International Journal of Industrial Engineering, 19(5), 232-240, 2012.

ISSN 1943-670X © INTERNATIONAL JOURNAL OF INDUSTRIAL ENGINEERING

AN EPQ MODEL WITH VARIABLE HOLDING COST

Hesham K. Alfares Systems Engineering Department, King Fahd University of Petroleum & Minerals,

Dhahran 31261, Saudi Arabia. Email: alfares@kfupm.edu.sa

Instantaneous order replenishment and constant holding cost are two fundamental assumptions of the economic order quantity (EOQ) model. This paper presents modifications to both of these basic assumptions. First, non-instantaneous order replenishment is assumed, i.e. a finite production rate of the economic production quantity (EPQ) model is considered. Second, the holding cost per unit per time period is assumed to vary according to the length of the storage duration. Two types of holding cost variability with longer storage times are considered: retroactive increase and incremental increase. For both cases, models are formulated, solutions algorithms are developed, and examples are solved. Keywords: Economic production quantity (EPQ), Variable holding cost, Production-inventory models.

(Received 13 Apr 2011; Accepted in revised form 28 Oct 2011) 1. INTRODUCTION In the classical economic order quantity (EOQ) model, the replenishment of the order is assumed to be instantaneous, i.e. the production rate is implicitly assumed infinite. In practice, many orders are manufactured gradually, at a finite rate of production. Even if the orders are purchased, the procurement and receipt of these orders is seldom instantaneous. Therefore, economic production/manufacturing quantity (EPQ/EMQ) models are more representative of real life. Moreover, the assumption of a constant holding cost for the entire duration of storage may not be always realistic. In many practical situations, such as in the storage of perishable items, longer storage periods require additional specialized equipment and facilities, resulting in higher holding costs. This paper presents an EPQ inventory model with a finite production rate and a variable holding cost. In this model, the holding cost is assumed to be an increasing step function of the storage duration. Two types of time-dependent holding cost functions are considered: retroactive increase, and incremental increase. Retroactive holding cost increase means that the holding cost of the last storage period applies to all previous storage periods. Incremental holding cost increase means that increasingly higher holding costs apply only to later storage periods. For each of these two types, optimal solutions algorithms are developed to minimize the total cost per unit time. Several EOQ and EPQ models with variable holding costs proposed in the literature consider holding cost to be a function of the amount or value of inventory. Only few EOQ-type models assume the holding cost to vary in relation to the inventory level. Muhlemann and Valtis-Spanopoulos (1980) revise the classical EOQ formula, assuming the holding cost to be an increasing function of the average inventory value. Their justification is that the greater the value of inventory, the higher the cost of financing it. Mao and Xiao (2009) construct an EOQ model for deteriorating items with complete backlogging, considering the holding cost as a function of the on-hand inventory. A solution procedure is developed, and the conditions are specified for the existence and uniqueness of the optimal solution when the total holding cost function is convex. Moon et al. (2008) develop mixed integer programming models and genetic algorithm heuristic solutions to minimize the maximum EOQ storage space requirement for both finite and infinite time horizons. Some inventory models have built-in flexibility, allowing the holding to be a function of either the inventory level or storage time. Goh (1994) considers an EOQ-type single-item inventory system with a stock-dependent demand rate and variable holding cost. Giri and Chaudhuri (1998) construct an EOQ-type inventory model for a perishable product with stock-dependent demand and variable holding cost. Considering two types of variation of the holding cost per unit, both Goh (1994) and Giri and Chaudhuri (1998) treat holding cost either as: (i) a non-linear continuous function of the time in storage, or (i) a non-linear continuous function of the amount of inventory. In several EOQ-type models, the holding cost is assumed to be a continuous function of storage time. For a non-linearly deteriorating item, Weiss (1982) considers the holding cost per unit as a non-linear function of the length of storage duration. Optimal order quantities are derived for deterministic and stochastic demands, and for both finite and infinite time horizons. Giri at al. (1996) develop a generalized EOQ model for deteriorating items with shortages, in which both the demand rate and the holding cost are continuous functions of time. The optimal inventory policy is derived assuming a finite planning horizon and constant replenishment cycles. Ferguson et al. (2007) apply Weiss (1982) formulas to approximate optimal order quantities for grocery store perishable goods, using regression to estimate the holding cost curve parameters. Alfares (2007) introduces the notion of holding cost variability as a discontinuous step function of storage time, with two types of holding cost increase. As the storage time extends to the next time period, the new (higher) holding cost can be

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applied either retroactively to all storage periods, or incrementally to the new (latest) period only. Urban (2008) extends Alfares (2007) work by allowing non-zero end inventory for each cycle, and shifting the objective from cost minimization to profit maximization. Singh et al. (2009b) present an EOQ inventory model with stock-dependent demand and partial backlogging of unsatisfied demand. Their model considers deteriorating items, inflation, and incrementally increasing holding cost function of storage time. In several published EPQ models, the holding cost is assumed to be a continuous nonlinear function of time. Goh (1992) presents three models of stock-dependent demands inventory systems: (1) a basic model, (2) a model with planned shortages, and (3) an EPQ model with non-instantaneous receipt of orders. Singh et al. (2009a) develop an EPQ inventory model with time-varying demand rate, demand-dependent production rate, and permissible delay in payments. Two possibilities are considered for the holding cost: (i) a constant value, and (ii) a constant component plus a time-dependent component. Sarfaraz (2009) formulates deterministic and stochastic EPQ models, dividing the inventory holding costs into two parts: one proportional to the average dollar value of inventory, and another proportional to the maximum inventory. Tripathy et al. (2010) develop an EPQ model for items with linear deterioration rate in which the holding cost is assumed to vary with time. Fully deteriorated items are discarded, partially deteriorated items are sold with a discount, and shortages are not allowed. Chiu and Chiu (2005) analyze an EPQ model with the possibility of producing defective items that require additional work to be repaired. While other EPQ models consider holding cost variation as a continuous nonlinear function of storage time, this paper uses the discontinuous step functions proposed by Alfares (2007). Two types of discontinuous step functions are used to represent holding cost variation over storage time: retroactive and incremental. The problem is further defined and the objective specified next in Section 2. The model and the optimal solution algorithm for retroactive holding cost are presented in Section 3. The corresponding model and algorithm for incremental holding cost are given in Section 4. Some application insights for practitioners are given in Section 5. Finally, conclusions and suggestions for future research are provided in Section 6. 2. PROBLEM DESCRIPTION AND MODEL In this paper, the main objective is to determine the minimum-cost production-inventory control plan for an EPQ system with and a time-dependent holding cost. Assuming non-instantaneous order receipt, the orders are produced/received gradually at a finite production rate during the uptime phase of the production cycle. Assuming a step holding cost function of storage time, the holding cost per unit of the item per unit time is higher for longer storage periods. The mathematical model developed below takes variable unit holding costs into consideration to determine the optimal production-inventory control policy. As stated earlier, the holding cost per unit per unit time is assumed to be an increasing step function of the storage duration. The storage time is divided into a number of intervals with different holding costs. Two types of holding cost step function increase are considered: retroactive, and incremental. In the retroactive type, the higher holding cost of the last storage interval is applied to all storage intervals. In the incremental type, the specific holding cost of each time interval, including the last interval, is applied only to units stored in that particular interval. 2.1 Notation The following notation is used to represent the EPQ model under consideration:

q(t) = the inventory level at time t D = demand rate P = constant production rate during the uptime phase (0 ≤ t ≤ t1), P > D n = number of distinct time intervals with different holding cost rates t = time from the start of the cycle at t = 0 t1 = duration of the first (uptime) phase of the cycle τi = end time of interval i, where i = 1, 2 ,…n, τ0 = 0, and τn = ∞ K = ordering cost per order h(t) = holding cost per unit per time period at time t, h(t) = hi if τi – 1 ≤ t ≤ τi T = cycle time, i.e. time between producing two successive lots of size Q e = holding cost ending interval for the cycle, such that τe– 1 ≤ T < τe u = holding cost interval that contains t1, such that τu– 1 ≤ t1 < τu Q = order quantity or lot size = Pt1 =DT

2.2. Assumption and limitations

1. The demand rate D is a known constant value. 2. The production rate P is a known constant value, where P > D.

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234

3. The holding cost increases with longer storage durations (h1 < h2 < ... < hn). 4. Shortages are not allowed. 5. A single item is considered.

2.3. The objective function The objective of the model is to minimize the total cost TC of the production-inventory system per unit time. This total cost is composed of the ordering cost and the holding cost. Since one order at a cost K is made per cycle T, the ordering cost per cycle is simply K/T. For each time period t, the holding cost is the product of the holding cost h(t) and the inventory level q(t). Dividing the holding cost by cycle time T, and adding the ordering cost per cycle, the total cost per time unit is given by:

TC(T) = ∫+T

dttqthTT

k

0

)()(1

... (1)

3. RETROACTIVE HOLDING COST Retroactive holding cost increase means that the holding cost of the last storage period applies retroactively to all storage periods. If the cycle ends in interval i, (ti –1 ≤ T ≤ ti), then the holding cost rate hi is applied to all intervals 1, 2, ..., i. In this case, the total cost per unit time TC is given as:

TC(T) = ,)(0∫+T

i dttqTh

TK ti –1 ≤ T ≤ ti

... (2)

The inventory level function q(t) has two forms during the two phases of the cycle. During the uptime phase (0 ≤ t ≤ t1), the function is q(t) = (P – D)t. During the downtime phase, (t1 ≤ t ≤ T), the function becomes q(t) = (T – t)D. Substituting these functions into (2) and integrating, the total cost per unit time can be expressed as:

TC(T) = ,2

)(PDPTDh

TK i −+ ti –1 ≤ T ≤ ti

... (3)

Minimizing the above equation leads to the following version of the conventional EPQ formula:

T = )(

2DPDh

KP

i −

... (4)

3.1 Retroactive solution algorithm

1. Use (4) to calculate the cycle time T for each hi. If T is in the correct range (i.e., τi –1 ≤ T ≤ τi), consider the cycle time as realizable TR.

2. For each TR, use (3) to calculate the total cost TC(TR). 3. For each interval i, let Ti = τi and use (3) to calculate the total cost TC(Ti). 4. Choose the value of T that gives the lowest TC(T). Use the relationship Q = TD to determine the corresponding

production lot size Q. 3.2 Example 1 The following demand and cost values are obtained from a local food industry company. Three types of food storage facilities with different holding costs are applicable to three distinct storage periods: short-term, medium-term, and long-term.

D = 1000 per year P = 1500 per year K = $850 per order h1 = $30 per unit per year 0 < t ≤ 0.2 year h2 = $40 per unit per year 0.2 < t ≤ 0.4 h3 = $50 per unit per year 0.4 < t Assume the holding cost increase is retroactive. Step 1(a). Substituting h1 = 30 into (4), we get

T = )000,1500,1)(000,1(30

)500,1)(850(2−

= 0.412

Since T is not in the range of h1 (0 < t ≤ 0.2), it is not realizable. Step 1(b). Substituting h2 = 40 into (4), we get

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235

T = )000,1500,1)(000,1(40

)500,1)(850(2−

= 0.357

Since T is in the range of h2 (0.2 < t ≤ 0.4), it is realizable. Step 1(c). Substituting h3 = 50 into (4), we get

T = )000,1500,1)(000,1(50

)500,1)(850(2−

= 0.319

Since T is not in the range of h3 (t > 0.4), it is not realizable. Step 2. Use (3) to calculate the total cost TC(T) for TR.

TC(0.357) = )500,1(2

)500,1500,1)(000,1)(357.0(40357.0850 −

+ = $4,760.95

Step 3. Use (3) to calculate the total cost for each Ti = τi.

TIC(0.2) = )500,1(2

)500,1500,1)(000,1)(2.0(302.0

850 −+ = $5,250.00

TIC(0.4) = )500,1(2

)500,1500,1)(000,1)(4.0(404.0

850 −+ = $4,791.67

Step 4. The optimum cycle time is T = 0.357 year (4.28 months), with an annual cost of $4,760.95. The corresponding production lot size is:

Q = 0.3571(1,000) = 357 4. INCREMENTAL HOLDING COST If the holding cost is an incremental step function of storage time, then higher storage costs apply only to storage in later time intervals. The ordering cost per cycle does not change. However, the holding cost is obtained by multiplying the individual holding cost rate hi for each time interval i (τi – 1 ≤ t ≤ τi) by the total inventory stored during the same interval. Assuming the cycle ends in interval e (τe – 1 ≤ T ≤ τe), the total cost per unit time TC is given by:

∑ ∫=

+=e

ii

i

dttqhTT

KTTC1 1-i

)(1)(τ

τ

... (5)

Or

∑=

Φ+=e

iiihTT

KTTC1

1)( ... (6)

Where Φi is the total inventory stored during time interval i (τi – 1 ≤ t ≤ τi), which is defined as follows:

∫=Φi

dttqi

τ

τ 1-i

)( ... (7)

During each cycle, the inventory-level function q(t) has one form during the uptime phase (0 ≤ t ≤ t1) and another during the downtime phase (t1 ≤ t ≤ T). Therefore, the integral in (7) depends on the locations (values) of the end points of interval i (τi – 1 and τi) with respect to end points of the cycle (t1 and T). Since (τi – 1 < τi) and (t1 < T), there are five distinct cases for the relationship between the four values (τi – 1, τi, t1, T). From these five cases (a, b, c, d, and e) shown in Figure 1, the following expressions for Φi can be developed.

,2

))(( )(2

12

1-i

−−−=−=Φ ∫ ii

iDPtdtDP

i τττ

τ

τi – 1 < τi ≤ t1 < T ... (8a)

,2)(21

)()(

2222

1

1

1

1-i 1

⎥⎦

⎤⎢⎣

⎡−+−−=

−+−=Φ

−

∫ ∫

PTDTDDPD

dtDtPttdtDP

iii

t

ti

i

τττ

τ

τ

τi – 1 ≤ t1 ≤ τi ≤ T

... (8b)

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236

[ ],)()(21

)()(

22221

1

1

1-i 1

TDPDPPDP

dtDtPttdtDP

i

t T

ti

−+−=

−+−=Φ

−

∫ ∫

τ

τ τi – 1 ≤ t1 < T ≤ τi

... (8c)

[ ],)(22

)( 22111

1-i

iiiii TDdtDtPti

τττττ

τ

−+−=−=Φ −−∫ t1 ≤ τi – 1 < τi

≤ T

... (8d)

( ) ,2

)(2

11

1-i

−−=−=Φ ∫ i

T

iTDdtDtPt τ

τ

t1 ≤ τi – 1 ≤ T ≤ τI ... (8e)

To minimize the total cost TC(T), equation (6) is differentiated with respect to T and then the derivative is set equal to zero, producing the following equation:

0')(1

2 =+−=ʹ′ ∑=

e

iiihT

KTCTI φ ... (9)

Where

dTTd i

i)/(' Φ

=φ ... (10)

Again, using equations (8a)-(8e), five different expressions for φiʹ′ are developed depending on the locations of the interval end points (τi and τi) with respect to the cycle end points (t1 and T).

2

21

2

2))(('

TDP ii

i−−−

−=ττ

φ τi – 1 < τi ≤ t1 < T ... (11a)

2

22221

2

2)('

PTTDPPDPD ii

i−−−

= −ττφ τi – 1 ≤ t1 ≤ τi ≤ T

... (11b)

2

221

22

2)()('

PTPPDDPDT i

i−−−

= −τφ τi – 1 ≤ t1 < T ≤ τi ... (11c)

2

21

2

2)('

TD ii

i−−

=ττ

φ t1 ≤ τi – 1 < τi ≤ T ... (11d)

2

21

2

2)('

TTD i

i−−

=τ

φ t1 ≤ τi – 1 ≤ T ≤ τi ... (11e)

EPQ Model

237

Figure 1. Five locations for the holding cost interval [τ i, τ i] in relation to the cycle end points (t1 and T).

4.1 Incremental solution algorithm 1. Substitute the minimum and maximum values of hi (i.e., h1 and hn) into (4) to find the range of values of T (and hence e).

Next, use the relationship t1 = TD/P to find the range of values of t1 (and hence u). 2. For every feasible combination of t1 and T (u and e), substitute the appropriate terms from (11a)-(11e) into equation (9).

Solve the equation to find the value of T and subsequently t1. 3. If both t1 and T fall in the correct time intervals (u and e, respectively) used in formulating and solving (9), then the

solution is considered realizable. 4. Substituting the appropriate terms from (8a)-(8e), use (6) to calculate the total cost TC(T) for each realizable solution.

q(t)

(P – D)t

t1 T t

(T – t)D

(d) t1 ≤ τi – 1 < τi ≤ T

q(t)

(P – D)t

t1 T t

(T – t)D

(c) τi – 1 ≤ t1 < T ≤ τi

q(t)

(P – D)t

t1 T t

(T – t)D

(b) τi – 1 ≤ t1 ≤ τi ≤ T

q(t)

(P – D)t

t1 T t

(T – t)D

τi-1 τi (a) τi – 1 < τi ≤ t1 < T

τi-1 τi

τi τi-1 τi-1 τi

q(t)

(P – D)t

t1 T t

(T – t)D

(e) t1 ≤ τi – 1 ≤ T ≤ τi τi τi -1

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238

5. Choose the solution which has the minimum total cost per unit time TC(T) and then calculate the optimal lot size by: Q = TD.

4.2 Example 2 Using the same data given in Example 1 for the local food industry company, we now assume that the holding cost increase is incremental. Step 1: Substitute the values of h1 and h3 into (4) to find the range of values of t1 and T.

Tmax = )000,1500,1)(000,1(30

)500,1)(850(2−

= 0.412, t1,max = 500,1

)412.0(000,1 = 0.275

Tmin = )000,1500,1)(000,1(50

)500,1)(850(2−

= 0.319, t1,min = 500,1

)319.0(000,1 = 0.213

Clearly, T may fall either in the second interval (0.2 < t ≤ 0.4) or the third interval (t > 0.4), thus e = 2 or 3. However, t1 falls only in the second interval, thus u = 2.

Step 2 (a): Assuming both t1 and T fall the second interval, then substituting terms from (11a) and (11c) into (9), we get:

[ ] 02.0}500,1000,1(500,1{}000,1)000,1(500,1{)500,1(2

402

)02.0)(000,1500,1(30850

22222

2

22

2

=−−−+

−−−−

TT

TT

Solving gives: t1 = 0.2236 T = 0.3354

Since both t1 and T fall in the second interval (0.2 < t ≤ 0.4), the solution is realizable. Step 2 (b): Assuming t1 falls in the second interval while T falls in the third interval, then substituting terms from (11a), (11b) and (11e) into (9), we get:

[ ]

02

)4.0)(000,1(50

000,1}500,1)000,1(500,1{2.0)4.0)(000,1(500,1)500,1(2

402

)02.0)(000,1500,1(30850

2

22

222222

2

22

2

=−

+

−−−+

−−−−

TT

TT

TT

Solving gives t1 = 0.243 T = 0.3645 Since T is not in the third interval (t > 0.4), the solution is not realizable. Step 3: Since we have only one realizable solution (T = 0.3354), it is the optimal solution. The corresponding optimal lot size is calculated as Q = 0.3354(1,000) = 335. Substituting the appropriate terms from (8a) and (8c), equation (3) is used to calculate the optimal total cost TC(0.3354).

[ ] 14.472,4$)3354.0)(500,1(2

}000,1000,1(500,1{3354.0}500,1)000,1(500,1{2.040

)02.0)(000,1500,1()3354.0(2

303354.0850

2222

22

=−+−

+

−−+=TC

5. INSIGHTS FOR PRACTITIONERS This paper presents new variations of the EPQ model, in which the holding cost increases incrementally or retroactively with the storage duration. Typical applications include the storage of food and other perishable items. Storing such items may be classified into short-, medium-, and long-term, requiring successively more advanced and expensive storage techniques and facilities. In order to apply the models of this paper in practice, several implementation steps are required.

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239

The first step is to check whether the assumptions of the model, listed in section 2.2, are satisfied in the given production control context. The holding per unit time should increase in discrete time periods, as a step function of the storage duration. The second step is to determine whether the holding cost increases retroactively or incrementally in order to apply the appropriate solution algorithm. In the third implementation step, all the values of the model parameters (D, P, K, h1, . . . hn) need to be collected, measured, or calculated. In the fourth step, the corresponding algorithm is used to determine the appropriate values of Q, T, and C(T). Finally, the solution is implemented and adjustments are made if necessary. An important insight for practitioners is the significant cost saving obtained from the proposed variable holding cost EPQ model. To illustrate this point, we apply the conventional constant holding cost EPQ model to the data of examples 1 and 2. Ignoring holding cost variability and assuming a constant holding cost of $30/unit/year leads to Q = 412, t1 = 0.275, and T = 0.412. For retroactive holding cost, the total cost TC(0.412) would be $5,496.44, compared to $4,760.95 in Example 1. For incremental holding cost, the total cost would be $4,568.93, compared to $4,472.14 in Example 2. 6. CONCLUSIONS An EPQ-type production-inventory system with finite production rate and a variable holding cost has been presented. The holding cost is assumed to be an increasing step function of the length of storage time. Two forms of holding cost variability with longer storage time have been analyzed: retroactive increase, and incremental increase. For each of these two forms, models have been formulated, optimal solution algorithms have been developed, and illustrative examples have been solved. There are many directions for future research based on the new EPQ model. The model can be extended to allow variable ordering costs and variable production rates. Other possibilities include incorporating planned shortages and considering capacity limitations on the production rate or storage space. Finally, the model could be enhanced by including stochastic considerations, such as probabilistic demand and probabilistic production rate due to random machine failure and repair. 7. REFERENCES 1. Alfares, H.K. (2007). Inventory model with stock-level dependent demand rate and variable holding cost. International

Journal of Production Economics, 108: 259-265. 2. Chiu, Y-S.P. and Chiu, S.W. (2005). The finite production model with the reworking of defective items. International

Journal of Industrial Engineering: Theory, Applications and Practice, 12: 15-20. 3. Ferguson, M.; Jayaraman, V. and Souza, G.V. (2007). Note: An application of the EOQ model with nonlinear holding

cost to inventory management of perishables. European Journal of Operational Research, 180: 485-490. 4. Giri, B.C. and Chaudhuri, K.S. (1998). Deterministic models of perishable inventory with stock-dependent demand

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BIOGRAPHICAL SKETCH

Dr. Hesham Alfares is Professor in the Systems Engineering Department of King Fahd University of Petroleum & Minerals, Saudi Arabia. He has a PhD in Industrial Engineering from Arizona State University. He has been as a visiting scholar at MIT, University of North Carolina, Loughborough University, University of Nottingham, and the University of Warwick. He won research grants from the British Council and the Fulbright Foundation. Dr. Alfares has more than 70 refereed journal and conference papers and a U.S. patent. He served as editorial board member for three international journals and as program committee member for 23 international conferences.