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Deterministic inventory model with two levels of storage, a linear trend in demand and a fixed time horizon
Computers & Operations Research 28 (2001) 1315}1331
Deterministic inventory model with two levels of storage, a linear trend in demand and a "xed time horizon S. Kar *, A.K. Bhunia , M. Maiti Haldia Institute of Technology, Debhog-721657, Haldia Midnapore, West Bengal, India Department of Applied Mathematics, Vidyasagar University, Midnapore 721102, India Received 1 February 1999; received in revised form 1 October 1999; accepted 1 March 2000
Abstract A deterministic inventory model is developed for a single item having two separate storage facilities (owned and rented warehouses) due to limited capacity of the existing storage (owned warehouse) with linearly time-dependent demand (increasing) over a "xed "nite time horizon. The model is formulated by assuming that the rate of replenishment is in"nite and the successive replenishment cycle lengths are in arithmetic progression. Shortages are allowed and fully backlogged. As a particular case, the results for the model without shortages are derived. Results are illustrated with two numerical examples. Scope and purpose Throughout the world, the production of food grains is periodical. Normally, in countries where state control is less, the demand of essential food grains is lowest at the time of harvest and goes up to the highest level just before the next harvest. This phenomenon is very common in developing third world countries where most of the people are landless or marginal farmers. At the time of harvest, they share some grain/product with landowners and as soon as the small inventory is exhausted, they are forced to buy food grains from the open market. As a result, demand for food grains increases with time in a period along with the number of the people whose initial stock of food grains gets exhausted. In this paper, a two-storage inventory model with time-dependent demand and "xed time horizon is developed and solved by a mathematical programme based on gradient method. This methodology of model development and its solution are quite general and it can be applied to inventory models of any product whose production is periodical and demand increases linearly with time. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Inventory management; Two storage; Deterioration; Finite time horizon
* Corresponding author. Tel.: #91-3224-52900; fax: #91-3224-52800. E-mail address: [email protected] (S. Kar). 0305-0548/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 0 0 ) 0 0 0 4 2 - 3
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1. Introduction It is well known to all that the harvest of food grains like paddy, rice, wheat, etc. is periodical. In the rural areas of India, a large number of landless people (small farmers and land labourers) live with small tracts of land and produce food grains by cultivating either their own land and/or the land of a landlord by sharing at a certain ratio. For various reasons, some of them are forced to sell a part of their product and buy grains from the market towards the end of the production cycle. Therefore, the rate of demand for food grains remains partly constant and increases partly with time for a "xed time horizon (i.e. for a calendar year). In the last few years, inventory problems involving time variable demand patterns have received attention from several researchers. This type of problem was "rst discussed by Stanfel and Sivazlian [1]. But, there is no speci"c assumption concerning demand in their problem. Next, Silver and Meal [2] established an approximate solution technique of a deterministic inventory model with time-dependent demand. Donaldson [3] developed an optimal algorithm for solving the classical no-shortage inventory model analytically with linear trend in demand over a "xed time horizon. However, the solution procedure requires a lot of computational time and cannot easily be employed to determine the values of the optimal decision variables. To remove the computational and conceptual complexity of Donaldson's [3] optimal analytic approach, several researchers employed search methods for solving the problem. Among them, Silver [4], Henry [5], Phelps [6], Buchanan [7], Mitra et al. [8], Ritchie [9] and others are worth mentioning, but none of them considered shortages. Considering shortages, Dave [10], Deb and Chaudhuri [11], Goyal et al. [12], Datta and Pal [13], and Horiga [14,15] developed an exact/heuristic solution following Donaldson's or alternative approaches. Among these papers, the solution procedure was computationally complicated except Datta and Pal where it was very simple and easy to calculate the values of the decision variables. They assumed that the successive replenishment cycles were diminishing by a constant amount, i.e. the lengths of successive replenishment cycles are in arithmetic progression (AP). Other recent papers of related topic were written by Chung and Ting [16], Goyal et al. [17], Teng et al. [18], Bhunia and Maiti [19], Chakrabarti and Chaudhuri [20] and others. The important problem associated with the inventory maintenance is to decide where to stock the goods. This problem does not seem to have attracted the attention of researchers in this "eld. In the existing literature, it is found that classical inventory models generally deal with a single storage facility. The basic assumption in these models is that the management owned a storage with unlimited capacity. In the "eld of inventory management, this is not always true. When an attractive price discount for bulk purchase is available or the cost of procuring goods is higher than the other inventory related costs or there are some problems in frequent procurement or the demand of items is very high, management then decides to purchase (or produce) a huge quantity of items at a time. These items cannot be stored in the existing storage, viz., the owned warehouse (OW) with limited capacity. Then for storing the excess of items, a (sometimes more than one) warehouse is hired on a rental basis. This rented warehouse (RW) may be located near the OW or a little away from it. It is generally assumed that, the holding cost in the RW is greater than the same as in OW. Hence, the items are stored "rst in OW and only excess stock is stored in the RW. Further, the items of RW are transferred to OW in a continuous release pattern to meet the demand until the stock level in the RW is emptied and then the items of the OW are released.
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Recently, several authors have considered extensions of the basic two warehouse inventory model discussed by Hartely [21]. Sarma [22] developed a deterministic inventory model with in"nite production rate and two levels of storage. Murdeshwar and Sathi [23] made an extension to the case of "nite production rate. Dave [24] recti"ed the errors and gave a complete solution for the model given by Sarma [23] and Murdeshwar and Sathi [23]. In the above models, the analysis is carried out without taking shortages. Further, Goswami and Chaudhuri [25] considered two-storage models with and without shortages, allowing time-dependent demand (linearly increasing). Bhunia and Maiti [26] developed the same inventory model correcting and modifying the assumptions of Goswami and Chaudhuri [25]. The above models are developed assuming bulk release pattern. Also, by considering constant demand, Sarma [27] presented a model for deteriorating items with an in"nite replenishment rate allowing for shortages. Pakkala and Achary [28] developed the same with "nite replenishment for the case of a continuous release pattern in the rented warehouse. In their analysis, the transportation cost for transferring the items from the RW to the OW was not taken into account. Recently, Bhunia and Maiti [29] studied a two warehouse inventory model for deteriorating items considering linearly time-dependent demand and shortages. The model is developed for an in"nite time horizon, but the entire cycle over the "rst period cannot repeat after completion of the "rst cycle. It can be subsequently repeated with a changed value of the constant part of the linear time-dependent demand. The present problem is also concerned with an inventory system having two-storage facilities and a linearly increasing time dependent demand pattern for a prescribed "nite time horizon. Here, the replenishment cycle lengths are in AP as in Datta and Pal [13]. In the existing literature of inventory models for a "xed time horizon, replenishment cost was taken to be the same. However, it is well known that for increasing demand, the lot size of successive cycles increase. Therefore, the replenishment cost should not be the same for successive replenishments. Again, this cost includes clerical and administrative costs, telephone charges, telegrams, transportation cost, loading and unloading costs, etc. Of these costs, only transportation cost, loading and unloading costs are dependent on the lot size. Generally, big merchants purchase the items in terms of round lots, i.e. rail wagons, full trucks, etc. But, in a developing country, like in India, the path of the movement of commodities for sale is as follows:
In these countries, retailers generally purchase the commodities, especially food grains, in terms of lots (some particular amount of quantities) depending upon the capital available to them. After
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that they join together to hire a truck/small transporting vehicle to transport the commodities to the selling place. The same type of procedure is followed by small retailers for purchase from retailers (country side), with the exception that here the mode of transport may be bullock cart, country boat, cycle van, auto rickshaw, etc. Keeping this scenario in mind, we have formulated the model assuming replenishment cost is a linear function of lot size. As a result, this cost is linearly dependent on the lot size which is included in the analysis of the proposed model. We formulate and analyze the model for the case of continuous release pattern for transferring the items from the RW to the OW. The transportation cost for transferring the goods from the RW to the OW is taken into account whereas it was neglected in Sarma [27] and Pakkala and Achary [28]. Backlogged shortages are allowed. As a special case, the case without shortages is also discussed. Finally, two numerical examples are considered to illustrate the results. Also, a sensitivity analysis on the optimal reorder quantity and the total cost has been presented for the variations of some parameters in the shortages as well as no-shortages case.
2. Assumptions and notations To develop the proposed model the following assumptions and notations are used. (i) The system operates for a prescribed period of H units of time (planning horizon). At time t"0 and t"H, inventory level is zero. (ii) The demand rate f (t) at any instant t is a linear function of t such that, f (t)"a#bt, a, b)0, 0)t)H. (iii) The lead time is constant. (iv) ¹ is the total time that elapses up to and including the ith cycle (i"1, 2,2, m), where G m denotes the total number of replenishment to be made during H. Clearly, ¹ "0 and ¹ "H. K (v) t is the total time that elapses up to and including the consumption period of the ith cycle G (i"1, 2,2, m). (vi) The consumption period of the ith cycle (i"1, 2,2, m) is K (0)K)1) fraction of the ith cycle length (¹ !¹ ). Then, G G\
K¹ #(1!K)¹ , i"1, 2,2, (m!1), G G\ t" G H, i"m. (vii) Shortages, if any, are completely backlogged and cleared as soon as a fresh stock arrives. Shortages are not allowed in the "nal cycle. (viii) ¹ is the length of the "rst replenishment cycle and u is the rate of reduction of the successive cycle lengths. (ix) The replenishment rate is in"nite and replenishments are instantaneous. At every replenishment, a variable lot size q is ordered so as to raise the initial inventory level of the ith cycle to G the order level of S units (i"1, 2,2, m) after ful"lling the backlogged quantity of the earlier G cycle, i.e. (i!1)th cycle.
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(x) The replenishment cost F for the ith cycle (i"1, 2,2, m) is partly constant and partly G dependent on the lot size during that cycle and is of the following form:
(xi) (xii)
(xiii) (xiv) (xv)
F "A#pq , G G where A and p are the constant part of the replenishment cost for the ith cycle and the additional cost per unit of items, respectively. The OW has limited capacity of = units and the RW has unlimited capacity. The holding cost h , h per unit per unit time for items in the RW and the OW, respectively, and the shortage cost C per unit per unit time are known and constant during the planning time horizon H. R , I (i"1, 2,2, m) be the mean stock level of inventory carried in the RW and the OW, G G respectively. E (i"1, 2,2, m) be the total amount of back order quantities over the period (t , ¹ ). G G G For transferring the goods from the RW to the OW, the transportation cost is C per unit. R
3. Model description and analysis Let the on-hand inventory levels in the OW and the RW be = and S !=, respectively, at time G t"¹ after ful"lling the back order quantities (q !S ). It is assumed that the quantities in the G\ G G OW are consumed only after consuming the quantities kept in the RW. In both warehouses, the stock declines due to demand only. After time t"¹ , the stock level in the RW decreases due to G\ demand and falls to zero at time t"t . Then in the OW, the stock level decreases after t"t and G G falls to zero at t"t . The shortages begin to accumulate after time t"t and continue up to the G G time t"¹ when the next lot arrives. This entire cycle repeats m times during (0, H). The pictorial G representation of the inventory system of the RW and the OW is given in Fig. 1. Our problem is to determine the optimal reorder and shortage points and hence to determine the optimal values of m, n and K which minimize the total cost over the time horizon (0, H). The stock depletion at OW during ¹ )t)¹ , i"1, 2,2, m is due to the demand only of G\ G the items which are disposed continuously as they come to the selling point. Therefore, the rate of change of stock at RW at any instant t (due to stock depletion) is equal to the demand at that time. Let Q (t) denote the amount in inventory at time t during the ith cycle [¹ )t)¹ , G G\ G i"1, 2,2, m] (see Fig. 1). The di!erential equation governing the system during the ith cycle is dQ (t)/dt"!f (t), ¹ )t)¹ , i"1, 2,2, m G G\ G subject to the condition Q (t)"0 at t"t , i"1, 2,2, m. G G The solution of (1) is
RG
f (u) du, ¹ )t)¹ , i"1, 2,2, m. G\ G R Then, from (2), the inventory level S at the beginning of the ith cycle is given by G RG S" f (u) du, i"1, 2,2, m. G G\ 2 Q (t)" G
(1)
(2)
(3)
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Fig. 1.
The lot size q for the ith cycle is given by G 2G\ RG q "S # f (u) du" f (u) du, i"1, 2,2, m, G G RG\ RG\ where t "0. Using the condition, Q (t)"= at t"t in (2), the following relation can be derived: G G RG =" f (u) du, i"1, 2,2, m RG or
(4)
="a(t !t )#b(t!t )/2 G G G G or bt #2at !(2at#bt!2=)"0, i"1, 2,2, m. G G G G The above equation gives the only admissible solution t "[!a#(a#b(2at #bt!2=]/b. G G G
(5)
(6)
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According to the earlier assumption, the (i#1)th replenishment time ¹ can be expressed as G ¹ "i¹!i(i!1)/2, i"0, 1, 2,2, m. (7) G The length of ith cycle is ¹ !¹ "¹!(i!1), i"1, 2,2, m. G G\ As the sum of the lengths of m replenishment cycles is H,
(8)
K [¹!(i!1)]"H. G On simplifying the above relation, we have (9)
¹"(m!1)/2#H/m.
The average amount in inventory in the RW during the ith cycle i.e. during the interval (¹ , t ) is G\ G RG R" Q (t) dt!=(t !¹ ), i"1, 2,2, (m!1) (10) G G G G\ G\ 2 and that in OW during (¹ , t ) is G\ G RG I "=(t !¹ )# Q (t) dt. (11) G G G\ G G R Also
R " K
RK
2K\
Q (t) dt!=(t !¹ ) K K K\
(12)
and
& Q (t) dt. (13) K RK The total amount of back order quantities over the period (t , ¹ ) is given by G G 2G E " (¹ !t)(a#bt) dt, i"1, 2,2, (m!1). (14) G G RG For transferring the stocks of RW to OW, the transportation cost ; during the ith cycle is given by G ; "C (S !=) G R G "C [a(t !¹ )#b(t!¹ )/2!=], i"1, 2,2, (m!1) (15) R G G\ G G\ and I "=(t !¹ )# K K K\
; "C (S !=) K R K "C [a(H!¹ )#b(H!¹ )/2!=]. R K\ K\
(16)
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The total cost of the proposed inventory system during the planning horizon H is K\ K\ C"mA#p q # [h R #h I #C E #; ]#h R #h I #; G G G G G K K K G G K\ "mA#p(aH#bH/2)# [(h !h )at /2#bt /3#=¹ G G G\ G #h a¹ /2#b¹ /6!(at #bt/2)¹ #h (at/2#bt/3) G\ G\ G G G\ G G #C a¹ (¹ !t )!(a!b¹ )(¹!t)/2!b(¹!t)/3 G G G G G G G G #C a(t !¹ )#b(t!¹)/2!=]#(h !h )at /2#bt /3#=¹ R G G\ G G K K K\ #h a¹ /2#b¹ /6!(aH#bH/2)¹ #h (aH/2#bH/3) K\ K\ K\ #C a(H!¹ )#b(H!¹ )/2!=. R K\ K\
(17)
According to the earlier assumptions, t is a function of K, ¹ where ¹ is a function of ¹ and ¹ is G G G another function of m and . As a result, the cost function C in (17) is a composite function of three variables m, and K of which m is a discrete variable and , K are continuous. Let it be C(m, , K). Our problem is to determine the values of m, and K which minimize C. For a given value m ('1) of m, the optimum values of and K which minimize the cost function C are the solutions of the equations C/"0 and C/K"0,
(18)
provided these values of and K satisfy the conditions: C/'0, C/K'0 and (C/) (C/K)!(C/K)'0 Eqs. (18) are two simultaneous equations in and K. The optimal values of say, (m ) and K say, K(m ) can be obtained from (18) by the Newton}Raphson method for m"m ('1). The corresponding optimal value of C is C(m , (m ), K(m ))"[CH(m ), say] which can be calculated from (17). Now putting m "2, 3, 4,2, etc., one can easily calculate CH(2), CH(3),2, etc. For m"1, the system reduces to a single period of "nite time horizon without shortages. In this case, the cost for the period H is independent of and K and is given by CH(1)"A#p(aH#bH/2)#(h !h )(at /2#bt /3)#h (aH/2#bH/3) #C (aH#bH/2!=), R where
(19)
t "[!a#(a#b(2aH#bH!2=]/b. (20) The minimum value of CH(1), CH(2), 2 is the optimal cost and the corresponding values of m, (m) and K(m) are their optimal values.
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3.1. Particular case If no shortage in inventory is allowed in any cycle, the length of the stock-out integral [t , ¹ ] G G should be zero. In that case, ¹ !t "(1!K)¹ !(1!K)t "0. This implies that K"1 for all cycles. The G G G G no-shortage situation can be obtained by letting K"1,
¹ , i"1, 2,2(m!1), i.e. t " G G H, i"m.
(21)
When shortages are not allowed, the total cost function (total cost for the entire time horizon H) can be obtained from (17), by substituting t "¹ and is given by G G K\ C"mA#p(aH#bH/2)# [(h !h )at/2#bt/3#=¹ G G G\ G #h a¹ /2#b¹ /6!(a¹ #b¹/2)¹ #h (a¹/2#b¹/3) G\ G\ G G G\ G G #C a(¹ !¹ )#b(¹!¹ )/2!=], R G G\ G G\
(22)
where t "[!a#(a#b(2a¹ #b¹!2=)]/b, i"1, 2,2, m. G G G
(23)
The optimal number of replenishments, optimal replenishment times and the minimum value of C can be obtained by following the same procedure as in the with shortage case.
4. Numerical illustration The proposed method is illustrated for both with shortage and without shortage cases by considering two examples. 4.1. Example I. (with shortage case) Let A"$15, p"$0.5, a"50, b"3, h "$0.5, h "$0.3, C "$3, ="25 quintal, C "$0.3, R H"8 months. Eqs. (18) are solved for and K(0(K(1) by the Newton}Raphson Method for di!erent integral values of m . Then putting these values for m, and K in Eq. (17), the corresponding values of C are obtained. Computed results are displayed in Table 1. From Table 1, it is seen that the total cost C becomes minimum, i.e. C"$510.29 for m "11. Hence, the optimal values of m , , K and ¹ become m "11, "0.023768, K"0.772758, ¹"0.846111 month.
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Table 1 Optimal solution for `with shortagea case m
K
¹
C
1 2 3 4 5 6 7 8 9 10 11 12 13
} 1.071684 0.418281 0.216467 0.130739 0.087075 0.062094 0.046495 0.036132 0.028953 0.023768 0.019904 0.016950
1 0.843421 0.835503 0.827612 0.819743 0.811889 0.804006 0.796176 0.788389 0.780570 0.772758 0.764951 0.757194
8 4.535842 3.084947 2.324701 1.861477 1.551021 1.329138 1.162733 1.033418 0.930287 0.846111 0.776140 0.717087
1128.30 833.50 672.50 599.98 561.11 538.39 524.72 516.62 512.10 510.32 510.29 511.64 514.04
Denotes the optimal solution.
4.2. Example II (without shortage case) Avoiding the shortage cost, we take the same parameters as in Example I above. In this case, results are shown in Table 2 which gives the minimum total cost as $540.77 for m "11, "0.014076, ¹"0.797654 month.
Table 2 Optimal solution for ano shortagea case m
¹
C
2 3 4 5 6 7 8 9 10 11 12 13
0.352326 0.162222 0.093328 0.060843 0.042989 0.032138 0.025054 0.020176 0.016674 0.014076 0.007210 0.001055
4.176163 2.828889 2.139993 1.721686 1.440807 1.239271 1.087690 0.969592 0.875035 0.797654 0.733190 0.678686
884.60 719.07 642.00 599.60 574.36 558.85 549.40 543.97 541.35 540.77 541.71 543.82
Denotes the optimal solution.
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Comparing the optimal solutions of Examples I and II, it is observed that the optimal system cost increases signi"cantly in the no-shortage case. However, the time period of the "rst replenishment cycle is greater when shortages are permitted. But, the reorder numbers of both the inventory system (shortage and no-shortage case) are same. Hence from an economic point of view, inventory backlogging is thus bene"cial as the system cost is considerably reduced by allowing shortages.
5. Sensitivity analysis Using the numerical examples I and II mentioned earlier, a sensitivity analysis is performed to study the e!ect of changes of the values of some parameters on the optimal reorder number mH and the optimal total system cost CH holding the values of the other parameter values as same. The results shown in Tables 3}4 are self-explanatory. Table 3 E!ects of changes in some parameters on the optimal reorder number and the optimal system cost in the `with shortagea case % change in parameters value
m
Total cost
% change in system cost
!20 !10 10 20
8 9 12 13
395.09 451.85 570.25 631.93
!22.58 !11.45 #11.75 #23.84
Owned ware-house capacity =
!20 !10 10 20
10 10 11 11
528.07 519.10 501.21 492.34
#03.48 #01.73 !01.78 !03.52
Location parameter of demand `aa
!20 !10 10 20
9 10 11 12
438.39 474.30 545.64 581.08
!14.08 !07.05 #06.93 #13.87
!20 !10 10 20
10 10 11 11
493.07 503.34 518.73 527.17
!03.37 !01.36 #01.65 #03.31
!20 !10 10 20
10 10 11 11
503.88 507.41 512.70 514.76
!01.25 !00.56 #00.47 #00.88
Parameters
Time horizon H
Shape parameter of demand `ba
Shortage cost C
Denotes the optimal solution.
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Table 4 E!ects of changes in some parameters on the optimal reorder number and the optimal system cost in the `no shortagea case % change in parameters value
m
Total cost
% change in system cost
!20 !10 10 20
9 10 12 13
418.10 478.49 604.94 670.10
!22.68 !11.52 #11.87 #24.08
Owned ware-house capacity =
!20 !10 10 20
10 11 11 12
560.74 550.92 530.84 521.03
#03.69 #01.88 !01.84 !03.65
Location parameter of demand `aa
!20 !10 10 20
10 10 11 12
463.24 502.21 579.23 617.40
!14.34 !07.13 #07.11 #14.17
!20 !10 10 20
10 10 11 11
522.35 531.56 549.99 559.21
!03.41 !01.70 #01.70 #03.41
Parameters
Time horizon H
Shape parameter of demand `ba
Denotes the optimal solution.
6. Applicability of the model The usual supply chain of commodities has been indicated in the `Introductiona. It may be a point of interest to know, who will be the user of the proposed model in the said supply chain? Normally, big merchants and wholesalers do not have direct contact with the common customers. For this reason, they operate the business with an o$ce and a big warehouse. On the contrary, both the retailer and small retailer have to deal directly with the ordinary customers and maintain a decorative show-room to impress and motivate the customers in order to boost their sale. They, therefore, adapt two-level storage facilities to run their business smoothly. Hence, the proposed model is applicable for the business of retailers and small retailers. In the developing countries, like India, etc., the two levels of storage facilities were seldomly used by the business community, they normally use the age old single warehouse system. Now-a-days, due to globalisation of the market with the introduction of multinationals in the business, there is a trend among the business houses specially retailers and small retailers of di!erent multinational products to compete with each other for sale and as a result, they use the decorative show-room to boost their items in addition to a separate warehouse. Another reason for the adoption of the present model is the crisis of having large space in the important market places, like super market, corporation market, etc. So, from the
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economical point of view, the proposed model will be more bene"cial to the business houses as it gives better business to them.
7. Concluding remarks In this paper, we have discussed the inventory problem for a single item with linear trend in demand (increasing) and shortages over a "xed and "nite time horizon. The replenishment cost is taken to be dependent on the lot size of the current replenishment. This feature of the model makes it the most general among the ones referred to in the introduction. Another feature here is that we have taken the transportation cost to transfer the items of the RW to the OW in a continuous release pattern, which is not considered by others. The present model is applicable for food grains like paddy, rice, wheat, etc., as the demand of the food grains increases with time for a "xed time horizon, i.e., for a calendar year. It is also applicable for other items where the demand is dependent linearly with time. The case of decreasing demand is also of great importance for a "nite horizon model. The results obtained in this article are also valid for linearly decreasing demand. For future development of research, one can extend the model developed in this article by incorporating more realistic situations, such as, multiple items and quantity discount policies.
Appendix For the development of the probabilistic model, the following assumptions are taken: (i) g(x ) is the probability density function for demand x which is a random variable during the G G ith cycle (¹ , ¹ ) (0)x (R). This demand is supposed to occur in a uniform pattern during G\ G G the ith cycle. Other assumptions are same as those given in Section 2. Two cases may occur in this model depending on the relative values of the demand x . We study G these cases separately. Case I: When shortages do not occur. Let Q (t) be the inventory level at any time t(¹ )t)¹ ) VG G\ G of the system and x is the demand during the "xed interval (¹ , ¹ ) so that the demand rate is G G\ G x /(¹ !¹ ) during the period. G G G\ Then the di!erential equation describing the system are given by dQ (t)/dt#x /(t !¹ )"0, ¹ )t)¹ , VG G G G\ G\ G
i"1, 2,2, m.
(A.1)
Using initial conditions, we have the solution as Q (t)"S !x (t!¹ )/(¹ !¹ ), i"1, 2,2, m. VG G G G\ G G\ Since shortages do not occur we have Q (¹ )*0 or x )S . VG G G G
(A.2)
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The lot size q for the ith cycle is given by G q "S . (A.3) G G Again, from Q (t)"= at t"t , we have VG G t "¹ #(S !=)/x . (A.4) G G\ G G Here, the total number of items R (x ) and I (x ) carried in inventory in the RW and the OW, G G G G respectively, are given by
R (x )" G G
RG
2G\
Q (t) dt!=(t !¹ ), x )S , i"1,2,2, m!1 VG G G\ G G
(A.5)
and
I (x )"=(t !¹ )# G G G G\
2G G
R
Q (t) dt, x )S , i"1,2,2, m!1. VG G G
(A.6)
Also,
R (x )" K G
RK
2K\
Q (t) dt!=(t !¹ ), x )S VG K K\ G G
(A.7)
and
& Q (t) dt, x )S . VG G G RK The total number of shortages during this period is I (x )"=(t !¹ )# K G K K\
(A.8)
M (x )"0, x )S . (A.9) G G G Case II: when shortages occur. In this case, the system is the same as that of the deterministic model except that the demand x which is a random variable with probability density function f (x ) G G during the scheduling period (¹ , ¹ ). G\ G Then the di!erential equations describing the system are given by dQ (t)/dt#x /(t !¹ )"0, ¹ )t)t , VG G G G\ G\ G dQ (t)/dt#x /(¹ !t )"0, t )t)¹ , i"1, 2,2, m. VG G G G G G Using initial conditions, we have the solutions as
S !x (t!¹ )/(¹ !¹ ), ¹ )t)t , G G\ G G\ G G Q (t)" G . VG x (t !t)/(¹ !¹ ), t )t)¹ , i"1, 2,2, m G G G G\ G G As Q (t )"0, we have VG G S "x (t !¹ )/(¹ !¹ ). G G G G\ G G\ Since shortages occur, we have Q (¹ )(0 or x 'S . VG G G G
(A.10) (A.11)
(A.12)
(A.13)
S. Kar et al. / Computers & Operations Research 28 (2001) 1315}1331
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The lot size q for the ith cycle is given by G q "x (t !t )/(¹ !¹ ). (A.14) G G G G\ G G\ Therefore, the total number of items R (x ) and I (x ) carried to inventory in the RW and the G G G G OW, respectively are given by
R (x )" G G
RG
2G\
Q (t) dt!=(t !¹ ), x 'S , i"1, 2,2, m!1 VG G G\ G G
(A.15)
and
I (x )"=(t !¹ )# G G G G\
2G G
R
Q (t) dt, x 'S , i"1, 2,2, m!1. VG G G
(A.16)
Also,
R (x )" K G
RK
2K\
Q (t) dt!=(t !¹ ), x 'S VG K K\ G G
(A.17)
and
& Q (t) dt, x 'S . VG G G RK The total number of shortages during this period is I (x )"=(t !¹ )# K G K K\
M (x )" G
(A.18)
2G
[!Q (t)] dt, VG RG "x (¹ !t ), x 'S , i"1, 2,2,(m!1). (A.19) G G G G G Therefore, the expected total cost of the proposed inventory system during the planning horizon H is
K\ K\ 1G 1G C"mA#p q (x )g(x ) dx # q (x )g(x ) dx # h R (x )g(x ) dx G G G G G G G G G G G G G G 1 G G 1G I (x )g(x ) dx # I (x )g(x ) dx # R (x )g(x ) dx #h G G G G G G G G G G G G G 1G 1G 1K #C M (x )g(x ) dx #C (S !=) #h R (x )g(x ) dx G G G G R G K K K K K 1G 1K I (x )g(x ) dx # I (x )g(x ) dx # R (x )g(x ) dx #h K K K K G K K K K K K K K 1K 1K #C (S !=). (A.20) R K
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The optimal values of m and S (i"1, 2,2, m) can be found out by using the same procedure of G the deterministic model.
References [1] Stanfel LE, Sivazlian BD. Analysis of systems in operations. Englewood Cli!s, NJ: Prentice-Hall, 1975. [2] Silver EA, Meal HC. A heuristic for selecting lot-size quantities for the case of a deterministic time varying demand rate and discrete opportunities for replenishment. Production Inventory Management 1973;14:64}74. [3] Donaldson WA. Inventory replenishment policy for a linear trend in demand-an analytical solution. Operation Research Quarterly 1977;28:663}70. [4] Silver EA. A simple inventory decision rule for a linear trend in demand. Journal of Operation Research Society 1979;30:71}5. [5] Phelps RI. Optimal inventory rule for a linear trend in demand with a constant replenishment period. Journal of Operation Research Society 1980;31:439}42. [6] Mitra A, Cox JF, Jesse RR. A note on determining order quantities with a linear trend in demand. Journal of Operation Research Society 1984;35:439}42. [7] Buchanan JT. Alternative solution methods for the inventory inventory replenishment problem under increasing demand. Journal of Operation Research Society 1980;31:615}20. [8] Henery RJ. Inventory replenishment policy for increasing demand. Journal of Operation Research Society 1979;30:611}7. [9] Ritchie E. The EOQ for linear increasing demand: a simple optimal solution. Journal of Operation Research Society 1984;35:949}52. [10] Dave U. A deterministic lot-size inventory model with shortages and a linear trend in demand. Naval Research Logistics 1989;36:507}14. [11] Deb M, Chaudhuri KS. A note on the heuristic for replenishment of trended inventories considering shortages. Journal of Operation Research Society 1987;38:459}63. [12] Goyal SK, Morin D, Nebebe F. The "nite horizon trended inventory replenishment problem with shortages. Journal of Operation Research Society 1992;43:1173}8. [13] Datta TK, Pal AK. A note on a replenishment policy for an inventory model with linear trend in demand and shortages. Journal of Operation Research Society 1992;43:993}1001. [14] Horiga M. The inventory replenishment problem with a linear trend in demand. Computer Engineering 1993;24:143}50. [15] Horiga M. The inventory lot-sizing problem with continuous time-varying demand and shortages. Journal of Operation Research Society 1994;45:827}37. [16] Chung KJ, Ting PS. A heuristic for replenishment of determining items with a linear trend in demand. Journal of Operation Research Society 1993;44:1235}41. [17] Goyal SK, Horiga MA, Alyan A. The trended inventory lot-sizing problem with shortages under a new replenishment policy. Journal of Operation Research Society 1996;47:1286}95. [18] Teng F, Chen M, Yeng H. An optimal recursive method for various inventory replenishment models with increasing demand and shortages. Naval Research Logistics 1997;44:791}806. [19] Bhunia AK, Maiti M. An inventory model of deteriorating items with lot-size dependent replenishment cost and a linear trend in demand. Applied Mathematical Modelling 1999;23:301}8. [20] Chakrabarti T, Chaudhuri KS. An EOQ model for deteriorating items with a linear trend in demand and shortages in all cycles. International Journal of Production Economics 1997;49:205}13. [21] Hartely RV. Operations research } a managerial emphasis. Santa Monica, CA: Goodyear Publishing Company, 1976. p. 315}7. [chapter 12]. [22] Sarma KVS. A deterministic inventory model with two levels of storage and an optimum release rule. Opsearch 1983;29:175}80. [23] Murdeshwar TM, Sathe YS. Some aspects of lot-size models with two levels of storage. Opsearch 1985; 22:255}62.
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[24] Dave U. On the EOQ models with two levels of storage. Opsearch 1988;25:190}6. [25] Goswami A, Chaudhuri KS. An economic order quantity model for items with two levels of storage for a linear trend in demand. Journal of Operation Research Society 1992;43:157}67. [26] Bhunia AK, Maiti M. A two warehouse inventory model for a linear trend in demand. Opsearch 1994;31:318}29. [27] Sarma KVS. A deterministic order-level inventory model for deteriorating items with two storage facilities. European Journal of Operation Research 1987;29:70}2. [28] Pakkala TPM, Achary KK. A deterministic inventory model for deteriorating items with two warehouses and "nite replenishment rate. European Journal of Operation Research 1992;57:71}6. [29] Bhunia AK, Maiti M. A two warehouses inventory model for deteriorating items with a linear trend in demand and shortages. Journal of Operation Research Society 1997;49:287}92.
S. Kar is a Lecturer in Mathematics at Haldia Institute of Technology, Midnapore, West Bengal, India. His areas of research work includes Inventory Management with Fuzzy and Stochastic Environments. A.K. Bhunia is a Lecturer in Mathematics at Haldia Institute of Technology, Midnapore, West Bengal, India. His research areas includes Inventory management and Development of Interval Analysis and its application in Inventory Management. Dr. Bhunia received his Ph.D. degree from the Vidyasagar University, Midnapore. M. Maiti is a professor in the Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, West Bengal, India. His "elds of research are Elasticity and Catastrophic Theory, Inventory Management in Fuzzy and Stochastic Environments, Development of Interval Analysis and its application in Inventory Management. Dr. Maiti received his Ph.D. degree from the University of Calcutta.
Deterministic inventory model with two levels of storage, a linear trend in demand and a "xed time horizon S. Kar *, A.K. Bhunia , M. Maiti Haldia Institute of Technology, Debhog-721657, Haldia Midnapore, West Bengal, India Department of Applied Mathematics, Vidyasagar University, Midnapore 721102, India Received 1 February 1999; received in revised form 1 October 1999; accepted 1 March 2000
Abstract A deterministic inventory model is developed for a single item having two separate storage facilities (owned and rented warehouses) due to limited capacity of the existing storage (owned warehouse) with linearly time-dependent demand (increasing) over a "xed "nite time horizon. The model is formulated by assuming that the rate of replenishment is in"nite and the successive replenishment cycle lengths are in arithmetic progression. Shortages are allowed and fully backlogged. As a particular case, the results for the model without shortages are derived. Results are illustrated with two numerical examples. Scope and purpose Throughout the world, the production of food grains is periodical. Normally, in countries where state control is less, the demand of essential food grains is lowest at the time of harvest and goes up to the highest level just before the next harvest. This phenomenon is very common in developing third world countries where most of the people are landless or marginal farmers. At the time of harvest, they share some grain/product with landowners and as soon as the small inventory is exhausted, they are forced to buy food grains from the open market. As a result, demand for food grains increases with time in a period along with the number of the people whose initial stock of food grains gets exhausted. In this paper, a two-storage inventory model with time-dependent demand and "xed time horizon is developed and solved by a mathematical programme based on gradient method. This methodology of model development and its solution are quite general and it can be applied to inventory models of any product whose production is periodical and demand increases linearly with time. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Inventory management; Two storage; Deterioration; Finite time horizon
* Corresponding author. Tel.: #91-3224-52900; fax: #91-3224-52800. E-mail address: [email protected] (S. Kar). 0305-0548/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 0 0 ) 0 0 0 4 2 - 3
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1. Introduction It is well known to all that the harvest of food grains like paddy, rice, wheat, etc. is periodical. In the rural areas of India, a large number of landless people (small farmers and land labourers) live with small tracts of land and produce food grains by cultivating either their own land and/or the land of a landlord by sharing at a certain ratio. For various reasons, some of them are forced to sell a part of their product and buy grains from the market towards the end of the production cycle. Therefore, the rate of demand for food grains remains partly constant and increases partly with time for a "xed time horizon (i.e. for a calendar year). In the last few years, inventory problems involving time variable demand patterns have received attention from several researchers. This type of problem was "rst discussed by Stanfel and Sivazlian [1]. But, there is no speci"c assumption concerning demand in their problem. Next, Silver and Meal [2] established an approximate solution technique of a deterministic inventory model with time-dependent demand. Donaldson [3] developed an optimal algorithm for solving the classical no-shortage inventory model analytically with linear trend in demand over a "xed time horizon. However, the solution procedure requires a lot of computational time and cannot easily be employed to determine the values of the optimal decision variables. To remove the computational and conceptual complexity of Donaldson's [3] optimal analytic approach, several researchers employed search methods for solving the problem. Among them, Silver [4], Henry [5], Phelps [6], Buchanan [7], Mitra et al. [8], Ritchie [9] and others are worth mentioning, but none of them considered shortages. Considering shortages, Dave [10], Deb and Chaudhuri [11], Goyal et al. [12], Datta and Pal [13], and Horiga [14,15] developed an exact/heuristic solution following Donaldson's or alternative approaches. Among these papers, the solution procedure was computationally complicated except Datta and Pal where it was very simple and easy to calculate the values of the decision variables. They assumed that the successive replenishment cycles were diminishing by a constant amount, i.e. the lengths of successive replenishment cycles are in arithmetic progression (AP). Other recent papers of related topic were written by Chung and Ting [16], Goyal et al. [17], Teng et al. [18], Bhunia and Maiti [19], Chakrabarti and Chaudhuri [20] and others. The important problem associated with the inventory maintenance is to decide where to stock the goods. This problem does not seem to have attracted the attention of researchers in this "eld. In the existing literature, it is found that classical inventory models generally deal with a single storage facility. The basic assumption in these models is that the management owned a storage with unlimited capacity. In the "eld of inventory management, this is not always true. When an attractive price discount for bulk purchase is available or the cost of procuring goods is higher than the other inventory related costs or there are some problems in frequent procurement or the demand of items is very high, management then decides to purchase (or produce) a huge quantity of items at a time. These items cannot be stored in the existing storage, viz., the owned warehouse (OW) with limited capacity. Then for storing the excess of items, a (sometimes more than one) warehouse is hired on a rental basis. This rented warehouse (RW) may be located near the OW or a little away from it. It is generally assumed that, the holding cost in the RW is greater than the same as in OW. Hence, the items are stored "rst in OW and only excess stock is stored in the RW. Further, the items of RW are transferred to OW in a continuous release pattern to meet the demand until the stock level in the RW is emptied and then the items of the OW are released.
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Recently, several authors have considered extensions of the basic two warehouse inventory model discussed by Hartely [21]. Sarma [22] developed a deterministic inventory model with in"nite production rate and two levels of storage. Murdeshwar and Sathi [23] made an extension to the case of "nite production rate. Dave [24] recti"ed the errors and gave a complete solution for the model given by Sarma [23] and Murdeshwar and Sathi [23]. In the above models, the analysis is carried out without taking shortages. Further, Goswami and Chaudhuri [25] considered two-storage models with and without shortages, allowing time-dependent demand (linearly increasing). Bhunia and Maiti [26] developed the same inventory model correcting and modifying the assumptions of Goswami and Chaudhuri [25]. The above models are developed assuming bulk release pattern. Also, by considering constant demand, Sarma [27] presented a model for deteriorating items with an in"nite replenishment rate allowing for shortages. Pakkala and Achary [28] developed the same with "nite replenishment for the case of a continuous release pattern in the rented warehouse. In their analysis, the transportation cost for transferring the items from the RW to the OW was not taken into account. Recently, Bhunia and Maiti [29] studied a two warehouse inventory model for deteriorating items considering linearly time-dependent demand and shortages. The model is developed for an in"nite time horizon, but the entire cycle over the "rst period cannot repeat after completion of the "rst cycle. It can be subsequently repeated with a changed value of the constant part of the linear time-dependent demand. The present problem is also concerned with an inventory system having two-storage facilities and a linearly increasing time dependent demand pattern for a prescribed "nite time horizon. Here, the replenishment cycle lengths are in AP as in Datta and Pal [13]. In the existing literature of inventory models for a "xed time horizon, replenishment cost was taken to be the same. However, it is well known that for increasing demand, the lot size of successive cycles increase. Therefore, the replenishment cost should not be the same for successive replenishments. Again, this cost includes clerical and administrative costs, telephone charges, telegrams, transportation cost, loading and unloading costs, etc. Of these costs, only transportation cost, loading and unloading costs are dependent on the lot size. Generally, big merchants purchase the items in terms of round lots, i.e. rail wagons, full trucks, etc. But, in a developing country, like in India, the path of the movement of commodities for sale is as follows:
In these countries, retailers generally purchase the commodities, especially food grains, in terms of lots (some particular amount of quantities) depending upon the capital available to them. After
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that they join together to hire a truck/small transporting vehicle to transport the commodities to the selling place. The same type of procedure is followed by small retailers for purchase from retailers (country side), with the exception that here the mode of transport may be bullock cart, country boat, cycle van, auto rickshaw, etc. Keeping this scenario in mind, we have formulated the model assuming replenishment cost is a linear function of lot size. As a result, this cost is linearly dependent on the lot size which is included in the analysis of the proposed model. We formulate and analyze the model for the case of continuous release pattern for transferring the items from the RW to the OW. The transportation cost for transferring the goods from the RW to the OW is taken into account whereas it was neglected in Sarma [27] and Pakkala and Achary [28]. Backlogged shortages are allowed. As a special case, the case without shortages is also discussed. Finally, two numerical examples are considered to illustrate the results. Also, a sensitivity analysis on the optimal reorder quantity and the total cost has been presented for the variations of some parameters in the shortages as well as no-shortages case.
2. Assumptions and notations To develop the proposed model the following assumptions and notations are used. (i) The system operates for a prescribed period of H units of time (planning horizon). At time t"0 and t"H, inventory level is zero. (ii) The demand rate f (t) at any instant t is a linear function of t such that, f (t)"a#bt, a, b)0, 0)t)H. (iii) The lead time is constant. (iv) ¹ is the total time that elapses up to and including the ith cycle (i"1, 2,2, m), where G m denotes the total number of replenishment to be made during H. Clearly, ¹ "0 and ¹ "H. K (v) t is the total time that elapses up to and including the consumption period of the ith cycle G (i"1, 2,2, m). (vi) The consumption period of the ith cycle (i"1, 2,2, m) is K (0)K)1) fraction of the ith cycle length (¹ !¹ ). Then, G G\
K¹ #(1!K)¹ , i"1, 2,2, (m!1), G G\ t" G H, i"m. (vii) Shortages, if any, are completely backlogged and cleared as soon as a fresh stock arrives. Shortages are not allowed in the "nal cycle. (viii) ¹ is the length of the "rst replenishment cycle and u is the rate of reduction of the successive cycle lengths. (ix) The replenishment rate is in"nite and replenishments are instantaneous. At every replenishment, a variable lot size q is ordered so as to raise the initial inventory level of the ith cycle to G the order level of S units (i"1, 2,2, m) after ful"lling the backlogged quantity of the earlier G cycle, i.e. (i!1)th cycle.
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(x) The replenishment cost F for the ith cycle (i"1, 2,2, m) is partly constant and partly G dependent on the lot size during that cycle and is of the following form:
(xi) (xii)
(xiii) (xiv) (xv)
F "A#pq , G G where A and p are the constant part of the replenishment cost for the ith cycle and the additional cost per unit of items, respectively. The OW has limited capacity of = units and the RW has unlimited capacity. The holding cost h , h per unit per unit time for items in the RW and the OW, respectively, and the shortage cost C per unit per unit time are known and constant during the planning time horizon H. R , I (i"1, 2,2, m) be the mean stock level of inventory carried in the RW and the OW, G G respectively. E (i"1, 2,2, m) be the total amount of back order quantities over the period (t , ¹ ). G G G For transferring the goods from the RW to the OW, the transportation cost is C per unit. R
3. Model description and analysis Let the on-hand inventory levels in the OW and the RW be = and S !=, respectively, at time G t"¹ after ful"lling the back order quantities (q !S ). It is assumed that the quantities in the G\ G G OW are consumed only after consuming the quantities kept in the RW. In both warehouses, the stock declines due to demand only. After time t"¹ , the stock level in the RW decreases due to G\ demand and falls to zero at time t"t . Then in the OW, the stock level decreases after t"t and G G falls to zero at t"t . The shortages begin to accumulate after time t"t and continue up to the G G time t"¹ when the next lot arrives. This entire cycle repeats m times during (0, H). The pictorial G representation of the inventory system of the RW and the OW is given in Fig. 1. Our problem is to determine the optimal reorder and shortage points and hence to determine the optimal values of m, n and K which minimize the total cost over the time horizon (0, H). The stock depletion at OW during ¹ )t)¹ , i"1, 2,2, m is due to the demand only of G\ G the items which are disposed continuously as they come to the selling point. Therefore, the rate of change of stock at RW at any instant t (due to stock depletion) is equal to the demand at that time. Let Q (t) denote the amount in inventory at time t during the ith cycle [¹ )t)¹ , G G\ G i"1, 2,2, m] (see Fig. 1). The di!erential equation governing the system during the ith cycle is dQ (t)/dt"!f (t), ¹ )t)¹ , i"1, 2,2, m G G\ G subject to the condition Q (t)"0 at t"t , i"1, 2,2, m. G G The solution of (1) is
RG
f (u) du, ¹ )t)¹ , i"1, 2,2, m. G\ G R Then, from (2), the inventory level S at the beginning of the ith cycle is given by G RG S" f (u) du, i"1, 2,2, m. G G\ 2 Q (t)" G
(1)
(2)
(3)
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Fig. 1.
The lot size q for the ith cycle is given by G 2G\ RG q "S # f (u) du" f (u) du, i"1, 2,2, m, G G RG\ RG\ where t "0. Using the condition, Q (t)"= at t"t in (2), the following relation can be derived: G G RG =" f (u) du, i"1, 2,2, m RG or
(4)
="a(t !t )#b(t!t )/2 G G G G or bt #2at !(2at#bt!2=)"0, i"1, 2,2, m. G G G G The above equation gives the only admissible solution t "[!a#(a#b(2at #bt!2=]/b. G G G
(5)
(6)
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According to the earlier assumption, the (i#1)th replenishment time ¹ can be expressed as G ¹ "i¹!i(i!1)/2, i"0, 1, 2,2, m. (7) G The length of ith cycle is ¹ !¹ "¹!(i!1), i"1, 2,2, m. G G\ As the sum of the lengths of m replenishment cycles is H,
(8)
K [¹!(i!1)]"H. G On simplifying the above relation, we have (9)
¹"(m!1)/2#H/m.
The average amount in inventory in the RW during the ith cycle i.e. during the interval (¹ , t ) is G\ G RG R" Q (t) dt!=(t !¹ ), i"1, 2,2, (m!1) (10) G G G G\ G\ 2 and that in OW during (¹ , t ) is G\ G RG I "=(t !¹ )# Q (t) dt. (11) G G G\ G G R Also
R " K
RK
2K\
Q (t) dt!=(t !¹ ) K K K\
(12)
and
& Q (t) dt. (13) K RK The total amount of back order quantities over the period (t , ¹ ) is given by G G 2G E " (¹ !t)(a#bt) dt, i"1, 2,2, (m!1). (14) G G RG For transferring the stocks of RW to OW, the transportation cost ; during the ith cycle is given by G ; "C (S !=) G R G "C [a(t !¹ )#b(t!¹ )/2!=], i"1, 2,2, (m!1) (15) R G G\ G G\ and I "=(t !¹ )# K K K\
; "C (S !=) K R K "C [a(H!¹ )#b(H!¹ )/2!=]. R K\ K\
(16)
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The total cost of the proposed inventory system during the planning horizon H is K\ K\ C"mA#p q # [h R #h I #C E #; ]#h R #h I #; G G G G G K K K G G K\ "mA#p(aH#bH/2)# [(h !h )at /2#bt /3#=¹ G G G\ G #h a¹ /2#b¹ /6!(at #bt/2)¹ #h (at/2#bt/3) G\ G\ G G G\ G G #C a¹ (¹ !t )!(a!b¹ )(¹!t)/2!b(¹!t)/3 G G G G G G G G #C a(t !¹ )#b(t!¹)/2!=]#(h !h )at /2#bt /3#=¹ R G G\ G G K K K\ #h a¹ /2#b¹ /6!(aH#bH/2)¹ #h (aH/2#bH/3) K\ K\ K\ #C a(H!¹ )#b(H!¹ )/2!=. R K\ K\
(17)
According to the earlier assumptions, t is a function of K, ¹ where ¹ is a function of ¹ and ¹ is G G G another function of m and . As a result, the cost function C in (17) is a composite function of three variables m, and K of which m is a discrete variable and , K are continuous. Let it be C(m, , K). Our problem is to determine the values of m, and K which minimize C. For a given value m ('1) of m, the optimum values of and K which minimize the cost function C are the solutions of the equations C/"0 and C/K"0,
(18)
provided these values of and K satisfy the conditions: C/'0, C/K'0 and (C/) (C/K)!(C/K)'0 Eqs. (18) are two simultaneous equations in and K. The optimal values of say, (m ) and K say, K(m ) can be obtained from (18) by the Newton}Raphson method for m"m ('1). The corresponding optimal value of C is C(m , (m ), K(m ))"[CH(m ), say] which can be calculated from (17). Now putting m "2, 3, 4,2, etc., one can easily calculate CH(2), CH(3),2, etc. For m"1, the system reduces to a single period of "nite time horizon without shortages. In this case, the cost for the period H is independent of and K and is given by CH(1)"A#p(aH#bH/2)#(h !h )(at /2#bt /3)#h (aH/2#bH/3) #C (aH#bH/2!=), R where
(19)
t "[!a#(a#b(2aH#bH!2=]/b. (20) The minimum value of CH(1), CH(2), 2 is the optimal cost and the corresponding values of m, (m) and K(m) are their optimal values.
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3.1. Particular case If no shortage in inventory is allowed in any cycle, the length of the stock-out integral [t , ¹ ] G G should be zero. In that case, ¹ !t "(1!K)¹ !(1!K)t "0. This implies that K"1 for all cycles. The G G G G no-shortage situation can be obtained by letting K"1,
¹ , i"1, 2,2(m!1), i.e. t " G G H, i"m.
(21)
When shortages are not allowed, the total cost function (total cost for the entire time horizon H) can be obtained from (17), by substituting t "¹ and is given by G G K\ C"mA#p(aH#bH/2)# [(h !h )at/2#bt/3#=¹ G G G\ G #h a¹ /2#b¹ /6!(a¹ #b¹/2)¹ #h (a¹/2#b¹/3) G\ G\ G G G\ G G #C a(¹ !¹ )#b(¹!¹ )/2!=], R G G\ G G\
(22)
where t "[!a#(a#b(2a¹ #b¹!2=)]/b, i"1, 2,2, m. G G G
(23)
The optimal number of replenishments, optimal replenishment times and the minimum value of C can be obtained by following the same procedure as in the with shortage case.
4. Numerical illustration The proposed method is illustrated for both with shortage and without shortage cases by considering two examples. 4.1. Example I. (with shortage case) Let A"$15, p"$0.5, a"50, b"3, h "$0.5, h "$0.3, C "$3, ="25 quintal, C "$0.3, R H"8 months. Eqs. (18) are solved for and K(0(K(1) by the Newton}Raphson Method for di!erent integral values of m . Then putting these values for m, and K in Eq. (17), the corresponding values of C are obtained. Computed results are displayed in Table 1. From Table 1, it is seen that the total cost C becomes minimum, i.e. C"$510.29 for m "11. Hence, the optimal values of m , , K and ¹ become m "11, "0.023768, K"0.772758, ¹"0.846111 month.
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Table 1 Optimal solution for `with shortagea case m
K
¹
C
1 2 3 4 5 6 7 8 9 10 11 12 13
} 1.071684 0.418281 0.216467 0.130739 0.087075 0.062094 0.046495 0.036132 0.028953 0.023768 0.019904 0.016950
1 0.843421 0.835503 0.827612 0.819743 0.811889 0.804006 0.796176 0.788389 0.780570 0.772758 0.764951 0.757194
8 4.535842 3.084947 2.324701 1.861477 1.551021 1.329138 1.162733 1.033418 0.930287 0.846111 0.776140 0.717087
1128.30 833.50 672.50 599.98 561.11 538.39 524.72 516.62 512.10 510.32 510.29 511.64 514.04
Denotes the optimal solution.
4.2. Example II (without shortage case) Avoiding the shortage cost, we take the same parameters as in Example I above. In this case, results are shown in Table 2 which gives the minimum total cost as $540.77 for m "11, "0.014076, ¹"0.797654 month.
Table 2 Optimal solution for ano shortagea case m
¹
C
2 3 4 5 6 7 8 9 10 11 12 13
0.352326 0.162222 0.093328 0.060843 0.042989 0.032138 0.025054 0.020176 0.016674 0.014076 0.007210 0.001055
4.176163 2.828889 2.139993 1.721686 1.440807 1.239271 1.087690 0.969592 0.875035 0.797654 0.733190 0.678686
884.60 719.07 642.00 599.60 574.36 558.85 549.40 543.97 541.35 540.77 541.71 543.82
Denotes the optimal solution.
S. Kar et al. / Computers & Operations Research 28 (2001) 1315}1331
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Comparing the optimal solutions of Examples I and II, it is observed that the optimal system cost increases signi"cantly in the no-shortage case. However, the time period of the "rst replenishment cycle is greater when shortages are permitted. But, the reorder numbers of both the inventory system (shortage and no-shortage case) are same. Hence from an economic point of view, inventory backlogging is thus bene"cial as the system cost is considerably reduced by allowing shortages.
5. Sensitivity analysis Using the numerical examples I and II mentioned earlier, a sensitivity analysis is performed to study the e!ect of changes of the values of some parameters on the optimal reorder number mH and the optimal total system cost CH holding the values of the other parameter values as same. The results shown in Tables 3}4 are self-explanatory. Table 3 E!ects of changes in some parameters on the optimal reorder number and the optimal system cost in the `with shortagea case % change in parameters value
m
Total cost
% change in system cost
!20 !10 10 20
8 9 12 13
395.09 451.85 570.25 631.93
!22.58 !11.45 #11.75 #23.84
Owned ware-house capacity =
!20 !10 10 20
10 10 11 11
528.07 519.10 501.21 492.34
#03.48 #01.73 !01.78 !03.52
Location parameter of demand `aa
!20 !10 10 20
9 10 11 12
438.39 474.30 545.64 581.08
!14.08 !07.05 #06.93 #13.87
!20 !10 10 20
10 10 11 11
493.07 503.34 518.73 527.17
!03.37 !01.36 #01.65 #03.31
!20 !10 10 20
10 10 11 11
503.88 507.41 512.70 514.76
!01.25 !00.56 #00.47 #00.88
Parameters
Time horizon H
Shape parameter of demand `ba
Shortage cost C
Denotes the optimal solution.
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Table 4 E!ects of changes in some parameters on the optimal reorder number and the optimal system cost in the `no shortagea case % change in parameters value
m
Total cost
% change in system cost
!20 !10 10 20
9 10 12 13
418.10 478.49 604.94 670.10
!22.68 !11.52 #11.87 #24.08
Owned ware-house capacity =
!20 !10 10 20
10 11 11 12
560.74 550.92 530.84 521.03
#03.69 #01.88 !01.84 !03.65
Location parameter of demand `aa
!20 !10 10 20
10 10 11 12
463.24 502.21 579.23 617.40
!14.34 !07.13 #07.11 #14.17
!20 !10 10 20
10 10 11 11
522.35 531.56 549.99 559.21
!03.41 !01.70 #01.70 #03.41
Parameters
Time horizon H
Shape parameter of demand `ba
Denotes the optimal solution.
6. Applicability of the model The usual supply chain of commodities has been indicated in the `Introductiona. It may be a point of interest to know, who will be the user of the proposed model in the said supply chain? Normally, big merchants and wholesalers do not have direct contact with the common customers. For this reason, they operate the business with an o$ce and a big warehouse. On the contrary, both the retailer and small retailer have to deal directly with the ordinary customers and maintain a decorative show-room to impress and motivate the customers in order to boost their sale. They, therefore, adapt two-level storage facilities to run their business smoothly. Hence, the proposed model is applicable for the business of retailers and small retailers. In the developing countries, like India, etc., the two levels of storage facilities were seldomly used by the business community, they normally use the age old single warehouse system. Now-a-days, due to globalisation of the market with the introduction of multinationals in the business, there is a trend among the business houses specially retailers and small retailers of di!erent multinational products to compete with each other for sale and as a result, they use the decorative show-room to boost their items in addition to a separate warehouse. Another reason for the adoption of the present model is the crisis of having large space in the important market places, like super market, corporation market, etc. So, from the
S. Kar et al. / Computers & Operations Research 28 (2001) 1315}1331
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economical point of view, the proposed model will be more bene"cial to the business houses as it gives better business to them.
7. Concluding remarks In this paper, we have discussed the inventory problem for a single item with linear trend in demand (increasing) and shortages over a "xed and "nite time horizon. The replenishment cost is taken to be dependent on the lot size of the current replenishment. This feature of the model makes it the most general among the ones referred to in the introduction. Another feature here is that we have taken the transportation cost to transfer the items of the RW to the OW in a continuous release pattern, which is not considered by others. The present model is applicable for food grains like paddy, rice, wheat, etc., as the demand of the food grains increases with time for a "xed time horizon, i.e., for a calendar year. It is also applicable for other items where the demand is dependent linearly with time. The case of decreasing demand is also of great importance for a "nite horizon model. The results obtained in this article are also valid for linearly decreasing demand. For future development of research, one can extend the model developed in this article by incorporating more realistic situations, such as, multiple items and quantity discount policies.
Appendix For the development of the probabilistic model, the following assumptions are taken: (i) g(x ) is the probability density function for demand x which is a random variable during the G G ith cycle (¹ , ¹ ) (0)x (R). This demand is supposed to occur in a uniform pattern during G\ G G the ith cycle. Other assumptions are same as those given in Section 2. Two cases may occur in this model depending on the relative values of the demand x . We study G these cases separately. Case I: When shortages do not occur. Let Q (t) be the inventory level at any time t(¹ )t)¹ ) VG G\ G of the system and x is the demand during the "xed interval (¹ , ¹ ) so that the demand rate is G G\ G x /(¹ !¹ ) during the period. G G G\ Then the di!erential equation describing the system are given by dQ (t)/dt#x /(t !¹ )"0, ¹ )t)¹ , VG G G G\ G\ G
i"1, 2,2, m.
(A.1)
Using initial conditions, we have the solution as Q (t)"S !x (t!¹ )/(¹ !¹ ), i"1, 2,2, m. VG G G G\ G G\ Since shortages do not occur we have Q (¹ )*0 or x )S . VG G G G
(A.2)
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The lot size q for the ith cycle is given by G q "S . (A.3) G G Again, from Q (t)"= at t"t , we have VG G t "¹ #(S !=)/x . (A.4) G G\ G G Here, the total number of items R (x ) and I (x ) carried in inventory in the RW and the OW, G G G G respectively, are given by
R (x )" G G
RG
2G\
Q (t) dt!=(t !¹ ), x )S , i"1,2,2, m!1 VG G G\ G G
(A.5)
and
I (x )"=(t !¹ )# G G G G\
2G G
R
Q (t) dt, x )S , i"1,2,2, m!1. VG G G
(A.6)
Also,
R (x )" K G
RK
2K\
Q (t) dt!=(t !¹ ), x )S VG K K\ G G
(A.7)
and
& Q (t) dt, x )S . VG G G RK The total number of shortages during this period is I (x )"=(t !¹ )# K G K K\
(A.8)
M (x )"0, x )S . (A.9) G G G Case II: when shortages occur. In this case, the system is the same as that of the deterministic model except that the demand x which is a random variable with probability density function f (x ) G G during the scheduling period (¹ , ¹ ). G\ G Then the di!erential equations describing the system are given by dQ (t)/dt#x /(t !¹ )"0, ¹ )t)t , VG G G G\ G\ G dQ (t)/dt#x /(¹ !t )"0, t )t)¹ , i"1, 2,2, m. VG G G G G G Using initial conditions, we have the solutions as
S !x (t!¹ )/(¹ !¹ ), ¹ )t)t , G G\ G G\ G G Q (t)" G . VG x (t !t)/(¹ !¹ ), t )t)¹ , i"1, 2,2, m G G G G\ G G As Q (t )"0, we have VG G S "x (t !¹ )/(¹ !¹ ). G G G G\ G G\ Since shortages occur, we have Q (¹ )(0 or x 'S . VG G G G
(A.10) (A.11)
(A.12)
(A.13)
S. Kar et al. / Computers & Operations Research 28 (2001) 1315}1331
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The lot size q for the ith cycle is given by G q "x (t !t )/(¹ !¹ ). (A.14) G G G G\ G G\ Therefore, the total number of items R (x ) and I (x ) carried to inventory in the RW and the G G G G OW, respectively are given by
R (x )" G G
RG
2G\
Q (t) dt!=(t !¹ ), x 'S , i"1, 2,2, m!1 VG G G\ G G
(A.15)
and
I (x )"=(t !¹ )# G G G G\
2G G
R
Q (t) dt, x 'S , i"1, 2,2, m!1. VG G G
(A.16)
Also,
R (x )" K G
RK
2K\
Q (t) dt!=(t !¹ ), x 'S VG K K\ G G
(A.17)
and
& Q (t) dt, x 'S . VG G G RK The total number of shortages during this period is I (x )"=(t !¹ )# K G K K\
M (x )" G
(A.18)
2G
[!Q (t)] dt, VG RG "x (¹ !t ), x 'S , i"1, 2,2,(m!1). (A.19) G G G G G Therefore, the expected total cost of the proposed inventory system during the planning horizon H is
K\ K\ 1G 1G C"mA#p q (x )g(x ) dx # q (x )g(x ) dx # h R (x )g(x ) dx G G G G G G G G G G G G G G 1 G G 1G I (x )g(x ) dx # I (x )g(x ) dx # R (x )g(x ) dx #h G G G G G G G G G G G G G 1G 1G 1K #C M (x )g(x ) dx #C (S !=) #h R (x )g(x ) dx G G G G R G K K K K K 1G 1K I (x )g(x ) dx # I (x )g(x ) dx # R (x )g(x ) dx #h K K K K G K K K K K K K K 1K 1K #C (S !=). (A.20) R K
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The optimal values of m and S (i"1, 2,2, m) can be found out by using the same procedure of G the deterministic model.
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[24] Dave U. On the EOQ models with two levels of storage. Opsearch 1988;25:190}6. [25] Goswami A, Chaudhuri KS. An economic order quantity model for items with two levels of storage for a linear trend in demand. Journal of Operation Research Society 1992;43:157}67. [26] Bhunia AK, Maiti M. A two warehouse inventory model for a linear trend in demand. Opsearch 1994;31:318}29. [27] Sarma KVS. A deterministic order-level inventory model for deteriorating items with two storage facilities. European Journal of Operation Research 1987;29:70}2. [28] Pakkala TPM, Achary KK. A deterministic inventory model for deteriorating items with two warehouses and "nite replenishment rate. European Journal of Operation Research 1992;57:71}6. [29] Bhunia AK, Maiti M. A two warehouses inventory model for deteriorating items with a linear trend in demand and shortages. Journal of Operation Research Society 1997;49:287}92.
S. Kar is a Lecturer in Mathematics at Haldia Institute of Technology, Midnapore, West Bengal, India. His areas of research work includes Inventory Management with Fuzzy and Stochastic Environments. A.K. Bhunia is a Lecturer in Mathematics at Haldia Institute of Technology, Midnapore, West Bengal, India. His research areas includes Inventory management and Development of Interval Analysis and its application in Inventory Management. Dr. Bhunia received his Ph.D. degree from the Vidyasagar University, Midnapore. M. Maiti is a professor in the Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, West Bengal, India. His "elds of research are Elasticity and Catastrophic Theory, Inventory Management in Fuzzy and Stochastic Environments, Development of Interval Analysis and its application in Inventory Management. Dr. Maiti received his Ph.D. degree from the University of Calcutta.