Economic lot sizing with the consideration of random machine unavailability time

Computers & Operations Research 27 (2000) 335}351

Economic lot sizing with the consideration of random machine unavailability time N.E. Abboud!,*, M.Y. Jaber!, N.A. Noueihed" !Engineering Management Graduate Program, American University of Beirut, Faculty of Engineering and Architecture, 850 Third Avenue, 18th yoor, New York, NY 10022-6297, USA "Department of Mathematics, Hagazian University, Beirut-Lebanon Received 1 January 1998; received in revised form 1 October 1998; accepted 1 February 1999

Abstract This paper extends the work of Abboud and Salameh (Production and Inventory Management 28 (1987) 38}45) by assuming random machine unavailability time and where shortages are allowed. In multiple-item intermittent production runs, the machine might not be available to restart the production of the next batch for a speci"c item. This situation would result in shortages where items are either backordered or lost. The optimum operating inventory doctrine is obtained by trading o! procurement cost per unit time, the inventory carrying cost per unit time, as well as the shortage cost per unit per unit time, so that their sum will be a minimum. Examples illustrating the calculation procedure are provided. Scope and purpose There are numerous production situations where a production facility (e.g., a machine) is used intermittently to produce lot sizes of a certain product. Upon the completion of production a run, the facility may not be available for a random amount of time due to several reasons, such as: the facility is being used to produce other items and their production schedules are not known in advance; the facility needs to be maintained and the maintenance time is random due to unforeseen circumstances; or perhaps the facility is leased by di!erent manufacturers and the demand for the facility is random. As a result, the facility may not be available when it is needed, and stockout situations will arise. It is of interest then to determine the optimal production lot size that minimizes the setup cost, inventory carrying cost, and shortage cost per unit time. Furthermore, we need to determine how the overall inventory cost function is in#uenced by the nature of the random variable that represents the unavailability time of the production facility. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Lot sizing; Machine unavailability; Backorders; Lost sales

* Corresponding author. Tel.: #961-1-350000 x3636; fax: #961-1-744462. E-mail address: [email protected] (N.E. Abboud) 0305-0548/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 9 ) 0 0 0 5 5 - 6

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1. Introduction An inventory system is a system in which three kinds of costs are signi"cant. These costs are the cost of replenishing inventories, the cost of carrying inventories and the cost of incurring shortages. The implication is that shortages are very expensive and must be avoided. When there is customer demand for an item which cannot be met immediately there are shortages, and each customer can either wait for the item to come into stock, in which case there are backorders, or he can withdraw his order and go to another supplier, in which case there are lost sales. The economic lot size problem with the consideration of shortages has been treated in Refs. [2}10]. The lost sales shortage case in a production-inventory system is treated in Refs. [7,11}16]. Kim et al. [17] examined setup reduction on machine availability. They found that setup reduction reduces total inventory cost (excluding shortage cost) and frees up machine production time. Kavusturucu and Gupta [18] developed a methodology for the analysis of tandem manufacturing systems where a machine takes a vacation of random duration every time the corresponding station becomes empty. A machine becomes unavailable, for reasons such as processing secondary jobs or being repaired. Kavusturucu and Gupta [18] assumed a "nite bu!er to cover for demand of primary items during machine vacations. Saad and Gindy [19] classi"ed disturbances in manufacturing systems into two groups: internal and external. They gave examples of internal disturbances, such as unavailability of machines (e.g. breakdown, corrective maintenance, preventive maintenance), tools/"xtures (e.g. not requested, not earmarked for job, shared by other resources), transport (e.g. breakdown, etc.), and operators (e.g. sickness, holiday, disputes, etc.). Apart from the work of Abboud and Salameh [1], none of the references surveyed above treated the case of shortages as a result of machine unavailability. This paper extends the work of Abboud and Salameh [1] by assuming the unavailability time, ¹, is a random variable that has a probability density function f(t). Two general mathematical models are developed to describe the lost sales and the backordering shortage cases. These models are investigated under two di!erent types of distributions; one is upper-bounded (e.g., uniform) and the other is bounded (e.g., exponential). The optimum operating inventory doctrine is obtained by trading o! procurement cost per unit time, the inventory carrying cost per unit per unit time, as well as the shortage cost per unit per unit time, so that their sum will be a minimum. Examples illustrating the solution procedures are provided. The rest of the paper is organized as follows. Section 2 presents the mathematical models developed for the economic manufactured quantity (EMQ) model where machine unavailability time is assumed to be a random variable with a known distribution. Two shortage situations are treated: the lost sales and the backordering cases. Section 3 illustrates the solution procedures for the mathematical models developed in Section 2. It also provides sensitivity analysis of the optimal solutions determined to changes in the values of the di!erent parameters associated with the inventory system described. Section 4 is for summary and conclusions.

2. The mathematical model De"ne K as the set-up cost per production cycle, h as the inventory carrying cost per unit per unit time, s as the shortage cost per unit of time, r as the consumption rate of units per unit time,

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Fig. 1. The behavior of inventory level over time for the economic manufacturing quantity model with no shortages.

and p as the production rate of units per unit time (p'r). Fig. 1 illustrates the classical production cycle with no shortages, where t and t are the times required to accumulate and deplete 1 2 a maximum inventory of I units, respectively. The total cost of producing q units, ¹C (q), is 1 expressed as hq2 ¹C (q)"K# (1!r/p) 1 2r and the total cost per unit time, ¹C; (q), is expressed as 1 Kr hq ¹C; (q)" # (1!r/p). 1 q 2

(1)

(2)

As argued in Abboud and Salameh [1], if t extends for a period less than ¹, then the total cost 2 per unit of time, ¹C; (q), is expressed as 2 pK h(p!r)q2 sMpr¹!q(p!r)N ¹C; (q)" # # . (3) 2 q#p¹ 2r(q#p¹) r(q#p¹) The economic manufacturing quantity that minimizes (3), qH, is given by setting the "rst derivative 2 of (3) equal to zero and solving for q, to give

S

2(rK#sp¹) qH"!p¹# (p¹)2# , 2 h(1!r/p)

(4)

whereas, when ¹)t , the economic manufacturing quantity that minimizes (2) is given by the 2 classical square root formula as

S

2Kr qH" . 1 h(1!r/p)

(5)

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Fig. 2. The behavior of inventory level over time for the economic manufacturing quantity model with shortages; the lost sales case.

The solutions provided by (2) and (3) are not necessarily all feasible. Eq. (2) is feasible when q3[0, r¹/(1!r/p)], and (3) is feasible when q3[r¹/(1!r/p),R). The optimal solution, qH, is obtained by following the solution procedure presented in Abboud and Salameh (1987). Next, we extend the above formulation to develop a more general production-inventory model that accounts for random machine unavailability time. Random unavailability can cause stockout situations, and we examine two shortage doctrines: the lost sales and the backordering cases. The lost sales and the backordering models are investigated under two di!erent types of distributions; one is upper-bounded (e.g., uniform) and the other is unbounded (e.g., exponential). 2.1. The lost sales case Shortages occur when machine unavailability time, ¹, is greater than the inventory depletion time, t , that is, ¹'t . In this model it is assumed that the on-hand inventory at the beginning of 2 2 each cycle is zero, ¹ follows a general probability density function, f(t), and the un"lled demand is lost when ¹'t . The lost sales case is illustrated in Fig. 2. 2 The expected total cost per cycle is the sum of (1) and the expected total shortage cost per cycle. That is

P

hq2 = ¹C(q, t)"K# (1!r/p)#sr (t!t ) f (t) dt 2 2r t2

(6)

where t "I/r"q(1!r/p)/r. When ¹)t , the expression in (6) reduces to (1). In Fig. 2 the cycle 2 2 length is the sum of the production time, t , and the unavailability time, ¹. Hence, the expected 1 cycle length is expressed as

P

q q = ¸(¹)"t #¹" #¹" #t # (t!t ) f (t) dt 1 2 2 p p t2

(7)

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Alternatively, we can express (7) as

P

q = (t!q(1!r/p)/r) f (t) dt. ¸(q)" # r q(1~r@p)@r

(8)

The expected total cost per unit time is given by dividing (6) by (8), to give K#(hq2/2r) (1!r/p)#sr:= (t!q(1!r/p)/r) f (t) dt q(1~r@p)@r ¹C;(q)" . g/r#:= (t!q(1!r/p)/r) f (t) dt q(1~r@p)@r

(9)

The cost model developed in (9) will be further analyzed for two particular cases of f (t). In the "rst case we assume that the random machine unavailability time, ¹, follows a uniform distribution, whereas in the second case we assume ¹ follows an exponential distribution. 2.1.1. The uniform distribution case Assume that the unavailability time, ¹, is a random variable that is uniformly distributed over the interval [0, b], where its probability density function, f (t), is given as

G

f (t)"

1/b,

0)t)b

0,

otherwise

(10)

As illustrated in Fig. 1, there is no stockout situation when ¹)t ; where b)t Nq*br/(1!r/p) 2 2 and the total expected cost is given by (1). However, if ¹'t and q(br/(1!r/p), then by 2 substituting (10) for f (t) in (9), the total cost function will reduce to

A

B

hq2(1!r/p) q2(1!r/p)2 q(1!r/p) b K# #sr ! # 2r r 2 2br2 ¹C; (q)" . 2 q2(1!r/p)2 q b # # p 2 2br2

(11)

The optimal-feasible value of q that minimizes (11) is determined by setting the "rst derivative of (11) equal to zero, d¹C; (q)/dq"0, and solving for q. Note that d¹C; (q)/dq is of a quadratic 2 2 form that holds two roots one of which is negative and it is discarded. The positive root of d¹C; (q)/dq, qH, is given as 2 2

C

qH" 2

D G BA

CA

s(1!r/p) h ~1 K(1!r/p) hb ] # ! # br br p 2

A

#2]

B A

B

hb K(1!r/p) 2 1!r/p 2 ] ! 2 r br

BD H

hr s srb K sb(1!r/p) # ] # # p 2 b p(1!r/p) 2p

1@2

(12)

For the case where q*br/(1!r/p), we simply set qH"qH from (5). 2 1 It is now necessary to demonstrate that (11) holds a unique minimum when (12) is substituted in (11). To do this, two observations are made. First, (11) approaches sr#2K/b as q approaches zero

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and sr#bhr/(1!r/p) as q approaches in"nity, where (11) is positive in both cases. Second, the limit of the "rst derivative of (11) is negative as q approaches zero from the right-hand side. That is, lim `d¹C; (q)/dqP!2(spb#2K)/pb2(0. This indicates that (12) is the global minimum q?0 2 that minimizes (11). From the above discussion, a three-step solution procedure to determine qH is illustrated as follows: Step 1: Compute q "br/(1!r/p). 0 Step 2: Compute qH and qH from (5) and (12), respectively. 1 2 Step 3: Let qH"qH if ¹C; (qH)(¹C; (qH); else set qH"qH. Determine the optimal production 3 1 1 1 2 2 3 2 quantity, qH, from the following:

G

q 0 qH qH" 1 qH 2 q 3

if qH'q and qH(q , 1 0 2 0 if minMqH, qHN'q , 1 2 0 if maxMqH, qHN(q , 1 2 0 if qH(q and qH'q . 1 0 2 0

2.1.2. The exponential distribution case Assume that the unavailability time, T, is a random variable that is exponentially distributed with mean 1/j. Then f (t) is given as f (t)"je~jt for j'0

(13)

The expected total cost per unit time is determined by substituting (13) for f (t) in (9) to give hq2(1!r/p) sr # e~jq(1~r@p)@r K# j 2r . ¹C;(q)" q 1 # e~jq(1~r@p)@r r j

(14)

It can be shown that the above cost function is not convex. However, any simple line search method over q'0 is guaranteed to converge to the global optimal solution qH. To show that this is true, we "rst make use of the following two lemmas. Lemma 1. Let f (x) be a real-valued function satisfying the following properties: (1) f (x) is continuous on [0,R,) and diwerentiable on [0,R). (2) lim f (x)"#R. x?`= (3) f (x) is decreasing over [0, x ) for some x '0. 0 0 (4) The equation f @(x)"0 has at most two solutions on (0,R). Then there exists a value x '0 with f (x ) the minimum global value of f (x) on [0,R), and x being the 1 1 1 unique global minimizer. Proof. Let u'f (0), by condition (2) we can "nd ¹'x , such that f (x)*u, for x'¹. By 0 condition (1), the continuity of f (x) on [0, ¹] guarantees the existence of the minimum value `ma of

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f (x) where m(f (0)(u is achieved by a point x . Condition (3) guarantees that 0(x (¹. Now 1 1 the di!erentiability of f (x) guarantees that f @(x )"0. Finally, suppose that by (4) there is another 1 value x 'x such that f @(x )"0. Since two successive local minima must include a local 2 1 2 maximum in between, and by condition (2), a local minimum followed by a local maximum must be followed by a local minimum. Then x does not correspond to a local extreme value and x is 2 1 a unique global minimizer. Lemma 2. Let f (x) and g(x) be diwerentiable over the open interval (a, b), and let h(x)"f (x)!g(x). If m is the number of distinct intersection points between the graphs of f (x) and g(x), and n is the number of distinct solutions of h@(x)"0 over (a, b), then m)n#1. Proof. Let (x , y ) and (x , y ) be two consecutive intersection points between the graphs of f (x) i i i`1 i`1 and g(x) with x (x . Clearly, h(x) is di!erentiable and h(x )"h(x )"0. By Rolle's theorem i i`1 i i`1 there exists at least one c within x (c (x such that h@(c )"0. Since there are at least m!1 i i i i`1 i consecutive pairs of intersection points, there are at least m!1 distinct solutions of h@(x)"0. This implies m!1)n and m)n#1. Lemma 3. The cost function in (14) has a unique global minimum. Proof. See the appendix for the proof. 2.2. The backordering case In this section we assume that demand is not immediately met, but rather backordered. Fig. 3 illustrates the behavior of the inventory level for this case. Let I denote the maximum inventory level attained in each production cycle. The total holding inventory cost per production cycle, HC(I), is given as hI2 HC(I)" . 2r(1!r/p)

(15)

Fig. 3. The behavior of inventory over time for the economic manufacturing quantity model with shortages, the backordering case.

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As shown in Fig. 3, backorders occur when ¹'t , where t "I/r. Then the total expected 2 2 shortage cost per cycle, SC(I), is given as

GP

cr ] SC(I)" 2(1!r/p)

H

= 2 (t!I/r) f (t) dt , I@r

(16)

where c is the backordering cost per unit of time. Thus, the expected total cost per unit time, ¹C;(I), is given as

G

H

hI2 cr K# # ](:= (t!I/r) f (t) dt)2 I@r 2r(1!r/p) 2(1!r/p) , ¹C;(I)" ¸(I)

(17)

where ¸(I) is the expected cycle length and is given as

G

P

H

I 1 I = ¸(I)" #max , (t!I/r) f (t) dt . p!r r (1!r/p) I@r

(18)

2.2.1. The uniform distribution case Following the same procedure as in Section 2.1.1, for I/r(b, the expected total cost per unit time, ¹C;(I), is expressed as

G H

H

I2 I b 2 hI2 cr ! # K(1!r/p)# # ] 2br2 r 2 2 2r , ¹C;(I)" b I2 # 2 2br2

G

(19)

whereas, for I/r*b, (19) reduces to Kr(1!r/p) hI ¹C;(I)" # I 2

(20)

which has a minimum at

S

IH"

2Kr(1!r/p) . h

(21)

2.2.2. The exponential distribution case Following the same procedure as in Section 2.1.2, the expected total cost per unit time, ¹C;(I), is expressed as

G

HA

B

hI2 cre~2Ij@r I e~Ij@r ~1 ¹C;(I)" K(1!r/p)# # ] # . 2r 2j j r

(22)

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Although all our experiments have shown that (22) appears to have a unique global minimum that can be easily found with a line search method, we have not been able to rigorously prove it thus far.

3. Numerical results Consider a production situation where the production rate, p, is 120 units per unit time; the consumption rate, r, is 80 units per unit time; the holding cost, h, is $20 per unit per unit time; the shortage cost, s, is $30 per unit of unmet demand; and the set-up cost, K, is $100 per production cycle. Assuming that the unavailability time is uniformly distributed over the interval [0,2], we will follow the solution procedure stated earlier for the lost sales case. This yields q "480 units, 0 qH"49 units, and qH"182 units. Since maxM49, 182N(480, then qH"qH"182 units; see Fig. 4. 1 2 2 If, instead, T is exponentially distributed with parameter j"1, then we can resort to spreadsheets and the dichotomous search method to minimize (14). Table 1 shows the evolution of the search, where B"0.5](q #q )!0.05 and C"0.5](q #q )#0.05. The search con.*/ .!9 .*/ .!9 verged to the optimal value qH"177 units. Suppose now we allow for backordering during stockouts, where the backordering cost, c, is $30 per unit per unit time. Keeping the rest of the cost parameters as before with T being uniformly distributed over [0,2], we minimize (21) using the dichotomous search technique as shown in Table 2, where the optimal value of I is 49 units. Fig. 5 illustrates graphically the behavior of the cost curves for the two cases where I/r(b and I/r*b. Finally, if T is exponential with parameter j"1, then as shown in Table 3, I"49 will minimize the cost function in (24). Numerous examples were performed, where we have varied all the parameters of the production-inventory model. Fig. 6 shows the logical behavior of the cost components as the lot size quantity varies for the lost sales case model with unavailability time being uniformly distributed. That is, by increasing q, the setup and shortage costs per unit time decrease, while the holding cost per unit increases. It is interesting to note that as long as the setup cost K is not signi"cantly large, then the shortage and the holding costs will dominate the setup cost. As a result, the optimal production quantity will

Fig. 4. The behavior of the total cost per unit time for both the traditional EMQ model, ¹C; , and the modi"ed EMQ 1 model, ¹C; , when unavailability time is uniformly distributed. 2

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N.E. Abboud et al. / Computers & Operations Research 27 (2000) 335}351 Table 1 Search for optimal lot size quantity when machine unavailability time, T, is exponentially distributed; the lost sales case q .*/

B

¹C;(B)

¹C;(C)

C

q .!9

1 1 1 126 126 157 173 173 177 177 177 177

500.45 250.73 125.86 188.29 157.08 172.69 180.49 176.59 178.54 177.56 177.08 176.83

1698.10 1022.31 1007.13 951.05 956.58 949.24 949.11 948.91 948.94 948.91 948.90 948.90

1698.41 1022.48 1006.87 951.09 956.50 949.23 949.12 948.90 948.95 948.91 948.90 948.90

500.55 250.83 125.96 188.39 157.18 172.79 180.59 176.69 178.64 177.66 177.18 176.93

1000 501 251 251 188 188 188 181 181 179 178 177

Table 2 Search for maximum inventory level when machine unavailability time, T, is uniformly distributed; the backordering case q .*/

B

¹C;(B)

¹C;(C)

C

I

20 20 20 20 20 35 42 46 48 48 49

259.95 139.98 79.99 49.99 35.00 42.50 46.24 48.12 49.06 48.59 48.82

2380.15 1406.38 726.55 559.30 604.59 568.07 560.45 559.09 559.00 559.00 558.99

2380.81 1407.49 727.49 559.35 603.90 567.78 560.33 559.06 559.02 558.99 558.99

260.05 140.08 80.09 50.09 35.10 42.60 46.34 48.22 49.16 48.69 48.92

500 260 140 80 50 50 50 50 50 49 49

.!9

not be far from the lot size that sets the holding cost equal to the shortage cost. This is particularly true for the case where the unavailability time is uniformly distributed, and backordering is allowed. As Fig. 7 shows, with backordering, the optimal production quantity stays almost constant as K varies from 0 to 7500. For higher values of K, the lot size quickly converges to the EMQ solution where the setup cost per unit time dominates the shortage cost.

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Fig. 5. The behavior of the total cost per unit time for both the classical EMQ model, ¹C; , and the modi"ed EMQ 1 model, ¹C; , when unavailability time is uniformly distributed and backordering is allowed. 2

Table 3 Search for maximum inventory level with the assumption that machine unavailability time, T, is exponentially distributed; the backordering case I .*/

B

¹C;(B)

¹C;(C)

C

I

1 1 1 1 32 48 48 48 48 49 49

250.45 125.73 63.36 32.18 47.77 55.57 51.67 49.72 48.75 49.23 48.99

2481.25 1158.31 627.61 653.13 594.41 601.77 595.52 594.29 594.18 594.19 594.17

2482.29 1159.36 628.05 652.38 594.37 601.99 595.62 594.32 594.17 594.20 594.18

250.55 125.83 63.46 32.28 47.87 55.67 51.77 49.82 48.85 49.33 49.09

500 251 126 63 63 63 56 52 50 50 49

.!9

Fig. 6. The behavior of the cost components as q varies, for the case of lost sales and ¹ uniformly distributed.

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Fig. 7. Behavior of optimal lot size quantity as K changes with ¹ being uniformly distributed.

4. Summary and conclusions This paper extended the work of Abboud and Salameh [1] by assuming that machine unavailability time is a random variable with a known distribution, and scheduling is not an issue. The models developed treated two shortage situations where unful"lled demand orders are either lost or backordered. The unavailability time may arise from the fact that the machine is being used to produce di!erent products without knowing before hand the production schedules of those products. Another possibility could be that the machine has to be maintained after producing a lot and the maintenance time is random. In analyzing the random aspect of machine unavailability, two types of distributions were discussed; the uniform distribution which has a "nite upper bound, and the exponential which has no upper bound. The purpose behind the di!erent types of distributions is to show how the cost structure and the solution procedure would be in#uenced by the range of the unavailability time. An obvious extension to the work presented in this paper is to incorporate safety stocks as protection measure against stockout situations. With safety stocks we would need to deal with an additional decision variable which determines the optimal safety stock levels. Furthermore, it would be interesting to incorporate the concept of learning and forgetting when production breaks are random. There are numerous results in the literature that deal with learning and forgetting (see Badiru [20], Jaber and Bonney [21,22], and Arzi and Shtub [23]). Very few, however, analyze such a concept in a probabilistic environment.

Appendix Lemma A.1. The cost function given in (14) has a global minimum. Proof. Without any loss of generality, let j"1, r"1, and A"(1!r/p)/r, and rewrite (14) as Ah K# q2#se~Aq 2 ¹C;(q)" . q#e~Aq

(A.1)

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It su$ces to show that the function satis"es the four conditions of Lemma 1, keeping in mind that 0(A(1 and s'h. We will not distinguish between the number of distinct intersection points between the curves representing the two functions f (q) and g(q), and the distinct solutions of the equation f (q)"g(q). The "rst derivative of (A.1) is

C

e~Aq ¹C;@(q)"

D C

D

A2hq2 Ahq2 #A(h!s)q#AK!s # !K 2 2 . (q#e~Aq)2

(A.2)

We can easily show that conditions (1) and (2) of Lemma 1 are satis"ed since ¹C;@(q) is continuous on [0,R), and lim ¹C;@(q)"(A!1)(K!s)(0. q?0` Hence, ¹C;@(q)(0 on the interval [0, x ) for some x '0, and conditions (3) of Lemma 1 holds. o o By setting ¹C;@(q)"0 in (A.2), we get

C

e~Aq

D

Ahq2 A2hq2 #A(h!s)q#AK!s "! #K 2 2

(A.3)

Let m be the number of solutions of (A.3). Then, from Lemma 2, m)n#1, with n being the number of solutions over [0,R) of

G C

d A2hq2 e~Aq #A(h!s)q#AK!s dq 2

DH G

H

d Ahq2 " ! #K dq 2

which reduces to e~Aq[A2hq2!A(h#s)q#AK!h]"hq.

(A.4)

Let ¹(q)"e~Aqp(q)"hq where p(q)"A2hq2!A(h#s)q#AK!h. A careful analysis of the graphical behavior of ¹(q) will show that its curve cuts the line `hqa in at most one point on [0,R). Setting ¹@(q)"0NAp"p@, or equivalently ¸(q)"!A2hq2#A[3h#s]q!(AK#s)"0.

(A.5)

From (A.5), two possible cases can arise and are explained as follows. Case 1: Eq. (A.5) has no roots, or exactly one root. In this case, ¸(q)(0 and ¹@(q)(0. Over [0,R),¹(q) is monotone decreasing, whereas hq is monotone increasing, and their graphs intersect by at most one point. Case 2: Eq. (A.5) has two roots q and q . Since q #q "(3h#s)/A'0 and q q " 1 2 1 2 1 2 (AK#s)/A'0, then 0(q (q . 1 2 Based on cases 1 and 2, and the observation that lim ¹(q)"#R and lim ¹(q)"0`, q?~= q?`= there are four graphical forms of ¹(q) as shown in Figs. 8}11. If ¹(q )!hq (0, i"1, 2 holds, then the graph of ¹(q) cuts the line hq in at most one point. We i i notice that q satis"es ¹@(q)"0, or Ap(q )"p@(q )Np(q )"p@(q )/A"2Ahq !h!s, which in turn i i i i i i

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Fig. 8. AK!h'0, p(q) has no roots.

Fig. 9. AK!h'0, p(q) has one root.

implies that ¹(q )!hq "e~Aqi[2Ahq !h!s]!hq . Consider for a*0 the functions i i i i ha . S(a)"e~Aa and G(a)" 2Aha!h!s Since h#s G(a)(0, 0)a( 2Ah

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349

Fig. 10. AK!h'0, p(q) has two roots.

Fig. 11. AK!h(0, p(q) has two roots.

and 1 h#s G(a)' , a' 2A 2Ah then the graph of S(a) will be strictly above the graph of G(a) on [0, (h#s)/(2Ah)), and strictly below that graph on ((h#s)/2Ah,R). See Figs. 12 and 13, respectively. Now it su$ces to show that S(a)"1/(2A) has either no solution on [0,R), or has exactly one solution which lies before (h#s)/(2Ah). If A'1/2, then e~Aa"1/(2A)Na"ln(2A)/A. However, ln(2A)/A(1/A"2h/(2Ah)((h#s)/(2Ah)

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N.E. Abboud et al. / Computers & Operations Research 27 (2000) 335}351

Fig. 12. Case when 1/(2A))1.

Fig. 13. Case when 1/(2A)'1.

Therefore,

G

'0 e~Aa[2Aha!h!s]!ha S(a)!G(a)" is 2Aha!h!s (0

if 0)a((h#s)/(2Ah), if a'(h#s)/(2Ah)

and then e~Aa[2Aha!h!s]!ha(0 for ha*0N¹(q )!h(q )(0, i"1, 2. i i Eq. (A.4) has at most one solution and (A.3) has at most two solutions by Lemma 2. Therefore condition (4) of Lemma 1 holds for the ¹C;(q) function and it has a unique global minimum qH on [0,R).

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