Mixture inventory model in fuzzy demand with controllable lead time

Transcription

1 Mixture inventory model in fuzzy demand with controllable lead time Jason Chao-Hsien Pan Department of Industrial Management National Taiwan University of Science and Technology Taipei 106 Taiwan R.O.C. Ming-Feng Yang Department of Information Management Yu Da College of Business No. 168, Hsueh-fu Rd. Tanwen Village, Chaochiao Township Miaoli County, 361 Taiwan R.O.C. Abstract The goal of this paper is to find the optimal order quantity and lead time simultaneously, and then minimizing the total inventory cost in fuzzy annual demand. In the past, most of the publications assume that the annual demand is deterministic. However, there are many uncertain factors in real world. Therefore, it can be described by fuzzy theory. In this study, we use the concept of fuzziness to join the mixture inventory system and construct the solution procedure to find the optimal order quantity and lead time. In the model, we use the signed distance, a ranking method for fuzzy numbers, to estimate the annual demand, and to derive the corresponding optimal solution. Numerical example is included to illustrate the procedures of the solution. Keywords : Inventory, fuzzy theory, lead time, signed distance. yang6029@pchome.com.tw Journal of Statistics & Management Systems Vol ), No. 2, pp c Taru Publications

2 32 J. C. H. PAN AND M. F. YANG 1. Introduction In most of the past studies Naddor [1, Tersine [2, and Vollmann [3, lead time is generally treated as a constant or a stochastic variable and cannot be controlled, provided general guidelines to discuss methods of reducing the lead time to promote the performance of an inventory system. Liao and Shyu [ proposed an inventory model with predetermined order quantity and normally distributed demand in which the lead time can be decomposed into several components and is the only variable to minimize the expected total cost. Ouyang et al [5 took the stock shortages into consideration and treated the total stockout as a mixture of backorders and lost sales based on Ben-Daya and Raouf s [6 research. Pan et al [7 developed an optimal reorder point inventory model with variable lead time and backorder discount. However, we noted that the previous researchers assumed the demand as crisp value or variable with a known probability distribution or in proportion of some parameters. In the crisp inventory model, the annual demand is a fixed constant and the backorder rate is assumed as a variable in proportion to the price discount offered by the supplier. In real environments, there are many uncertain factors which can affect the values of the annual demand. Recently, many inventory problems Chang [8, Chang et al [9, Chen and Wang [10, Park [11, and Yao and Lee [12 have introduced the concept of fuzzy theory. In this article, we use the fuzzy theory concept to deal with the mixture inventory model. Furthermore, it is assumed that the inventory lead time is controllable and the crashing cost can be represented as a function of reduced lead time and the quantities in the orders. The model incorporates the fuzziness of annual demand that use Yao and Wu s [13 ranking method for fuzzy number to defuzzy, and then find the corresponding optimal solution. 2. Preliminaries Before considering our inventory problems, we firstly introduce some definitions and properties related to this study and most of these can be found in Chang [8, Kaufmann and Gupta [1, Pu and Liu [15, and Zimmermann [16. Definition 1. The fuzzy set B = a, b, c), where a < b < c defined on R, is called the triangular fuzzy number, if the membership function of B is

3 MIXTURE INVENTORY MODEL 33 given by x a)/b a), a x b, µ Bx) = c x)/c b), b x c, 0, otherwise. 1) Definition 2. The fuzzy set [a α, b α defined on R, 0 α 1, is called an α -level fuzzy interval if the membership function of [a α, b α is given by µ [aα,b α = { α, α x b, 0, otherwise. 2) Definition 3. Let B be a fuzzy set on R, and 0 α 1, then the α -cut, Bα), of B consists of points x such that µ Bx) α ; that is, Bα) = {x µ B α}. Decomposition principle. Let B be a fuzzy set on R and 0 α 1 and its Bα) = [B L α), B U α) is a closed interval, then it follows that B = αbα) = [B L α) α, B U α) α 3) or where µ Bx) = αc Bα) x) = 1) αbα) is a fuzzy set with membership function { α, x Bα), µ αbα) x) = 0, otherwise. µ [BL α) α,b U α) α x), ) 2) C Bα) x) is a characteristic function of Bα); namely, C Bα) x) = { 1, x Bα), 0 x / Bα). For any a, b, c, d, k R, a < b, and c < d, the interval operations are as follows: 1) [a, b+)[c, d = [a + c, b + d. 2) [a, b )[c, d = [a d, b c. 5)

4 3 J. C. H. PAN AND M. F. YANG { [ka, kb, k > 0, 3) k )[a, b = [kb, ka, k < 0. Furthermore, for a > 0, and c > 0, ) [a, b )[c, d = [ac, bd. [ a 5) [a, b )[c, d = d, b. c Definition. For any a R, define the signed distance from a to 0 as d 0 a, 0) = a. If a > 0, then the distance from a to 0 is a = d 0 a, 0); if a < 0, the distance from a to 0 is a = d 0 a, 0). This is the why d 0 a, 0) is referred as the signed distance from a to 0. Let Λ be the family of all fuzzy sets B defined on R with which the α -cut Bα) = [B L α), B U α) exists for every α [0, 1, and both B L α) and B U α) are continuous functions on α [0, 1. Then, for any B Λ, we have B = [B L α) α, B U α) α. 6) From Definition, the signed distance of two end points, B L α) and B U α), of the α -cut Bα) = [B L α), B U α) of B to the origin 0 is d 0 B L α), 0) = B L α) and d 0 B U α), 0) = B U α), respectively. We define the signed distance from 0 to the interval [B L α), B U α) to be d 0 [B L α), B U α), 0) = [d 0 B L α) α, 0) + d 0 B L α) α ), 0)/2 = B L α), B U α))/2. 7) Since crisp interval [B L α), B U α) has a one-to-one correspondence with α -level fuzzy interval [B L α) α, B U α) α, it is natural to define the signed distance from α -level fuzzy interval [B L α) α, B U α) α to 0 1 as d[b L α) α, B U α) α, 0 1 ) = d[b L α) α, B U α) α, 0) = B L α), B U α))/2. 8) Moreover, for B Λ, since the above function is continuous on 0 α 1, we can use the integration to obtain the mean value of the signed distance as follows: 1 d[b L α) α, B U α) α, )dα = 1/2 B L α) + B U α))dα. 9) 0 0

5 MIXTURE INVENTORY MODEL 35 Thus, from equations 6) and 9), we have the following definition. d B, ) = d[b L α) α, B U α) α, 0 1 )dα 0 1 = 1/2 B L α) + B U α))dα. 10) 0 According to equation 10), we obtain the following property. Property 1. For a triangular fuzzy number distance from à to 0 1 is defined as à = a, b, c) Λ, the signed dã, 0 1 ) = 1 a + 2b + c). 11) Furthermore, for two fuzzy sets B, D Λ, where B = [B L α) α, B U α) α, D = [D L α) α, D U α) α, and k R, we have 1) B+) D = [B L α) + D L α)) α, B U α) + D U α)) α. 2) B ) D = [B L α) D L α)) α, B U α) D U α)) α. 12) [kb L α)) α, kb U α)) α, k > 0, 3) k 1 ) B = [kb L α)) α, kb U α)) α, k < 0, 01, k = 0. Property 2. For two fuzzy sets B, D Λ and k R, 1) d B+) D, 0 1 ) = d B, 0 1 ) + d D, 0 1 ). 2) d B ) D, 0 1 ) = d B, 0 1 ) d D, 0 1 ). 13) 3) d k 1 ) D, 0 1 ) = kd D, 0 1 ). 3. Mixture inventory model in fuzzy demand with controllable lead time 3.1 Notation and assumptions The notation used in this paper is listed as follows: D L Q R Annual demand per year; Length of lead time; Order quantity; Reorder point;

6 36 J. C. H. PAN AND M. F. YANG A h W W 0 q p X Ordering cost per order; Annul inventory holding cost per unit; Backorder penalty per unit; The gross marginal profit per unit; Allowed probability of inventory shortage during an order cycle; The backorder ratio; The demand during the lead time; µ The average of the daily demand; σ k SS Br) The standard deviation of the daily demand; The safety factor of the inventory; Safety stock; The expected backorder quantity per cycle; RL) The total crashing cost per cycle; TC OC HC SC CC The annul expected total cost; The annul expected ordering cost; The annul expected holding cost; The annul expected shorting cost; The annul expected crashing cost. The following assumptions are made on the models in this paper. 1. Inventory is continuously reviewed and replenishments are made whenever the inventory level falls to the reorder point r. 2. The demand during lead time X has finite mean µl and standard deviation σ L. 3. The reorder point r = µl + kσ L, where k is the safety factor.. The price of the inventory per unit is fixed. 5. The lead time consist of n mutually independent components. The ith component has a normal duration T i and a minimum duration t i, i = 1, 2,..., n. 6. The crashing cost per unit time c i is in proportion to the reduced lead time and the order quantity. Vojosevic et al [17 let a i be the fixed cost and b i be the variable cost per unit product per ith component lead time reduced. Thus, the crashing cost per unit time c i can be expressed as c i = a i + b i Q, i = 1, 2,..., n, where a i > 0 and b i > 0.

7 MIXTURE INVENTORY MODEL Let L i be the length of lead time with component 1, 2,..., i crashed to their minimum values, and L i can be expressed as L i = n T j i T j t j ). Thus, the lead time crashing cost RL) per replenishment cycle is given by i 1 RL) = a i + b i Q)L i 1 L) + a i + b i Q)T j t j ) for L L i, L i 1 ). 8. For any two crashing cost lines c i = a i + b i Q and c j = a j + b j Q, where a i > a j, and b i > b j and i = j, there is an intersection point Q s such that c i = c j. There are at most nn 1)/2 points and they can be arranged so that Q s 0 Qs 1 Qs 2 Qs m Q s m+1, where Q s 0 = 0, Qs m+1 = and m n 1)/2. For any order quantity range Qi s, Qs i+1 ), c i s are arranged such that c 1 c 2 c n. The lead times are crashed one at a time starting from the component with the least c i, and so on. The lead time demand X is assumed to be normal distribution with mean µl and standard deviation σ L. Since shortage is allowed, the expected inventory shortage at the end of a cycle is given by Br) = σ Lψk), where ψk) = φk) k[1 Φk), and φ, Φ are the standard normal distribution function and cumulative distribution function, respectively Park [11. For backorder ratio p, the expected number of backorders per cycle is pbr), the expected demand lost per cycle is 1 p)br), and the annual stockout cost is D/Q[W + W 0 1 p)br) Pan and Hsiao [18. Therefore, the expected net inventory level at the end of a cycle is r µl + 1 p) Br)), and at the beginning of the cycle is Q + r µl + 1 p) Br)). Consequently, the annul expected total cost contains ordering cost, holding cost, stockout cost and the crashing cost; and it can be represented by TC = OC + HC + SC + CC = AD [ Q Q + h + r µl + 1 p)br) 2 + D Q [W + W 01 p)br)br)

8 38 J. C. H. PAN AND M. F. YANG Where + D Q [ i 1 a i + b i Q)L i 1 L) + a j + b j Q)T j 1). 1) OC = A D Q ; 15) [ Q HC = h + r µl + 1 p)br) ; 16) 2 SC = D Q [W + W 01 p)br) ; 17) CC = D Q [ i 1 a i + b i Q)L i 1 L) + a j + b j Q)T j t j ), L L i, L i 1. 18) Hence, the expected annual total cost can be represented as: FQ, L) = AD [ Q Q + h 2 + kσ L + 1 p)br) + D Q [W + W 01 p)br) + D Q [ i 1 a i + b i Q)L i 1 L) + a j + b j Q)T j t j ), L L i, L i 1 ). 19) For convenience, we denote the total cost function TCQ, L) by FQ, L). Now we fuzzify D to be a triangular fuzzy number, D = D 1, D, D + 2 ), where 0 < 1 < D and 0 < 2, 1 and 2 are determined by the decision-makers. In this model, the total annual expected cost is a fuzzy value also, which is express as FQ, L) = A D Q + h [ Q 2 + kσ L + 1 p)br) + D Q [W + W 01 p)br) + D [ a Q i + b i Q)L i 1 L)+ i 1 a j + b j Q)T j t j ), L L i, L i 1. 20)

9 MIXTURE INVENTORY MODEL 39 Then, we defuzzify F by using the signed distance method. The signed distance of F to 0 1 is given by d F, 0 1 ) = A/Q)d D, 0 1 ) + h[q/2 + kσ L + {h1 p)+1/q[w + W 1 1 p)d D, 0 1 )}σ Lψk) [ i 1 + 1/Q a i + b i Q)L i 1 L) + a j + b j Q)T j t j ) d D, 0 1 ). 21) Where d D, 0 1 ) is measured as follow. From property 1, the signed distance of fuzzy number D to 0 1 is d D, 0 1 ) = 1/[D 1 ) + 2D + D + 2 ) = D + 1/ 2 1 ). 22) Therefore, substituting the result of 22) into 21), we have FQ, L) F, 0 1 ) ) A = D + ) ) 2 1 Q + h Q 2 + kσ L { + h1 p) + 1 Q [W + W 01 p) [ i 1 a i + b i Q)L i 1 L) + D a j + b j Q)T j t j ) )} σ Lψk) + 1 Q D + ) ) Utilizing the classical optimization technique, we can take the partial derivatives of FQ, L) with respect to Q and L respectively, we can obtain dfq, L) = A dq Q 2 D + ) h 2 1 Q 2 [W + W 01 p) and dfq, L) dl D [ i 1 a i L i 1 L) + )σ Lψk) 1Q 2 D a j T j t j ) = 1 1 hkσ L { h1 p) Q [W + W 01 p) ) 2)

10 350 J. C. H. PAN AND M. F. YANG D + )} 2 1 σ L 1 a 2 ψk) i + b i Q) D + ) ) Q For any given safety factory k, ψk) > 0 so = 2A D d 2 FQ, L) dq 2 and Q 3 D ) d 2 FQ, L) dl 2 = 1 hkσ L D ) + 2 Q 3 [W + W 01 p) σ Lψk) + 2RL) Q 3 D + ) 2 1 > 0 26) { h1 p) + 1 Q [W + W 01 p) )} σ L 3 2 ψk) < 0. 27) Hence, for fixed L L i, L i 1, FQ, L) is convex in Q, and for fixed Q, FQ, L) is concave in L L i, L i 1. That is, for fixed Q, the minimum annual expected total cost will occur at the end point of the interval [L i, L i 1. Solving for Q by setting Eq 2) to zero, we obtain [ 2 D + ) 2 A + [W + W 0 1 p)σ Lψk) 1 [ + a i L i 1 L) + i 1 a j T j t j ) 1 Q 2 =. 28) h For the lead time, according to the degree of crashing, the lower and upper limits of Q can be constructed from 28). If all of the components are completed in normal durations, then the crashing cost is zero. On the other hand, if all of the components are crashed to their minimum duration, then the fixed crashing cost is Q min = n a j T j t j ). Hence, we let } t j [ 2 D + ){ 2 1 n A + [W + W 0 1 p)σψk) h ) and [ 2 D + ) 2 A + [W + W 0 1 p)σψk) 1 n T j + n a i T j t j ) 1 2 Q max =. 30) h

11 MIXTURE INVENTORY MODEL 351 From 28), 29) and 30), it follows that Q min Q Q max. If Qi 1 s Q min Qi s and Q s j 1 Q max Q s j, where i j, then Q lies in a range between Qi 1 s, Qs i ) and Qs j 1, Qs j ). The component with the least one c j will be crashed first, and next the second. The crashing priorities are different in various order quantity ranges. For instance, we consider the three crashing cost lines, c 1 = a 1 + b 1 Q, c 2 = a 2 + b 2 Q, and c 3 = a 3 + b 3 Q which a 1 < a 2 < a 3, b 1 > b 2 > b 3 with their intersection points Q s 1, Qs 2 and Qs 3. If Q falls into the interval [0, Q s 1, since c 1 is the least of the three crashing cost per unit time, component 1 has the highest priority to be crashed, then next comes component 2, and component 3 is the last. If Q falls into the interval [Q s 1, Qs 2, component 2 is the first to be crashed, then is component 1, followed by component 3, and so forth. Since the annual expected total cost Fy) is convex in Q for any fixed L L i, L i 1 ), given safety factor k, backorder rate p and order quantity range Qi 1 s, Qs i ), we can use Eq 28) to find out Q. If Qs i Q, then the quantity Qi s along with these three equations can be applied to replace Q. For Q Qi 1 s, then Qs i 1 is used instead of Qs i. From all of the above, we can construct the following solution procedure to find the optimal values of Q and L. Algorithm 1. Step 1. For annual demand D do a) fuzzify the annual demand D to be a triangular fuzzy number. D 1, D, D + 2 ). b) the decision-makers decision i, for i = 1, 2. Then, use the signed distance method to defuzzify the fuzzy number D. Step 2. Compte the intersection points Q s of arbitrary two crashing cost lines c i = a i = b i Q and c j = a j + b j Q, for all i and j, where a i > a j, b i > b j and i = j. Arrange these intersection points as: Q s 0 Qs 1 Qs 2 Q s m Q s m+1, where we define Qs 0 = 0, Qs m+1 =. a) Use 29) and 30) to compute Q min and Q max. b) Find the values of e and f, make Q s e 1 Q min Q s e and Q s f 1 Q max Q s f.

12 352 J. C. H. PAN AND M. F. YANG Step 3. For i = e, e + 1,..., 1, do a) rearrange c j such that c 1 c 2 c n, j = 0, 1, 2,..., n. b) use Eq 28) to compute Q r j, j = 0, 1, 2,..., n. c) if Q r j Qs i 1, define Qr j = Qs i 1 ; if Qs i Q s j, let Qr j = Qs i. d) use Eq 23) and Q r j to compute the annual expected total cost FQ r j, L j), j = 0, 1, 2,..., n. e) find FQ i, L i ) = Min{FQ r j, L j), j = 0, 1, 2,..., n}. Step. Find FQ, L ) = Min{FQ i, L i ), i = e, e + 1,..., f }. Step 5. Q, L ) is the optimal solution of this model. Remark 1. If 1 = 2 =, then 22) reduce to d D, 0 1 ) = D. The estimation of the annual demand in fuzzy sense 22) is the same of as the crisp case. Therefore, the crisp annual demand can be seen as a special case of the fuzzy model. Besides, when 1 = 2 = the optimal order quantity reduces to A + [W + W 0 1 p)σ Lψk) [ 2D [ + a i L i 1 L) + i 1 a i T j t j ) 1 Q 2 =. 31) h Numerical example We consider an inventory problem with the following data: D = 600 units/year, A = $200 per order, h = $20 per unit year, W = $50 per unit, W 0 = $150 per unit, σ = 7 units/week, k = 0.85, and the lead has three components with data shown in Table 1. The demands corresponding to the three different stages of the item life cycle are: The solution procedure proceeds as follows. Find the intersection points of crashing cost lines. There are three intersection points, Q s 1 = 100, Q s 2 = 25.9, and Qs 3 = , and order quantity ranges are R 10, 100), R 2 101, 25), R 3 26, 1357), R 1358, ). The crashing priorities are different in each range.

13 MIXTURE INVENTORY MODEL 353 Table 1 Lead time data of numerical Example 1 Lead time component, i Normal duration, T i days) Minimum duration, t i days) Unit fixed crashing cost, a i $/day) Unit variable crashing cost, b i $/day) For range R 1 0, 100), component 1 is the first to be crashed, then is component 2, and 3 is the last. For R 2 101, 25), component 2 is the first, next is component 1, followed by 3. For R 3 26, 1357), the crash sequence is components 2, 3, 1; and R 1358, ) it becomes 3, 2, 1. Consequently, the values of Q min, Q max, 1, 2, d D, 0 1 ) and the respective values of e and f are found for the backorder rate p equals 0, 0.5, 0.8, 1 and presented in Table 2. Table 2 shows that the optimal solutions for each p lies in R 2 101, 25) since both of its e and f values equal 2. In Table 3 we compare the results of fuzzy case and crisp case. Note that in practice, 1 and 2 are determined by the managers or decision makers due to the uncertain factors of the inventory problem. The L value in Table 3 indicates that the lead times are expressed in weeks Ben-Daya and Raouf [6, Pan and Hsiao [18 and Pan et al [7. For example, L = 3 means all the three components are crashed, and L = 8 says that no component is crashed. The number marked with is the optimal solution in crisp sense. Then, the relative variation between fuzzy case and crisp case for the optimal order lot size and the minimum expected annual cost can be measured respectively as follows: Rel Q = [Q f Q c )/Q c 100%, Rel F = {[FQ f ) FQ c )/FQ c )} 100%. The computational results are listed in the last two columns of Table 3. From Table 3, we observe that I) If p and lead time L are the same, when 1 < 2, the estimate of annual demand d D, 0 1 ) > D = 600). In this case, Q f > Q c and FQ f ) > FQ c ), which result in Rel Q > 0 and Rel F > 0. Further, if the modulus 1 2 decrease, both Rel Q and Rel F decrease, which means that the smaller the difference between 1 and 2 is, the smaller the variation between fuzzy and crisp case.

14 35 J. C. H. PAN AND M. F. YANG Table 2 The values of Q min, Q max, e and f in Example 1 p D 1 2 d D, 0 1 ) Q min e Q max f

15 MIXTURE INVENTORY MODEL 355 Table 2 The solution result of Example 1 p D 1 2 d D, 0 1 ) Q FQ, L) Rel Q%) Rel F%) The number marked with is the optimal solution in crisp sense

16 356 J. C. H. PAN AND M. F. YANG II) When 1 > 2, the estimation of annual demand d D, 0 1 ) < D. Then, Q f < Q c and FQ f ) < FQ c ), which result in Rel Q < 0 and Rel F < 0. Similarly, as 1 2 increase, both Rel Q and Rel F increase. III) When 1 = 2 = 0 or 1 = 2 =, the fuzzy case becomes the crisp one, and the optimal order quantity can derive from Eq 31).. Conclusions In the model, we suppose that the inventory problem with fuzzy demand is represented by triangular fuzzy number. Numerical example is performed to investigate the result of our proposed model. In practice, the uncertainties of product demand are inherent. Many researchers assume that these uncertainties could be constant based on the concept of probability theory. However, using probability distribution to solve the uncertain factor needs historical data that may be mismanaged. In this situation, it is not appropriate to use probability theory. The value of this study is using fuzzy theory to solve the uncertainties from these perspectives. Eventually, we would like to indicate that the benefit of using fuzzy model to solve our problem. It can be conformed that the practice of using fuzzy model is better than the crisp model. It is the reason why the fuzzy model can keep the uncertainties and match the real situations better. Although we can not prove that the fuzzy model can reduce more inventory cost than the crisp model in this study. However, using the concept of fuzzy theory is able to provide an alternative method of inventory problem for managers. References [1 E. Naddor 1966), Inventory Systems, John Wiley, New York. [2 R. J. Tersine 1982), Principles of Inventory and Material Management, North Holland, New York. [3 T. E. Vollmann, W. L. Berry and D. C. Whybark 1992), Manufacturing Planning and Control Systems, 3rd edn., Irwin, Chicago. [ C. J. Liao and C. H. Shyu 1991), Analytical determination of lead time with normal demand, International Journal of Operations and Production Management, Vol. 11, pp

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