European Journal of Operational Research 174 (2006) 861–873 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Two-warehouse inventory model with deterioration under FIFO dispatching policy Chun Chen Lee
*
Department of Accounting, Soochow University, 56, Sec. 1, Kuei-Yang Street, Taipei 100, Taiwan Received 3 November 2004; accepted 21 March 2005 Available online 3 January 2006
Abstract In most of the literatures on two-warehouse inventory decision models, the last-in-first-out (LIFO) dispatching policy has always been assumed. This presumption, however, is not in line with the actual practice of most business entities. To enhance the freshness of merchandise or goods, businesses commonly follow the first-in-first-out procedure (FIFO). This inconsistency forms the base and main motivation for our research. In this paper, Pakkala and Achary’s two-warehouse LIFO model is first modified and then a FIFO dispatching two-warehouse model with deterioration is proposed. Comparison of the two models indicated that the FIFO model is less expensive to operate than LIFO, if the mixed effects of deterioration and holding cost in RW are less than that of OW. Finally, numerical examples are provided to investigate and examine the impact that various parameters have on policy choice. Ó 2005 Published by Elsevier B.V. Keywords: Inventory; Deterioration; Two-warehouse; FIFO; LIFO
1. Introduction The classical economic order quantity (EOQ) model is formulated by considering three inventory costs to achieve a minimum system cost. These costs are the procurement cost, carrying cost and shortage cost. One of the unrealistic assumptions is that items are not perishable while in storage. However, there are items such as highly volatile substances, radioactive materials, fresh goods, etc., in which the rate of deterioration is higher. Loss from deterioration should not be ignored. Ghare and Schrader [4] were the first to
*
Tel.: +886 2225 38244. E-mail address:
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0377-2217/$ - see front matter Ó 2005 Published by Elsevier B.V. doi:10.1016/j.ejor.2005.03.027
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consider issues regarding on-going deterioration of inventory. Since then, research for deterioration of inventory has been extensively examined by many researchers from time to time. Raafat [14] and Goyal and Giri [6] have made excellent reviews of these models. In various situations, the degree of deterioration depends on the preservation of inventory in the facility and its environmental conditions which are available in the warehouse. An interesting research topic incorporating deterioration effect in inventory decision involves the situations in which there are two storage facilities. Sarma [15] is the first to discuss the two-warehouse inventory model with deterioration. In his model, a single inventory item is first stored in the owned warehouse (OW), with limited capacity, and any additional quantity to be stored in the rented warehouse (RW). An infinite replenishment rate is considered in this model with uniform scheduling period and shortage allowance. Other authors, e.g. Benkherouf [2], Bhunia and Maiti [3], Goswami and Ghaudhuri [5], and Lee and Ma [8] proposed the two-warehouse models when demand is a function of time either with or without the consideration of deterioration. Pakkala and Achary [10] extended Sarma’s model to the case of finite replenishment rate with shortage. All of the above mentioned research models are commonly referred to as continue release model, assuming that inventory is to be released directly and continuously in each warehouse. Murdeshwar and Sathe [9], Pakkala and Achary [11] considered bulk release model which inventory in RW must first be transferred to OW before its release to the customer. It is generally assumed that the RW offers better preserving facilities than the OW, therefore it charges a higher holding cost. The two-warehouse models discussed above naturally adopt the LIFO (last-in-firstout) inventory flow. Under such circumstances, inventories are first stored in OW with overflows going to RW. But when retrieving for consumption, it is always from RW when available before retrieving from OW. However, we believe such rule needs to be further investigated when applying to a real world situation. First, in the RW, particular in a public warehouse, a professional vendor who specializes in the warehousing operation would carry a lower operating cost due to well equipped set ups, learning effect of trained worker, and the economics of scale from high volume. Second, as competition increases between warehouse facilities in real world, their ability to offer valued added service with completive lower price is becoming more and more necessary. Many businesses have gotten into or expanded their use of public warehouse because of cheap shipping or other financial reasons (Anonymous, [1]). Finally, a critical point of inventory decision for perishable products, to allow later stored inventory in RW to be dispatched last means a greater risk of deterioration of inventory. The cost of deteriorated inventory and related opportunity cost may far exceed the cost saving benefit derived from the warehouse rent. In the real world, maintaining a FIFO rule of inventory flow has been the common practice of most managers. In fact, Pierskalla and Roach [12] have shown that a FIFO issuing policy is optimal for perishable and deteriorating inventories in a single warehouse setting with unlimited capacity. In this paper, Pakkala and Achary’s [10] LIFO model with finite replenishment rate will be reconsidered. We propose a FIFO two-warehouse model that inventory in OW, which is stored first, will be consumed before those in RW based on the above considerations that the true holding cost in RW is not necessarily higher than in OW. Before making comparisons between the two models, in Section 3.1, we made a modification to Pakkala and Achary’s model to allow their model to be more complete. Furthermore, the proposed assumption of a predetermined cycle time will also be relaxed to a more general approach which is to let cycle time be part of decision and to determine both order level and backorder level simultaneously. In the final section, various parameter analyses are implemented to examine the impact on policy choice.
2. Notations and assumptions The two-warehouse inventory models proposed in this research are based on the following notations and assumptions:
C.C. Lee / European Journal of Operational Research 174 (2006) 861–873
D P C1 C2 C3 Ti Ii(t) F, H b, a W R B
863
demand rate which is a constant constant production rate, P > D cost of a deteriorated item shortage cost per unit inventory per unit of time in shortage unit set up cost time period in a production cycle of stage i, i = 1, . . . , 6 inventory level at time t during time period Ti holding cost held in the RW and OW respectively deteriorating rates in RW and OW respectively, 0 < b, a < 1 capacity of the OW maximum inventory level in RW maximum shortages allowed
The following assumptions are adopted in this study: 1. Lead time is zero and shortages are allowed. 2. The rented warehouse RW has unlimited capacity. 3. Inventory items are stored in RW only after OW is fully utilized. Once stored, these items are assumed not to be relocated. For convenience to differentiate between the models, each time stage Ti under LIFO and FIFO policy is further denoted by P TLi and TFi, i = 1, P. . . , 6. Denote TL and TF as total production cycle time for the two policies, then T L ¼ i T Li , and T F ¼ i T Fi . Also, inventory level during each stage i for the two models are set as ILi(t) and IFi(t).
3. The models 3.1. Modified LIFO inventory model The inventory in a production system with LIFO dispatching policy is depicted in Fig. 2. The inventory cycle can be divided into six parts, named TLi, i = 1, . . . , 6. Initially, BL units of backorders are carried over
Inventory level RL
W
Time
BL
Fig. 1. Inventory level of Pakkala and Achary’s two-warehouse LIFO model.
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Inventory level RL IL4(t) W
IL5(t) IL3(t)
IL6(t)
IL2(t)
IL7(t)
Time TL1 BL
TL2
TL3
TL4
IL1(t)
TL5
TL6 IL8(t)
Fig. 2. Inventory level of modified two-warehouse LIFO model.
from the previous cycle. The production run starts at the beginning of TL1 and, while production and demand happen simultaneous, backorders are made up within TL1 at the rate of P D. During TL2, inventory items in OW are built from 0 up to W units with a deterioration rate of a. Any production quantity exceeding this level must be stored in RW. During TL3, inventory items in RW are built from 0 to RL units but with a deterioration rate of b. Meanwhile, in OW, inventory level will be depleted because of deterioration in stock with a rate of a. In (11) of Pakkala and Achary [10], the available spaces in OW released from deteriorated inventory items during TL3 are assumed not to be reutilized (see Fig. 1). However, under the proposed assumption that H < F, the system cost will obvious be higher if it is not reutilized, and vice versa. This short coming will be modified in this paper before making a comparison between FIFO and LIFO in the next section. The production run stops at the end of TL3, and RL units of inventory items in RW are depleted in TL4. The remaining inventory items in OW are then depleted in TL5 by demand and deterioration. Finally, BL units of backorders are accumulated at the end of TL6 by a rare of D, which completes the production cycle. In this system, the management seeks to find the optimal levels of both RL and BL. The differential equations describing the inventory level at any time in the production cycle are given as follows: dI L1 ðtÞ=dt ¼ P D;
0 6 t 6 T L1 ;
dI L2 ðtÞ=dt þ aI L2 ðtÞ ¼ P D; dI L3 ðtÞ=dt ¼ 0;
0 6 t 6 T L2 ;
0 6 t 6 T L3 ;
dI L4 ðtÞ=dt þ bI L4 ðtÞ ¼ P D aW ; dI L5 ðtÞ=dt þ bI L5 ðtÞ ¼ D; dI L6 ðtÞ=dt þ aI L6 ðtÞ ¼ 0; dI L7 ðtÞ=dt þ aI L7 ðtÞ ¼ D; dI L8 ðtÞ=dt ¼ D;
0 6 t 6 T L3 ;
0 6 t 6 T L4 ; 0 6 t 6 T L4 ; 0 6 t 6 T L5 ;
0 6 t 6 T L6 .
Using the boundary conditions that IL1(TL1) = 0, IL2(0) = 0, IL4(0) = 0, IL5(TL4) = 0, IL6(0) = W, IL7(TL5) = 0, and IL8(0) = 0, the above equations can be solved respectively as follows: I L1 ðtÞ ¼ ðP DÞðT L1 tÞ; 0 6 t 6 T L1 ; I L2 ðtÞ ¼ ðP DÞð1 eat Þ=a; 0 6 t 6 T L2 ;
ð1Þ ð2Þ
C.C. Lee / European Journal of Operational Research 174 (2006) 861–873
I L3 ðtÞ ¼ W ;
ð3Þ
0 6 t 6 T L3 ; bt
I L4 ðtÞ ¼ ðP D aW Þð1 e Þ=b; 0 6 t 6 T L3 ; I L5 ðtÞ ¼ D ebðT L4 tÞ 1 =b; 0 6 t 6 T L4 ; I L6 ðtÞ ¼ W eat ; 0 6 t 6 T L4 ; I L7 ðtÞ ¼ D ½eaðT L5 tÞ 1 =a; 0 6 t 6 T L5 ; I L8 ðtÞ ¼ Dt; 0 6 t 6 T L6 .
ð4Þ ð5Þ ð6Þ ð7Þ ð8Þ
Now, the inventory items held in RW and OW for a production cycle are Z T L3 Z T L4 I L4 ðtÞdt þ I L5 ðtÞdt ¼ ½P T L3 DðT L3 þ T L4 Þ aW T L3 =b; G1 ¼ 0
865
ð9Þ
0
and G2 ¼
Z
T L2
I L2 ðtÞdt þ
0
Z
T L3
I L3 dt þ
0
Z
T L4
I L6 ðtÞdt þ 0
Z
T L5
I L7 ðtÞdt
0
¼ ½P T L2 DðT L2 þ T L5 Þ þ aW T L3 =a.
ð10Þ
Denote G3 the inventory items deteriorated per cycle, G3 = bG1 + aG2: G3 ¼ P ðT L2 þ T L3 Þ DðT L2 þ T L3 þ T L4 þ T L5 Þ.
ð11Þ
The total amount of shortages in the production cycle is Z 0 Z 0 G4 ¼ I L1 ðtÞdt þ I L8 ðtÞdt ¼ ðP DÞT 2L1 þ DT 2L6 =2. T L1
T L6
Denote TLB = TL1 + TL6, using IL1(0) = IL8(TL6), from (1) and (8), TL1 and TL6 can be expressed as functions of TLB: T L1 ¼ DT LB =P
and
T L6 ¼ ðP DÞT LB =P .
ð13Þ
We then have G4 ¼ DðP DÞT 2LB =2P . The total system cost per unit of time for LIFO policy is TC L ¼ ð1=T L ÞðFG1 þ HG2 þ C 1 G3 þ C 2 G4 þ C 3 Þ ¼ ð1=T L ÞfF ½P T L3 DðT L3 þ T L4 Þ aW T L3 =b þ H ½P T L2 DðT L2 þ T L5 Þ þ aW T L3 =b þ C 1 ½P ðT L2 þ T L3 Þ DðT L2 þ T L3 þ T L4 þ T L5 Þ þ C 2 DðP DÞT 2LB =ð2P Þ þ C 3 g. Now that IL2(TL2) = W, TL2, which is a constant can be derived from (2): 1 P D T L2 ¼ ln . a P D aW
ð14Þ
ð15Þ
Using IL4(TL3) = IL5(0), from (4) and (5), we get TL4 in terms of TL3: T L4 ¼
1 ðP aW Þ ðP D aW ÞebT L3 ln . b D
ð16Þ
Also, using IL7(0) = IL6(TL4) from (6) and (7), TL5 can be derived as a function of TL4 (also function of TL3):
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T L5
1 aW eaT L4 ¼ ln 1 þ . a D
ð17Þ
Therefore, the total cost per unit time can then be expressed explicitly in terms of TL3 and TLB. The optimal value of TL3 and TLB must satisfy the following two necessary conditions: oTC/oTL3 = 0 and oTC/ oTLB = 0. After rearrangement, we can obtain aF F F dT L4 H dT L5 dT L4 dT L5 W H D C1 þ TC 1 þ þ þ ðP DÞ C 1 þ D C1 þ ¼0 b b b dT L3 a dT L3 dT L3 dT L4 ð18Þ and C 2 DðP DÞ T LB TC ¼ 0; P where
ð19Þ
dT L4 P D aW ¼ dT L3 ðP aW ÞebT L3 ðP D aW Þ and dT L5 aW ðP D aW ÞeaT L4 . ¼ dT L3 ðD þ aW eaT L4 Þ½ðP aW ÞebT L3 ðP D aW Þ The total system cost in (14) is a complicated nonlinear function in terms of TL3 and TLB and not easy to solve analytically. Through an enormous amount of numerical analyses, we have found that the total cost function shows convexity with respect to TL3 and TLB. By applying numerical subroutine DNEQNF in IMSL, the optimal value of TL3 and TLB can be obtained from (18) and (19). Now that IL1(0) = B, and that IL4(TL3) = RL from (1), (13) and (4), BL ¼
DðP DÞ T LB ; P
RL ¼
ðP D aW Þð1 ebT L3 Þ . b
There the optimal production policy, i.e., BL and RL , can be easily derived after the optimal solutions T L3 and T LB are obtained. Theorem 1. Modified LIFO two-warehouse model always has a lower cost than Pakkala and Achary’s LIFO model if H aF/b > 0. Proof. Denote TCP as average total cost of Pakkala and Achary’s (11). Let TP1 = T t1, and TPi = ti1 ti2, for i = 2, . . . , 6. After variables and parameters transformation, Pakkala’s (11) can expressed as TC P ¼ ð1=T P ÞfH ½P T P2 DðT P2 þ T P5 Þ=a þ F ½P T P3 DðT P3 þ T P4 Þ=b þ C 1 ½P ðT P2 T P3 Þ 2
DðT P2 þ T P3 þ T P4 þ T P5 Þ þ C 2 DðP DÞðT P1 þ T P6 Þ =ð2P Þ þ C 3 g.
ð20Þ
For our convenience and without loss of generality, assuming that TPi = TLi for i = 1, . . . , 6. From (14) and (20), cost difference between modified LIFO and Pakkala and Achary’s LIFO model is given by T C L T C P ¼ W T L3 ðH aF bÞ. Since WTL3 > 0, if a is not significantly less than b, modified LIFO model will have a lower cost than Pakkala and Achary’s LIFO model under their assumption that H < F. h
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3.2. FIFO model In a system with FIFO dispatching policy, inventory items in OW that is stored first will first be released for consumption before that of RW. After the end of TF3 (see Fig. 3), when production stops, inventory items in RW will remain in storage but with a deterioration rate b. Any demands are withdrawn from OW until the inventory items in OW are completely consumed, thereafter withdrawing from RW. Other inventory fluctuations and the decision objectives are all to be the same as those in a LIFO system. The differential equations describing inventory behavior for IFi, for i = 1, 2 and 8, are the same as LIFO model and can be obtained from (1), (2) and (8). Inventory level IFi, for i = 3, . . . , 7, are described as follows: dI F3 ðtÞ=dt þ aI F3 ðtÞ ¼ 0; 0 6 t 6 T F3 ; dI F4 ðtÞ=dt þ bI F4 ðtÞ ¼ P D; 0 6 t 6 T F3 ; dI F5 ðtÞ=dt þ bI F5 ðtÞ ¼ 0; 0 6 t 6 T F4 ; dI F6 ðtÞ=dt þ aI F6 ðtÞ ¼ D; 0 6 t 6 T F4 ; dI F7 ðtÞ=dt þ bI F7 ðtÞ ¼ D;
0 6 t 6 T F5 .
Using the boundary conditions that IF3(0) = W, IF4(0) = 0, IF5(0) = IF4(TF3), IF6(TF4) = 0, IF7(TF5) = 0, one can obtain following inventory level functions: I F3 ðtÞ ¼ W eat ;
ð21Þ
0 6 t 6 T F3 ;
I F4 ðtÞ ¼ ðP DÞð1 e I F5 ðtÞ ¼ ðP DÞ½1 e
bT
Þ=b;
bT F3
e
ð22Þ
0 6 t 6 T F3 ;
bt
ð23Þ
0 6 t 6 T F4 ;
=b;
I F6 ðtÞ ¼ D½e
aðT F4 tÞ
1=a;
0 6 t 6 T F4 ;
ð24Þ
I F7 ðtÞ ¼ D½e
bðT F5 tÞ
1=b;
0 6 t 6 T F5 .
ð25Þ
The inventory holding in RW and OW are S 1 ¼ ½P T F3 DðT F3 þ T F5 Þ=b and
S 2 ¼ ½P T F2 DðT F2 þ T F4 Þ=a.
ð26Þ
The total inventory deteriorated and shortages are S 3 ¼ bS 1 þ aS 2 ¼ P ðT F2 þ T F3 Þ DðT F2 þ T F3 þ T F4 þ T F5 Þ
and
S 4 ¼ DðP DÞT 2FB =ð2P Þ;
where TFB = TF1 + TF6. Inventory level IF5(t)
RF IF4(t)
IF7(t)
W IF3(t) IF2(t)
IF6(t)
Time TF1 BF
IF1(t)
TF2
TF3
TF4
TF5
TF6 IF8(t)
Fig. 3. Inventory level of two-warehouse FIFO model.
ð27Þ
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Finally, total system cost per unit of time under FIFO dispatching policy is TC F ¼ ð1=T F ÞðF S 1 þ HS 2 þ C1 S 3 þ C 2 S 4 þ C 3 Þ ¼ ð1=T F ÞfF ½P T F3 DðT F3 þ T F5 Þ=b þ H ½P T F2 DðT F2 þ T F4 Þ=a þ C 1 ½P ðT F2 þ T F3 Þ DðT F2 þ T F3 þ T F4 þ T F5 Þ þ C 2 DðP DÞðT FB Þ2 =ð2P Þ þ C 3 g.
ð28Þ
In (28), TF2 is a constant that should have no difference with TL2, we have T F2 ¼ T L2 ¼
1 P D ln . a P D aW
ð29Þ
By using IF3(TF3) = IF6(0) and IF7(0) = IF5(TF4), from (21), (23), (24) and (25), the value of TF4 and TF5 can be derived respectively as 1 aW eaT F3 1 ðP DÞð1 ebT F3 ÞebT F4 T F4 ¼ ln 1 þ and T F5 ¼ ln 1 þ . ð30Þ a b D D Let TF3 and TFB be the two decision variables of (28). The optimal value of TF3 and TFB must satisfy the two necessary conditions: oTC/oTF3 = 0 and oTC/oTFB = 0. After rearrangement, we have F H dT F4 F dT F5 dT F4 dT F5 D C1 þ TC 1 þ þ ðP DÞ C 1 þ D C1 þ ¼ 0; ð31Þ b a dT F3 b dT F3 dT F3 dT F3 and C 2 DðP DÞ T FB TC ¼ 0; P
ð32Þ
where dT F4 aW ; ¼ dT F3 DeaT F3 þ aW
dT F5 ðP DÞðaW þ DeðabÞT F3 Þ . ¼ dT F3 ðaW þ DeaT F3 Þ½DebT F4 þ ðP DÞð1 ebT F3 Þ
Furthermore, BF can be derived as BF ¼ RF ¼
ðP D aW Þð1 e b
bT F3
Þ
DðP DÞ T FB , P
and by using IF4(TF3) = RF, from (22)
.
Therefore, the optimal production policy under FIFO dispatching, i.e., BF and RF , can also be derived after the optimal solutions T F3 and T FB are obtained. From (14) and (28), one interesting observation is shown between the two policies. Theorem 2. If the two warehouses have the same deterioration rate, i.e., a = b, then TCF > TCL for H < F, otherwise TCF > TCL if H < F, or TCF < TCL if H > F. Proof. Let T3 and TB be the decision objectives of the two models. We want to prove that if F < H and a = b, then TCF < TCL for any combinations of TC (T3, TB). First, let TL3 = TF3, TLB = TFB. The following Lemmas will hold: Lemma 1. TL4 + TL5 = TF4 + TF5. Lemma 2.
aW D
T L3 þ T F4 > T L5 .
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Proof of Lemma 1 (i) Add TL4 to both sides of (17), we have 1 aW aT L4 1 1 aW e þ eaT L4 . T L5 þ T L4 ¼ ln 1 þ þ lnðeaT L4 Þ ¼ ln a D a a D Substitute the value of TL4 in (16) to above equation and after simplification 1 P ðP D aW ÞebT L3 T L4 þ T L5 ¼ ln . a D (ii) Add TF4 to both sides of (30), we have bT F4 1 ðP DÞð1 ebT F3 ÞebT F4 1 1 De þ ðP DÞð1 ebT F3 Þ T F5 þ T F4 ¼ ln 1 þ . þ ln ebT F4 ¼ ln b b b D D Substitute the value of TF4 in (30) to above equation and after simplification 1 P ðP D aW ÞebT F3 T F4 þ T F5 ¼ ln ¼ T L4 þ T L5 . b D (iii) Note also that, from (29) TL2 = TF2, we hence have T LB þ T L2 þ T L3 þ T L4 þ T L5 ¼ T FB þ T F2 þ T F3 þ T F4 þ T F5 ;
i.e., T L ¼ T F .
Proof of Lemma 2. Denote r = aW/D > 0, and let T 0F4 ¼ 1a lnð1 þ rÞ. Define f ðT F3 Þ ¼ T F4 þ rT F3 T 0F4 , from (30)
aT f ðT F3 Þ ¼ 1a lnð1 þ reaT F3 Þ 1a lnð1 þ rÞ þ rT F3 ¼ 1a ln 1þre1þr F3 þ rT F3 , h i aT F3 þreaT F3 Þ where f (0) = 0, f 0 ðT F3 Þ ¼ rð1e1þreaT > 0 ðby eaT F3 < 1Þ. F3 Which implies that f (TF3) > 0 for TF3 > 0. We hence have T F4 þ rT F3 > T 0F4 . Furthermore, from (17) T L5 ¼ 1a lnð1 þ reaT L4 Þ, which implies T L5 < T 0F4 < T F4 þ rT F3 . h Proof of Theorem 2. From (14), (28), and the two lemmas 1 F H ðDT L4 DT F5 þ aW T L3 Þ þ ½DT L5 DT F4 aW T L3 TC F TC L ¼ TF b a 1 F ¼ ðDT L4 DT F5 þ aW T L3 þ DT L5 DT F4 aW T L3 Þ TF b ðH F Þ ½DT L5 DT F4 aW T L3 þ a D ðF H Þ aW T L3 þ T F4 T L5 ¼ > 0; provided that F > H . TF a D The above theorem implies that, in facing policy choice, if the two warehouses have similar preservation conditions that the inventory deterioration are nearly the same, then the policy choice solely depends on the difference in inventory holding cost, namely H and F. FIFO policy will be less expensive when HF, otherwise, LIFO is suggested. Undoubtedly, when the two warehouses have all the same parameters i.e., b = a, and F = H, these two policies should have no difference, i.e., TCF > TCL. h
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3.3. Choice from one-warehouse system (L1) and two-warehouse system (L2) Let the two warehouses be utterly no difference, i.e., a = b, H = F, total cost of different dispatching policy in (14) and (28) of L2 can both be reduced to the following expression: TC F ¼ TC L ¼ ð1=T ÞfðC 1 þ F =bÞ½P ðT 2 þ T 3 Þ DðT 2 þ T 3 þ T 4 þ T 5 Þ þ C 2 DðP DÞðT 1 þ T 6 Þ2 =ð2P Þ þ C 3 g.
ð33Þ
After certain variable simplification, expression in (33) is the same as Raafat et al. [13], which is an economic production quantity model for deteriorating items with unlimited warehouse space. Denote TCL1 to be the average total cost of L1 system, we have TCF = TCL = TCL1. Furthermore, let W = 0 and RW be the sole warehouse under consideration. By using the fact that TL2 = TL5 = 0 [substitute W = 0 into (15) and (17)], same result in (33) can also be obtained from (14) of our modified LIFO model. Or, similarly, from (28)–(30), one can derive TCL1 from TCF. Under the assumption that OW is to be utilized first, L2 system will not necessarily be used if it is economically less than L1. The following algorithm can be employed to determinate between the systems choices for the two policies. Step 1. First solve L1 in (33). Step 2. Calculate and denote RL1 the optimal maximum inventory level of L1. Step 3. If RL1 is less than W, L1 will be used. Otherwise, when RL1 > W , compare TC L2 with the boundary cost on L1 at W, i.e., TCL1(W). L2 will be used if TC L2 < TC L1 ðW Þ, otherwise L1(W) is the optimal solution.
4. Illustrative example The following parameters are used to illustrate the application of the two models. The production capacity is 32 000 units per year; the demand rate is 8000 units per year; other related factors are as follows: shortage cost is $8 per unit per year; deterioration cost is $20 per unit; OW capacity is 1200 units. In Table 1, in order to make comparison of the deteriorating effect on policy selection, holding cost in the two warehouses are assumed to be equal, i.e., (H, F) = (2, 2), deterioration rate in RW be fixed at 0.06. Denoted r = a/b, by increasing the value r (increase a), total cost would increase under both policies. From Table 1, we can observe that, if r = 1, both policies will utterly bear no difference and have the same decision as Theorem 2 has shown. In fact, the selection of policy depends on the value of r when there are no material differences in the holding cost between the two warehouses. If r < 1, when deterioration effect in OW is smaller than in RW, LIFO is suggested in order to avoid a higher cost in RW due to a higher invenTable 1 Comparison of policy by varying deterioration rate r
FIFO RF
0.1 0.5 1 2 4
þW
2305.8 2311.4 2317.7 2328.4 2342.1
LIFO BF
TC F
RL þ W
BL
TC L
882.6 902.5 927.1 975.7 1070.4
7061.3 7219.9 7416.7 7805.2 8563.3
2497.7 2419.3 2317.7 2100.7 1588.6
837.2 878.0 927.1 1018.5 1170.8
6697.5 7024.1 7416.7 8147.8 9366.3
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tory deterioration. On the other hand, if r > 1, FIFO is preferred to LIFO. Defining cost penalty DTC% = (TCL TCF)/TCF, in the case of relatively higher deterioration rate in OW, namely r = 2, and r = 4, the cost penalty of using LIFO policy are 4.39% and 9.37% respectively. By letting (a, b) = (0.0625, 0.05), i.e., r = 1.25, other parameters remain the same, Table 2 shows the impact that H and F have on the optimal policy. We have the following observations: 1. Under the L2 system, FIFO policy will always suggest a lower total cost than LIFO when F 5 H. While for H = 8, the optimal policy suggests the L1 policy, it is unnecessary to make any differentiation. 2. The higher the value of holding cost in H and F, the higher the value of TCF and TCL. While it is indicated that TCL is more sensitive to a change in H than a change in F, and vice versa, TCF is more sensitive to the change of F. When F increases, it is expected that FIFO has a higher increase in total cost than LIFO, as it implies higher holding cost in RW, since inventory items have to be carried longer than that of LIFO. 3. Under the assumption that OW is to be stored first, any changes in RW parameters (F or b) will not change the decision from L2 to L1, or vice versa, from L1 to L2 in both two models. From Table 2, for example, for H = 2 (where RL1 > W ), L2 is suggested as the optimal solution except when F = 8
Table 2 Comparison of the difference in policy under varying holding cost combination H
F
FIFO
LIFO
RF ðþW Þ
BF
TC F
RL ðþW Þ
BF
TC L
2
2 4 8
2417.7 1715.9 1200.0(W)
915.8 925.6 932.2
7326.8 8044.8 8365.7
2370.2 1957.1 1646.7
926.0 961.7 992.2
7408.6 7694.3 7938.1
4
2 4 8
2429.5 1721.3 1200.0(W)
996.5 1084.8 1091.2
7971.7 8678.2 8973.8
1967.8 1684.1 1481.3
1073.9 1089.9 1105.5
8591.4 8719.4 8820.7
8
2,4,8
1097.2(L1)
1268.9
10151.2
1097.2(L1)
1268.9
10151.2
Table 3 Analysis of change in various parameters has on policy choice 0
0
TC L
TC F
DTC 0 %
Policy suggest
0
W /W
0.5 2
8075.2 8729.7
7549.7 8729.7
6.96% 0%
FIFO L1
P 0 /P
0.5 2
7100.9 9223.3
6858.4 8404.7
3.54% 9.74%
FIFO FIFO
D 0 /D
0.5 2
6668.3 9792.3
6241.2 9314.6
6.84% 5.13%
FIFO FIFO
C 01 =C 1
0.5 2
8170.0 9244.5
7462.7 8792.9
9.48% 5.14%
FIFO FIFO
C 02 =C 2
0.5 2
7360.4 9456.7
7008.6 8620.8
5.02% 9.69%
FIFO FIFO
C 03 =C 3
0.5 2
6170.6 11782.5
5936.9 10908.3
3.94% 8.01%
FIFO FIFO
872
C.C. Lee / European Journal of Operational Research 174 (2006) 861–873
where L1 is to be used but at full capacity (W). Similarly, in LIFO, when F increase (under H = 2 and 4 where L2 is used) it would not reverses back to L1 system. In fact, only as F ! 1, L1(W) would be the optimal solution of LIFO, a similar result has been shown in (12–56) of Hartley [7]. The sensitivity analysis, with respect to other parameters on the total system cost is examined. The results are summarized in Table 3. The following inference may be drawn from Table 3. 1. The range of DTC 0 % is from 3.54% to 9.74%. The average value of DTC 0 % is about 6.68%. 2. The value of DTC 0 % is more sensitive to the parameter of subset P, C1, C2, C3, and less sensitive to parameter D. 3. The higher the value of subset W, D, C1, the smaller the value of DTC 0 %, but the higher the value of subset P, C 2, C3, the higher the value of DTC 0 %. 4. Changes in the parameter subset W, P, D, C1, C2, C3 do not change the optimal dispatching policy.
5. Summary and conclusions Previous literature on two-warehouse inventory model has always assumed that inventory holding cost in RW is higher than OW. This resulted in a LIFO flow of inventory that items in RW must be consumed prior to OW to avoid higher holding cost. This assumption is not necessarily true in the real world because RW is a specialized operation faced with severe competition that the opportunity to gain lower holding cost than OW is higher. Most important for managers that deal with perishable products using FIFO, rather than LIFO, is a common accepted practice of making sure that the products are dispatched at its maximum freshness. In this paper, a two-warehouse inventory model with the FIFO dispatching policy for deteriorating inventory items was proposed. It has been proven that when deterioration rate is the same in the two warehouses, FIFO is less expensive than LIFO provided that holding cost in RW will be lower than OW. In addition, Pakkala and Achary’s two-warehouse LIFO model has been sufficiently modified to be more complete. The modified LIFO model has proven to have a lower cost than Pakkala and Achary’s model under their assumption that H < F, when a is not significantly less than b. Numerical analysis have indicated {a, b, H, F} are the key set of factors in choosing LIFO or FIFO. Particularly, when RW parameters {b, F} are superior to that of OW {a, H}, in this case FIFO would be employed rather than LIFO. From the analysis, it was pointed out that TCL is more sensitive to a change in H than a change in F, and to the contrary, TCF is more sensitive to a change in F. Other parameters such as {P, D, W, C1, C2, C3} would have impacted solely on the magnitude but not in the directions between the two policies.
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