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Research Article
 

Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts



Ming-Cheng Lo, Jason Chao-Hsien Pan, Kai-Cing Lin and Jia-Wei Hsu
 
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ABSTRACT

This study investigates the impact of safety factor on the continuous review inventory model involving controllable lead time with mixture of backorder discount and partial lost sales. The objective is to minimize the expected total annual cost with respect to order quantity, backorder price discount, safety factor and lead time. A model with normal demand is also discussed. Numerical examples are presented to illustrate the procedures of the algorithms and the effects of parameters on the result of the proposed models are analyzed.

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  How to cite this article:

Ming-Cheng Lo, Jason Chao-Hsien Pan, Kai-Cing Lin and Jia-Wei Hsu, 2008. Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts. Journal of Applied Sciences, 8: 528-533.

DOI: 10.3923/jas.2008.528.533

URL: https://scialert.net/abstract/?doi=jas.2008.528.533

INTRODUCTION

Competition features of 90’s have evolved into Time-Based Manufacturing (TBM). The two time elements considered in TBM are the replenishment lead time to supply a product to the customer for a specific order and the time to develop a product from concept to final market delivery. Therefore, reducing lead time on product supply is the strategic objective of the TBM company (Bockerstette and Shell, 1993). Although the time compression will inevitably raise the cost, a customer will pay a premium to the supplier who can furnish its product faster and more reliably than the competition and the premium may be respectable.

Silver et al. (1998) defined the replenishment lead time as the time elapsed from the moment at which it is decided to place an order, until it is physically on the shelf to satisfy customer demands. Although lead time can be a constant or a random variable, it is often treated as a prescribed parameter (Silver et al., 1998) thus not controllable. However, lead time can be reduced at extra cost and shorter lead time is the primary driver to achieving customer satisfaction in successful TBM operations (Bockerstette and Shell, 1993). The benefits resulting from reduced lead time include lower cost, less waste and less obsolescence, greater flexibility to response to change, closely linked organization priorities to customers’ needs, improved service, quality and reliability and substantially accelerated supply system improvements (Blackburn, 1991). Tersine (1994) and Vollmann et al. (1992) attributed the replenishment lead time mostly to manufacturing considerations and addressed some guidelines for its reduction. Liao and Shyu (1991) suggested that lead time could be decomposed into n components each having a different crashing cost for reduction. Ben-Daya and Raouf (1994) generalized the Liao and Shyu model (1991) by considering both lead time and the order quantity as decision variables.

Ouyang et al. (1996) extended the Ben-Daya and Raouf’s model (1994) to include a mixture of backorder and lost sales in the model by assuming a predetermined service level with both reorder point and backorder rate being fixed. Moon and Choi (1998) suggested that it was not appropriate to include the service constraint if the shortage cost was explicitly specified and claimed that a better solution could be obtained by allowing the reorder point to be variable. Hariga and Ben-Daya (1999) also revised Ouyang et al. (1996) model by including the reorder point as a decision variable. Pan and Hsiao (2001) presented two inventory models where shortage was allowed with controllable back ordering.

This study considers a continuous review inventory system in which the lead time is controllable and can be decomposed into several components each having a crashing cost function. In addition, shortage is permitted and the total amount of stockout is a combination of backorder and lost sale. It is further assumed that the patient customers with outstanding orders during the shortage period are offered a backorder price discount and consequently the backorder ratio is proportional to the magnitude of this discount (Pan and Hsiao, 2001). Since the shortage cost is explicitly included, the reorder point is also treated as a decision variable in this study (Moon and Choi, 1998).

There is form of lead time demand considered following a normal distribution in the study. In this models, the objectives are to simultaneously optimize the order quantities, back order discounts, reorder points and lead times such that the total expected annual costs are minimized. Furthermore, an iterative algorithm is applied to find the optimal solution for the case where the lead time demand follows a normal distribution. Our model serves as a pioneering work investigating the effects backorder discounts and safety factor have on the integrated inventory system.

NOTATIONS AND ASSUMPTIONS

The following notations are used throughout the study.

L = The length of lead time (decision variable)
Q = The order quantity (decision variable)
πx = The backorder price discount offered by the supplier per unit (decision variable)
k = The safety factor (decision variable)
r = The reorder point
π0 = The gross marginal profit per unit
D = The average demand per year
A = The fixed ordering cost per order
h = The inventory holding cost per unit per year
β = The backorder ratio
β0 = The upper bound of the backorder ratio
φ = The standard normal distribution;
Φ = The standard normal cumulative distribution function
SS = The safety stock
B(r) = The expected shortage of a cycle
R(L) = The total crashing cost of a cycle

The assumptions made in the research are defined as follow:

The reorder point r = expected demand during lead time + safety stock, that is, r = DL + kσ√L, where k is a safety factor
The lead time L has n mutually independent components, where the ith component has a normal duration Ti and a minimum duration ti, i = 1, 2, ..., n and a crashing cost per unit time ai. These ai’s are arranged such that a1 ≤ a2 ≤ ... ≤ an. The lead times are crashed that should be first on component 1 and then component 2 and so on.

Let Li denote the length of lead time with component 1, 2, ...n, i crashed to their minimum values, then Li can be expressed as:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
and L0 is not cashed the length of lead time. Thus, the lead time crashing cost R(L) per replenishment cycle is given by:
Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
The backorder ratio β is variable and is in proportion to the backorder price discount offered by the supplier per unit πx; thus, β = β0πx0, for 0 <β0≤1, 0≤πx ≤ π0 (Pan and Hsiao, 2001)

NORMAL DEMAND MODEL

The problem under study is to minimize the following total expected annual cost:

EAC(Q, πx, k, L) = OC + HC + SC + CC

Where:
OC = Stands for ordering cost
HC = Holding cost
SC = Shortage cost
CC = Crashing cost

The annual expected ordering cost can be expressed as:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

The lead time demand X is assumed to follow the normal distribution with mean μL and standard deviation σ√L. Since shortage is allowed, the expected inventory shortage at the end of a cycle is given by:

E(X-r)+ = B(r) = σ√LΨ(k),

Where:
Ψ(k) = φ(k)-k[1 - Φ(k)]
φ, Φ = Standard normal distribution and cumulative distribution function, respectively (Ravindran et al., 1987)

For backorder ratio β, the expected inventory at the end of a cycle is βB(r) and the expected lost sale is (1-β)B(r). Therefore, the average inventory of a cycle is:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

Hence, the expected annual holding cost is:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

The backorder price discount of a cycle is βπxB(r) and the expected lost sales is (1-β)B(r), so the profit lost due to shortage is π0(1-β)B(r) and the expected annual shortage cost can be expressed as:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

Following the definition of Li, the lead time crashing cost R(L) per replenishment cycle is given by:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

Hence, the expected annual crashing cost is:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

Consequently, the total expected annual cost EAC(Q, πx, k, L) is:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(1)

Taking the first partial derivatives of EAC(Q, πx, k, L) with respect to Q, πx, k and L, respectively, it follows that:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(2)

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(3)

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(4)

and

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(5)

Setting Eq. 3 to zero and solving for πx, it follows that:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(6)

Setting Eq. 2 to zero and substituting Eq. 6 into 2 to solve for Q, thus:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(7)

Setting Eq. 4 to zero and substituting Eq. 6 into 4 to solve for k, then:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(8)

It can be shown from Eq. 8 that hQ/D≤ π0, so Φ(k)≥0.5 holds for nonnegative safety factor k. Consequently, the value of πx derived in Eq. 6 will automatically lie between 0 and π0 as specified in assumption 4th.

For fixed values of Q, πx and k, EAC(Q, πx, k, L) is concave in the range L∈(Li, Li-1], since:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(9)

For fixed L∈(Li, Li-1], denote the values of Q, πx and k found from Eq. 6-8 by Q*, πx* and k*, respectively. In addition, for fixed L∈(Li, Li-1], the determinant of Hessian matrix of EAC(Q, πx, k, L) is positive definite at (Q*, πx*, k*) as shown in Appendix.

The following algorithm can be used to find the optimal values of the order quantity, backorder discount, reorder point and lead time.

Step 1: For i = 0, 1, 2, ..., n

(i) Set kio = 0 (implies Ψ(kio) = 0.39894)
(ii) Substitute Ψ(kio) into Eq. 8 to evaluate Qio
(iii) Use Qio to determine Φ(kin) from Eq. 8, then find kin from Φ(kin) by checking the normal table. Let kio = kin
(iv) Repeat (ii) and (iii) until no change occurs in the values of Qi and ki. Denote these resulting solutions by Qi and ki.
(v) Use Qi and Eq. 7 to compute the backorder price discount πxi.
(vi) Use Eq. 1 to compute the expected total annual cost EAC(Qi, πxi, ki, Li).

Step 2: Set EAC(Q*, πx*, k*, L*) = Min{EAC(Qi, πxi, ki, Li), i = 0, 1, 2, ..., n }.

Step 3: (Q*, πx*, k*, L*) is a set of optimal solutions.

NUMERICAL EXAMPLE AND ITS SENSITIVITY ANALYSIS

Consider an inventory system with the following data: D = 600 units/year, A = $200 per order, h = $20 per unit per year, π0 = $150 per unit, σ = 7 unit/week and the lead time has three components with data shown in Table 1 (Pan and Hsiao, 2001). Apply the proposed algorithm to solve the problem for the upper bound of the backorder ratio β0 = 0.95, 0.80, 0.65, 0.50, 0.35 and 0.20, respectively. The resulting optimal solutions are summarized in Table 2. Also included in Table 2 are the results obtained from the associated Pan and Hsiao model (2001) by setting k fixed at 0.85, along with the corresponding saving on the total expected annual cost of the proposed model over that of Pan and Hsiao (2001). It is interesting to observe that the saving increases as β0 decreases.

Next, we study the effect of change in the model parameters such as D, A, h, σ and π0 under β0 = 0.5, the optimal order quantity (Q* = 121), the optimal backorder price discount (πx* = 77.0157), the optimal safety factor (k* = 1.88), the optimal lead time (L* = 4) and the corresponding total cost (EAC* = EAC(Q*, πx*, k*, L*) = 2947.72) keeping the same parameter values in Example. The sensitivity analysis is performed by changing each of the parameter by -50, -25, +25 and +50%, taking one at a time while keeping remaining unchanged. The results are shown in Table 3. Let EAC(δ) denote the percentage difference between the new total expected annual cost obtained from changing the values of these factors and the original total expected annual cost, that is, EAC(δ) = (the new total expected annual cost-the original j total expected annual cost)/(the original total expected annual cost).

Makes the sensitivity analysis in view of various parameters, the obtained data result has the following discovery.

k*, Q* and EAC* all increases while πx* decreases with an increase in the value of the model parameter demand rate D. The obtained results show that πx* and k* are lowly sensitive and Q* and EAC* are middling sensitive to changes in the value of D. Moreover, Li* is unchanged in D.
πx*, Q* and EAC* increases while k* decreases with an increase in the value of the model parameter A. The obtained results show that πx* and k* are lowly sensitive and Q* and EAC* are middling sensitive to changes in the value of A. Moreover, Li* is unchanged in A.
πx* and EAC* increases while Q* and k* decreases with an increase in the value of the model parameter h. The obtained results show that πx* and k* are lowly sensitive and Q* and EAC* are middling sensitive to changes in the value of h. Moreover, Li* is unchanged in h.

Table 1: Lead time data of the examples
Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

Table 2: Summary of the results for example (Li in weeks)
Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

Table 3: Effect of changes in the parameters of the continuous review inventory models
Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

πx*, Q* and EAC* increases while k*and Li* decreases with an increase in the value of the model parameter σ. The obtained results show that πx* and k* are lowly sensitive and Q* and EAC* are middling sensitive to changes in the value of σ. Moreover, Li* is highly sensitive to changed in σ.
πx*, k* and EAC* all increase with an increase in the value of the model parameter π0. The obtained results show that EAC* are lowly sensitive, k* are middling sensitive and πx* highly sensitive to changes in the value of π0. Moreover, Li*and Q* is unchanged in π0.

CONCLUSIONS

This study extends the Pan and Hsiao model (2001) to study the impact of safety factor on the continuous review inventory model involving controllable lead time and backorder price discount with mixture of backorder and partial lost sales. The objective is to minimize the expected total annual cost by simultaneously optimizing order quantity, backorder price discount, safety factor and lead time. It is found that the expected total inventory cost tends to decrease as the upper bound of the backorder ratio increases while all the other parameters stay fixed. The results of the illustrative example also show that the expected total inventory cost and the safety factor increase for a given the upper bound of the backorder ratio decreases. The savings of the total cost on considering safety factor that is decision variable are demonstrated in the examples as well.

APPENDIX

The Hessian matrix H of EAC(Q, πx, k, L) for a given value of L is:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(10)

Where:
Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

and

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

Next, evaluate the principal minor of H at point (Q*, πx*, k*). The first principal minor of H is

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(11)

Since from Eq. 7 that πx* = hQ*/2D + π0/2, the second principal minor of H is:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(12)

Therefore, after substituting πx* from Eq. 6 and Φ(k*) from Eq. 8, we obtain

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

The third principal minor of H is:

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts
(13)

Where, F(k*) = β0(4-β0)φ(k*)Ψ(k*) - 2β0(1-Φ(k*))2

Image for - Impact of Lead Time and Safety Factor in Mixed Inventory Models with Backorder Discounts

For ∀ k* ∈ [0, ∞) and 0 < β0 ≤1, F(k*) is positive. Hence, we have | H33 | >0. Consequently, the results from 11-13 show that the Hessian matrix H is positive define at (Q*, πx*, k*).

REFERENCES
1:  Blackburn, J.D., 1991. Time-Based Competition: The Next Battleground in American Manufacturing. 1st Edn., Richard Irwin, Inc., Homewood, Illinois.

2:  Ben-Daya, M. and A. Raouf, 1994. Inventory models involving lead time as a decision variable. J. Oper. Res. Soc., 45: 579-582.
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3:  Bockerstette, J.A. and R.L. Shell, 1993. Time Based Manufacturing. 1st Edn., McGraw Hill, New York.

4:  Hariga, M.A. and M. Ben-Daya, 1999. Some stochastic inventory models with deterministic variable lead time. Eur. J. Operat. Res., 113: 42-51.
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5:  Liao, C.J. and C.H. Shyu, 1991. Analytical determination of lead time with normal demand. Int. J. Operat. Res. Prod. Manage., 11: 72-78.
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6:  Moon, I. and S. Choi, 1998. A note on lead time and distribution assumptions in continuous review inventory models. Comput. Operat. Res., 25: 1007-1012.
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7:  Ouyang, L.Y., N.C. Yen and K.S. Wu, 1996. Mixture inventory model with backorders and lost sales for variable lead time. J. Operat. Res. Soc., 47: 829-832.
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8:  Pan, J.C. and Y.C. Hsiao, 2001. Inventory models with backorder discounts and variable lead time. Int. J. Syst. Sci., 32: 925-929.
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9:  Ravindran, A., D.T. Phillips and J.J. Solberg, 1987. Operations Research. Principle and Practices. 1st Edn., John Wiley, New York.

10:  Silver, E.A., D.F. Pyke and R. Peterson, 1998. Inventory Management and Production Planning and Scheduling. 3rd Edn., Wiley, New York, ISBN: 9780471119470, Pages: 754.

11:  Tersine, R.J., 1994. Principles of Inventory and Material Management. 1st Edn., North Holland, New York.

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