Economic Ordering Quantity Model with Lead Time Reduction and Backorder Price Discount for Stochastic Demand

Transcription

1 American Journal of Applied Sciences 6 (3): , 9 ISSN Science Publications Economic Ordering uantity Model with Lead Time Reduction and Bacorder Price iscount for Stochastic emand Ming-Cheng Lo Ching Yun University, 9, Chien-Hsin Road, Jung-Li 3, Taiwan, Republic of China Abstract: This study considered reorder point on the continuous review inventory model under controllable lead time with miture of bacorder price discounts and partial lost sales. We developed a continuous review inventory model where the lead time, the order quantity, bacorder discount and safety factor were considered as the decision variables of a miture of bacorders and lost sales inventory model. The objective was to minimize the epected total annual cost with respect to related decision variables. The purpose model with lead time demand distribution was unnown. The author applies a minima distribution free procedure to find the optimal solution and numerical eample was included to illustrate the solution procedure of the proposed algorithms. Key words: Lead time, bacorder price discounts, reorder point, minima distribution free procedure INTROUCTION It is important to manage inventory quantity for modern enterprises. Although the time condensation will predictably raise the cost, a customer will pay remuneration to the supplier who can provide its product faster and more firm than the competition and the price payment may be praiseworthy. Silver and Peterson [8] defined the replenishment lead time as the time beyond from the moment at which it is decided to place an order, until it is physically on the shelf to satisfy customer demands. While Naddor [], Silver and Peterson [8] consider that lead time can be a constant or a random variable, it is often treated as a prescribed parameter and not controllable. However, lead time can be reduced at etra cost and shorter lead time is the primary driver to achieving customer satisfaction in successful TBM operations []. The benefits resulting from reduced lead time include lower cost, less waste and less obsolescence, greater fleibility to response to change, closely lined organization priorities to customers needs, improved service, quality and reliability and substantially accelerated supply system improvements []. Tersine [] and Vollmann et al. [] attributed the replenishment lead time mostly to manufacturing considerations and addressed some guidelines for its reduction. Liao and Shyu [5] suggested that lead time could be decomposed into n components each having a different crashing cost for reduction. Ben-aya and Raouf [] widespread the Liao and Shyu [5] model by considering both lead time and the order quantity as decision variables. It is related [,,,3] are all focus on the reorder point or safety factor as a decision variable. We solve the problem by applying the minima distribution-free approach, originally disseminated by Gallego and Moon [3]. Moon and Choi [8] and Moon and Yun [9] apply distribution-free approach to solving the purpose models. Ouyang and Wu [5] relaed the assumption on the form of cumulative distribution function of the lead time demand and applied the minima distribution free procedure to solve the problem. Moon and Silver [] and Silver and Moon [9] study a single period replenishment problem under the distribution free situation. On the other hand, this study considers a continuous review inventory system in which the lead time is controllable and can be decomposed into several components each having a crashing cost function. In addition, shortage is permitted and the total amount of stocout is a combination of bacorder and lost sale. It is further assumed that the patient customers with outstanding orders during the shortage period are offered a bacorder price discount and consequently the bacorder ratio is proportional to the magnitude of this discount [6]. Since the shortage cost is eplicitly included, the reorder point is also treated as a decision variable in this study []. There is a form of lead time demand that assumes a distribution free in the study. In this model, the objectives are to simultaneously optimize the order quantities, bac order discounts, reorder points and lead Corresponding Author: Ming-Cheng Lo, epartment of Business Administration, Ching Yun University, 9, Chien-Hsin Road, Jung-Li, 3, Taiwan, Republic of China 387

2 Am. J. Applied Sci., 6 (3): , 9 times such that the total epected annual costs are minimized. Finally, a particular case is deduced, which has been previously available and a numerical illustrative eample is added. ISTRIBUTION FREE MOEL In a distribution free model, the lead time demand assumption is relaed to any distribution function by only assuming that the distribution function F belongs to the class Ω with finite mean µl and variance σ L. Again, the epected total annual cost of the distribution free model can be epressed as EAC (,,, L) OC HC SC CC. While the epected ordering and epected crashing costs are of the same forms as those of the normal demand model, the epected shortage at the end of a cycle can be epressed by B(r) E[X-r] and the corresponding epected total annual holding cost is: HC h{ σ L ( )E[X r] } The epected annual shortage cost is: SC [ ( ) ]E[X r] min ma EAC (,,, L) >,L> F Ω Gallego and Moon [3] proved that the following inequality holds for any F Ω: E(X r) σ L (r µ L) (r µ L) () Substituting r µl σ L into the model, the problem can be reduced to minimize: EAC (,,, L) A h L σ h σ for L < L L i L( ) a i(li L) a j(tj t i i (3) Taing the partial derivatives of EAC (,,, L) with respect to,, and L in each time interval (L i, L i ], we obtain: Therefore, the epected total annual cost for distribution free model can be presented by: EAC(,,, L) A h L E[X r] σ [ ]E[X r] i a i (L i L) a j (T j t j ) A h L L () σ σ Ψ [ ]E[X r] for L < L i a i (L i L) a j (T j t j ) i Li Since the form of the lead time demand distribution is not nown, the minima criterion is applied to find the least favorite distribution function in Ω for each (,,, L) and then find the optimal values of,, and L that minimize the epected total annual cost. In mathematical symbolization, the problem under investigation can be epressed as: 388 () EAC(,,,L) A h σ L a L L a T t i ( ) i i j j j EAC(,,, L) h σ L EAC(,,,L) h hσ L σ L( ) () (5) (6)

3 h L EAC(,,,L) σ L hσl ai Am. J. Applied Sci., 6 (3): , 9 (7) However, for fied, and, EAC(,,, L) is concave in L (L i, L i ], since: h EAC(,,,L) L σ L hσ L < (8) For fied L (L i, L i ], the determinant of Hessian matri of EAC(,,, L) is positive definite at (,, ) as proved in Appendi. Setting equation (5) to zero and solving for, it follows that: h (9) Setting Eq. to zero and substituting (9) into () to solve for, it follows that: / A σ L i a i(li L) a j(tj t h h σ L ( ) / () Solving for by setting Eq. 6 to zero and substituting (9) into (6), we have: h h h α h ( ) () 389 The following algorithm is developed to find the optimal values of the order quantity, bacorder price discount, safety factor and lead time for the problem under study. Step : For i,,,, n (i) Set io (ii) Substitute io into () to evaluate io (iii) Utilize io to determine in from (). Let io in (iv) Repeat (ii) and (iii) until no change occurs in the values of i and i. enote the resulting solutions by i and i (v) For i,,,, n, use equation (3) to compute the epected total annual cost EAC( i, i, i, L i ) Step : Set EAC(,,, L ) Min{EAC( i, i, i, L i ), i,,,, n}. Step 3:,,, L ) is a set of optimal solutions. NUMERICAL EXAMPLE Consider an inventory system with the following data: 6 units/year, A $ per order, h $ per unit per year, $5 per unit, 7 unit/wee and the lead time has three components with data shown in Table [6], ecept that the probability distribution of the lead time demand is unnown. Apply the proposed algorithm to solve the problem for the upper bound of the bacorder ratio.95,.8,.65,.5,.35 and.. The results are summarized in Table. Net, compare the performance of distribution-free approach against those of normal distribution. For eample, consider the results with.5. Let ( N, N, N, L i N ) (,77.57,.88,) denote the optimal solution set obtained in the normal distribution case [3] and EAC N ( N ) be the associated cost where N represents the normal distribution. On the other hand, let ( U, U, U, L i U ) (6, 77.3,.9, 3) denote the optimal solution set found in the general distribution case where U denote the distribution free case with associated cost EAC N ( U ) and EAC N ( U ) is the annual epected cost of using ( U, U, U, L i U ) when the real demand distribution is normal. Then the additional cost of EAC N ( U )-EAC N ( N ) EAC N (6, 77.3,.9, Table : Lead time data of the eamples Lead time component I 3 Normal duration T i (days) 6 Minimum duration t i (days) Unit fied crashing cost a i ($/day).. 5.

4 Am. J. Applied Sci., 6 (3): , 9 Table : Summary of the results for eample (L i in wees) The proposed model The Pan and Hsiao [6] model (.85) Savings (%)(() L i EAC( ) () L i EAC( ) () -())/() Table 3: Summary of computational results EAC U EAC N EAC N EAC N EAC N ( U ) ( U, U, U, L U i ) ( U ) ( U ) ( N ) /EAC N ( N ).95 (, 77.39,.3, 3) (, 77.,.38, 3) (5, 77.,.3, 3) (6, 77.3,.9, 3) (6, 77.,.5, 3) (7, 77.5,.6, 3) )-EAC N (, 77.57,.88, ) $ $8. is the largest amount that one would be willing to pay for the nowledge of the distributional form of lead time demand. This quantity can be regarded as the Epected Value of Additional Information (EVAI) [3]. The results for.95,.8,.65,.5,.35 and. in Table 3 reconfirm the robustness of distribution free approach that has been proven in recent related studies []. CONCLUSIONS This research studies distribution free model having both mean and variance nown and finite. We consider the impact of safety factor on the continuous review inventory model involving controllable lead time and bacorder price discount with miture of bacorder and partial lost sales. The objective is to minimize the epected total annual cost by simultaneously optimizing order quantity, bacorder price discount, safety factor and lead time. Algorithms are developed to find the optimal solutions for the model and eample are provided to illustrate the procedures of the algorithms. We may obtain one conclusion when the upper bound of the bacorder ratio increases, bacorder ratio and safety factor and total epected annual cost decreases. If the buyer permits seller bigger bacorder price discount, will be allowed to promote the service level and to reduce total epected annual cost. APPENIX The Hessian matri H of EAC(,,, L) for a given value of L is: 39 EAC(,,,L) EAC(,,,L) EAC(,,,L) EAC(,,,L) EAC(,,,L) EAC(,,,L) H EAC(,,,L) EAC(,,,L) EAC(,,,L) Where: EAC(,,, L) A 3 3 σ L 3 a L L a T t i ( ) i i j j j EAC(,,,L) σ L Ψ() h EAC(,,,L) 3 / σ σ L EAC(,,,L) EAC(,,,L) h σ L EAC(,,,L) EAC(,,,L) L( )) ()

5 and EAC(,,,L) EAC(,,,L) σ L( ) Net, we proceed to evaluate the principal minor of H at point (,, ). The first principal minor of H is: H A Am. J. Applied Sci., 6 (3): , σ L i 3 a i ( L i L ) a j ( T j t j ) > The second principal minor of H is: H σ L (3) i A a i(li L) a j(tj t () σ L > By substituting from (9) and (/( ) / ) from (), the third principal minor of H is: σ L i H33 A a i(li L) a j(tj t σ L( ) h L σ σ ( ) 3/ σ L L 39 h σ L h σ L ( ) σ L ( ) 8 3 / h σ L 3 σ L σ L ( ) 5 8 where h G( ) ( ) 3/ 3/ 3 G( ) ( ) ( ) (5) (6) For [, ) and <, G( ) is positive. Hence, we have H 33 >. Therefore, from (3) to (5), it is concluded that the Hessian matri H is positive define at point (,, ). NOTATION AN ASSUMPTIONS The following notation is used throughout the study. L r A Length of lead time (decision variable) Order quantity (decision variable) Bacorder price discount offered by the supplier per unit (decision variable) Safety factor (decision variable) Reorder point Gross marginal profit per unit Average demand per year Fied ordering cost per order

6 h Inventory holding cost per unit per year µ Average demand rate in units/day Bacorder ratio Upper bound of the bacorder ratio φ Standard normal distribution Standard normal cumulative distribution function SS Safety stoc B(r) Epected shortage of a cycle R(L) Total crashing cost of a cycle The assumptions made in the study are listed below: The reorder point r epected demand during lead time safety stoc, that is, r L σ L, where is safety factor The lead time L has n mutually independent components. The ith component has a normal duration T i and a minimum duration t i, i,,, n and a crashing cost per unit time a i. The a i s are arranged such that a a a n. The lead times are crashed one component at a time starting with the one of least c i and so on Let L i denote the length of lead time with component,,, i crashed to their minimum values, then L i can be epressed as L i Am. J. Applied Sci., 6 (3): , 9 n T j i (T j j t. Thus, the lead time crashing cost R(L) per replenishment cycle is given by R(L) a i i (L i L) a (T t ) j j j, for L (L i, L i ] [6] The bacorder ratio is variable and is in proportion to the bacorder price discount offered by the supplier per unit, thus, /, for [6] <, REFERENCES. Blacburn, J.., 99. Time-Based Competition: The Net Battleground in American Manufacturing. Richard Irwin, Inc., Homewood, Illinois.. Ben-aya, M. and A. Raouf, 99. Inventory models involving lead time as a decision variable. J. Operat. Res. Soc., 5: Gallego, G. and I. Moon, 993. The distribution free newsboy problem, review and etensions. J. Operat. Res. Soc., : Hariga, M.A. and M. Ben-aya, 999. Some stochastic inventory models with deterministic variable lead time. Eur. J. Operat. Res., 3: Liao, C.J. and C.H. Shyu, 99. Analytical determination of lead time with normal demand. Int. J. Operat. Res. Prod. Manage., : Liao, C.J., 99. Optimal control of jobs for production systems. Comput. Ind. Engine, : Moon, I. and S. Choi, 995. The distribution free newsboy problem with baling. J. Operat. Res. Soc., 6: Moon, I. and S. Choi, 997. istribution free procedures for Mae-To-Order (MTO), Mae-In- Advance (MIA) and composite policies. Int. J. Prod. Econ., 8: Moon, I. and W. Yun, 997. The distribution free job control problem. Comput. Ind. Eng., 3: Moon, I. and S. Choi, 998. A note on lead time and distribution assumptions in continuous review inventory models. Comput. Operat. Res., 5: 7-.. Moon, I. and E. Silver,. The multi-item newsvendor problem with a budget constraint and fied ordering costs. J. Operat. Res. Soc., 5: Naddor, N., 966. Inventory System. John Wiley, New Yor. 3. Lo, M.C., J.C. H. Pan, K.C. Lin and J.W. Hsu, 8. Impact of lead time and safety factor in mied inventory models with bacorder discounts, J. Applied Sci., 8: Ouyang, L.Y., N.C. Yen and K.S. Wu, 996. Miture inventory model with bacorders and lost sales for variable lead time. J. Operat. Res. Soc., 7: Ouyang, L.Y. and K.S. Wu, 998. A minima distribution free procedure for mied inventory model with variable lead time. Int. J. Prod. Econ., 56: Pan, J.C. and Y.C. Hsiao,. Inventory models with bacorder discounts and variable lead time. Int. J. Syst. Sci., 3: Ravindran, A.,.T. Phillips and J.J. Solberg, 987. Operations Research, Principle and Practices. John Wiley, New Yor. 8. Silver, E.A. and R. Peterson, 985, ecision System for Inventory Management and Production Planning. John Wiley, New Yor. 9. Silver, E., I. Moon and E. Silver,. The multiitem single period problem with an initial stoc of convertible units. Eur. J. Operat. Res., 3: Tersine R.J., 99. Principles of inventory and material management. North Holland, New Yor.. Vollmann, T.E., W.L. Berry and.c. Whybar, 99. Manufacturing Planning and Control System. 3rd Edn. Irwin, Chicago.