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Yugoslav Journal of Operations Research 21 (2011), Number 2, 293-306 DOI: 10.2298/YJOR1102293H CONTINUOUS REVIEW INVENTORY MODELS UNDER TIME VALUE OF MONEY AND CRASHABLE LEAD TIME CONSIDERATION Kuo-Chen HUNG Department of Logistics Management, National Defense University, Taiwan, R.O.C. Kuochen.hung@msa.hinet.net Received: April 2008 / Accepted: October 2011 Abstract: A stock is an asset if it can react to economic and seasonal influences in the management of the current assets. The financial manager must calculate the input of funds to the stock intelligently and the amount of money cycled through stocks, taking into account the time factors in the future. The purpose of this paper is to propose an inventory model considering issues of crash cost and current value. The sensitivity analysis of each parameter, in this research, differs from the traditional approach. We utilize a course of deduction with sound mathematics to develop several lemmas and one theorem to estimate optimal solutions. This study first tries to find the optimal order quantity at all lengths of lead time with components crashed at their minimum duration. Second, a simple method to locate the optimal solution unlike traditional sensitivity analysis is developed. Finally, some numerical examples are given to illustrate all lemmas and the theorem in the solution algorithm. Keywords: Inventory model, crashable lead time, time value of money. MSC: 90B05. 1. INTRODUCTION From the perspective of financial management, stocks often comprise a very large proportion of a balance sheet. Funds invested in stock cannot be used elsewhere because they are not liquid assets. They become liquid only when the stocks are sold. Considering capital running factors, stocks must be turned over fast, so enterprises must 294 K.C. Hung / Continious Review Inventory Models determine appropriate inventory policies in order to reduce idleness of the stocks, and dead and scrap stocks in order to sell and produce effectively. Studying inventory models and considering time and value, Moon and Yun [13] employed the discounted cash flow approach to fully recognize the time value of money and constructed a finite planning horizon EOQ model in which the planning horizon is a random variable. Jaggi and Aggarwal [8], in order to discuss an optimal replenishment policy with an infinite planning horizon, reported that a deteriorating product under the impact of a credit period did not allow shortages. Bose et al. [2] and Hariga [6] developed two inventory models, which incorporated the effects of inflation and time value of money with a constant rate of deterioration and time proportional demand. Moon and Lee [12] investigated the effect of inflation and time-value of money in an inventory model with a random product life cycle. Wee and Law [20] employed the concepts of inflation and the time value of money in a model where demand is pricedependent and shortages allowed. Chung and Tsai [3] derived an inventory model for deteriorating items with the demand of linear trends and shortages during the finite planning horizon, considering the time value of money. Sun and Queyranne [19] investigated general multi-product, multi-stage production and an inventory model using the net time value of money with its total cost as the objective function. Balkhi [1] considered a production lot size inventory model with deteriorated and imperfect products, taking into account inflation and the time value of money. Moon et al. [9] developed inventory models for ameliorating and deteriorating items with a time-variant demand pattern over a finite planning horizon, taking into account the effects of inflation and the time value of money. Shah [17] derived an inventory model by assuming a constant rate of deterioration of units in an inventory and the time value of money under the conditions of permissible delay in payments. Wee et al. [21] developed an optimal replenishment inventory strategy to consider both ameliorating and deteriorating effects, taking into account the time value of money and a finite planning horizon. Both the amelioration and deterioration rate were assumed to follow Weibull distribution. Dey et al. [5] considered an inventory model for a deteriorating item with time dependent demand and interval-valued lead-time over a finite time horizon. The inflation rate and time value of money are taken into account. In addition, Ji [9] constructed a general framework of an inventory system for non-instantaneous deteriorating items with shortages, the time value of money, and inflation. Das et al. [4] developed a two-warehouse production-inventory model for deteriorating items considered under inflation and the time value of money over a random time horizon. Hou et al. [7] presented an inventory model for deteriorating items with a stock-dependent selling rate under inflation and the time value of money over a finite planning horizon. However, Kumar Maiti [10] also has developed an inventory model incorporating customers’ credit-period dependent dynamic demand, inflation, and the time value of money, where the lifetime of the product is imprecise in nature. In the recent studies, decomposing the lead time into several crashing periods is a controllable approach to lead time reduction. Ouyang et al. [15] constructed a variable lead time from a mixed inventory model with backorders and lost sales. In this article, we extend the inventory model of Ouyang et al. [15]. When the distribution of lead time demand is normal, we consider the time value of a continuous review inventory model with a mixture of backorders and lost sales. 295 K.C. Hung / Continious Review Inventory Models This paper is organized as follows. In the next section, we define the notation of the inventory model and its assumptions. In section 3, first we construct the inventory model, taking into account the time value. Then we prove that the total expected annual cost is piece-wisely concave down with respect to lead time, and convex in order quantities. We apply a simple method to develop four lemmas and one theorem, and locate the optimal solution for constructing the procedure of solving a replenishment policy in section 4. This approach differs from the traditional methods. In section 5, numerical examples are offered to illustrate our algorithm. Section 6 summarizes the article and presents some conclusions. 2. NOTATION AND ASSUNPTIONS We use the following notation and assumptions to develop inventory models with crashing component lead time and the time value of money. A : Fixed ordering cost per order. D : Average demand per year. h : Inventory holding cost per item per year. L : Lead time that has n mutually independent components. The i th component has a minimum duration a i and normal duration bi with a crashing cost ci per unit time under the assumption c1 ≤ c 2 ≤ " ≤ c n . The components of L are crashed one at a time, starting from the component of the least ci and so forth. Hence, the range for L n is from ∑a j =1 n j to ∑b j =1 j . L j : The length of lead time with components 1, 2,…, j are crashed to their n n n i =1 i =1 t = j +1 minimum durations. We define L0 = ∑ bi and Ln = ∑ ai and L j = Ln + ∑ bt − at , for j = j ,..., n − 1 . Since b j > a j , it follows that L j −1 > L j , for j = 1,..., n . R ( L) : The lead time crashing cost per cycle for a given L ∈ [ Li , Li −1 ] is given by i −1 R ( L) = ci ( Li −1 − L) + ∑ ct (bt − at ) . t =1 Q : Order quantity. Q j : The optimal order quantity when lead time is L j . X : Lead time demand that follows a normal distribution with mean μ L and standard derivation σ L . r : Reorder point. Since r = expected demand during lead time + safety stock, r = μ L + kσ L . Inventory is continuously reviewed. Replenishments are made whenever the inventory level falls to the reorder point r . q : Allowable stockout probability during L . 296 K.C. Hung / Continious Review Inventory Models k : Safety factor that satisfies P( X > r ) = P( Z > k ) = q , Z representing the standard normal random variable. B (r ) : Expected shortage at the end of the cycle. We quote the results of Ouyang et al. [15], B (r ) = σ LΨ (k ) where Ψ (k ) = ϕ (k ) − k [1 − Φ (k ) ] as φ, where Φ denotes the standard normal probability density function and cumulative distribution. β : The fraction of the demand during the stockout period that will be backordered. π : Fixed penalty cost per unit short. π 0 : Marginal profit per unit. θ : The interest rate per year. 3. MATHEMATICAL FORMULATION First, we study the total expected annual cost of the inventory model with backorders and lost sales for variable lead time. We quote the Equation (2) of Ouyang et al. [15], for L ∈ [ Ln , L0 ] , who derived the total expected annual cost, EAC (Q, L) , without considering the time value of money as follows: EAC (Q, L) = EAC j (Q, L) (1) for L ∈ ⎡⎣ L j , L j −1 ⎤⎦ , with j = 1, 2,..., n . We rewrite the total expected annual cost as EAC j (Q, L) = h D D Q + R j ( L) + p ( L ) + Ω( L ) 2 Q Q (2) Where Ω( L) = hσ [ k + (1 − β )Ψ (k ) ] L , p( L) = σ [π + (1 − β )π 0 ] Ψ (k ) L + A and j −1 R j ( L) = c j ( L j −1 − L) + ∑ ct (bt − at ) t =1 for L ∈ ⎡⎣ L j , L j −1 ⎤⎦ . Secondly, we consider the inventory model, taking into account the time value. The expected net inventory level, just before the order arrives, is kσ L + (1 − β ) B (r ) , K.C. Hung / Continious Review Inventory Models 297 and the expected net inventory at the beginning of the cycle is Q + kσ L + (1 − β ) B (r ) . Therefore, the expected average inventory level is Q + kσ L + (1 − β ) B(r ) − Dt for ⎡ Q⎤ t ∈ ⎢ 0, ⎥ . Hence, the inventory carrying cost for the first cycle equals ⎣ D⎦ Q D ∫ ⎡⎣Q + kσ t =0 L + (1 − β ) B (r ) − Dt ⎤⎦he −θ t dt (3) Q Q −θ ⎛ ⎞ h D ⎛ −θ Q⎞ = ⎡⎣ kσ L + (1 − β ) B(r ) ⎤⎦ ⎜⎜ 1 − e D ⎟⎟ + h 2 ⎜⎜ e D − 1 + θ ⎟⎟ θ D⎠ θ ⎝ ⎝ ⎠ We adopt the discounted cash flow approach following Moon and Yun [14]. At the beginning of each cycle will be cash outflows for the ordering cost, stockout cost and lead time crashing cost. Therefore, the total relevant cost for the first cycle is A + (π + π 0 (1 − β ))σ Ψ (k ) L + R( L) + + Q −θ ⎛ ⎞ h⎡ kσ L + (1 − β ) B(r ) ⎤⎦ ⎜⎜ 1 − e D ⎟⎟ ⎣ θ ⎝ ⎠ Q Dh ⎛ −θ D Q⎞ − 1 + θ ⎟⎟ . e ⎜ 2 ⎜ D⎠ θ ⎝ Referring to Silver and Peterson [18], we get that the time value of money of the expected total relevant cost over an infinite time horizon, C (Q, L) , is given by 1 1− e −θ Q D ⎢ A + (π + π 0 (1 − β ))σψ (k ) L + R ( L) ⎥ ⎣ ⎦ 1 + 1− e −θ Q D Q Q −θ Dh −θ Q ⎪⎫ ⎪⎧ h ⎨ [ r − μ L + (1 − β ) B (r ) ] (1 − e D ) + 2 (e D − 1 + θ ) ⎬ D ⎪⎭ θ ⎪⎩θ We can rewrite C ( Q, L ) as follows: C (Q, L) = f ( L) 1− e Q −θ D + g ( L) + h θ Qθ 2 1− e −θ (4) Q D for 0 < Q < ∞ and 0 ≤ L < ∞ , where f ( L) = p ( L) + R ( L), p( L) = σ [π + (1 − β )π 0 ] Ψ (k ) L + A, for L ∈ ⎡⎣ L j , L j −1 ⎤⎦ , j = 1,..., n, R ( L) = R j ( L), j −1 R j ( L) = c j ( L j −1 − L) + ∑ ct (bt − аt ), g ( L) = t =1 Ω( L) θ − Dh θ2 298 K.C. Hung / Continious Review Inventory Models and Ω( L) = hσ [ k + (1 − β )Ψ (k ) ] L . Third, we use R ( L) to denote the crashing cost. We have that R ( L) = R j ( L) where j −1 R j ( L) = c j ( L j −1 − L) + ∑ ct (bt − at ) for L ∈ ⎡⎣ L j , L j −1 ⎤⎦ , t =1 with j = 1, 2,..., n . Since R j ( L) is a linear decreasing function on ⎡⎣ L j , L j −1 ⎤⎦ , we get L j −1 − L j = b j − a j j −1 j t =1 t =1 and R j ( L j ) = c j ( L j −1 − L j ) + ∑ ct (bt − at ) = ∑ ct (bt − at ) = R j +1 ( L j ) , it follows that R ( L) is a piece-wise linear decreasing and continuous function on [ Ln , L0 ] . At the points { L j : j = 1, 2,..., n − 1} , R ( L) has different slopes c j and c j +1 of the tangent line from the right and left, respectively. Hence, R ( L) is not differentiable at those points, so we must divide the domain of L from [ Ln , L0 ] into subintervals ⎡⎣ L j , L j −1 ⎤⎦ , with j = 1, 2,..., n . According to Rachamadugu [16], in order to compare our results with the previous model of Ouyang et al. [15], we use A(Q, L) = θ C (Q, L) , an alternate but equivalent measure. A(Q, L) represents the equivalent uniform cash flow stream that ⎛ generates the same C (Q, L ) . From lim θ ⎜ − Dh + h 2 2 θ →0 ⎜ θ ⎝ θ Qθ 1− e Q −θ D ⎞ ⎟ = h Q , we have ⎟ 2 ⎠ h D D Q + Ri ( L) + p ( L) + Ω( L) . 2 Q Q That is equation (2) for the total expected annual cost of Ouyang et al. [15]. Hence, we extend their model. Now, we begin to find the minimum value of the total expected annual cost C (Q, L) for 0 < Q < ∞ and 0 ≤ L < ∞ . Taking the first and second partial derivatives of C (Q, L) with respect to L gives lim A(Q, L) = θ →0 299 K.C. Hung / Continious Review Inventory Models ∂ C (Q, L) = ∂L hσ 2θ L σ Ψ (k ) ⎡ ⎤ − cj ⎥ + ⎢ (π + π 0 (1 − β )) 2 L ⎣ ⎦ 1 1− e −θ Q D (5) [ k + (1 − β )Ψ (k )] and ∂ 2 C (Q, L) −σ (π + π 0 (1 − β ))ψ (k ) = + − hσ [k + (1 − β )ψ (k )] . Q −θ ∂ L2 ⎞ 3 ⎛ 4θ L3 4 L ⎜1 − e D ⎟ ⎝ From (6) ⎠ ∂ 2 C (Q, L) < 0 , C (Q, L) is concave in L ∈ ⎡⎣ L j , L j −1 ⎤⎦ . Hence, we can ∂ L2 n n ⎧ ⎫ reduce the minimum problem from ⎨C (Q, L) : ∑ ai ≤ L ≤ ∑ bi , 0 ≤ L < ∞ ⎬ to the i =1 i =1 ⎩ ⎭ boundary of each piece-wise defined domain as {C (Q, L) : L = L j , for j = 0,1,", n, 0 < Q < ∞} . Fixing L = L j , with j = 0,1,..., n, taking the first and second partial derivative of C (Q, L j ) with respect to Q , gives f (Lj ) ∂ C (Q, L j ) −θ e = Q 2 D ⎛ ∂Q −θ ⎞ D ⎜⎜ 1 − e ⎝ ⎟⎟ ⎠ Q −θ D + h θ Q D Q Q −θ D e D Q 2 −θ ⎛ ⎞ D e 1 − ⎜⎜ ⎟⎟ ⎝ ⎠ 1− e −θ −θ and Q −θ ⎛ ⎞ −θ Q D e D 1 e + ⎜ ⎟⎟ Q ∂ 2 C (Q, L j ) h −θ D θ 2 ⎜⎝ ⎠ ( ) f L e = + j Q 3 D D2 ⎛ ∂ Q2 −θ ⎞ ⎜⎜1 − e D ⎟⎟ ⎝ ⎠ Q Q ⎞ −θ D Q ⎛ − 2 +θ ⎜ 2 +θ ⎟e D⎠ D ⎝ . Q 3 −θ ⎛ ⎞ ⎜⎜ 1 − e D ⎟⎟ ⎝ ⎠ 2− x , for x > 0 . Hence, we know that 2+ x the second term of the second partial derivative is positive, so C (Q, L j ) is convex in Rachamadugu [10] derived that e− x > Q ∈ (0, ∞) with the minimum point at Q j such that θ eD Qj −1− θ D Qj = θ2 Dh f (Lj ) (7) 300 K.C. Hung / Continious Review Inventory Models θ Q θ Q for 0 ≤ Q < ∞ . We know that φ (Q) is a strictly D increasing function from φ (0) = 0 to lim φ (Q ) = ∞ . Therefore, given an L j , there exists Let φ (Q) = e D − 1 − Q →∞ θ a unique point Q j satisfying e D Qj −1 − θ D Qj = θ2 Dh f (L j ) . We have shown that C (Q, L) is concave down in L ∈ ⎡⎣ L j , L j −1 ⎤⎦ . In addition, for L = L j , with j = 0,1,..., n, C (Q j , L j ) is concave up in Q . So the minimum problem is to consider the points (Q j , L j ) for j = 0,1,..., n .We construct an algorithm as follows. (i) Find the local minimum points (Q j , L j ) for j = 0,1,..., n along the boundaries of each subinterval. (ii) For each point (Q j , L j ) , evaluate the total expected annual cost C (Q j , L j ) for j = 0,1,..., n . (iii) Solve the minimum of {C (Q j , L j ) : j = 0,1," , n} . 4. MONOTONIC PROPERTY AND PROPOSITIONS We determine a criterion to reduce the computation of finding the local minimum for the inventory model. In addition, we construct a new function as the difference of the total expected annual cost function evaluated at two adjacent local minimum points. Then we verify if it is an increasing function of the fraction of backorders. Therefore, we can reduce the calculation for locating the optimal solution. Our purpose in this section is to develop a procedure that eliminates the need to compute the exact values of {Q j : j = 0," , n} and {C (Q j , L j ) : j = 0," , n} . We establish a criterion to compare Q j and Q j −1 implicitly. Moreover, we change the value of β to investigate the sensitive analysis of backordered ratio per cycle. Our new method significantly reduces the amount of computation. First, we offer such a criterion that we can implicitly compare Q j with Q j −1 . All the proofs for the Lemmas and the theorem are in the Appendix. Lemma 1: Given a backordered fraction ratio β , then Q j < Q j −1 ⇔ ci ( Li −1 + Li ) < σ [π + (1 − β )π 0 ] Ψ (k ) . Secondly, we state the monotone property between C (Qi , Li ) and Q j . 301 K.C. Hung / Continious Review Inventory Models Lemma 2: For a given β , if Q j < Q j −1 , then C (Qi , Li ) < C (Qi −1 , Li −1 ) . From the Table 2 of Ouyang et al. [15], if Q j > Q j −1 , we know that there is no regulation between C (Q j , Li ) and C (Q j −1 , Li −1 ) . However, if we treat Q j ( β ) and Q j −1 ( β ) as functions of β , then we can still measure the difference between C (Q j ( β ), Li ) and C (Q j −1 ( β ), Li −1 ) . Lemma 3: For a given interval β ∈ [ β 0 , β1 ] , if Q j ( β ) ≥ Q j −1 ( β ) , then for β ∈ [ β 0 , β1 ] , C (Q j ( β ), Li ) − C (Qi −1 ( β ), Li −1 ) is an increasing function of β . Here, we show the monotone property of Q j ( β ) ≥ Q j −1 ( β ) with respect to β . Lemma 4: Given a fixed interval β ∈ [ β 0 ,1] . β0 , if Q j ( β 0 ) ≥ Q j −1 ( β 0 ) , then Q j ( β ) ≥ Q j −1 ( β ) for the Finally, we derive a criterion to compare C (Qi −1 ( β ), Li −1 ) with C (Qi ( β ), Li ) . Theorem 1: If C (Qi ( β 0 ), Li ) > C (Qi −1 ( β 0 ), Li −1 ) and Qi ( β 0 ) ≥ Qi −1 ( β 0 ) for a fixed β 0 , then C (Qi ( β ), Li ) > C (Qi −1 ( β ), Li −1 ) for the interval β ∈ [ β 0 ,1] . 5. NUMERICAL EXAMPLES The following numerical examples explain how the above Lemmas and the theorem simplify the solution procedure. Using the numerical example from Ouyang et al. [15], we have the following data: D = 600 units/year, k = 0.845 , A = $200 /per order, h = $20 /per item per year, π = $50 /per unit short, π 0 = $150 /per unit, σ = 7 units/per week, q = 0.2 (in this situation, from the normal distribution, we find k = 0.845 and ψ (k ) = 0.110 ), and the lead time has three components with data shown in Table 1. We assume that the interest rate θ = 0.1 . Following the solution algorithm, we obtain Table 2. When β = 1 in Table 2, we slightly change the decimal expression of Qi , so apparently it implies Q2 (1) < Q1 (1) . Table 1: Lead time data Lead time 0 1 2 3 component, i Li 8 6 4 3 R( Li ) 0 5.6 22.4 57.4 Normal duration, bi (days) 20 20 16 Minimum duration, a i (days) 6 6 9 bi − ai 2 2 1 2.8 8.4 35 (weeks) Unit crashing cost, ci ($/week) 302 K.C. Hung / Continious Review Inventory Models Table 2: Summary of solutions ( L j in weeks) β=0 j β = 0.5 β = 0.8 β=1 Qj C (Q j , L j ) Qj C (Q j , L j ) Qj C (Q j , L j ) Qj C (Q j , L j ) 0 239 52579.28 191 42293.07 161 36160.27 142.02 32088.04 1 223 48811.71 181 39921.80 156 34616.81 139.34 31092.47 2 208 44979.04 174 37734.19 153 33406.93 139.19 30530.02 3 206 44235.12 176 37960.58 158 34210.40 146.53 31716.41 Considering the cases for β = 0, 0.5, 0.8 and 1, we use Table 3 to evaluate ci ( Li −1 + Li ) along with σ [π + (1 − β )π 0 ] Ψ (k ) . Table 3: Data for comparison c j ( L j + L j −1 ) j 1 2 3 14.78 37.38 130.62 When β =0 β σ [π + (1 − β )π 0 ] Ψ (k ) 0 0.5 0.8 1 154.2 96.38 61.68 38.55 , we find ci ( Li −1 + Li ) < σ [π + (1 − β )π 0 ] Ψ (k ) for all i = 1, 2,3 . By Lemma 1, we get Qi (0) < Qi −1 (0) for all i = 1, 2,3 For all i = 1, 2,3 , Lemma 2 implies C (Qi (0), Li ) < C (Qi −1 (0), Li −1 ) , so the optimal solution is (Q3 (0), L3 ) = (206,3) . When β = 0.5, 0.8 and 1, we find ci ( Li −1 + Li ) < σ [π + (1 − β )π 0 ] Ψ (k ) for all i = 1, 2 . Thus, by Lemma 2, we have C (Qi ( β ), Li ) < C (Qi −1 ( β ), Li −1 ) when β = 0.5, 0.8 and 1 with i = 1, 2 . Therefore, we need to calculate only min C (Qi ( β ), Li ) instead of min C (Qi ( β ), Li ) in order to get the i = 2, 3 i = 0,1,2,3 β = 0.5, 0.8 and 1. Furthermore, we find optimal solution for C (Q3 (0.5), L3 ) = 34960 > 37734 = C (Q2 (0.5), L2 ) and Q3 (0.5) = 176 > 173 = Q2 (0.5) . Using Theorem 1, we can conclude that min EAC (Qi ( β ), Li ) = EAC (Q2 ( β ), L2 ) for i = 2, 3 β = 0.5, 0.8 and 1. Consequently, Lemmas 1, 2, 3 and 4, and Theorem 1, can simplify the solution procedure. With our criterion, it is very easy to compare the local minimum