International journal of computer integrated manufacturing , tập 23, số 5, 2010

International Journal of Computer Integrated Manufacturing Vol 23, No 5, May 2010, 391–401 Prevention of resource trading fraud in manufacturing grid: a signalling games approach Haijun Zhanga,b*, Yefa Hua and Zude Zhoua a School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China; bDepartment of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA (Received 13 March 2009; final version received 14 January 2010) In the manufacturing grid resource market, the information asymmetry for buyers and sellers is a common situation With resource information superiority, some resource service providers (RSPs, sellers) often make the pooling equilibrium, so that resource service demanders (RSDs, buyers) cannot recognise high-quality resources owing to imperfect information Hence, the authors propose a signalling games approach to prevent resource trading fraud in the manufacturing grid In support of the architecture of resource negotiation and trading, RSDs can get more accurate information about resource quality based on the collateral currency promised by RSPs In this way, RSDs can more accurately identify resource quality This method focuses on preventing low-quality RSPs from sending out an incorrect signal suggesting high resource quality to entice RSDs to purchase the low-quality resource Simulation results indicate that the game theoretical model has a reasonable and perfect Bayesian separating equilibrium, from which RSPs not initiatively deviate Keywords: manufacturing grid; singling games; the perfect Bayesian equilibrium; trading fraud Introduction With the rapid development of advanced manufacturing technology, information technology, and the global market, modern manufacturing enterprises are facing sustainable, variable, and unpredictable competition The traditional quality and price competitive model in manufacturing has turned into a service, technology, and time competitive model In this situation, enterprises cannot continue the traditional mode of ‘big and all-inclusive projects’; they must adopt a new ‘small and specialised’ thought processes Modern enterprises must emphasise professionalism and standardisation In the new economy, most enterprises adopt a win–win strategy for cooperation Thus, a common platform must be developed – one that promotes the sharing of manufacturing resources and service The manufacturing grid (MGrid) (Qui et al 2004, Shi et al 2007) is a new concept and one of the next generations of manufacturing models It has been proposed to meet the cooperative demand of the manufacturing industry It enables geographically distributed manufacturing resources to be connected through the internet using grid technology With the MGrid platform, common resources and services (including human, equipment, material, and software) can be shared Enterprise competitiveness can be enhanced in the MGrid owing to the shortened *Corresponding author Email: haijun@whut.edu.cn ISSN 0951-192X print/ISSN 1362-3052 online Ó 2010 Taylor & Francis DOI: 10.1080/09511921003642113 http://www.informaworld.com development and manufacturing time and minimised cost The MGrid tries to achieve the sharing of geographically distributed manufacturing resources and services through the reconfigurable manufacturing processes (also called ‘virtual organisations’) Therefore, enterprises in the MGrid can achieve the goal of TQCSEF (Time, highest Quality, lowest Cost, best Service, cleanest Environment, and greatest Flexibility) With the technical support of Web services (Mockford 2004) and Grid (Foster et al 2001), MGrid has constructed a huge manufacturing-oriented resource and service market Enterprises and even individuals can acquire manufacturing resources and services from the MGrid resource market as conveniently as they obtain water, electricity, and internet information MGrid users can easily pay for resources or services (such as the material, design, or machinery they consume) with the records and payment functions provided by the MGrid core middleware Participants in the MGrid market are divided into two categories: they are either resource service providers (RSPs) or resource service demanders (RSDs) Resource quality is the basis of both the RSD’s payment and the RSP’s pricing The aim of both RSPs and RSDs is to maximise their own profits As in the commodity market, the resource information asymmetry is for RSPs and RSDs in the MGrid 392 H Zhang et al resource market With resource information superiority, some RSPs send out a confusing signal (or inaccurate information) regarding resource quality, which may entice RSDs to buy or use the low-quality resource to make the pooling equilibrium When this happens, RSDs cannot recognise a high-quality resource due to imperfect information, and RSPs make an abnormal profit This paper focuses on preventing RSPs who have low-quality resources from sending out untrue information Therefore, resource information asymmetry induces two important issues in the MGrid resource trading: (1) how to display resource quality information for RSPs in order to attract the attention of RSDs and (2) how RSDs can obtain accurate information about the resource quality from RSPs MGrid resource trading involves the individual behaviours of RSPs and RSDs In free market economics, whether RSPs provide high-quality resources or low-quality resources is a result of the rational behaviours of both RSPs and RSDs The authors adopt the signalling game theory in the information economics to analyse individual behaviours of both RSPs and RSDs The signalling game theory was chosen because it focuses on decision making and the related market equilibrium given the interaction between decision makers The rest of the paper is organised as follows: Section investigates the related work about MGrid Section describes the MGrid resource negotiation and trading model based on perfect Bayesian Nash equilibrium Section simulates the model and analyses the results Section shows an industrial case for the model The conclusion and future work are given in Section Related work The conception, system platform, Open Grid Services Architecture (OGSA) and Web Services Resource Framework (WSRF) of MGrid have been investigated by several authors (Li et al 2007, Tao et al 2008a, Zhang et al 2008) The application demands of grid technology in e-science, e-government, e-entertainment, e-education, e-business, and manufacturing industries have also been studied In such MGrid environments, RSPs and RSDs have different goals, objectives, strategies, and supply-and-demand patterns Moreover, both resources and end-users are geographically distributed with different time zones The economic approach provides a fair basis in successfully managing the decentralisation and heterogeneity that is present in human economies In an economic-based approach, the resource scheduling is made dynamically at runtime and is driven and directed by the end-users’ requirements Pricing based on user demand and resource supply is the main driver in the competitive, economic MGrid resource market Since MGrid uses the internet as a carrier for providing remote manufacturing services, MGrid can be used to share manufacturing resources in a seamless manner for cooperative problem solving The resource price is adjusted, according to the law of supply and demand Price fluctuation reflects the market’s supply-and-demand dynamics, and the optimal resource allocation occurs at the supply-anddemand equilibrium The application of economic theory in computer system resource allocation can be traced back to 1968, when Sutherland proposed the auction mechanism for resource allocation in the PDP-1 computer (Sutherland et al 1968) Since then, the economic theory has been primarily used for solving the load balance of computer clusters and distributed systems Ferguson (1996) investigated the application of general equilibrium theory and Nash equilibrium in the distributed computer resource management Waldspurger (1992) designed and developed the Spawn system, which is the market-oriented scheduling system for a group of heterogeneous computers connected to the internet Bogan (1994) used the market mechanism to allocate the central processing unit (CPU) time and proposed the CPU leasing agreement per unit of time Bredin (1998) proposed a trusted third-party arbiter to prevent fraud in transactions between mobile agents, which is a little similar to the notary public in Section 3.3 However, this paper focuses on how to let the collateral currency represent the quality level of resources and/or services truly In recent years, studies related to the application of economic theory in grid resource allocation have become very popular (Buyya et al 2002, Subramoniam et al 2002) Abramson (2002) studied the resource management, scheduling, and computational economics in a grid environment and designed the grid architecture for computational economy (GRACE) Buyya (2009) later developed a grid application software toolkit based on Globus–Nimrod/G Wolski (2000, 2001) proposed that the relative value of a resource varies with the supply-and-demand and allocated dynamic resources using an auction model in G-commerce project The Popcorn project focused on formulating computer service time as currency (London 1998) Computing grid researchers have carried out an indepth study on the application of economic theory in a large-scale, dynamic system–Grid They achieve pretty good results that now provide the references for MGrid International Journal of Computer Integrated Manufacturing More and more researchers are becoming aware of the importance of economics in MGrid Wu et al (2005) presented a trading support manufacturing resource sharing model similar to P2P and deployed core services at every node responsible for aggregating manufacturing resource information Giovanni (2002) investigated the economic convenience in front of dedicated manufacturing systems depending on the competitive market conditions They proposed a theoretical model that can locate market conditions and make scope economy manufacturing systems less profitable than dedicated manufacturing ones, and set some general criteria to guide the entrepreneur in making wise investment decisions regarding these kinds of manufacturing investments Lu et al (2007) defined the competitive equilibrium of Pareto optimal in the MGrid resource optimal allocation, in order to realise profit maximisation for RSPs and efficiency maximisation for RSDs However, insufficient research has been done on how to display resource quality information for RSDs and how to obtain accurate information regarding resource quality from RSPs There are two traditional ways of solving this issue First is the centralised trust system (Resnick and Zeckhauser 2002), which assumes that a few of the entities record the history of entities that participate in the network, and calculate and publish the results of the creditability evaluation for every entity Second is the distributed trust system (Cornelli et al 2002), which assumes that every entity calculates the credibility of a certain entity in the network, according to the behaviour evaluations provided by others Information is generally static in the centralised trust system, while network communication in the distributed system makes it difficult to calculate credibility The MGrid environment has the following characteristics: resource distribution and sharing, selfsimilitude, dynamics and diversity, autonomist and multiplicity of management, and highly abstract and transparency (Tao et al 2009) The resource management and cooperation in MGrid is more complex and difficult than other types of distributed information systems Therefore, the static information in the centralised trust system and the complex communication in the distributed system are not suitable for the MGrid environment It needs a simple and flexible approach for solving the above-mentioned issues for its resource market The resource trading process in MGrid is essentially a game in which RSPs and RSDs evaluate each other’s behaviour and characteristics, and then choose the strategies that maximise their own individual benefits Current state-of-the-art grid technology uses game theory for grid resource optimised allocation 393 (Riky et al 2008, Tao et al 2008b) The authors introduce the game theory into the resource trading process in MGrid and propose an MGrid resource trading model based on the perfect Bayesian Nash equilibrium In this way, the authors analyse the trading process using a dynamic model of incomplete information This model makes RSPs tend not to cheat and RSDs determine the quality of the resource according to the collateral currency promised by RSPs This is a new method of solving the above issues of MGrid resource trading The key is to prevent RSPs from sending out an improperly high quality signal MGrid resource negotiation and trading model 3.1 The model of MGrid resource market Consider resource trading in the MGrid environment: If RSDs need some manufacturing resource or service, they search for resources and inquire about prices in the MGrid market, according to the catalogue of resource RSPs can also register and maintain resource and service information through the MGrid portal The information in the MGrid market is uniformly stored and managed by the MGrid index information server (MGIIS), as shown in Figure Suppose RSPs who provide the same type of resources but at different levels of quality, all charge the same price P The RSPs know the level of quality of resources they offer (but the RSDs not), and they promise a collateral currency F If a RSD argues that the purchased resource does not meet contract specifications, then the RSP compensates the RSD by paying F Therefore, RSDs would like to purchase resources with higher F, given a constant P The goal of this paper is to establish an effective signalling game model in which RSPs are motivated to make a promise of a collateral currency F that accurately reflects the quality of their resource In signalling game theory, the quality of resource can be regarded as the types of RSPs given by ‘nature’, and the collateral currency F as the signalling according to their types RSDs infer the quality of a given resource from F and choose one RSP as their supplier Then, the dynamic signalling game model of incomplete information would be established in order to solve the above-mentioned issues in MGrid resource trading 3.2 Signalling games Signalling games (Myerson 1997) are dynamic games in which information transfer is viewed as the signal They are incomplete information games with two players, a sender and a receiver The type of sender (i.e the quality of resources that RSPs provide) is private 394 Figure H Zhang et al The MGrid resource market model information; the type of receiver (i.e RSDs prefer to buy resources with a higher collateral currency, given a constant price) is public information The two players receive payoffs depending on the type of sender, the message chosen by the sender, and the action chosen by the receiver The sequence of signalling games is as follows: (1) The sender has a certain type y Y, which is given by nature Y ¼ {y1, , yK} is the set of sender types The sender knows its own y while the receiver does not know the type of the sender The receiver only knows the prior probability p ¼ p(y) where the type of the sender is y, Skp(yk) ¼ (2) Based on his knowledge of his own type, the sender chooses to send a message (m) from a set of possible messages M ¼ {m1, , mJ} (3) The receiver observes the message (m) but not the type of the sender (y), and gets the posterior probability p~ ¼ p~ (y j m) from the prior probability p ¼ p (y) according to the Bayes rule Then, the receiver chooses an action a A from a set of feasible actions A ¼ {a1, , aH} (4) The payoff functions of the sender and the receiver are u1(m, a, y) and u2(m, a, y), respectively The equilibrium concept that is relevant for signalling games is Perfect Bayesian equilibrium Perfect Bayesian equilibrium is a refinement of Bayesian Nash equilibrium, which is an extension of Nash equilibrium of incomplete information games Perfect Bayesian equilibrium is the equilibrium concept relevant for dynamic games of incomplete information 3.3 Architecture of MGrid resource negotiation and trading For MGrid resource trading, an architecture for MGrid resource negotiation and trading is designed in the section The architecture is sufficiently general to accommodate different economic models used for resource trading and service access cost determination (see Figure for more details) This consists of four parts: RSDs, grid brokers, core grid middleware, and RSPs The features of GRACE (Abramson et al 2002) have been extended to focus on preventing resource trading fraud (such as the credit management module) On the basis of the resource negotiation and trading architecture, the resource trading process is as follows (shown in Figure 3): (1) RSPs and RSDs login to MGIIS (2) RSPs and RSDs deposit money into the Grid Bank In this study, the authors use a ‘prepaid’ payment mechanism (3) RSPs publish their resource information (i.e process capability, price, duration, and collateral currency) in the MGrid market through the MGrid portal to attract RSDs RSDs can also search and check resource information through the portal (4) RSPs sign a contract with RSDs In the meantime, the contract must be legally notarised and the notary public takes over both deposits in the Grid Bank 395 International Journal of Computer Integrated Manufacturing Figure The architecture of resource negotiation and trading in MGrid (5) If a trade is successful, the RSD’s deposit will be transferred into the RSPs account by the notary public Otherwise, the RSP’s deposit is transferred into the RSD’s account as compensation No matter what, the credit system records this trading information about RSDs and RSPs for later credit inquiry 3.4 Model analysis According to Taguchi et al (2005), quality implies low failure rate, low energy consumption, long service life, high efficiency, and low damage to users For convenience, assume the quality level of the resource is q (0 q 1), where higher q indicates higher resource quality and q ¼ means that the resource has a zero failure rate The authors employ the exponential distribution for the failure rate G(q) ¼ e–kq, where k Rþ such that k RSPs charge P for each resource and promise a collateral currency F The cost of a resource is given by C(q) ¼ a1e–b1q þ a2e–b2q (Juran et al 1999), where a1 refers to the loss of defective resource as q ! 0, a2 refers to the cost of the resource as q ! 1, and b1, b2 are the coefficients of the function From a practical perspective, P ! C(q) 0, F The action of RSDs is to determine the trading probability Prob(Á) In general, RSDs consider two factors: quality and price (that is, the cost performance) At the same P, the higher F that RSPs promise, the higher qualities of resource RSDs assume Therefore, RSDs prefer to trade with RSPs who have > Therefore, the the highest F, such that dProbðFÞ dF formulae of Prob(Á) is given by ProbðFÞ ¼ m qeðFÞ þ n; P ð1Þ where Prob(F) 1, m 0,0 n 1, m and n are the adjustment coefficients q~(F) is the estimation of resource quality after the RSD observed compensation price F The MGrid resource trading model based on perfect Bayesian Nash equilibrium is described as follows: (I) ‘Nature’ chooses one resource quality q according to a certain prior probability density, and informed RSPs know it as their types (II) RSPs fix F (III) RSDs observe F without knowledge of q, and then determine the trading probability Prob(F) (IV) The payoff function of RSPs is U (q, F, Prob (F)) Define the benefit as the added value of profit, not considering the funding rate If there is no trading, the benefit is zero Suppose that RSPs are risk-neutral; therefore, the expected benefit of RSPs is Uðq; F; ProbðFÞÞ ¼ ½ðP À F À CðqÞÞ Á GðqÞ þ ðP À CðqÞÞ Á ð1 À GðqÞފ Á ProbðFÞ 396 H Zhang et al Figure The flow chart of resource trading in MGrid Simplify it and get the following function: Uðq; F; ProbðFÞÞ ¼ ½P À F Á GðqÞ À Cðqފ Á ProbðFÞ Furthermore, the price F should satisfy U(q,F, Prob(F)) ! 0, so the constraint of F can be obtained as follows: F P À CðqÞ GðqÞ ð2Þ 3.5 Model solution As a pure strategies equilibrium, the above model may have the solutions of a pooling equilibrium, either separating equilibrium or semi-separating equilibrium The goal of this study is to get the solution of separating equilibrium, that is, the higher quality of resource, the more likely RSPs are to promise a higher compensation price Based on perfect Bayesian Nash equilibrium, RSPs and RSDs plan optimal reactions in (II) and (III), against the possible strategies of their opponents for each potential type of their own The reaction of a RSP depends on their type Hence, an RSP’s reaction reveals some information about their type RSDs can infer their opponent’s type or revise the prior probability, and then choose the optimal reaction RSPs know that their reactions will be known or utilised by RSDs, and therefore try to choose reactions that most benefit themselves Suppose that there exists the partial derivative of U to F, and let @U @F ¼ 0, International Journal of Computer Integrated Manufacturing @U ¼ ½P À F Á GðqÞ À Cðqފ @F   dProbðFÞ Â À GðqÞ Á ProbðFÞ ¼ dF ð3Þ If RSPs promise the collateral currency F for their resource, u˜(q j F) is the posterior probability of q after RSDs observe the price F Then the expected value of R1 the resource quality for RSDs is qeðFÞ ¼ q~ uðq j FÞdq u˜(q j F) can be calculated according to the Bayes formula – a theorem that is valid in all common interpretations of probability – in the following way: u~ðq j FÞ ¼ R 1PðF j qÞuðqÞ PðF j qÞuðqÞdq Regarding the information set F observed by RSDs, the belief of RSDs u˜(q j F) satisfies R1 e u ðq j FÞdq ¼ For separating equilibrium, RSDs can infer q from F because F(q) is the optimal reaction of RSPs who provide a resource of quality q So there is de qðFðqÞÞ=dq ¼ In fact, there are P(q j F(q)) ¼ 1and P(q0 j F(q)) for the separating equilibrium,where q0 6¼Rq R1 0 Therefore, R 0qeðFÞ ¼ q ueðq j FÞdq ¼ q Pðq j FðqÞÞdq ¼ q dðq À qÞdq ¼ q, where d is the Kronecker function   de q dFðqÞ de q dFðqÞ À1 Á ¼ 1; ¼ ð4Þ dF dq dF dq By combining Equations (1), (3), and (4), the following formula can be obtained: @U m½P À F Á GðqÞ À Cðqފ ¼ Á @F  À1 P  q  dFðqÞ À GðqÞ Á m þ n ¼ dq P 397 Next, the RSP’s strategy is analysed (as shown in Figure 4) The curve L1 represents the inequality (2) and the curve L2 represents the Equation (3) The two curves intersect at M (q0,F3) When the quality of the resource provided by a RSP (q) is less than q0, the expected profit U of the RSP is less than zero at the optimal compensation price F* Therefore, the authors not consider there are resources with the quality q q0 In Figure 4, F1 represents the maximum of the collateral currency which satisfies the inequality (2); F2 represents the optimal value of the compensate price that satisfies Equation (3) RSPs with resource quality q can gain the maximum profit by promising a compensation price F2 The curve L2 in Figure implies that F* should range from F3 to F4 Therefore, prob(F) ¼ when F F3, while Prob(F) ¼ when F F4 Therefore, the trading probability is given by F > F4 < 1; ProbðFÞ ¼ m eqðPFÞ þ n; F3 F F4 : 0; F < F3 3.6 Model analysis (1) The model identifies users as belonging to one of two parties (either RSPs or RSDs) in the MGrid resource market The strategy of all RSDs is the same, and that of all RSPs is also the same (2) The model assumes that RSPs with different quality resources all charge the same price, because RSDs would have a better idea of a particular resource’s level of quality if they were able to compare RSPs’ prices (3) For convenience of analysis, the authors assume the quantity of the trading resource is one in the model However, the model is the same for other quantities By combining C(q) and G(q) with the above formula and solving the differential equation, the expected value F is obtained  P kq a1 Fà ¼ yðqÞ ¼ m e À eqðkÀb1 Þ k k À0 b1 a2 eqðkþb2 Þ þ C1 À ðmq þ nPÞ ð5Þ k þ b2 where C1 is the integration constant There is a game corresponding to C1 Let q~(F) ¼ q ¼ y–1(F*), under the observation of rational expectations (two parties can identify the type accurately), the payoff function of RSPs is Uðq; F; ProbðFÞÞ ¼ ½P À F Á GðqÞ À CðqފÁ   à m Á yÀ1 ðFÃ Þ P þ n Figure curves The compensation price & resource quality 398 H Zhang et al (4) The model takes into account the cost of a resource, given different actual qualities, while the literature (Li et al 2005) does not Because the different quality resources have different costs, obviously, the profit is different at the same price Model simulation The authors employ MATLAB V7.6.0 as the simulation tool and personal computer (Intel Pentium CPU 3.40 GHz and 2GB memory, operation system: Windows Vista Enterprise) as the simulation platform 4.1 Simulation parameter set A machine plant (RSD) wants to purchase a certain part, for which all machine tool plants (RSPs) charge P ¼ 13.50 However, the quality of the parts provided by different machine tool plants may vary So, the machine plant plays games with machine tool plants In the MGrid resource market, machine tool plants promise different compensation prices based on the quality of their part The other parameters are C1 ¼ 1.00, m ¼ 13.095; n ¼ 0.03 According to the historical data of products, a1 ¼ 2.64, b1 ¼ 0.98, a2 ¼ 0.31, b2 ¼ 3.35, k ¼ 5.00 can be obtained by means of statistical computing The cost of a part is given by C(q) ¼ 2.64e–0.98q þ 0.31e3.35q and the failure rate is given by G(q) ¼ e–5q According to the above resource trading model, the expected profit distribution for the different resource qualities and the different compensation prices is shown in Figure In order to validate the model, q is set to be 0.38, 0.68, and 0.98, respectively Figure shows the 36 Figure The expected profit distribution for RSPs expected profits of machine tool plants at the collateral currency F for every two dollars on the closed interval [0,F1], when the quality of resource provided by the machine tool plant is 0.38 The machine tool plants can obtain the maximum profit (Umax ¼ 2.15) at the collateral currency F2 ¼ 34.59, while q is 0.38 Figure shows the 28 expected profits at the collateral currency F for every ten dollars on the closed interval [0,F1], when q is 0.68 The machine tool plants achieve the maximum profit (Umax ¼ 4.35) at the collateral currency F2 ¼ 84.82 for q ¼ 0.68 Figure shows the 29 expected profits at the collateral currency F for every ten dollars on the closed interval [0,F1] for q ¼ 0.98 The machine tool plants achieve the maximum profit (Umax ¼ 2.73) at the collateral currency F2 ¼ 193.89 for q ¼ 0.98 Figure q ¼ 0.38 The expected profit distribution of RSPs at International Journal of Computer Integrated Manufacturing 4.2 Analysis of simulation results (1) When the collateral currency F promised by a RSP is the optimal value F2 corresponding to q, the RSP achieves the maximal profit (2) The optimum value F2 increases with the resource quality, and RSPs always pursue their own individual profit maximisation in signalling games RSDs should be able to infer the resource quality q from F correctly (3) When F is greater than F1 or less than F3, the expected profit is less than zero Figure q ¼ 0.68 Figure The expected profit distribution of RSPs at The RSOS system of MBRSSP-MGrid 399 These results are feasible No one would like to purchase a resource that has a very low collateral currency Conversely, RSPs may take risks by paying RSDs more collateral currency in the case of resource breakdown if the collateral currency is too high The simulation results indicate that the game theoretical model has a reasonable and perfect Bayesian separating equilibrium, from which RSPs would not initiatively deviate This means that in pursuit of maximal profits, RSPs prefer to provide RSDs with accurate information about their resource’s quality (i.e the compensation price) initially and faithfully, given the strategy of RSDs (i.e the trading probability) in the Figure q ¼ 0.98 The expected profit distribution of RSPs at 471 International Journal of Computer Integrated Manufacturing Notation for the reverse manufacturing Y The fraction of direct shipment EP The degree of implementing information technology application HR Holding cost per unit per unit time during the remanufacturing process FCL Fixed cost including cleaning and disassembly cost during the collecting process CCL Variable cost including cleaning and disassembly cost during the collecting process ADF Fixed component-life-cycle design cost ratio for the green design BDV Variable component-life-cycle design cost ratio for the green design CD Component design cost for the green design Rj Reliability of the sub-function j Fm Fixed cost during the remanufacturing process Cm Variable cost during the remanufacturing process dm The arrival rate of the failed returned items FCV Fixed convertibility cost CCV Variable convertibility cost dC The convertibility parameter for technology evolution of the returned items Frp Fixed repair cost Crp Variable repair cost l1 The arrival rate of the returned-items needing to be repaired Sav Salvage for the unusable items after cleaning, sorting and disassembly x The green handling ratio of the reusable and take-back items RS The ratio for remanufacturing process after cleaning, disassembly and sorting rm The ratio for the items needing to be remanufactured during the remanufacturing distribution 2.2 Figure Time-weighted inventory (TWI) and supplier’s inventory level for the production inventory deteriorating model differential equations governing the inventory level can be presented as follows: dGS1 ðt1 Þ ¼ l À y Á I1 ðt1 Þ dt1 dGS2 ðt2 Þ ¼ Ày Á I2 ðt2 Þ dt2 dIb ðtÞ ¼ ÀD À y Á Ib ðtÞ dt t1 T1 ; ð1Þ t2 T2 ; ð2Þ t Tb : ð3Þ Using the various boundary conditions, GS2(T2) ¼ Nq, GS1 (0) ¼ and Ib(Tb) ¼ 0, the solutions of these differential equations are: Model development The total cost of an integrated production-inventorydistribution model is derived considering the relevant costs with information technology investment when implementing direct shipment and JIT delivery (Figure 2) The model assumes that a successive batch arrives at the store as soon as the previous batch has been depleted Figure depicts the behaviour of the inventory level of the proposed model and the supplier’s time-weighted inventory (TWI) The supplier’s production advances the first batch of the buyer’s need until the production lot satisfies the buyer’s demand during production up-time This paper uses the supplier’s TWI to derive the saw-tooth holding cost instead of the traditional holding cost, which uses the average inventory concept As shown in Figure 4, the l GS1 ðt1 Þ ¼ f1 À exp ðÀy Á t1 Þg y ð4Þ GS2 ðt2 Þ ¼ Nq exp ½y Á ðT2 À t2 ފ ð5Þ D fexp ½yðTb À tފ À 1g y ð6Þ and Ib ðt Þ ¼ From Equation (6), the delivery size is: q¼ D ½exp ðyTb Þ À 1Š y ð7Þ The following equality is derived from the boundary conditions GS1(T1) ¼ GS2(0): f1 À exp ðÀy Á T1 Þgðl=yÞ ¼ Nq exp ½y Á T2 Š 472 C.-J Chung and H.-M Wee when y 55 and Tj 1, j ¼ 1,2, exp (7yT1) and exp (7yT2) can be replaced by 1–yT1 þ (yT1)2/2! and 17yT2 þ (yT2)2/2! respectively (Table 1) Assuming a very small deterioration rate y, and neglecting the terms equal or higher than (y2Tj), j ¼ 1,2, one has: n o À þ y Á T1 À ðÀy Á T1 Þ2 =2 Á ðl=yÞ ¼ Nq½1 þ y Á T2 Š ð8Þ Multiply both sides of Equation (8) by (1 þ yT1/2) and rearrange the terms, one has: Nq½1 þ y Á T2 Š à T1 ¼  l À yQp ð9Þ Substituting (9) and Nq ¼ ND y ½exp ðyTb Þ À 1Š % NDTb ½1 þ yTb =2Š into T1 þ T2 ¼ ðN À 1ÞTb þ q=l and solving for T2 and T1, one has: Tb ½ðN À 2ÞDð1 þ yTb Þ À 2lðN À 1ފ NDyTb ð2 þ yTb Þ þ 2l sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( ) ! NDyTb yTb 1À 1À 1þ ½yT2 ðyT2 À 2Þ À 2Š T1 ¼ y l 2.2.1 Investment in production-distribution planning In the integrated production inventory system, the reduction of inventory level, which minimises the system’s total cost and optimises the system, is one of the important operational control issues The investment in distribution planning by implementing direct shipment is to reduce the inventory level Assume the investment total cost to increase the fraction of implementing direct shipment to an upper value HY be K If no capital is spent, then Y ¼ LY Y is assumed to increase exponentially with increasing K until a technological upper value limit HY is reached, the relationship between Y and K can be represented as: Y ¼ LY þ ðHY À LY Þ½1 À exp ðÀbKފ Since the set-up cost of distribution planning in the supply chain system is considered, the system’s investment in distribution planning is: T2 ¼ ð10Þ Property For a fixed N; N > 1; lim T1 ¼ y!0 lim T2 ¼ y!0 Tb DN ; l T b ð N À 1Þ ðl À D Þ l ð11Þ With a limited value on T1 and T2, the proof of Equation (11) is obtained It is noted that T1 in Equation (9), derived by the traditional differential method, is the same as T1 in Equation (10) From property 1, the optimal number of deliveries must be equal or larger than When N ¼ 1, the integrated production inventory model is the lot for lot policy Property indicates that the proposed integrated model is reduced to the production inventory system without deterioration Table ð12Þ CK ¼ K ð1 þ sYÞ N ð13Þ 2.2.2 The set-up of information technology application in the integrated supply chain The study by Mukhopadhyay et al (1995) showed that information technology has a significant influence on reducing the percentage of the material usage and the total cost Mukherji et al (2006) showed that the investment in upgrades is one of the critical factors for the new information technology applications The system’s expenditure of ITA consists of the informationtechnology set-up cost and the obsolete inventory cost 2.2.3 The buyer’s cost function As shown in Figure 4, since a single-order multipledelivery contract is considered, the buyer’s ordering and holding cost per unit time is: A NHb ð1 À YÞ þ T T ZTb Ib ðtÞdt ð14Þ t¼0 Absolute percentage errors for neglecting the terms higher than the third term in Taylor’s series yTj Error in exp(-yTj) (%) Error in exp(yTj) (%) yTj Error in exp(-yTj) (%) Error in exp(yTj) (%) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.00002 0.00002 0.00014 0.00013 0.00046 0.00044 0.00110 0.00104 0.00216 0.00201 0.00377 0.00344 0.00603 0.00543 0.00906 0.00804 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.01300 0.01136 0.01797 0.01547 0.02410 0.02043 0.03152 0.02633 0.04038 0.03323 0.05082 0.04119 0.06297 0.05029 473 International Journal of Computer Integrated Manufacturing The buyer’s unit deteriorating cost is Cdb and the buyer’s total deteriorating cost is: N Cdb ðq À DTb Þ T & ' NCdb D ¼ ½exp ðyTb Þ À 1Š À DTb y T DcðT; NÞ ¼ ð15Þ 2.2.3.1 The buyer’s set-up of information technology application Through ITA, the buyer will gain more benefits from faster processing and information exchange accuracy The variable set-up cost, KbI, is proportional to: (1) the fraction of direct shipment Y; (2) total production quantity Q (¼Nq); (3) the degree, EP, of implementing ITA The total set-up cost including the buyer’s set-up cost AbI, change-management cost, Kbc, of utilising ITA and the cost reduction, RbI, from ITA benefit can be derived as follows: CbIT ðT; N; YÞ ¼ AbI þ KbI NqY  ½1 À exp ðÀab EP Tb ފ T þ Kbc NqEP Y À RbI NqEP Y ð16Þ 2.2.3.2 The obsolete inventory cost As in Mukhopadhyay et al (1995), the obsolete inventory cost influences information technology performance The investment in ITA is one of the key techniques to reduce the obsolete inventory The obsolete inventory cost includes the fixed handling cost of the obsolete inventory and the variable cost proportional to the degree of implementing ITA error and the fraction of non-direct shipment parts One has: Cbob ðT; N; YÞ ¼ Obo þ Kbo Nq Á rbo Á fEP Á Z T ð17Þ shipment policy is utilised, the buyer is assumed to incur the order handling cost with regard to the buyer’s warehousing function Therefore, the order handling cost of the buyer is proportional to the amount of nondirect shipment items in the buyer’s warehouse and can be derived as follows: Cbhn ðT; N; YÞ ¼ ð18Þ 2.2.4 Forward manufacturing of the supplier Blanchard and Fabrycky (1997) proposed that the systematical engineering process of a product consists of three stages: (1) conceptual, preliminary and detail design; (2) production construction; (3) operational use and system support It is a lifecycle stage consideration for a product In the development of this study, the relevant cost functions are derived in sequence with a point of view of the systematical engineering process 2.2.4.1 Green product design value Imposing extended producer responsibility on manufacturers is a means to achieve a critical leverage point between environment and business benefits Manufacturers have the unique ability to facilitate product recovery and remanufacture by designing products with components suitable for reuse and that are easy to disassemble Through green product design, suitable materials are selected and those decisions (such as employing easily recyclable materials and avoiding unusual materials, components and hazardous materials) can reduce the negative impact on the environment (Fishbein 2000, Toffel 2002) The supplier’s product design cost, Y(T), is a function of T, where T is the product lifecycle Although there are many parameters influencing the design cost and production cost of a product, from the productdesign life point of view, it is appropriate to take it as a function increasing with product lifecycle The supplier’s product design cost is: where fEP ¼ (1–bbEpTb) and Z ¼ (1–zbY) 2.2.3.3 Order handling cost In SCM, the integrated information system can decide how to pick the order, based on available inventory to optimise transportation costs and maximise customer service level As shown in Mason et al (2003), the information integration of warehousing and transportation system through information technology is one of the keys to coordinate the physical flow along the supply chain Therefore, the transportation and order handling cost in warehousing function should be considered when the ITA is utilised In this study, since the direct- Obh þ Kbh Nq Á ð1 À YÞ T CD YðTÞ ¼ T ( ADF À M Y Á þ af EP T þ BDV Rj ) ð19Þ j¼1 where Rj is the reliability of subcomponent j Simplifying the analysis, it is assumed that there are two major parts for the component in which R1 is the reliability of power supply and R2 is the reliability of subcomponent consisting of some parallel subfunctions For the first term, it is assumed that design cost increases and is proportional to the degree of implementing ITA, Ep 474 C.-J Chung and H.-M Wee 2.2.4.2 Production construction As can be seen in Figure 4, the supplier’s holding cost: ¼ ½bold line areaŠ Á HS =T T Z1 ZT2 HS < ¼ GS1 ðt1 Þ dt1 þ GS2 ðt2 Þdt2 T : t1 ¼0 management cost ASI and cost reduction RSI of IT application benefit are as follows: CSIT ðT; N; YÞ ¼ ASI þ KSI QP Y  ½1 À exp ðÀaS EP Tފ T ð24Þ þ KSc Qp EP À RSI QP EP Y t2 ¼0 ÀTb qð1 þ þ Á Á Á þ ðN À 1ÞÞg ð20Þ The supplier’s set-up cost of the production operations per cycle is CS T ð21Þ The unit item cost is UiS and the supplier’s total item cost per cycle is UiS lT1 T ð22Þ Assuming the number of inspection is the same as the number of deliveries and the inspection cost occurs at the start of the cycle, then the inspection cost is:   Cif þ NCiv þ Cins lT1 INm ðT; NÞ ¼ ð23Þ T N 2.2.4.3 The supplier’s set-up of information technology application The supplier sets up the information system, which provides coordination between the supplier’s and buyer’s operational process The information system makes up the majority of the supplier’s expenditure The utilisation of ITA for both supplier and buyer costs more than the individual player with a simple information technology application investment, but such information systems can be a long-term advantage reducing labour-intensive and operational errors The supplier and the buyer with coordinated information-technology linkages may experience some conflict; however, each partner’s efficiency will benefit the supply chain As mentioned in the study of Mukherji et al (2006), the cost of conflict, i.e the change-management cost, results from the modified operational processes or upgraded functions regarding procurement, production or supporting activities Assuming the supplier’s variable set-up cost, KSI, is proportional to the fraction of direct shipment Y, production quantity QP (¼lT1) and the degree of implementing ITA, EP increases exponentially due to the complexity of multiple information system options Consequently, the total set-up cost including the total supplier’s fixed set-up cost ASI, the change- 2.2.4.4 The obsolete inventory cost The obsolete inventory cost, which includes the fixed and variable handling cost, is proportional to production lot size, the fraction of manual-labour error and the degree of implementing ITA The obsolete inventory cost can be reduced through applying the information system and is as follows: CSob ðT; NÞ ¼ OSo þ KSo QP rSo ð1 À bSo EP TÞ T ð25Þ 2.2.4.5 The variable labour cost In practical business, in addition to the benefit from the error reduction, another benefit from the utilisation of ITA is the reduction in the number of employees who compare the supplier’s service bill of materials (BOM) with the buyer’s orders The variable labour cost occurs and is proportional to the degree of utilising ITA as follows: ! OSe CSem ¼ À KSe nSe EP ð26Þ T 2.2.4.6 Transportation cost considering the ratio of direct shipment Y As demonstrated in the studies of Mason et al (2003), the transportation cost is considered in the proposed model Since the distribution planning using direct shipment is considered in the coordinative supplier–buyer business, the supplier incurs the transportation cost for all direct shipment items and non-direct shipment parts Therefore, the supplier’s transportation cost can be derived as follows:   OTF CTrp ðT; N; YÞ ¼ OTr þ þ NOTv T N þ KTr lT1 Á ½mY þ Lð1 À Yފ ð27Þ 2.2.5 Reverse manufacturing of the supplier As seen in Figure 4, the bold-dashed line illustrates the collecting process and remanufacturing inventory level The differential equations of the collecting process and the collected item consuming process can be represented as follows: dcc ðtc Þ ¼ RC À ycc ðtc Þ; dtc tc T1 ð28Þ International Journal of Computer Integrated Manufacturing dcc2 ðtc2 Þ ¼ ÀPRC À ycc2 ðtc2 Þ ; T1 dtc2 tc2 T ð29Þ ZT F¼ KÀ1 ðlK Þdt ¼ ðl1 TÞK bt 475 ð37Þ dZR ðtR Þ ¼ PRC À y Á ZR ðtR Þ; T1 dtR tR T ð30Þ where Tb ¼ T/N and T2 þ Tb – q/l ¼ T–T1 Using the various boundary conditions, cc(T1) ¼ QPx, cc2(0) ¼ QPx, and ZR(T–T1) ¼ q, the solutions of these differential equations are:   RC cc ðtc Þ ¼ QP x À ½exp ðyðT1 À tc Þފ þ RC ð31Þ y y cc2 ðtc2 Þ ¼ PRC ½exp ðÀy Á tc2 Þ À 1Š þ QP xexp ðÀy Á tc2 Þ y ð32Þ  ZR ð t R Þ ¼  PRC qÀ ½exp fyðT À T1 À tR ÞgŠ þ PRC y y ð33Þ Using Equations (31), (32) and (33), the supplier’s holding cost of the collecting process, collected items’ consuming and remanufacturing process can be derived as follows: (1) holding cost of the collecting process HRc T ZT1 fcc ðtc Þgdtc ð34Þ (2) holding cost of collected-items’ consuming process HRc T TÀT Z fcc2 ðtc2 Þgdtc2 ð35Þ (3) holding cost of remanufacturing process HR T TÀT Z fZR ðtR ÞgdtR ð36Þ tR ¼0 In relation to repair cost, Weibull distribution is assumed for the component functional life and hazard rate, as commonly found in the literature Various research and product databases compiled by relevant industries in computer and electronics have shown this functional life distribution K is the shape parameter Assuming the hazard rate item is Weibull distribution, the mean failure rate is: The supplier’s repair cost per unit time with the form of K ¼ 1, when the partial functions are of malfunction, is: É Frp È Crp F Á QP xð1 À RS Þ þ T T É 1È Frp þ Crp l1 T Á QP xð1 À RS Þ ¼ T ð38Þ The remanufacturing cost is dependent on the cycle time It is considered that the probability of failure will occur and have an exponential function, as shown in the maintenance literature, such as Jayabalan and Chaudhuri (1992) and Kim and Hong (1997) It is assumed that the component has an exponential recondition distribution with a mean of 1/dm For the competitive and time-to-market driven industry, the development cycle at an electrical product company is based on a significant amount of reusable technology A continuous development cycle allows for the introduction of new technology on a regular basis in order to improve the capabilities of the next generation products (Winters et al 2004) Therefore, the cost of convertibility has a strong relationship with reusable technology It is straightforward to assume that the cost of convertibility is proportional to the time between technological evolutions The cost of convertibility is valid in that the new evolutions increase exponentially in various situations of product development It is assumed that the mean time between technological evolutions is in an exponential pattern The supplier’s total remanufacturing cost consists of cleaning, disassembling, remanufacturing and convertibility The supplier’s remanufacturing cost per unit time can be derived as: 1n Fm þ QP xð1 À RS Þrm Cm fFCL þ CCL QP xg þ T T o 1n  ½1 À exp ðÀdm Tފ þ FCV þ QP xð1 À RS Þ T o  CCV ½1 À exp ðÀdC Tފ ð39Þ The supplier’s salvage per unit time after cleaning, disassembling, sorting and identifying in the collecting distribution is: fSav Á QP xRS g T ð40Þ 476 C.-J Chung and H.-M Wee where 2.3 The green supply chain’s total relevant cost The total relevant cost of the GSC per unit time is: TCðT1 ; T2 ; Tb ; N; YÞ ¼ ftotal cost offorward manufacturingg þ ftotal cost of reverse manufacturingg where, Tb ½ðN À 2ÞDð1 þ yTb Þ À 2lðN À 1ފ T2 ¼ NDyTb ð2 þ yTb Þ þ 2l ð41aÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( ) ! NDyTb yTb T1 ¼ ½yT2 ðyT2 À 2Þ À 2Š 1À 1À 1þ y l T ¼ NTb ; and Nq % NDTb ½1 þ yTb =2Š ð41bÞ Note that from (41b), one can reduce the number of variables in TC from five to three Optimisation The purpose of this study is to derive the optimal number of inspection and deliveries, the optimal fraction of direct shipment quantity and the delivery cycle time by determining the optimal values of N, Y and Tb, which minimise total cost Assuming the degree of developing information technology EP is predetermined by supplier, K and Y are derived by an iterative method and cycle time T is NTb, the total cost function TC(T,N,Y,K) is reduced to TC(N,Y,Tb) Using Taylor’s series, one has T1and T2 in the forms of Tb: ! ND NDy 2ND þ 2ðl À DÞðN À 1Þ Tb þ 1þ T1 % Tb l l l ! ð N À 1Þ ð l À D Þ DyðN À 1Þ 2ðl À DÞN Tb þ 1À T2 % Tb l l l ð42Þ Since yTb is very small, substituting Equation (42) into the total cost function of Equation (41) and neglecting the cost terms higher than yT2b and for any given positive integer N and Y, the cost function can be written (41) as follows: TCNY ðTb Þ ¼ j1 ðNÞ þ j2 ðNÞTb þ j3 ðNÞT2b þ CK Tb ð43aÞ &   Cif þ OTF A þ CS þ NCiv þ NOTv þ j1 ðNÞ ¼ N !  M Y FCL þ Frp þ þ CD ADF þ BDV þ Rj FCV þ Fm j¼1  !' OSe þ OSo þ ASI þ þAbI þ OTr þ Obh þ Obo N  j2 ðNÞ & À Á CCV ðy À 2NdC Þ ¼ Dxð1 À RS Þ N Crp l1 þ Cm rm dm þ  ! PRC ND l D þ 2À þ ðHRc þ HR Þ 2l  D l   ! D Hb HRc N D 2x À RC þ D HR À þ þ l   2! l HS D D Cdb y ð N À 1Þ À þ þ þ  l' 2 ÀSav RS CCL þ Dxy þ þ DyfðUiS þ Cins Þg 2 ! Kbo rbo Zþ þ ND KTr ðmY þ Lð1 À YÞÞ þ Kbh ! ð1 À YÞ K E À ðRbI þ RSI ÞEP Y þ ND Sc P þKbc Y þ KSo rSo ðNÞ j3 ! & ÀCCV dC C m dm r m ðyþNdC Þþ ) Crp l1 y  ðyÀNdm Þþ    ! D PRC HR PRC Nx þNDy 1À þHRc À l l   ! l HRc N 2DRC Cdb xÀ þDy þ  l  2 ! HR D 2D D ND ðNþ1Þ 1À 1À þ À 2& l l  l   y Kbc þND Kbo rbo Z ÀEP bb þYy ÀRbI EP 2 y þ Kbh ð1ÀYÞþYEP ½NKSI aS þKbI ab '  ! HS yD D ðNÀ1Þ À ÀNKbo rbo bb Zb Š þ 2 l ( ! ðHS þ UiS þ Cins Þ À xðSav RS þ CCL Þ CK ¼ D þCCV xð1 À RS Þ ) ¼ ð1ÀRS ÞNDx ÀKSe nSe EP þ CD ADF af EP þ Kð1 þ sYÞ=N ð43bÞ where Z ¼ (1–zbY) 477 International Journal of Computer Integrated Manufacturing 3.1 Total cost function with respect to Tb when j2 is positive 3.2 Total cost function with regard to Tb when j2 is negative From Equations (41) and (43), for given N and Y, the first and second derivative of TC(N,Y,Tb) with regard to Tb can be derived as follows: When j2(N) is negative, a root-finding method of cubic function is proposed to solve the relevant solutions From Equation (44), multiply both sides of Equation (44) by T2b , one has: @TCðN; Y; Tb Þ dTCðTb jN; YÞ dTCNY ðTb Þ ¼ ¼ @Tb dTb dTb Àj1 ðNÞ ¼ þ j2 ðNÞ þ 2j3 ðNÞTb T2b À j1 ðNÞ þ j2 ðNÞT2b þ 2j3 ðNÞT3b ¼ ð44aÞ @ TCðN; Y; Tb Þ d2 TCðTb jN; YÞ d2 TCNY ðTb Þ ¼ ¼ @T2b dT2b dT2b 2j ðNÞ ¼ þ 2j3 ðNÞ ð44bÞ Tb Property For given N, N 1, when j1(N), j2(N) and j3(N) are strictly positive, the GSC inventory model has a unique solution and is convex in Tb Proof: From (44b), since j1(N), j2(N) and j3(N) are strictly positive, the second derivative of TC (N,Y,Tb) is: d2 TCNY ðTb Þ 2j1 ðNÞ ¼ þ 2j3 ðNÞ > dT2b T3b For N 1, TCNY (Tb) is convex in positive Tb When j1(N), j2(N) and j3(N) are strictly positive, TCNY (Tb) has a unique, finite minimum, given by the solution of: dTCNY ðTb Þ Àj1 ðNÞ ¼ þ j2 ðNÞ þ 2j3 ðNÞTb ¼ dTb T2b ð45Þ One has the iterative equation, Tb;nþ1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j1 ðNÞ ¼ j2 ðNÞ þ 2j3 ðNÞTb;n ð46aÞ where n is an index Starting from Tb,0 ¼ 0, Equation (46a) provides a spiral convergence to the optimal value of TÀb given N (see Á Mathews 1992) When j3 ðNÞ Á T2b approaches to zero, one has the near optimal solution: T^b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j1 ðNÞ=j2 ðNÞ ð46bÞ ð47Þ 3.2.1 Condition The optimal Tb is obtained by iterative procedure using the following equation, which provides a convergence: Tb;nþ1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2 ðNÞT2b;n j ðNÞ À ¼ 2j3 ðNÞ 2j3 ðNÞ ð48Þ 3.2.2 Condition From Equation (47), when T3b approaches to zero, the optimal Tb can be obtained as same as T^b ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p j1 ðNÞ=j2 ðNÞ and the solution procedure of optimal N value is similar to the condition of positive j2 Due to the complexity in solving the symbolic solutions of Tb and the integer N simultaneously, a solution procedure was developed to derive the relevant optimal values When the total cost function, TC, of the multiple-level production-inventory in a GSC is obtained, the iterative equations and procedure are used to derive the optimal solution considering the take-back product with component value design and reverse manufacturing 3.3 Optimal investment in information technology applications considering production-distribution planning To derive the optimal investment in distribution planning for the supplier, the proposed model develops a simple procedure when the effects of the take-back product with component value design and reverse manufacturing are considered When the total cost function, TC, of the multiple-level production-inventory in a GSC is obtained, the iterative equations and procedure are utilised With known Tb and N, from the first derivatives of Equation 12) and the total cost function of the chain model with regard to K and Y, one has the following results, respectively: Y ¼ LY þ ðHY À LY Þ½1 À exp ðÀbKފ dY ¼ bðHY À LY Þ½exp ðÀbKފ ¼ Y0 dK ð49aÞ 478 C.-J Chung and H.-M Wee dTCN;Tb ðYÞ ~ þ 1=N ¼ AY dK ð49bÞ where A~ ¼ DTb fN½Kbc À RbI EP À Kbh Š Á ð1 þ yTb =2Þ þ N½Kbo rbo Zb ðbb EP Tb À À yTb =2ފ þ N½KTr ðm À LÞ þ EP Tb ðKSI aS þ KbI ab ފ ÀðHb =2Þ Á ð1 þ yTb Þ þ s=ðDNTb Þg From Equation (49a), Equation (12) can be rewritten as: dY ¼ bðHY À YÞ dK ð50Þ Substituting Equation (50) in Equation (49b) and setting the first derivatives of the total cost function to zero, the optimal fraction of direct shipment and investment in improving distribution planning can be derived as:   YÃN;Tb ðKÞ ¼ HY þ ð51Þ NbA~ and Kà ðYN;Tb Þ ¼ ðÀ1=bÞ Á ½lnðHY À YÞ À lnðHY À LY ފ Figure The solution procedure of the proposed production inventory model ð52Þ Since HY is the maximum value of the fraction of direct shipment and constant b is positive, then the sign of A˜ is negative and the optimal condition is as follows: LY 3.4 YÃN;Tb ¼ HY þ NbA~ HY ð53Þ Solution procedure Due to the complexity in solving the symbolic solutions of Tb, Y and the integer N simultaneously, when N and Y are given, Equation (43), i.e TCNY (Tb), can be used to derive the optimum value of Tb instead of the total cost function TC(N,Y,Tb) The first step of the solution procedure is to initialise the investment in distribution planning K as KS and the integer N to derive Tb corresponding to N The optimal solution is obtained when the conditions of total cost and Hessian matrix are satisfied This paper provides a solution procedure as shown in Figure and the solution procedure is described as follows: Step Initialising YS ¼ Y0 and small value e,È and Y setting the KS ¼ K0 0,where Y0 $É j È K K Y HY g, K0 z jKL z KU , LY $ e1 ¼ 0.001 and e2 ¼ 0.1 Step For a range of N, using Equation (43), 1) When j2(N) 0, derive the Tb value using Equation (46) 2) When j2(N) 0, derive and determine the value Tb using Equation (48) Denote the value of Tb as Tb (N), which minimises TC(N,Y,Tb) with known Y Step Using N and Tb values derived from (1) or (2) of step 2, the values of new YN,Tb and new K can be derived by using Equations(51) and (52) Step If (new YN;T b À old YN;Tb ) e1 and (new K – old K) e2, then go to step 5, otherwise go to step Step The optimal values of N, Tb, and YN,Tb can be derived when the following conditions of Hessian matrix are satisfied: International Journal of Computer Integrated Manufacturing ð1Þ The Hessian matrix is H ¼ @ TC b @T @ TC @N@Tb @ TC @Y@Tb @ TC @Tb @N @ TC @N2 @ TC @Y@N @ TC @Tb @Y @ TC @N@Y @ TC @Y2 The first principal minor determinant of H, jH11j 0, the second principal minor determinant of H, jH22j and the third principal minor determinant of H, jH33j 0, 2) The total cost of the GSC inventory model is optimal when TC(Tb(N* – 1), N* – 1,Y*) ! TC(Tb(N*), N*,Y*) TC (Tb(N* þ 1),N* þ 1,Y*), and TC(Tb(N*),N*,Y*) is minimum value of the all TC(Tb(N),N,Y) for the given N value Step When the optimum of N and Tb are derived, find the optimal values of T1, T2 and T from Equations (10), (11) and (12) respectively Step With the delivery size from Equation (7), the total cost and PTCD can be found Example and sensitivity analysis A product manufacturer (the supplier) produces the components for the buyer With regard to the regulations of Waste Electrical and Electronic Equipment and Restriction of Hazardous Substances, the green issues are critical to the supplier’s business The product manufacturer engages in modifying the product design, produces a new model for the buyer and adjusts their business model for quick response and globalisation The company delivers the product to the buyer Goods in the supplier’s and the buyer’s warehouses deteriorate due to the nature of the Table 479 product and storage environment For joint benefit, the two players decide to cooperate closely Since the supplier–buyer system aims to cut cost, an operational process improvement is conducted and a fraction of final product, with Y, is delivered directly to the buyer’s customer (see Figure 3) A JIT multiple delivery is implemented to reduce the warehouse’s inventory level and lead time uncertainty Owing to legislations, reusability and recoverability of the product are considered The convertibility parameter and the factor of implementing ITA for designing components are incorporated in the proposed inventory model to consider their effects on decision making For simplicity of the design phase, this study assumes that there are two major component parts, where the reliability of the principle component is R1 and the reliability of subcomponent is R2 In order to reduce the operational error and diminish the waste at all levels, the manufacturer and the buyer determine to set up an efficient process control with the aid of ITA There are some relevant costs to be incurred, such as the set-up cost for ITA, obsolete inventory cost, management change cost, etc In the manufacturer’s standard recycle and reuse processes, the manufacturer needs to take back the used components Historical data indicate that the manufacturer takes back a small fraction of the used components from the end users After sorting and cleaning, a small ratio of the product is remanufactured From historical data, the parameter inputs are modified and provided by the component manufacturer to simplify the modelling of the replenishment problem of the cooperative players (Table 2) For the case when j1, j2 and j3 are strictly positive, the convexity of the GSC inventory model with green component value design is shown in Figure The values of the input parameters in the proposed production inventory model y ¼ 0.01 l ¼ 18000 units D ¼ 5200 units A ¼ $300 CS ¼ $500 Hb ¼ $1.4 HS ¼ $1.0 Cdb ¼ $4 UiS ¼ $3.5 OSe ¼ 400 KSe ¼ 200 nSe ¼ 30 rbo ¼ 0.5 bbo ¼ 0.1 bSo ¼ 0.1 OTF ¼ 3130 OTv ¼ 160 Cif ¼ $2800 Civ ¼ $90 Cins ¼ $0.7 R1 ¼ 0.998 R2 ¼ 0.99 ADF ¼ 600 BDV ¼ 40 KTr ¼ AbI ¼ 560 OSo ¼ 400 Obo ¼ 440 m ¼ 0.5 L ¼ 0.8 KSI ¼ 18 KbI ¼ 13 Ep ¼ 0.6 zb ¼ 0.5 dC ¼ 0.002 dm ¼ 0.002 Cm ¼ $2 CCV ¼ $2 Crp ¼ $2 Sav ¼ $7 CD ¼ $32 CCL ¼ $ HR ¼ $1.4 HRC ¼ $0.8 Obh ¼ 360 Kbh ¼ 10 KSc ¼ 30 Kbc ¼ 16 KTr ¼ OTr ¼ 460 aS ¼ 0.1 rm ¼ 0.4 RS ¼ 0.9 x ¼ 0.80 l1 ¼ 0.001 PRC ¼ 4000 units RC ¼ 3500 units Fm ¼ $1360 FCV ¼ $1360 FCL ¼ $1450 Frp ¼ $1350 RSI ¼ 22 RbI ¼ 20 KSo ¼ Kbo ¼ a ¼ 0.06 ab ¼ 0.1 af ¼ 0.06 480 C.-J Chung and H.-M Wee Using the fast solution procedure in section 3, the optimal solution considering deterioration is derived as ðNÃ; Tb Ã; QÃ; YÃ; KÃ; TCÃÞ ¼ (six times, 0.0794 years, 2477 units, 0.8640, $16,379.24, $152,229.95), respectively If deterioration approximates to zero, the solution is derived as in Table The percentage of total cost difference is used to analyse the sensitivity analysis of the input parameters, defined as: When the parameters value change, then the relationships among the known parameters, the decision variable and the percentage integrated total cost difference are derived The main conclusions drawn from the sensitivity analysis are as follows: PTCD ¼ ðTC À TCÃÞ=TC Ã: Figure Convexity of the green supply chain inventory model when the optimal values of Tb and N are derived Table (1) When the deterioration rate y and the green handling ratio of reusable and take-back items x increase, the delivery time interval Tb decreases In practice, when the deteriorating rate y increases and the more reusable products are taken back, the delivery time interval Tb should be shortened, i.e policy with more frequent deliveries may be implemented (Tables and 5) (2) When the convertibility parameter dC and green handling ratio x increase, the total cost tends to decrease The new technologies evolve frequently and the take-back items increase will benefit the GSC inventory system (Tables and 4) From the results of the sensitivity analysis, a managerial planning, strategy and technique should positively incorporate the new technology evolution, the green handling of reusable item, remanufacturing and recycling items in production-distribution planning, such as the direct shipment policy (Tables and 4) From the results of (1) and (2), the engineering designer should involve the new technologies in product design This is because the green handling ratio of reusable and take-back items increase will result in the decrease in the delivery time interval Tb and the reduction in the on-hand inventory level Sensitivity analysis when the deteriorating rate changes y N Tb(¼ T/N) Q Y TC 0.0001 0.001 {0.01} 0.05 0.1 0.0794 2476 0.8640 152218.82 0.0794 2476 0.8640 152219.83 0.0794 2477 0.8640 152229.95* 0.0793 2480 0.8640 152274.93 0.0793 2484 0.8641 15233.13 {} ¼ the base column *The optimal total cost Table Sensitivity analysis when the green handling ratio of reusable and take-back items changes x N Tb(¼ T/N) Q Y TC {} ¼ the base column *The optimal total cost 0.64 0.72 {0.80} 0.88 0.96 0.0795 2482 0.8640 153660.92 0.0794 2480 0.8640 152945.53 0.0794 2477 0.8640 152229.95* 0.0794 2474 0.8639 151514.20 0.0792 2471 0.8638 150798.26 481 International Journal of Computer Integrated Manufacturing (3) The total cost is positively sensitive to the supplier’s and the buyer’s change management costs {KSc, Kbc} and the design cost CD and is negatively sensitive to the supplier’s and the buyer’s unit cost reduction {RSI, RbI} When the supplier’s and the buyer’s change Table management costs {KSc, Kbc} increase, the optimal delivery time interval Tb decreases From the dispatch plan point of view, a manager may shorten the delivery-time interval Tb and use a multi-lot-sizing policy as a dispatch plan (Table and Figure 7) When Sensitivity analysis when convertibility parameter of technology evolution changes dC N Tb( ¼ T/N) Q Y TC 0.0002 0.0004 {0.002} 0.02 0.2 0.0794 2477 0.8640 152229.95 0.0794 2477 0.8640 152229.95 0.0794 2477 0.8640 152229.95* 0.0794 2477 0.8640 152229.76 0.0793 2477 0.8640 152210.37 {} ¼ the base column *The optimal total cost Table Sensitivity analysis when the design cost changes CD N Tb (¼ T/N) Q Y K TC PTCD 25.6 28.8 {32} 35.2 38.4 0.07421 2316 0.8601 15829.42 143207.38 75.93% 0.0768 2398 0.8622 16113.76 147793.00 72.91% 0.0794 2477 0.8640 16379.24 152229.95* 0.00% 0.0818 2553 0.8656 16628.18 156532.16 2.83% 0.0842 2627 0.8671 16862.52 160711.51 5.57% {} ¼ the base column *The optimal total cost Table Sensitivity analysis when the holding costs change Hb, HS, HR, HRC 1.12, 0.80, 1.12, 0.64 1.26, 0.90, 1.26, 0.72 {1.4, 1.0, 1.4,0.88} 1.54, 1.10, 1.54, 0.88 1.68, 1.20, 1.68, 0.96 N Tb (¼ T/N) Q Y K TC PTCD 0.0954 2481 0.8526 14844.16 151160.93 70.70% 0.0795 2481 0.8640 16382.26 151503.04 70.48% 0.0794 2477 0.8640 16379.24 152229.95* 0.00% 0.0792 2473 0.8640 16376.25 152956.53 0.48% 0.0791 2469 0.8639 16373.29 153682.79 0.95% {} ¼ the base column *The optimal total cost Table Sensitivity analysis when the fixed inspection cost changes Cif N Tb (¼ T/N) Q Y K TC PTCD {} ¼ the base column *The optimal total cost 2240 2520 {2800} 3080 3360 0.0949 2469 0.8525 14826.10 152408.21 0.12% 0.0793 2475 0.8639 16373.38 152131.90 70.06% 0.0794 2477 0.8640 16379.24 152229.95* 0.00% 0.0794 2479 0.8640 16385.09 152327.93 0.06% 0.0795 2480 0.8640 16390.94 152425.85 0.13% 482 C.-J Chung and H.-M Wee the change management cost {KSc, Kbc} increases, then the total cost increases sharply With this, an essential operation activities reduction of inventory management and Table efficient methods, i.e the cost reduction methodology using ITA, should be implemented (4) The delivery interval Tb is negatively sensitive to the unit holding cost {HS, Hb, HR, HRc}, the Sensitivity analysis when the variable inspection cost changes Civ N Tb (¼ T/N) Q Y K TC PTCD 72 81 {90} 99 108 0.0792 2036 0.8639 16365.66 152002.92 70.15% 0.0793 2035 0.8639 16372.46 152116.48 70.07% 0.0794 2477 0.8640 16379.24 152229.95* 0.00% 0.0794 2038 0.8640 16386.01 152343.32 0.07% 0.0952 2039 0.8527 14851.43 152833.25 0.40% {} ¼ the base column *The optimal total cost Table 10 Sensitivity analysis when the variable transportation cost changes OTv N Tb (¼ T/N) Q Y K TC PTCD 128 144 {160} 176 192 0.0791 2469 0.8638 16355.09 151826.09 70.27% 0.0792 2473 0.8639 16367.17 152028.17 70.13% 0.0794 2477 0.8640 16379.24 152229.95* 0.00% 0.0952 2476 0.8527 14850.18 152812.24 0.38% 0.0953 2479 0.8528 14860.17 152980.26 0.49% {} ¼ the base column *The optimal total cost Table 11 Sensitivity analysis when the supplier’s change management cost and the buyer’s change-management cost change KSCh KbCh N Tb (¼ T/N) Q Y K TC PTCD 24 14.4 27 16.2 {30} {18} 33 19.8 36 21.6 0.0913 2850 0.8830 19845.75 134211.64 711.84% 0.0847 2644 0.8747 18152.25 143535.97 75.71% 0.0794 2477 0.8640 16379.24 152229.95* 0.00% 0.0747 2337 0.8494 14453.66 160405.16 5.37% 0.0852 2216 0.8097 10735.52 168533.63 10.71% {} ¼ the base column *The optimal total cost Table 12 change Sensitivity analysis when the supplier’s unit obsolete inventory cost and the buyer’s unit obsolete inventory cost KSob Kbob N Tb (¼ T/N) Q Y K TC PTCD {} ¼ the base column *The optimal total cost 6.4 7.2 7.2 8.1 {8} {9} 8.8 9.9 9.6 10.8 0.0812 2536 0.8627 16190.98 148950.40 72.15% 0.0803 2505 0.8633 16286.27 150599.61 71.07% 0.0794 2477 0.8640 16379.24 152229.95* 0.00% 0.0785 2449 0.8646 16470.01 153842.03 1.06% 0.0776 2421 0.8651 16558.69 155436.44 2.11% 483 International Journal of Computer Integrated Manufacturing Table 13 Sensitivity analysis when the supplier’s unit cost reduction and buyer’s unit cost reduction change RSIT RbIT N Tb (¼ T/N) Q Y K TC PTCD 17.6 16 19.8 18 {22} {20} 24.2 22 26.4 24 0.0741 2314 0.8412 13539.13 161871.01 6.33% 0.0766 2391 0.8541 15027.36 157132.36 3.22% 0.0794 2477 0.8640 16379.24 152229.95* 0.00% 0.0824 2572 0.8718 17632.90 147145.74 73.34% 0.0858 2679 0.8782 18816.18 141858.18 76.81% {} ¼ the base column *The optimal total cost Table 14 Sensitivity analysis when the degree of implementing information technology application changes EP N Tb (¼ T/N) Q Y K TC PTCD 0.48 0.54 {0.60} 0.66 0.72 0.0789 2463 0.8462 14083.21 153725.20 0.98% 0.0791 2470 0.8563 15306.91 152978.11 0.49% 0.0794 2477 0.8640 16379.24 152229.95* 0.00% 0.0796 2484 0.8700 17334.16 151480.72 70.49% 0.0798 2491 0.8749 18195.35 150730.41 70.99% {} ¼ the base column *The optimal total cost Figure PTCD and change rate in the green supply chain inventory model Figure The fraction of direct shipment for productiondistribution policy and change rate in the green supply chain inventory model fixed inspection {Cif}, the supplier’s and buyer’s change management costs {KSc, Kbc} and the supplier’s and buyer’s unit obsolete cost {KSo, Kbo} (Tables 7, 8, 11 and 12) The delivery interval Tb is positively sensitive to the supplier’s and buyer’s unit cost reduction {RSI, RbI}, the variable inspection cost Civ and variable transportation cost OTv (Tables 9, 10 and 13) When the supplier’s and the buyer’s unit obsolete inventory costs increase, the delivery interval Tb should be shortened, i.e a frequent delivery policy is encouraged (5) The optimal number of deliveries N is sensitive to the parameters: the deterioration rate y, the fixed and variable transportation cost {OTF, OTv} and the fixed and variable inspection cost {Cif, Civ} When a dispatch plan is considered, there is a trade-off between the pair of fixed inspection and transportation costs, and the pair of variable inspection and transportation costs for determining the optimal number of deliveries (Tables 7, and 9) (6) When the enterprise increases the degree of implementing ITA, Ep, i.e when the more powerful ITA is involved, the amount of direct shipment items increases and the total cost tends to decrease In practical terms, the ITA benefits the production-distribution planning in a system tactical plan (Table 13) 484 C.-J Chung and H.-M 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