Zero Inventory/Production Control Policy for Manufacturing Systems Subject to Quality Deterioration

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1 Proceeings of the 0 International onference on Inustrial Engineering an Operations Management Istanbul, Turkey, July 3 6, 0 Zero Inventory/Prouction ontrol Policy for Manufacturing ystems ubject to Quality Deterioration Hanen Bouabi an Karem Dhouib Mechanical Engineering an Prouctique Department, MDT Laboratory École upérieure es ciences et Techniques e Tunis Montfleury, 000 Tunis, Tunisia Ali Gharbi Automate Prouction Engineering Department, P Laboratory École e Technologies upérieures Montréal, Québec H3-K3, anaa Abstract In this paper, we stuy the prouction an inventory control of a manufacturing system compose of an imperfect process. After a ranom perio operating in the in-control state, the manufacturing system switches to the out-ofcontrol state an starts proucing non-conforming items. The manufacturing system in-control horizons have a general eterioration istribution. Restoration actions with ranom urations are thus planne to restore the manufacturing system to the in-control state after a specific logistic perio. Restoration urations have a general probability istribution. Despite of numerous works publishe about the prouction an inventory control, an base on the just in time concept, this paper eals with the ynamic control of manufacturing system prouction rates uner a zero inventory policy. In this context, two prouction control policies were propose: the first one entails to stop immeiately the manufacturing activity when the system transits to the out-of-control state, the secon policy avocates continue proucing until the beginning of the restoration activity. The objective of the paper is to etermine the best control policy among the two propose ones base on the assessment of the overall incurre cost incluing restoration, non-conforming items an shortage costs. Analytical moels are propose an the expressions of the overall incurre costs are erive for each control policy. Keywors Manufacturing systems, prouction an inventory control, zero inventory policy, prouct quality alteration.. Introuction In the last few ecaes, prouction planning an control has been an active area of research to evaluate the optimality of numerous strategies. The objective was to reuce the inventory levels while ensuring the customer eman. Akella an Kumar (986) have introuce the Heging Point Policy, which entails that the buffer stock is buil-up with an excess prouction capacity, an then maintaine at its maximum level in orer to palliate for interruptions ue to breakowns. Other researchers have focuse on integration of systematic maintenance policies in conjunction with control prouction policy. The objective is to etermine the optimal parameters of the control law an the systematic maintenance policy minimizing the costs of corrective an preventive actions an those associate with inventory management (heung an Hausman 997, alameh an Jaber 000, Dohi et al. 00, Gharbi et al. 007, Berthaut et al. (0)). In the prouction planning an inventory control literature, manufacturing systems have been stuie uner various conitions. Most of the propose moels are base on the assumptions that the prouction process always prouces items with perfect quality. However, in practice, the quality of the prouct may not be acceptable because the prouction process coul eteriorate with time. Many authors have propose a novel strategies integrating quality eterioration in the prouction an inventory control policies (Ben-Daya 00, Rahoui et al. 009, Dhouib et al. 0). In orer to reuce the impact of proucing non-conforming items the majority of these policies consier the construction of a security stock. However, all these researches assume that the inventory level is strictly positive an o not consier the zero inventory policy avocate by the just in time 840

2 philosophy. In this paper, an base on the zero inventory strategy, two prouction control policies were propose: the first one entails to stop immeiately the manufacturing activity when the system transits to the out-of-control state, an the secon policy avocates continue proucing until the beginning of the restoration activity. For the propose policies, we evelop mathematical formulations moeling the overall incurre costs per time unit over an infinite horizon. The overall incurre cost inclues shortage, efective items an restoration actions costs. The next section presents the manufacturing system escription, notations an assumptions. ection 3 iscusses the ynamic an stochastic behavior of the manufacturing system. Mathematical moels evaluating the overall incurre costs per time unit over an infinite horizon are evelope in section 4. ection 5 exposes first a numerical example an then a sensitivity analysis to show the effectiveness of the propose methoology. Finally, section 5 contains a summary of the paper an some concluing remarks.. Description of the Prouction ystem, Notations an Assumptions Moel. Description of the prouction system The system uner stuy is a manufacturing system consisting of a single machine eicate to proucing a single prouct type to satisfy a constant an a continuous eman. This machine can be consiere as an aggregation of several machines. The manufacturing system is subject to ranom perturbations ue to the alteration of prouct quality uring the prouction phase. At the start, the manufacturing system begins in an in-control state proucing conforming items of acceptable quality. After a ranom operating perio (τ), the manufacturing process may shift from the in-control state to the out-of-control state an starts proucing a percentage (α) of non-conforming items. The manufacturing system allows instantaneous etection of the out-of-control state. The in-control state perio is a ranom variable with general probability istribution. A restoration action is starte ahea transition to the out-of-control state but after a constant elay, enote logistic elay perio. The logistic elay perio is require to prepare all resources necessary for restoration actions. During restoration actions, the prouction process is aborte. The restoration action elay is a ranom variable having a general probability istribution. The manufacturing system operates at a capacity equal to eman. In orer to reuce the impact of prouct quality eterioration an restoration elays on the eman satisfaction, the manufacturing cell can recover the nonconforming items by increasing its prouction rate uring the logistic elay perio. After a restoration action, the manufacturing facility transits to the as-goo-as-new in-control state an restarts proucing conforming items. Once restorations achieve, the prouction process is resume immeiately. During a restoration action, an inventory shortage occurs. In this situation, quantities require uring a shortage perio are not elivere after the manufacturing system restoration an are consiere as lost eman. The objective is to fin first the expecte overall cost incurre per time unit over an infinite spam an then eucing the best policy among the propose ones.. notations The following notations will be use: - : eman rate. - U max : prouction capacity of the manufacturing system efining its maximum prouction rate. - u(t) : controlle prouction rate of the manufacturing system at time t. - z(t) : prouction surplus at time t; if z(t) is positive, it represents inventory, while a negative value represents a shortage. -τ: continuous time ranom variable ientifying the istribution of the in-control state perios. - f(τ) (F(τ)) : ensity (istribution) function associate with the ranom variable τ. - α: percentage of non-conforming items prouce uring the out-of-control perio. - LDP: Logistic Delay Perio. - t r : continuous time ranom variable ientifying the istribution of the restoration action elays to bring the manufacturing system from the out-of-control to the in-control state. - h(t r) (H(t r)) : ensity (istribution) function associate with the ranom variable tr. - T i : total cost incurre by the prouction-inventory control policy i. - R : cost of restoration action. - : one-item shortage cost. 84

3 - : one-item raw material cost. - MO : manufacturing system operation cost per time unit. - F i : expecte total cost incurre uring prouction phase of the policy i. - R i : expecte total cost incurre uring restoration phase of the policy i. - i : expecte total cost incurre uring shortage phase of the policy i. - Li: expecte value of the prouction/restoration cycle length for policy i. - E(.) : expecte value of a ranom variable. - Ni : expecte cost of non-conforming items manufacture uring prouction phase of the policy i..3 Assumptions Other assumptions are mae as follows: The emans which cannot be satisfie are lost. The prouce items are imperishable with time The non-conforming items are not reinserte in the prouction process. All costs relate to restoration activities, quality of manufacture proucts an inventory management are known an constant. The out of control state is etecte instantly. The logistic elay perio is known an constant. 3. Manufacturing ystem Prouction ontrol uner Zero Inventory Policy The majority of control policies propose in the literature are base on the built up of a safety stock with a strictly positive level (Z max ). The mathematical formulations assessing the overall incurre costs of such strategies o not consier the case where Z max equals zero. However, an accoring to the just-in-time philosophy an a-value prouctivity concept, inventories are consiere costly categories of waste that shoul be reuce to the minimum if not eliminate (uzaki 987, Dear 988). In this section, we propose a prouction control policy base on zero inventory concepts. The zero inventory policy implies not to buil an not to maintain a security stock uring prouction phases. In effect, an in cases where managers aopt zero stock policy, they consent that it is better to suffer the costs of lost emans than cost ue to inventory holing. In this case, two ifferent policies can be evelope epening on the ecision about immeiately stopping or not prouction when the first non-conforming items appeare. 3. Policy : escription an mathematical formulation This policy avocates stopping immeiately the prouction if the manufacturing system switches to the out of control state. Therefore, this policy oes not allow the prouction of any efective item. The transition from the incontrol state to the out-of-control can occur at any instant of the prouction horizon. onsequently, for a constant eman rate, the relate prouction control policy is given by: u t if the manufacturing system is in the 'in-control' state = 0 elsewhere () Figure shows, uner policy, the evolution of the inventory level uring the prouction, the logistic, an the restoration phase. In fact the system starts proucing conforming items at a prouction rate equal to the eman rate. After a ranom perio, the manufacturing system transits to the out of control state an is force own. onsequently, a shortage eman is observe uring the logistic elay perio an the restoration action require to restore the manufacturing system to the as-goo-as-new operating state. The overall cost is compose of the incurre costs uring the prouction phase an those uring the shortage phase when the manufacturing system is shut own (Fig. ). Therefore, by the renewal theorem, the overall expecte cost (T ) is given by equation (), where F equals 0 since no non-conforming items are prouce uring the prouction phase. T F + L = () 84

4 z(t) Z max = 0 τ LDP t r restoration phase t Prouction phase hortage phase - Figure : Inventory evolution uring the prouction/shortage phase uner control policy Expecte cycle length (L ) The expecte length of a prouction/shortage cycle inclues the mean lifetime before transiting to the out-ofcontrol state, the logistic elay perio require to prepare the resources for the restoration action, an the mean restoration time. Hence, the expecte prouction/shortage cycle length is given by equation (3), where E(τ) is the mean life time of the manufacturing system in control state (Eq. 4), an E(t r ) is the mean uration of restoration actions (Eq. 5). L = E τ + LDP + µ (3) 0 E τ = τ f τ τ E t = t h t t r r r r 0 (4) (5) Expecte cost incurre uring shortage phase ( ) The expecte cost incurre uring restoration phase is given by equation (6). ( ) = + LDP + E t (6) R r Expecte overall incurre cost (T ) Therefore, the expecte overall incurre cost uner the policy is: T ( ) + LDP + E t R r = (7) L 3. Policy : escription an mathematical formulation ontrary to the policy, which stops immeiately the manufacturing process when the prouction system transits to the out of control state, policy permits the prouction uring the logistic elay perio. Uner this policy, the manufacturing system prouce a percentage of non-conforming items equal to α. In orer to respon to the eman, the manufacturing system fills the quantity of non-conforming items by increasing its prouction rate. Thus the prouction rate of the manufacturing cell increases from to ( + α ). This policy insures that less shortage items are recore compare to those recore uner policy. The prouction control uner policy is ictate by equation (8). 843

5 if the manufacturing system is in the 'in-control' state u( t) = ( + α ) if the manufacturing system is in the 'out-of-control' state 0 elsewhere (8) Accoring to the ynamic an stochastic behavior of the manufacturing system controlle by policy, figure shows the evolution of the inventory level uring the prouction, the logistic, an the restoration phases while the time evolving. z(t) Z max = 0 τ LDP t r t Prouction phase Restoration phase - Figure : Inventory evolution uring the prouction/restoration phase uner control policy The expecte overall cost is compose of the incurre costs uring the prouction phase an those uring the restoration phase (Eq. 9). T F + R L = (9) Expecte cycle length (L ) The expecte length of a prouction/shortage cycle inclues the mean lifetime before transiting to the out-ofcontrol state, the logistic elay perio require to prepare the resources for the restoration action, an the mean restoration time. omparing figures an, one can note that the expecte cycle lengths (L an L ) uner policies an are equals. Expecte cost incurre uring prouction phase (F ) The expecte cost incurre uring the prouction phase is equal to the cost of non-conforming items. The cost of a non-conforming item inclues the cost of the prouct raw material an the system operating cost. As soon as it shifts to the out-of-control state, the manufacturing system starts an continues to prouce non-conforming items uring the logistic elay perio. Thus, the expecte cost of nonconforming items prouce by the manufacturing system uner policy is given by: F = N = + α LDP (0) MO Expecte cost incurre uring restoration phase (R ) The expecte cost incurre uring the restoration phase is the sum of a restoration activity cost an those ue to non elivere items uring the restoration perio (Eq. ). R = + E t () R r Expecte overall incurre cost (T ) 844

6 Therefore, the expecte overall incurre cost uner the policy is: T N + + E t R r = () L 4. Numerical Example an ensitivity Analysis In this section, we first present a base case. Thus, we carrie out a sensitivity analysis in orer to show the impact of the input parameters on the etermination of the best policy. 4. Base case onsier a manufacturing system having an automatic machine, manufacturing a single prouct type, an responing to a constant an perioic eman of 0,60 units/month. The manufacturing system is subject to a ranom alteration of the prouct quality. The statistics show that the in-control state perios follow a Weibull istribution with shape an scale parameters.5 an, respectively; the mean lifetime in the in-control state of the manufacturing system is equal to month. After a ranom operating perio in the in-control state, the manufacturing process may shift to the out-of-control state an starts proucing non-conforming items with a percentage equal to % of the actual prouction rate. Upon transition to the out-of-control state, a preparation of the require resources for restoration with uration of 0.03 month is carrie out. Restoration actions have ranom urations following a Gamma istribution with shape an scale parameters an 40, respectively; the mean restoration time to the as-goo-as new in-control state is equal to 0.05 month. The following costs are also chosen consiering nevertheless realistic context: - hortage cost (): 300 ($/prouct unit) - Restoration cost (R): 0,000 ($/restoration action) - Raw material cost (): 500 ($/prouct unit) - Manufacturing system operating cost (MO): 50,000 ($/month) onsiering the previous input ata, the expecte overall costs incurre by policies an are respectively equal to 4,0,39.0 $/month an 74,8.6 $/month. Therefore, the best zero inventory/prouction control policy entails to prouce at the eman rate (0,60 units/month) while the manufacturing system is in the in-control state. Once the manufacturing system transits to the out-of-control state, it continuous proucing uring the logistic elay perio with a rate equal to 0,36 units/month. Accoring to these settings, the minimal overall expecte cost incurre by the optimal policy is 74,8.6 $/month. 4. ensitivity analysis In orer to analyze the superiority of policy compare to policy, expecte overall incurre costs equations () an (9) were compare. The analysis has shown that policy outperforms policy if an only if the following conition is respecte: + (3) Equation (3) shows that the relative superiorly of a specific policy epens on the prouct unit shortage cost, the percentage of non conforming items, an the cost of proucing a non-conforming unit which epens on the raw material cost, the manufacturing process operating cost, an the customer eman. onsiering the input parameter of the base case presente previously, equation (3) allows the assessment of the limiting value of the unit shortage cost implying no superiority of the consiere prouction control policies. This limiting conition is obtaine when s equals $/prouct unit. Table presents 7 configurations of cost parameters erive from the previous configuration (base case ), by changing them to higher an lower values, one at a time; the case is the manufacturing system configuration consiere in the previous numerical example where s equals $/prouct unit. Table highlights the consistency between the variation of each cost parameter an the optimal ecision variable an total cost. 845

7 ase Number 5, , , , , , , , , , , ,000 Table : ensitivity analysis for cost parameters Eq. 3 T T MO = α + Best Policy 73, ,578.4 or 3, , , , , , , , , ,44.95 α + 73, , onclusions Because of its great importance for manufacturing companies, several researches have been evote to the problem of moeling an optimizing the prouction planning an control of stochastic manufacturing systems. The prouction control policy is generally base on builing-up a security stock to palliate against ranom perturbations as manufacturing process failures an/or quality prouct egraation. However, the majority of these control policies an specifically the propose mathematical moels assume that safety stock level equals a strictly positive value. In toay s inustry environment, a huge effort is eploye in orer to reuce to the minimum if not to eliminate inventory which are the principal source of manufacturing waste. In the other han, an contrary to what happens in real inustrial context, most of the researches in the prouction planning an control fiel o not consier explicitly the impact of efective proucts an the process quality eterioration on the etermination of the best prouction control policy of manufacturing systems. This paper presents a prouction control policy of a manufacturing system incluing an imperfect process subject to a mono-prouct, constant, an continuous eman. The prouction control policy is base on the zero inventory strategy which entails to not builing a finishe prouct security stock uring prouction phase. Base on the zero inventory strategy, two prouction control policies were propose epening on the ecision about immeiately stopping or not the manufacturing process when the prouction system switches to the out-of-control state an starts proucing non-conforming items. Analytical moels have been propose for manufacturing systems having general istributions of the resience times in the in-control state an of the restoration perios. The analytical moels allow assessing the expecte overall costs incurre by each prouction control policy an which inclues shortage, restoration, an nonconforming items costs. The analytical moels allow etermining the best zero inventory/prouction control policy to implement base on minimum expecte overall cost. The analysis of the analytical moels shows that policy, which entails stopping immeiately the prouction when the manufacturing system transits to the out-of-control process, outperforms policy, which entails to continue proucing uring the logistic elay perio, if the prouct unit shortage cost is less than a specific limit value epening on the percentage of proucing non-conforming units an the cost incurre by proucing one nonconforming unit. The results reporte in this paper exten the theory of the prouction control policy of manufacturing systems incluing imperfect processes an subject to prouct quality eterioration. The propose approach coul be extene to take into account not only prouct quality eterioration but also equipment failures which forces the manufacturing process to be shut own. In this context, preventive maintenance policies coul also 846

8 be consiere in orer to attenuate the impact of catastrophic failures an the shift rate to the out-of-control state on the overall performance of manufacturing systems. References Akella, R., Kumar, P. R., Optimal control of prouction rate in a failure prone manufacturing system, IEEE Transactions on automatic control, vol. A-3, pp. 6-6, 986. Ben Daya, M., The economic prouction lot-sizing problem with imperfect prouction processes an imperfect maintenance, International Journal Prouction Economics, vol. 76, pp , 00. Berthaut, F., Gharbi, A., Dhouib, K., Joint moifie block replacement an prouction/inventory control policy for a failure-prone manufacturing cell, OMEGA-The International Journal of Management cience, vol. 39, pp , 0. heung, K.L., Hausman, W.H., Joint etermination of preventive maintenance an safety stocks in an unreliable prouction environment, Naval Research Logistics, vol. 44, pp. 57 7, 997. Dear, A., Working Towars Just-In-Time, Kogan Page, Lonon, 988. Dhouib, K., Gharbi, A., Ben Aziza, M. N., Joint Optimal Prouction ontrol/preventive Maintenance Policy for Imperfect Process Manufacturing ell, International Journal of Prouction Economics, vol. 37, pp. 6-36, 0. Dohi, T., Okamura, H., Osaki,., Optimal control of preventive maintenance scheule an safety stocks in an unreliable manufacturing environment, International Journal of Prouction Economics, vol. 74, pp , 00. Gharbi, A., Kenné, J.P., Beit, M., Optimal safety stocks an preventive maintenance perios in unreliable manufacturing systems, International Journal Prouction Economics, Vol. 07, pp , 007. Rahoui, M., Rezg, N., helbi, A., Integrate moel of preventive maintenance, quality control an buffer sizing for unreliable an imperfect prouction systems, International Journal of Prouction Research, vol. 47, pp , 009. alameh, M.K., Jaber, M.Y., Economic prouction quality moel for items with imperfect quality, International Journal of Prouction Economics, vol. 64, pp , 000. uzaki, K., The New Manufacturing hallenge: Techniques for ontinuous Improvement, Free Press, New-York,