Chapter 5. Inventory models with ramp-type demand for deteriorating items partial backlogging and timevarying

Transcription

1 Chapter 5 Inventory models with ramp-type demand for deteriorating items partial backlogging and timevarying holding cost

2 5.1 Introduction Inventory is an important part of our manufacturing, distribution and retail infrastructure where demand plays an important role for the best inventory policy. Researchers were engaged to develop the inventory models assuming the demand of the items to be constant, linearly increasing or decreasing demand or exponential increasing or decreasing with time, stock-dependent etc. Later, this has been realized that the above demand patterns do not precisely depict the demand of certain items such as newly launched fashion items, garments, cosmetics, automobiles etc; for which the demand increases with time as they are launched into the market and after some time it become constant. In order to consider the demand of such types, the concept of ramp-type demand is introduced. Ramp-type demand depicts a demand, which increases up to a certain time after which it stabilizes and become constant. Research on this field continues with (Mandal and Pal, 1998, Wu et. al, 1999, Wu and Ouyang, 2000, Wu, 2001) who studied inventory models with linearly increasing, up to its stabilization point, demand rate under various assumptions. In the above cited papers, the determination of the optimal replenishment policy requires the determination of the time point, when inventory level falls to zero. So the following two cases should be examined: (i) this time point occurs before the point where the demand is stabilized, and (ii) this time point occurs after the point where the demand is stabilized. Almost all of the researchers examine only the first case. (Deng et. al, 2007) reconsidered the inventory model of (Mandal and Pal, 1998 and Wu and Ouyang, 2000) and studied it exploring these two cases. (Skouri et. al, 2009) extended the work of (Deng et. al, 2007) by introducing a general ramp-type demand rate considering weibull distribution deterioration rate. Maintenance of inventories of deteriorating items is a problem of major concern in the supply chain of almost any business organizations. Many of the physical goods undergo decay or deterioration over time. The inventory lot-size problem for

3 deteriorating items is prominent due to its important connection with commonly used items in daily life. Fruits, vegetables, meat, photographic films, etc are examples of deteriorating products. Deteriorating items are often classified in terms of their lifetime or utility as a function of time while in stock. A model with exponentially decaying inventory was initially proposed by (Ghare and Schrader, 1963, Covert and Phillip, 1973). They developed an EOQ model with Weibull distributed deterioration rate. Thereafter, a great deal of research efforts have been devoted to inventory models of deteriorating items and the details is discussed in the review article by (Raafat,1991) and then by (Goyal and Giri, 2001). The review article of inventory models considering deterioration with shortages by (Karmakar and Dutta Choudhury, 2010) may also be considered. (Gupta and Vrat, 1986) developed an inventory model where demand rate is replenishment size (initial stock) dependent. They analyzed the model through cost minimization. (Pal et. al,1993, Datta and Pal,1990) have focused on the analysis of the inventory system which describes the demand rate as a power function, dependent on the level of the on hand inventory, the holding cost per unit item per unit time is taken as a constant in all these models. In most models, holding cost is known and constant. But holding cost may not be constant always. In generalization of EOQ models, various functions describing holding cost were considered by several researchers like (Naddor, 1966, Van Der Veen, 1967, Muhlemann and Valtis-Spanopoulos, 1980, Weiss, 1982 and Goh, 1994). (Giri and Chaudhuri, 1998) treated the holding cost as a non-linear function of the length of time for which the item is held in stock and as a functional form of the amount of the on-hand inventory. (Roy, 2008) developed an EOQ model for deteriorating items where deterioration rate and holding cost are expressed as linearly increasing functions of time and demand rate is a function of selling price and shortages are allowed and completely backlogged. (Paul et. al, 1996) considered twocomponent demand rate and extended the model by considering backlogged shortages and it is also assumed the storage space has a limited capacity.

4 Again in most of the above referred papers, complete backlogging of unsatisfied demand is assumed. In practice, there are customers who are willing to wait and receive their orders at the end of shortage period, while others are not. (Teng et. al, 2002) extended the work done by (Chang and Dye s, 1999 and Skouri and Papachristos, 2002), assuming backlogging rates any decreasing function of the waiting time up to the next replenishment. In the present work have extended the work of (Skouri et. al, 2009) as follows: (i) the holding cost is expressed as linearly increasing function of time, (ii) the deterioration rate is considered as constant. The model has been studied under the following different replenishment policies (i) starting without shortage (ii) starting with shortages. The paper has been arranged according to the following order. The notation and assumption used are given in section 5.2. The model starting with no shortages is studied in section 5.3.A and the corresponding one starting with shortages is studied in section 5.4. For each model the optimal policy is obtained. A numerical example highlighting the obtained results and a sensitivity analysis are given in section 5.5 and 5.6 respectively. The paper closes with concluding remarks in section Notations and assumptions The inventory model is developed under following notations: 1. T the constant scheduling period. 2. t 1 the time at which the inventory reaches to zero. 3. C 1 the holding cost per unit per time unit. 4. C 2 the shortage cost per unit per time unit. 5. C 3 the cost incurred from the deterioration of one unit.

5 6. C 4 the per unit opportunity cost due to lost sales. 7. the constant deterioration rate, where 0 < < 1 8. the parameter of the ramp-type demand function (time point) 9. the inventory level at time t [0,T] 10. be the total holding cost. The inventory model is developed under following assumptions: 1. The system operates for a prescribed period T units of time and the replenishment rate is infinite and lead time is zero. 2. Holding cost per unit time is time dependent and is assumed to be as = +, where > 0 and > reasing function of x with 0< / (x) / backlogging model. 4. The demand rate D(t) is a ramp-type function of time and is as follows: =, <,,, f(t) is a positive, continuous for t [0,T] 5.3.A Mathematical formulation of the model starting without shortages In this section the inventory model starting with no shortage is studied. The replenishment at the beginning of the cycle brings the inventory level up to Q. Due to deterioration and demand, the inventory level gradually depletes during the period

6 [0, ] and falls to zero at =. Thereafter shortages occur during [,T], which is partially backlogged. The backlogged demand is satisfied at the next replenishment. The inventory level,, 0 + =, 0, = (5.3.1) =,, = (5.3.2) The solutions of these differential equations are affected by the relation between and through the demand rate function. To continue the two cases: (i) < (ii) must be considered. Case (t 1 < ) In this case, equation (5.3.1) becomes + = 0, = (5.3.A.1) Equation (5.3.2) leads to the following two: =, µ, = (5.3.A.2) =, µ, = --- (5.3.A.3) The solutions of (5.3.A.1)-(5.3.A.3) are respectively =, (5.3.A.4) =, µ --- (5.3.A.5) =, µ --- (5.3.A.6)

7 The total amount of deteriorated items during [0, ] = --- (5.3.A.7) The total holding cost in the interval [0, ] is given as follows using (5.3.A.4) = = ( + ) = (5.3.A.8) Time weighted backorders due to shortages during [,T] is = [ ] = [ ] + [ ] = (5.3.A.9) The amount of lost sales during [,T] is = [1 ] + [1 ] The order quantity is Q= (5.3.A.10) The total cost in the interval [0,T] is the sum of holding, shortage, deterioration and opportunity costs and is given by = H(t)+C 2 I 2 +C 3 D+C 4 L = + + C 2 { + + } + C 3 { } + C 4 { [1 ] + [1 ] } --- (5.3.A.11)

8 Case (t 1 ) In this case equation (5.3.1) reduces to the following two: + =, 0, = --- (5.3.A.12) + =,, =0, --- (5.3.A.13) Equation (5.3.2) becomes =,, =0, --- (5.3.A.14) Their solutions are respectively, = +, (5.3.A.15) =,, --- (5.3.A.16) =, --- (5.3.A.17) The total amount of deteriorated items during [0, ] = (5.3.A.18) The total holding cost in the interval [0, ] is given as follows using (5.3.A.12) &(5.3.A.13) = = ( + ) + ( + ) = (5.3.A.19) Time weighted backorders due to shortages during [,T] is

9 = [ ] = --- (5.3.A.20) The amount of lost sales during [,T] is = [1 ] --- (5.3.A.21) The order quantity is Q= (5.3.A.22) The total cost in the interval [0,T] = H(t)+C 2 I 2 +C 3 D+C 4 L = C 2 +C C 4 [1 ] --- (5.3.A.23) Summarizing the total cost of the system over [0,T] =, <, 5.3.B The optimal replenishment policy of the model starting without shortages The existence of uniqueness, say, which minimizes the total cost function for the model starting without shortages. Although the argument of the functions, is constrained, we shall search for their unconstrained minimum. The first and second order derivatives of are, respectively

10 =, --- (5.3.B.1) = + where = (1 ) --- (5.3.B.2) It can be easily verified that, when 0 1, g(0)<0, g(t) = / ] + / > (5.3.B.3) The assumption (5.3.B.1) made for, ( non-increasing so / < 0 and + / 0) implies that g is strictly increasing. By assumption f and so the derivative vanishes at, with 0< <T, which is the unique root of g = (5.3.B.4) For this we have = + > 0 So that corresponds to unconstrained global minimum. For the branch, the first and second order derivatives are =, = 0, --- (5.3.B.5) Where the function ( ) is given by = (1 ) --- (5.3.B.6)

11 The inequality (5.3.B.5) follows from (5.3.B.3) and ensures the strict convexity of. When 0 1 and based on the properties of we conclude that vanishes at, with 0 < <, which is the unique root of = (5.3.B.7) Now an algorithm is proposed that leads to the optimal policy Step1 compute from equation (5.3.B.4) or (5.3.B.7) Step2 compare to Step 2.1 If < then the total cost function and the optimal order quantity are given by (5.3.A.11) and (5.3.A.10) Step 2.2 If then the total cost function and the optimal order quantity are given by (5.3.A.23) and (5.3.A.22) 5.4.A Mathematical formulation of the model starting with shortages The cycle now starts with shortages, which occur during the period [0, ] and are partially backlogged. After time a replenishment brings the inventory level up to Q. Demand and deterioration of the items depletes the inventory level during the period [,T] until this falls to zero at t = T. Again the two cases (i) < (ii) must be examined. Case ( < ) The inventory level, 0 satisfies the following differential equations: =, 0 1, (0) = 0, --- (5.4.A.1) + =,, =, --- (5.4.A.2)

12 + =,, I(T) = 0, --- (5.4.A.3) The solutions of (5.4.A.1) (5.4.A.3) are respectively =, (5.4.A.4) = +, --- (5.4.A.5) =, --- (5.4.A.6) The total amount of deterioration during [,T] is = (5.4.A.7) The total holding cost in the interval [,T] is = (5.4.A.8) Time weighted backorder during the time interval [0, ] is = --- (5.4.A.9) The amount of lost sales during the time interval [0, ] is = (5.4.A.10) The order quantity is / = (5.4.A.11)

13 Using similar argument as in the previous model, the total cost of this model is / = (5.4.A.12) Case The inventory level,, 0 =, 0, (0)=0, --- (5.4.A.13) =,, =, --- (5.4.A.14) + =,, =0, --- (5.4.A.15) The solutions are respectively, =, 0, --- (5.4.A.16) =,, --- (5.4.A.17) =,, --- (5.4.A.18) Total amount of deteriorated during [,T] is =, --- (5.4.A.19) The total holding cost in the interval [,T] is = (5.4.A.20)

14 Time weighted backorder during the time interval [0, ] is = + + ( ) --- (5.4.A.21) The amount of lost sales during the time interval [0, ] is = (5.4.A.22) The order quantity is / = (5.4.A.23) Using similar argument as in the previous model, the total cost of this model is / = ( ) (5.4.A.24) Summarizing the total cost of the system over [0,T] / = /, < /, 5.4.B The optimal replenishment policy of the model starting with shortages We derive the optimal replenishment policy of the model starting with shortages, we calculate the value of, say, which minimize the total cost function.

15 Taking the first order derivative of / and equating to zero gives ( + + ) / / = (5.4.B.1) If is a root of (5.4.B.1) for this root the second order condition for minimum is / > 0 i.e. ( + ( 1) + ) + +( + + ) + 2 / + // // + (0) / (0) > (5.4.B.2) Equating the first order derivative of / to zero gives ( + + ) + + / / ] + + / / ( )] = (5.4.B.3) If is a root of (5.4.B.3) for this root the second order condition for minimum is / > 0 i.e. + ( 1 + ) + ( + + ) + 2 / + // // + 2 / + // // + ( (0) / (0)) > (5.4.B.4)

16 Now an algorithm is proposed that leads to the optimal policy: Step1 Find the global minimum, for TC / 1 (t 1 ). This will be one of the following: a) a root from equation (5.4.B.1), which satisfies ( ) b) = c) = 0 The total cost function and the optimal order quantity are given by (5.4.A.12) and (5.4.A.11) Step2 Find the global minimum, for TC / 2 (t 1 ). This will be one of the following: a) a root of from equation (5.4.B.3), which satisfies (5.4.B.4) b) = c) = The total cost function and the optimal order quantity are given by (5.4.A.24) and (5.4.A.23) 5.5 Numerical example The input parameters are: = 15 ; = 5 ; = 20 ; = 1 ; =.01 ; = 0.001; = 0.2 ; = 1 ; = 3. ; =. ; Model starting without shortage: Using (5.3.A.4) or (5.3.A.7) the optimal value of is = >. The optimal ordering quantity is = (from 5.3.A.22) and the minimum cost is TC 2 = Rs Since the nature of the total cost function is highly non linear thus the convexity of the function shown graphically.

17 Figure 5.1: The graphical representation of the total cost function model starting without shortage. Model starting with shortages: Using (5.4.B.1) the optimal value of is = <. The optimal ordering quantity is = (from 5.4.A.11) and the minimum cost is TC * 1 = Rs Since the nature of the total cost function is highly non linear thus the convexity of the function shown graphically. Figure 5.2: The graphical representation of the total cost function model starting with shortages.

18 5.6 Sensitivity analysis Using the numerical example presented earlier, a sensitivity analysis is performed to explore the effect of change on some of the basic model s parameters to the optimal policy. TABLE 5.1 parameter Model starting without Model starting with shortages shortages Percentage TC Q TC / change(%) Q /

19 From the result of the above table, the following observations may be made. 1) For the negative change of the parameter, in both the models starting with shortages and without shortage, total cost will be increased. 2) Other parameters are less sensitive. 5.7 Conclusion In this paper, an order level inventory model for deteriorating items with time-varying holding cost has been studied. The model is fairly general as, the demand rate is any function of time up to the time-point of its stabilization (general ramp-type demand rate), and the backlogging rate is any non increasing function of waiting time, up to the next replenishment. Moreover the traditional parameter of holding cost is assumed here to be time varying. As the changes of the time value of money and in the price index, holding cost may not remain constant over time. It is assumed that the holding cost is linearly increasing function of time. The inventory model is studied under two different replenishment policies: (i) starting with no shortage and (ii) starting with shortages. Again if holding cost is constant then, the model starting with no shortage, = >, TC= Rs and Q= and for the model starting with shortage = <, TC= Rs and Q=

20 The total cost with constant holding cost is less than the total cost with time-varying holding cost, which is a realistic situation.