππβπΏπ (π)+ π·π (π + π0 ππβπΏπ (π)1 + ππβπΏπ (π))] + π·ππ (πΏ) .
(5)
In thismodel, under the SSMDpolicy, the cycle length forsupplier is ππ/π·. Thus, the setup cost per unit time for thesupplier is π΄ π π·/ππ (see for instance Figure 1). The average
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Mathematical Problems in Engineering 5
Quantity mQ/P
Time
Q/P
mQ
Q/D
Q
Accumulated inventory for the buyer
Accumulated inventory for the supplier
Q
(m β 1)Q/D
Figure 1: Inventory pattern under the SSMD policy (see for reference Ouyang et al. [38]).
inventory of the supplier can be written as [{ππ(π/π +(π β 1)(π/π·)) β π2π2/2π} β {(π2/π·)(1 + 2 + β β β + (π β1))}](π·/ππ) = (π/2)[π(1 β π·/π) β 1 + 2π·/π]. Hence, theholding cost per unit per unit time for the supplier becomesβπ (π·/2)[π(1 β π·/π) β 1 + 2π·/π].
In this model, there are two investments to reduce thetotal supply chain cost to make the supply chain moreprofitable. An investment is used to improve the quality ofproducts and another investment is used to reduce setupcost. We consider the concept of Porteus [37] for qualityimprovement πΌπ(π) = π ln(π0/π) for 0 < π β€ π0 and for setupcost reduction πΌπ΄(π΄) = π΅ ln(π΄0/π΄) for 0 < π΄ β€ π΄0. Hence,the supplierβs the total investment for quality improvementand setup cost reduction becomes as follows:
πΌ (π΄, π) = πΌπ (π) + πΌπ΄ (π΄) = πΊ β π ln π β π΅ lnπ΄, (6)where πΊ = π ln(π0) + π΅ ln(π΄0).
Using the concept of defective items, the expected annualtotal cost is
ππΆπ (π,π, π, πΏ) = πΆ (π,π, π, πΏ) + π π·πππ2 . (7)Therefore, the total expected cost per unit time for
supplier can be expressed as follows:
ππΆπ (π, π, π΄ π , π) = π΄ π π·ππ+ βπ π·2 [π(1 β π·π ) β 1 + 2π·π ]+ πΌ (πΊ β π ln π β π΅ lnπ΄ π )+ π π·πππ2
(8)
for 0 < π β€ π0 and 0 < π΄ β€ π΄0.
To make the profitable supply chain, an attempt of trade-credit policy is used. By using the trade-credit policy, buyersaves his/her total interest during the credit-period and thesupplier lost opportunity cost.We define the trade-credit costfor buyer offered by the supplier as follows:
ππ (π β π·π)2 ππ 2π· β π·2πππ2ππ2π·
β ππππππ· πβπΏπ (π)1 + ππβπΏπ (π) .(9)
Nowadays, for highly competitive business market, trans-portation cost is a major issue of the total operational costin SCM. For appropriate incorporation of transportation costinto the total annual cost function, it should identify the exacttransportation cost which relates the reality. In many SCMmodels, the transportation cost is only considered implicitlyas a part of fixed setup or ordering cost and thus, it is assumedto be the independent of the size of the shipment. In thissection, we address the case, where the transportation cost isexplicitly considered in the model. The structure of all-unit-discount transportation cost is adopted, which is similar toErtogral et al. [14] (see Table 2 for it).
Another attempt of transportation cost discount is con-sidered to make a SCM forever. For selling large quantities,the supplier offers a transportation cost discount to the buyer.In this model, the transportation cost is dependent onπ. Weconsider that the supplier offers the discount once the buyerplaces the order π units. Thus, the buyer orders quantityπ for the transportation cost discounts from the supplier.Besides, the supplier carries quantity π instead of π due tovarious reasons. However, this imperfect quantity π does notaffect the transportation cost discount condition. For a givenshipment of lot size π β [ππ,ππ+1), transportation cost perunit time is equal to πΆππ/(π/π·) = πΆππ·, which can be found
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6 Mathematical Problems in Engineering
Table 2: Structure of all-unit-discount transportation cost.
Range Unit transportation cost0 β€ π < π1 πΆ0π1 β€ π < π2 πΆ1π2 β€ π < π3 πΆ2... ...ππβ1 β€ π < ππ πΆπβ1ππ β€ π πΆπwhere πΆ1 > πΆ2 > β β β > πΆπ
by dividing the transportation cost per order cycle by theduration of the order cycle. The transportation cost can berepresented as
ππ· (π) =
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
πΆ0π·, π β [0,π1) ,πΆ1π·, π β [π1,π2) ,πΆ2π·, π β [π2,π3) ,... ...πΆππ·, π β [ππ,β) .
(10)
Hence, the expected annual total cost per unit timeincludes the receiving of uncertain quantity and the trans-portation cost for the SCM model with partial backorder,setup cost, quality improvement, and trade-credit. Therefore,this problem reduces to
min ππΆπ π (π, π, π, π΄ π , π, πΏ)= π΄ππ·π + βπ (π2 + ππβπΏ) + πβπΏπ (π) [βπ ππβπΏπ (π)1 + ππβπΏπ (π) + π·π (π + π0 ππβπΏπ (π)1 + ππβπΏπ (π))]
+ π·π [[ππ (πΏ π β πΏ) +πβ1βπ=1
ππ (ππ β π‘π)]] βπ·2πππ2ππ2π β π·ππππππ πβπΏπ (π)1 + ππβπΏπ (π) + π΄ π π·ππ
+ βπ π2 [π(1 β π·π ) β 1 + 2π·π ] + πΌ (πΊ β π ln π β π΅ lnπ΄ π ) + π π·πππ2 + ππ (π β π·π)2 ππ 2π + ππ· (π)
subject to 0 < π β€ π0,0 < π΄ β€ π΄0.
(11)
2.4. Solution Procedure. Now the optimum cost of the wholesupply chain model is calculated. To do that optimization,we initially ignore all constraints and calculate all the partialderivatives which are necessary for the optimization; thenall restrictions are applied on it. The values of all the partialderivatives are as follows:
πππΆπ πππ = 1π2 [βπ΄ππ· β π·πππ β π·π (πΏ) + πππππ·2π22
+ πππππ·π2π2 (1 + π) β ππππ π·2π22 β π΄ π π·π ] + βπ2
+ π π·ππ2 + βπ 2 [π(1 β π·π ) β 1 + 2π·π ] + ππππ 2 ,πππΆπ πππ = βππβπΏ + πβπΏπ3 [ βππ(1 + π) + π·ππ ]+ πβπΏ[ βπππ3(1 + π)2 + π·π0ππ3π (1 + π)2] β π·ππππππβπΏπ3π (1 + π)2 ,
πππΆπ πππ = βπΌππ + π π·ππ2 ,πππΆπ πππ΄ π = π·ππ β πΌπ΅π΄ π ,πππΆπ πππ = βπ΄ π π·π2π + βπ π2 (1 β π·π ) + π π·ππ2 ,πππΆπ πππΏ = 12βππππΏβ1/2 β π·πππ + (βππ
2πΏβ1/2π + π·π0π2
πππΏ2β π·ππππππ2πππΏ ) β π2 (1 + π)2 [βπππ (π)2π2
+ π·π0ππ (π)2π2π β π·πππππππ (π)π ] ,(12)
where π = π + π0π/(1 + π), π = ππβπΏπ(π), and π3 =Ξ¦(π) β 1.
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Mathematical Problems in Engineering 7
To obtain the global minimum solution of the supplychain model, the following second-order partial derivativesare used to calculate all minors:
π2ππΆπ πππ2 = 2π3 [π΄ππ· + π·πππ + π·π (πΏ) β πππππ·2π22
β πππππ·π2π2 (1 + π) + ππππ π·2π22 + π΄ π π·π ] ,
π2ππΆπ πππ2 = πβπΏπ (π) [ βππ(1 + π) + π·ππ ] + πβπΏ (Ξ¦ (π)β 1) [βπππβπΏ (Ξ¦ (π) β 1)(1 + π)β βππ2π2πΏ (Ξ¦ (π) β 1) π (π)(1 + π)2 ] + πβπΏ (Ξ¦ (π) β 1)β [(βπππβπΏ + π·π0ππβπΏπ )((Ξ¦ (π) β 1)(1 + π)2 )]+ πβπΏπ (π) [(βπππβπΏ + π·π0ππβπΏπ )( π (π)(1 + π)2β 2ππβπΏ ((Ξ¦ (π) β 1))2(1 + π)2 )] β π·ππππππβπΏπ (π)π (1 + π)2+ 2π·ππππππ2π2πΏ ((Ξ¦ (π) β 1))2π (1 + π)3 ,
π2ππΆπ πππ2 = πΌππ2 ,π2ππΆπ πππ΄ π 2 = πΌπ΅π΄ π 2 ,π2ππΆπ πππ2 = 2π΄ π π·π3π ,
π2ππΆπ πππΏ2 = β[14βππππΏβ3/2 + 12β π2πΏ5/2 (1 + π)3 {π·πππππππ (π)2π β βππ
2πΏβ3/2πβ π·π0π2πΏβ3/2ππ } + π2πΏ ((1 + π)2) (2βππ
2
ππΏ2+ 32 π·π0π
2
πππΏ2 β π·πππππ2π (π + (1 + π) ππΏπ ))] ,(13)
where π = π+π0π/(1+π), π = ππβπΏπ(π), and π3 = Ξ¦(π)β1.It is found that ππΆπ π(π, π, π, π΄ π , π, πΏ) is concave with
respect to πΏ as the second-order partial derivative ofππΆπ π(π, π, π, π΄ π , π, πΏ) with respect to πΏ which is negative asthe 2nd term is very smaller than the 1st term within theparenthesis; that is,
π2ππΆπ πππΏ2 = β[14βππππΏβ3/2 + 12β π2πΏ5/2 (1 + π)3 {π·πππππππ (π)2π β βππ
2πΏβ3/2πβ π·π0π2πΏβ3/2ππ } + π2πΏ ((1 + π)2) (2βππ
2
ππΏ2+ 32 π·π0π
2
πππΏ2 β π·πππππ2π (π + (1 + π) ππΏπ ))] < 0.
(14)
Thus, by taking the values of π, π, π, π΄ π , and π asconstant, ππΆπ π(π, π, π, π΄ π , π, πΏ) is concave with respect to πΏ.Hence, for constant values ofπ, π, π, π΄ π , andπ, theminimumexpected cost can be obtained from the end point of [πΏ π, πΏ πβ1].Thus, the optimal values ofπ, π, π, π΄ π , andπ can be obtainedfor given πΏ β [πΏ π, πΏ πβ1].Therefore, equating other four partialderivatives to zero, we can find the optimum values as
π = β [π΄ππ· + π·ππ/π + π·π (πΏ) β πππππ·2π2/2 β πππππ·π2π/2 (1 + π) + ππππ π·2π2/2 + π΄ π π·/π]βπ/2 + π π·ππ/2 + (βπ /2) [π (1 β π·/π) β 1 + 2π·/π] + ππππ /2 , (15)
Ξ¦ (π) = 1 β (1 + π)2 βππβπππ (1 + π) + π·ππ (1 + π)2 + βπππ + π·π0π β π·ππππππ , (16)π = 2πΌππ π·ππ, (17)π΄ π = πΌπ΅πππ· . (18)
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8 Mathematical Problems in Engineering
Lemma 1. For a given πΏ β [πΏ π, πΏ πβ1], ππΆπ π(π, π, π, π΄ π , π, πΏ)has the global minimum solution at the optimal values(πβ, πβ, πβ, π΄ π β).Proof. See Appendix.
It is a nonlinear program. Thus, the following algorithmis employed to obtain the optimum results.
Algorithm 2.Step 1. Setπ = 1 and input all parametric values.Step 2. For each πΏ π, π = 1, 2, . . . , π, perform Steps 2(a)β2(f).
Step 2(a). Set π΄ π π1 = 0, ππ π1 = 0, and ππ1 = 0 (impliesπ(ππ1) = 0.39894).Step 2(b). Substitute π(ππ1) into (15) and evaluateππ1.Step 2(c). Utilize ππ1 to calculate the value of Ξ¦(ππ2)from (16).
Step 2(d). For the value of Ξ¦(ππ2), find ππ2 from thenormal table and hence evaluate π(ππ2).Step 2(e). Utilize ππ1 to obtain ππ π2 and π΄ π π2 from (17)and (18).
Step 2(f). Repeat 2(b)β2(e) until no changes occur inthe values of ππ, ππ, ππ π, and π΄ π π; denote these valuesby (ππ, ππ, ππ, π΄ π π).
Step 3. Compare ππ π and π0 and π΄ π π and π΄ π 0, respectively.Step 3(a). If ππ < π0 and π΄ π π < π΄ π 0, then the solutionfound in Step 1 is optimal for the given πΏ π. We denotethe optimal solution by (πβπ , πβπ , πβπ , π΄ π βπ ); that is, if(πβπ , πβπ , πβπ , π΄ π βπ ) = (ππ, ππ, ππ, π΄ π π), go to Step 4.Step 3(b). If ππ β₯ π0 and π΄ π π < π΄ π 0, then for givenπΏ π, assume πβπ = π0 and utilize (15) (replace π by π0),(16), and (18) to obtain the new (ππ, ππ, π΄ π π) by similarprocedure like Step 1 (the solution is denoted by(ππ, ππ, π΄ π π)). If π΄ π π < π΄ π 0, then the optimal solutionis found; that is, if (πβπ , πβπ , πβπ , π΄ π βπ ) = (ππ, ππ, π0, π΄ π π),go to Step 4; otherwise, go to Step 3.
Step 3(c). If ππ < π0 and π΄ π π β₯ π΄ π 0, then for givenπΏ π, let π΄ π βπ = π΄ π 0 and utilize (15) (replace π΄ π byπ΄ π 0), (16), and (17) to obtain the new (ππ, ππ, ππ) bysimilar procedure like Step 1 (the solution is denotedby (ππ, ππ, ππ)). If ππ < π0, then the optimal solution isfound; that is, if (πβπ , πβπ , πβπ , π΄ π βπ ) = (ππ, ππ, ππ, π΄ π π), goto Step 4; otherwise, go to Step 3.
Step 3(d). If ππ β₯ π0 and π΄ π π β₯ π΄ π 0, go to Step 4.
Step 4. FindππΆπ π(πβπ , πβπ , ππ βπ ,π΄π βπ , πΏπ,π) andminπ=1,2,...,πππΆπ π(πβπ ,πβπ , ππ βπ , π΄ π βπ , πΏ π, π).Step 4(a). If ππΆπ π(πβπ , πβπ , ππ βπ , π΄π βπ , πΏπ , π) =minπ=1,2,...,πππΆπ π(πβπ , πβπ , ππ βπ , π΄ π βπ , πΏ π, π), then ππΆπ π(πβπ ,πβπ , ππ βπ , π΄ π βπ , πΏ π, π) is the optimal solution for fixedπ.
Step 5. Set π = π + 1. If ππΆπ π(πβπ, πβπ, ππ βπ, π΄ π βπ, πΏπ, π) β€ππΆπ π(πβπβ1, πβπβ1, ππ βπβ1, π΄ π βπβ1, πΏπβ1, π β 1), repeat Step 2.Otherwise go to Step 6.Step 6. Set ππΆπ π(πβπ, πβπ, ππ βπ, π΄ π βπ, πΏπ, π) = ππΆπ π(πβπβ1,πβπβ1, ππ βπβ1, π΄ π βπβ1, πΏπβ1, π β 1). Then (πβ, πβ, πΏβ, ππ β,π΄ π β, πβ) is the optimal solution and the optimal reorderpoint can be calculated from πβ = π·πΏβ + πβπβπΏβ, where πβdenotes the optimal reorder point.
3. Numerical Experiments
The input parameters are taken from Sarkar and Moon [18]and the rest of the values are taken fromSarkar andMajumder[39] (see Tables 3 and 4 for it) as follows:
π· = 600 units/year.π΄0 = $1500/setup.π΄π = $200/order.βπ = $100/unit/year.βπ = $80/unit/year.π = $5/unit.π0 = $10/unit.π = 1500 unit/year.π = $75/unit.π0 = 0.0002.π΅ = 5800.πΌ = 0.5 dollar/unit.π = 400.π = 7 units.π = 0.2 dollar/unit.π‘ππ = $0.1/unit.ππ = $2/unit.π = 3.
The optimal cost ππΆπ π = $1961.21/year, and the optimaldecision variable is πβ = 37.11, πβ = 1.89, πβ =.000004, π΄ π = $1076.35, π = 2, πΏ = 21days. It isclearly found that the optimum lot size belongs to themaximum range of transportation discount, which indicatesthat the supply chain is profitable forever for the purpose oftransportation discount with the trade-credit financing.
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Mathematical Problems in Engineering 9
Table 3: Lead time data.
Lead timecomponent π Normal durationππ (days) Minimumduration π‘π (days) Unit crashingcost ππ ($/day)1 20 6 0.42 20 6 1.23 20 9 5.0
Table 4: Transportation cost structure.
Range Unit transportation cost0 β€ π < 100 0.4100 β€ π < 200 0.25200 β€ π < 300 0.173000 β€ π 0.01Table 5: Sensitivity analysis.
Parameters Changes of parameters (in %) ππΆπ π (in %)π΄0
β10% β15.58β5% β7.58+5% 7.21+10% 14.09βπ
β10% β32.40β5% β16.10+5% 15.83+10% 23.71βπ
β10% β65.91β5% β32.15+5% 30.70+10% 60.07π΄π
β10% β7.45β5% β3.68+5% 3.58+10% 7.08π
β10% β16.58β5% β16.00+5% 15.18+10% 14.67πΌ
β10% β1.72β5% β8.36+5% 7.78+10% 2.123.1. Sensitivity Analysis. Sensitivity analysis for the total costof supply chain is executed with changing parameters byβ10%, β5%, +5%, and +10% in (Table 5). From the sensitivityanalysis results, the following can be concluded:
(i) The holding cost for supplier is themost sensitive costin the supply chain. Negative changes are more thanpositive changes; that is, when supplierβs holding costincreases total cost increases and vice versa. Its effectsare more in supply chain than any other parameters.
(ii) Theholing cost of buyer is 2ndmost sensitive compar-ing other costs of the supply chain. Negative changesare more than positive changes. Decreasing value of
β15 β10 β5 0(%)
5 10 15
TC
TCsc
β25
β20
β15
β10
β5
0
5
10
15
20
Figure 2: π versus total cost.
β15 β10 β5 0(%)
5 10 15β1111111
TC
TCsc
β10β8β6β4β2
02468
10
Figure 3: πΌ versus total cost.buyerβs holding cost affects more than the increasingvalue of buyerβs holding cost in the total supply chaincost.
(iii) From the sensitivity analysis, it is found that if initialsetup cost increases, total cost also increases. It followsthat negative and positive changes are almost similarfor two changes. Negative changes are slightly morethan positive change. Thus, this model consideredthe reduction of this setup cost by some investmentfunction and by the numerical study, the obtainedreduced setup cost with reduced total supply chaincost.
(iv) The increasing value of the buyerβs ordering costindicates the increasing value of the total cost. Bycomparing the changes within positive and negativedirection, two changes are similar. Positive and nega-tive percentage changes are almost same.
(v) If rework cost increases or decreases, then the totalcost increases or decreases and negative percentagechange and positive percentage change are almost thesame (see Figure 2).
(vi) The percentage changes for annual fractional costare less sensitive than rework cost. Total supply cost
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10 Mathematical Problems in Engineering
change increases for the increase of this parameter.This is the least sensitive parameter among all param-eters (see Figure 3).
4. Conclusions
The paper developed a supply chain model with a stochasticlead time demand, trade-credit policy, quality improvementof products, setup cost reduction of supplier, and variablebackorder rate. The backorder rate was lead time-dependent.The aim was to minimize the total supply chain cost withsimultaneous optimization of six decision variables as num-ber of shipments, lot size, lead time, setup cost of supplier,quality improvement parameters, and safety stock. Sarkarand Moon [18] did not consider supply chain model andwe extended their model with supplier-buyer supply chainmodel and trade-credit policy. Due to highly nonlinear costequation, we cannot obtain closed form solutions. We usedan improved algorithm to obtain the numerical results.Our results indicated that the cost was minimized basedon the existing literature. The limitation of the model wasthat we used constant demand for both buyer and supplier.The managers can use our suggested policy and can savemore funds. This model can be extended with the uncertaindemand along with multiechelon sustainable supply chainmodel. Several sustainability issues like water resources andenergy consumption can be added to make a new andimproved sustainable supply chain.
Appendix
Proof of Lemma 1. For given concave function πΏ β [πΏ π, πΏ πβ1]and π is integer, thus Hessian matrixπ» is calculated for thevariables πβ, πβ, πβ, π΄ π β as follows:π»
=[[[[[[[[[[[[[[[
π2ππΆπ π (β )ππβ2 π2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβππβ π2ππΆπ π (β )ππβππ΄ π βπ2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβ2 π2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβππ΄ π βπ2ππΆπ π (β )ππβππβ π2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβ2 π2ππΆπ π (β )ππβππ΄ π βπ2ππΆπ π (β )ππ΄ π βππβ π
2ππΆπ π (β )ππ΄ π βππβ π2ππΆπ π (β )ππ΄ π βππβ π
2ππΆπ π (β )ππ΄ π β2
]]]]]]]]]]]]]]]
, (A.1)
where ππΆπ π(β ) = ππΆπ π(πβ, πβ, πβ, π΄ π β, π, πΏ). The partialderivatives with respect to decision variables are obtained asfollows:
π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβ2 = 2πβ3 [π΄ππ·+ π·πππ + π·π (πΏ) β πππππ·
2π22 β πππππ·π2π2 (1 + π)
+ ππππ π·2π22 + π΄ π π·π ] ,
π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβ2 = πβπΏπ (π) [ βππ(1 + π)+ π·ππβ ] + πβπΏπ3 [βπππβπΏπ3(1 + π)β βππ2π2πΏπ3π (π)(1 + π)2 ]+ πβπΏπ3 [(βπππβπΏ + π·π0ππβπΏπβ )β ( π3(1 + π)2)] + πβπΏπ (π)β [(βπππβπΏ + π·π0ππβπΏπβ )β ( π (π)(1 + π)2 β 2ππ
βπΏ (π3)2(1 + π)2 )]β π·ππππππβπΏπ (π)πβ (1 + π)2 + 2π·ππππππ
2π2πΏ (π3)2πβ (1 + π)3 ,π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβ2 = πΌππβ2 ,π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππ΄ π β2 =
πΌπ΅π΄ π β2 ,π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβππβ= π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβππβ= 1πβ2 [βπ·πβπΏππ3 β 2π·π0πβπΏπ3π(1 + π)+ π·π0πβπΏπ3π2(1 + π)2 + π·π
2ππππππβπΏπ32 (1 + π)β π·π2ππππππβπΏπ3π2 (1 + π)2 ] ,
π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβππβ= π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβππβ = π π·π2 ,
π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβππ΄ π β= π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππ΄ π βππβ = β π·ππβ2 ,
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Mathematical Problems in Engineering 11
π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβππβ= π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβππβ = 0,
π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβππ΄ π β= π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππ΄ π βππβ = 0,
π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππβππ΄ π β= π2ππΆπ π (πβ, πβ, πβ, π΄ π β, π, πΏ)ππ΄ π βππβ = 0.
(A.2)
At the optimum values of the decision variables, the principalminors are calculated to confirm their positivity as follows.
For the 1st minor, one can obtain easily as
det (π»11) = det(π2ππΆπ π (β )ππβ2 ) = 2πβ3 [π΄ππ· + π·πππ+ π·π (πΏ) β πππππ·2π22 β πππππ·π
2π2 (1 + π)+ ππππ π·2π22 + π΄ π π·π ] > 0.
(A.3)
For 2nd minor, it is found as
det (π»22) = det[[[[π2ππΆπ π (β )ππβ2 π
2ππΆπ π (β )ππβππβπ2ππΆπ π (β )ππβππβ π2ππΆπ π (β )ππβ2
]]]]= ππ β π2,
(A.4)
where
π = 2πβ3 [π΄ππ· + π·πππ + π·π (πΏ) β πππππ·2π22
β πππππ·π2π2 (1 + π) + ππππ π·2π22 + π΄ π π·π ] ,
π = πβπΏπ (π) [ βππ(1 + π) + π·ππβ ]+ πβπΏπ3 [βπππβπΏπ3(1 + π) β βππ
2π2πΏπ3π (π)(1 + π)2 ]
+ πβπΏπ3 [(βπππβπΏ + π·π0ππβπΏπβ )β ( π3(1 + π)2)] + πβπΏπ (π)β [(βπππβπΏ + π·π0ππβπΏπβ )β ( π (π)(1 + π)2 β 2ππ
βπΏ (π3)2(1 + π)2 )]β π·ππππππβπΏπ (π)πβ (1 + π)2 + 2π·ππππππ
2π2πΏ (π3)2πβ (1 + π)3 ,π = 1πβ2 [βπ·πβπΏππ3 β 2π·π0πβπΏπ3π(1 + π)+ π·π0πβπΏπ3π2(1 + π)2 + π·π
2ππππππβπΏπ32 (1 + π)β π·π2ππππππβπΏπ3π2 (1 + π)2 ] .
(A.5)
Now,
β π·πβπΏππ3 β 2π·π0πβπΏπ3π(1 + π) + π·π0πβπΏπ3π2
(1 + π)2+ π·π2ππππππβπΏπ32 (1 + π) β π·π
2ππππππβπΏπ3π2 (1 + π)2= 2π·π0πβπΏπ3π2 β π·π2ππππππβπΏπ3π2 (1 + π)2+ π·π2ππππππβπΏπ3 β 4π·π0πβπΏπ3π β 2 (1 + π)π·πβπΏππ32 (1 + π) .
(A.6)
Again
π = πβπΏπ (π) [ βππ(1 + π) + π·ππβ ] + πβπΏπ3 [βπππβπΏπ3(1 + π)β βππ2π2πΏπ3π (π)(1 + π)2 ] + πβπΏπ3 [(βπππβπΏ + π·π0ππβπΏπβ )β ( π3(1 + π)2)] + πβπΏπ (π) [(βπππβπΏ + π·π0ππβπΏπβ )β ( π (π)(1 + π)2 β 2ππ
βπΏ (π3)2(1 + π)2 )] β π·ππππππβπΏπ (π)πβ (1 + π)2+ 2π·ππππππ2π2πΏ (π3)2πβ (1 + π)3 = πβπΏπ (π) [ βππ(1 + π) + π·ππβ ]+ πβπΏπ3 [βπππβπΏπ3(1 + π) β βππ
2π2πΏπ3π (π)(1 + π)2 ]+ 2π·ππππππ2π2πΏ (π3)2πβ (1 + π)3 β π·ππππππβπΏπ (π)πβ (1 + π)2
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12 Mathematical Problems in Engineering
+ [(βπππβπΏ + π·π0ππβπΏπβ )β (πβπΏ (π3)2 + πβπΏπ (π) π (π) β 2ππβπΏ (π3)2 πβπΏπ (π)(1 + π)2 )]= π + π,
(A.7)
where
π = πβπΏπ (π) [ βππ(1 + π) + π·ππβ ]+ πβπΏπ3 [βπππβπΏπ3(1 + π) β βππ
2π2πΏπ3π (π)(1 + π)2 ]+ 2π·ππππππ2π2πΏ (π3)2πβ (1 + π)3 β π·ππππππβπΏπ (π)πβ (1 + π)2 ,
π = (βπππβπΏ + π·π0ππβπΏπβ )β (πβπΏ (π3)2 + πβπΏπ (π) π (π) β 2ππβπΏ (π3)2 πβπΏπ (π)(1 + π)2 ) .
(A.8)
Thus, ππ > π2 whereπ = 2πβ3 [π΄ππ· + π·πππ + π·π (πΏ) β πππππ·
2π22β πππππ·π2π2 (1 + π) + ππππ π·
2π22 + π΄ π π·π ] > π,π = π + π > π.
(A.9)
Hence, det(π»22) > 0.For 3rd minor, the value is obtained as
π»33 =[[[[[[[[[
π2ππΆπ π (β )ππβ2 π2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβππβπ2ππΆπ π (β )ππβππβ π2ππΆπ π (β )ππβ2 π
2ππΆπ π (β )ππβππβπ2ππΆπ π (β )ππβππβ π2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβ2
]]]]]]]]]= [[[[[[
π π π π·π2π π 0π π·π2 0 πΌππβ2]]]]]]
= π π·π2 [βπ π·π2 π] + πΌππβ2π»22= π( πΌππβ2π β π
2π·2π24 ) β πΌππβ2 π2.
(A.10)
It is already proved that π > π; thus it is enough to show(πΌπ/πβ2)π β π 2π·2π2/4 > (πΌπ/πβ2)π; that is,πΌππβ2π β πΌππβ2 π > π
2π·2π24 βπΌππβ2 (π β π) > π
2π·2π24 βπ β π > π 2π·2π2πβ24πΌπ β
π β π β π 2π·2π2πβ24πΌπ > 0;
(A.11)
that is, det(π»33) > 0.Finally, for 4th minor, the optimum value is obtained as
π»44
=[[[[[[[[[[[[[
π2ππΆπ π (β )ππβ2 π2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβππβ π2ππΆπ π (β )ππβππ΄ π βπ2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβ2 π2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβππ΄ π βπ2ππΆπ π (β )ππβππβ π2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβ2 π2ππΆπ π (β )ππβππ΄ π βπ2ππΆπ π (β )ππ΄ π βππβ π
2ππΆπ π (β )ππ΄ π βππβ π2ππΆπ π (β )ππ΄ π βππβ π
2ππΆπ π (β )ππ΄ π β2
]]]]]]]]]]]]]
= βπ2ππΆπ π (β )ππβππ΄ π β[[[[[[[[
π2ππΆπ π (β )ππβππβ π2ππΆπ π (β )ππβ2 π
2ππΆπ π (β )ππβππβπ2ππΆπ π (β )ππβππβ π2ππΆπ π (β )ππβππβ π
2ππΆπ π (β )ππβ2π2ππΆπ π (β )ππ΄ π βππβ π2ππΆπ π (β )ππ΄ π βππβ π
2ππΆπ π (β )ππ΄ π βππβ
]]]]]]]]+ π2ππΆπ π (β )ππ΄ π β2 π»33
= π·ππβ2[[[[[[
π π 0π π·π2 0 πΌππβ2β π·ππβ2 0 0]]]]]]+ πΌπ΅π΄ π β2π»33
= πΌππ·2ππ2πβ4πβ2 + πΌπ΅π΄ π β2π»33.
(A.12)
First part is positive and π»33 is already shown greater thanzero.
Hence, det(π»44) > 0.From the above calculations, all principal minors of the
Hessian matrix are positive. Therefore, the Hessian matrixπ» is positively definite at (πβ, πβ, πβ, π΄ π β). Thus, total costfunction has a global minimum.
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper.
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Mathematical Problems in Engineering 13
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