Reorder Quantities for (Q, R) Inventory Models

Transcription

1 International Mathematical Forum, Vol. 1, 017, no. 11, HIKARI Ltd, Reorder uantities for (, R) Inventory Models Ibrahim Isa Adamu Department of Mathematics Moddibo Adama University of Technology, Yola P.M.B 076, Yola Adamawa State, Nigeria Copyright 017 Ibrahim Isa Adamu. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The reorder level R and the reorder quantity are the parameters to be decided in a continuous review inventory policy. Their optimal values can be approached through iterative methods, but these are tedious and inconvenient for control routines. A frequent practice is to set as the economic reorder quantity and compute R accordingly. Yet this practice may introduce a substantial cost penalty. Existing literature demonstrated attempts to rearrange conventional theoretical expressions to facilitate the use of numerical approximations to help find an optimal solution in a continuous review inventory policy, however, the cost function derived and used in the rearrangement were faulty. This paper re-visits the formulation of the cost function by stating explicitly necessary assumptions, and obtained the correct cost function. Expressions for the optimising parameters R & were re-obtained based on the correct cost function. The new optimal expressions for R & and the cost function were implemented. The result of this work showed that, the reviewed model gives lower inventory cost with higher optimal order quantity than the model under review, and hence the reviewed model performs better when compared to the model under review. Mathematics Subject Classification: Primary 9D40, 9D5; Secondary: 34D0 Keywords: Inventory, Continuous review, stochastic, Re-order quantity, Re-order level, Wilson s EO, backlogging, Demand, probabilistic

2 506 Ibrahim Isa Adamu Introduction Controlling stock (inventory) is a process and a method of total stock management. A lot of research efforts have been devoted to stock (inventory) control over the past few decades. Previous researches assume that the lead time is an incontrollable variable, unmet demand is always backordered, and so on. However, in reality these assumptions may not be tenable; this leads to the need to reconsider the inventory control problem. Some studies relevant to continuous review inventory models with stochastic demand were carried out by [9], [10], [8] and [3]. Moreover, many companies have recognized the importance of response time as a competitive weapon; and have used it as a means of differentiating themselves in the marketplace. In some previous studies, lead time is viewed as a prescribed constant or a random variable, which therefore is not subject to control. In many practical situations, lead time can be reduced at an added crashing cost; in other words, [1], [11]. In stochastic inventory control models under continuous review technique, the reorder level R and the order quantity are the parameters to be determined optimally, their optimal values have to be obtained iteratively in most cases, but this is tedious and inconvenient for control routines. In some cases, a practice of setting as the economic re-order quantity and computing R accordingly is employed, this practice may have a substantial cost penalty because was not obtained through optimal procedures, the problem is explicit expressions for and R cannot be obtained, [1]. [1] studied (, R) inventory model and proposed a non-iterative method of determining the optimal parameters of the model, their model uses numerical approximation rather than iteration. They obtained their model s cost function as: C,R = (S+sz) D + hc ( + R d), at optimal points, they obtained: = D(S+sz) and P hc r = hc sd Where: Reorder quantity R Reorder level P r = f(x)dx Probability of stock out R h Holding Cost D Annual average demand Z = (x R)f(x)dx Expected shortage per cycle R S Cost per order s Shortage cost per physical unit short P p = Z Expected shortage rate x Lead-time demand f(x) Density distribution function d Lead time mean demand σ Lead-time standard deviation

3 Reorder quantities for (, R) inventory models 507 g = ( d σ ) The modulus of the gamma distribution w = SD hc Economic order quantity (Wilson model) After mathematical rearrangement, [1] obtained a standardised expression for as; d = Z + ( Z ) + ( w dp r dp r d ) (1) This is the expression used in their numerical approximations. [5] attempted to apply the fuzzy set concepts to deal with uncertain backorders and lost sales. The purpose of their paper was to modify Moon and Choi S (1998) s continuous review inventory model with variable lead time and partial backorders by fuzzifying the backorder rate (or equivalently, fuzzifying the lost sales rate). They first considered the case where the lost sales rate is treated as the triangular fuzzy number. Then, through the statistical method for establishing the interval estimation of the lost sales rate, they constructed a new fuzzy number, namely statistic-fuzzy number. For each fuzzy case, they investigated a computing schema for the modified continuous review inventory model and develop an algorithm to find the optimal inventory strategy. [] in his work developed a mathematical model for continuous-review reorder quantity (r, ) inventory system. In his paper, he took into account the continuous-review reorder point-lot size under stochastic demand, with the backorders-lost sales mixture. To minimize the expected total cost per time unit, they then propose an exact algorithm and a heuristic procedure. The heuristic method exploits an approximated expression of the total cost function achieved by means of an ad hoc first-order Taylor polynomial. They carried out numerical experiments with a twofold objective. On the one hand they examine the efficiency of the approximated solution procedure, while on the other hand they investigated the performance of the maximum entropy principle in approximating the true lead-time demand distribution. [6] discussed the effects of ramp-type of demand on an EO model for noninstantaneous Weibull deteriorating items under time-dependent partial backlogging. They considered a retailer who purchases and sells a single product over an infinite time horizon who faces a ramp-type of demand. They allowed shortages with partially backlogging with time to the deterioration of the item distributed as Weibull. They considered the Total cost and the optimum number of replenishments as decision variables. They obtained the optimal replenishment policies that minimise the total cost and demonstrated the performance of the model with a numerical samples are explained in the following sections. [7] considered the management of safety stock in a coordinated single-vendor single-buyer supply chain under continuous review and Gaussian lead-time demand. They assumed the lead time to be controllable, and shortages not allowed. They followed value criterion by considering both inflation and time value of money. The aim was to present a novel approach to optimizing the safety

4 508 Ibrahim Isa Adamu stock in such system In their paper, they took a different perspective by putting the order quantity and the safety factor in functional dependence through the adoption of a specific parameter. More precisely, they expressed the service level as a function of the number of admissible stockouts per time unit and the order quantity. This allows optimizing the safety stock taking into account the constraint on the number of admissible stockouts per time unit. They present both exact and approximated minimization algorithms. Numerical examples were finally shown to illustrate the effectiveness of the approximation algorithm, and to investigate the sensitivity of the model with respect to variations in some fundamental parameters. In this paper, we revisit the paper by [1] in the face of the reviewed literatures above to correct the cost components to obtain the correct formulations. [1] gave their Model s cost function as; where; SD szd (S + sz) C,R = D + hc ( + R d) Total ordering cost component Total shortage cost component ch (( ) + r d) Total holding cost component The third (Total holding cost) component needs to be reformulated for the following reasons. i. Multiplying h(( ) + R d) by c is not correct, this is because c is a unit item cost and we don t use unit item cost of stock ordered in determining inventory holding cost, rather we use the unit item holding cost like storage cost, heating cost, cooling cost e.t.c. incurred in holding the inventory before depletion. ii. Using in the holding cost expression has no significance other than just to make it possible to use Wilson s EO model to approximate and simplify computations, you can use only when demand rate is known to be constant, i.e. stock is withdrawn from the inventory at a constant rate, in this case the average number of unit in inventory over time T is. New Model Formulation Assumptions 1) Shortage is allowed with backlogging ) Demand is probabilistic 3) Minimization of the average cost is the objective 4) Planning horizon is infinite 5) Replenishment is instantaneous 6) Cost of replenishment is independent of the quantity ordered 7) Lead time is finite

5 Reorder quantities for (, R) inventory models 509 Notations We maintain the same notations used by [1] as follows Reorder quantity R Reorder level P r = f(x) dx Probability of stock out R h Holding Cost c Item unit Cost D Annual average demand Z = (x R)f(x)dx Expected shortage per cycle R S Cost per order s Shortage cost per physical unit short P p = Z Expected shortage rate X Lead-time demand f(x) Density distribution function d Lead time mean demand σ Lead-time standard deviation g = (d σ) The modulus of the gamma distribution model) w = SD hc Economic order quantity (Wilson Derivation of the new model In deriving the mathematical model, we approach the problem by obtaining the individual cost components as follows; SD szd Total ordering cost component Total shortage cost component h( + r d) Total holding cost component Therefore the new cost function is the sum of the above cost components as follows; C,R = (S+sz) D + h( + R d) () The optimal (in this case minimum) value for C,R is obtained by setting to zero (0) its & R derivatives. Thus obtaining the new expressions for and P r as follows; = D(S+sz) h and P r = h sd (3) (4)

6 510 Ibrahim Isa Adamu Standard formulation Equation (3) & (4) can be rearranged to standardise computing procedure. From (3) we have = SD h + szd h = DSc hc + szd = ( w ) c + szd h. h Here w is the Wilson s EO = ( w ) c Therefore z ( w ) c = 0 P r + z : Since Ds h = /P P r from equation (3) r By completing squares on and solving for, we have = Z P r + ( z P r ) + ( w ) c (5) Dividing through by d to get the new standardised expression for, we have d = Z + ( z ) + ( w dp r dp r d ) c (6) Equation (6) is the equivalence of equation () in the model by [1]. Now, we apply the new model to the same Gamma distributed demands as follows; If lead time demand follows a ve exponential distribution, i.e. d = σ, then the modulus of the gamma distribution g = ( d σ ) = 1, this implies that z P r = d. Hence we have d = ( w d ) c (7) which is the equivalence of d = ( w d ) In [1] this expression gives the optimum directly, whatever the value of Pr. As in BC Vasconcelos and MP Marques (000), the ratio z/dp r for other distributions other than ve exponential, strongly depend on g, depends only slightly on P r within the usual range adopted in practice, say 5% < P r < 35%. As in [1], the values for z/dp r were computed through numerical approximations borrowed from [4]. After obtaining, the subsequent step is the computation of the reorder level R, we are going to adopt the same procedure as in [1] for determining.

7 Reorder quantities for (, R) inventory models 511 Hypothetical Example: We are going to use the same example as in [1] to enable us asses the performance of the model. Given a continuous review inventory system and let c = $15, S = $30, h = 0% s = $1, D = 1000 units p. a., d = 100, σ = 10. We are to compute and R using the new model Solution The computed values of w, g, z, dp r z dp r, are as follows w = 158, g = 0.7, z dp r = 1.30, z dp r = 0.65 Therefore, d = ( ) + (0.65) = = ( ) 100 = 503 Now computing P r from P r = h/sd, we have P r = = 0.08 Now, adapting Johnston approximations as follows: We assume gamma distribution, 0.5 < g < 1, P r < 0.5 R σ = A 3 + A 4 ln(p r ) + A 5 (P r ) + A 6 P r ln(p r ) where A 3 = g [ln(g)] g ln(g) A 4 = g ln(g) [ln(g)] g ln(g) A 5 = g g ln(g) A 6 = g g ln(g) g ln(g)

8 51 Ibrahim Isa Adamu R = [ (0.49) σ ln(0.7) {ln(0.7)} (0.7) ln(0.7)] + [ ln(0.7) {ln(0.7)} (0.7) ln(0.7)] ln(0.08) + [ ln(0.7)] (0.0064) + [ (0.7) ln(0.7) (0.7) ln(0.7)] 0.08 ln(0.08) = =.3358 Z Therefore R = = , = Z = 10.4 d C 503,80 = ( ) = $44.3 Discussion 1) The Cost functions, Reorder quantity and Probability of stock out This paper revisited the inventory problem by [1] and obtained the correct expressions for C,R, and P r, respectively as follows; C,R = (S+sz) D + hc( + R d), = D(S+sz) h and P r = h sd ) The standard formulated expressions This paper re-obtained the standardized expression, by [1], for based on the correct cost function as: Z + ( Z ) + ( w dp r dp r d ) d = Z + ( Z ) + ( w dp r dp r d ) c instead of d = 3) Calculating w and hence that, R and P r In this paper, the value of w was recomputed to obtained the correct value of w as w = 158 units instead of w = 141 as in [1]. The values of P r and R were also recomputed based on the new value w = 158 units to obtain P r = 0.08 and R 80. 4) The performance of the models i) The new model gives higher optimal order quantity of 508 units against 30 units from [1] model at the same reorder level. ii) The optimal cost of the inventory with the new model is $44.3 as against $1,56 using the same example by [1].

9 Reorder quantities for (, R) inventory models 513 iii) Clearly the new model gives lower inventory cost with higher optimal order quantity than the model by [1], which gives an inventory cost of $1,56 with an optimal order quantity of 30 units with the same example. Conclusion From the above discussion, we see that the result of this work demonstrated that, the reviewed model gives lower inventory cost with higher optimal order quantity than the model under review, and hence the reviewed model performs better when compared to the model under review. Acknowledgements. I wish to acknowledge Mr. Aliyu Mohammed of MAUTECH Yola, Adamawa state Nigeria for typesetting the document References [1] B.C. Vasconcelos and M.P. Marques, Reorder quantities for (, R) inventory models, Journal of the Operational Research Society, 51 (000), [] Davide Castellano, Stochastic Reorder Point-Lot Size (r,) Inventory Model under Maximum Entropy Principle, Entropy, 18 (016), no. 1, [3] J.K. Jha, K. Shanker, Two-echelon supply chain inventory model with controllable lead time and service level constraint, Computers & Industrial Engineering, 57 (009), [4] F.R. Johnston, An interactive stock control system with a strategic management role, Journal of the Operational Research Society, 31 (1980), [5] Liang-Yuh Ouyang, Hung-Chi Chang, The Variable Lead Time Stochastic Inventory Model with a Fuzzy Backorder Rate, Journal of the Operations Research Society of Japan, 44 (001), no. 1, [6] M. Valliathal, R. Uthayakumar, Optimal replenishment policies of an EO model for non-instantaneous Weibull deteriorating items with ramp-type of demand under shortages, Int. J. of Mathematics in Operational Research, 8 (016), no. 1, [7] Marcello Braglia, Davide Castellano and Marco Frosolini, A novel approach to safety stock management in a coordinated supply chain with control-

10 514 Ibrahim Isa Adamu lable lead time using present value, Applied Stochastic Models Bus. Ind., 3 (016), [8] I. Moon, S. Choi, A note on lead time and distributional assumptions in continuous review inventory models, Computers & Operations Research, 5 (1998), no. 11, [9] L.Y. Ouyang,N.C. Yeh and K.S. Wu, Mixture inventory model with backorders and lost sales for variable lead time, Journal of the Operational Research Society, 47 (1996), no. 6, [10] L.Y. Ouyang, B.R. Chuang, A minimax distribution-free procedure for stochastic inventory models with a random backorder rate, Journal of the Operations Research Society of Japan, 4 (1999), no. 3, [11] A. Ravindran, D.T. Phillips and J.J. Solberg, Operations Research: Principles and Practice, Wiley, New York, [1] F. Ye, X.J. Xu, Cost allocation model for optimizing supply chain inventory with controllable lead time, Computers & Industrial Engineering, 59 (010), Received: November 14, 016; Published: March 9, 017