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Process mean determination under constant raw material supply
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of OperationalResearch 99 (1997) 353-365
Theory and Methodology
Process mean determination under constant raw material supply J i n s h y a n g R o a n a, L i n g u o G o n g
b, K w e i T a n g b,. ,l
a Department of Business Administration, Soochow University, Taipei, Taiwan, ROC b Department oflnformation Systems and Decision Sciences, Louisiana State University, Baton Rouge, LA 70 803, USA
Received September 1994; revised October 1995
Abstract Setting the mean (target value) for a production process is an important decision for a producer when material cost is a significant portion of production cost. Because the process mean determines the process conforming rate, it affects other production decisions, including, in particular, pro.duction setup and raw material procurement policies. In this paper, we consider the situation in which the product of interest is assumed to have a lower specification limit, and the items that do not conform to the specification limit are scrapped with no salvage value. The production cost of an item is a linear function of the amount of the raw material used in producing the item, and the supply rate of the raw material is finite and constant. Furthermore, it is assumed that quantity discounts are available in the raw material cost and that the discounts are determined by the supply rate. Two types of discounts are considered in this paper: incremental quantity discounts and all-unit quantity discounts. A two-echelon model is formulated for a single-product production process to incorporate the issues associated with production setup and raw material procurement into the classical process mean problem. Efficient solution algorithms are developed for finding the optimal solutions of the model. © 1997 Elsevier Science B.V. Keywords: Production/operations management;Inventorytheory; Statistics
1. Introduction Setting the process mean (target value) of a production process is a classical problem in quality control. A typical situation considered in the problem is as follows. The product of interest has a performance variable with a lower specification limit, and the raw material requirement for producing the product is an increasing function of the performance variable. An item that meets this specification is
i Supported in part by National Science Foundation grant no. DDM 8857557. * Corresponding author.
classified as conforming and is sold at a fixed regular price, while the nonconforming item is sold at a reduced price, reprocessed, or scrapped, depending on the production and marketing environments. The product is produced by a production process with an adjustable mean and a constant variance. If the process mean is set too high, the probability of producing nonconforming items becomes small. The manufacturer saves the cost raised by nonconforming items at the expense of material cost. (The cost of extra material is sometimes referred to as 'give-away' COSt.) Consequently, a process mean should be determined based on a balance between production cost and economic consequences associated with conforming items and nonconforming items. Note that
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J. Roan et al./ European Journal of Operational Research 99 (1997) 353-365
the typical performance variable under study are weight, volume, count, or concentration. The problem is first addressed by Springer [11]. He considered a production situation where upper and lower specification limits are both presented and the performance variable follows a gamma distribution. Bettes [1] studied the problem of finding optimal values for both process mean and upper limit. All the nonconforming (lower than the lower limit or higher than the upper limit) items are reprocessed at a fixed cost. Hunter and Kartha [9] considered a product with a lower specification limit and discussed the situation where nonconforming items can be sold at a (constant) reduced price. Bisgaard, Hunter and Pallesen [2] argued that Hunter and Kartha's assumption concerning the selling price of the nonconforming items is not realistic and, therefore, modified the model by assuming that the selling price of the nonconforming items is a linear function of the performance variable. Carlsson [3] followed Hunter and Kartha's model except for assuming that the net income function is a piecewise linear function of the performance variable. Golhar [4] assumed that only the regular market (fixed selling price) is available for the conforming items and that the underfilled items are emptied and reprocessed at a fixed expense (reprocessing cost). Golhar and Pollock [5] extended Golhar's [4] model to include an artificial upper limit so that nonconforming items as well as the items larger than the upper limit are reprocessed. Golhar and Pollock [6] also studied the cost benefit of reducing the variance of the process. A detailed review of these and related studies was given by Tang and Tang [ 12]. The selection of process mean determines the process conforming rate and, thus, the production yield rate. As a result, other production decisions, such as production setup and raw material procurement policies, are also dependent on the process mean setting. Roan, Gong and Tang [10] incorporated the issues associated with production setup and raw material procurement into the process mean problem. It is assumed that the product of interest requires one major raw material, which is purchased from outside vendors at a fixed unit cost. The production cost of an item is a linear function of the amount of the raw material used in producing the item. A two-echelon model is formulated for jointly
determining the process mean, production lot size and raw material procurement policy in a singleproduct production process. Gong, Roan and Tang [7] extended the model by Roan et al. to consider quantity discounts in raw material cost. In this paper, we base on the model proposed by Roan et al. [10] to consider a situation wherein a constant supply rate of raw material is offered by a reliable source. This approach of receiving raw materials from a vendor is known as 'continuous replenishment', which has drawn much attention recently [8]. Basically, this approach is beneficial to both the manufacturer and the supplier: the manufacturer can practically eliminate the raw material inventory and avoid out-of-stock situations, and the supplier can better plan production and inventory of the raw material. A smooth and cooperative manufacturersupplier relation is essential for using this method. In addition, we also consider the situation in which quantity discounts, incremental and all-units quantity discounts, for raw material are offered by vendors. The organization of this paper is as follows. In the next section, assumptions are given and models are formulated for three cases in quantity discounts: no quantity discounts, incremental discounts, and allunits quantity discounts. In Section 3, analytical properties associated with the optimal solution are derived and solution procedures are proposed. In Section 4, numerical examples are given to illustrate the solution procedures, and in Section 5 a sensitivity analysis is carried out to compare the three cases in terms of quantity discounts. Finally, a brief summary of the results and a discussion of possible future extensions are given in the last section.
2. Model assumption and formulation Consider a product with a constant demand rate of D items per unit time. A production process with a production rate of r items per unit time is used to satisfy the demand. Let X denote the performance variable of interest, which is a measure of the raw material used in the production, such as weight and volume. Let L be the lower specification limit of X, so that an item is conforming if its X value is larger than or equal to L. Assume that the production process is stable and X follows a probability density
J. Roan et al. / European Journal of Operational Research 99 (1997) 353-365
function, f i x ) , with an adjustable mean /x and a constant variance cr 2. For given /x, the conforming rate of the production process is
355
Inventory q(rp-D) t _
_ _
(a) ~
p= f~f( x) dx. Assume that nonconforming items are scrapped with no salvage value. Consequently, for given /x, the yield rate of the production process is r- p items per unit time. It is assumed that all the demand will be satisfied in such a way that the expected total number of conforming items produced is equal to the total demand and no backlog is allowed. Note that use of the expected conforming items is reasonable, especially in high-speed production, because the production output can be treated as approximately constant. Otherwise, it may require considering safety stock or using other inventory models. Note that r. p has to be greater than or equal to D to ensure that the production capacity is large enough to meet the demand. Suppose the unit cost of the raw material is c; thus, cX is the material cost required for producing an item of the finished product. We further assume that the direct cost of producing an item is a linear function of the item's material cost:
g( X ) = b + acX, where b is the fixed production cost, and a is a constant larger than or equal to 1. This cost function implies that the production cost consists of a fixed cost and a variable cost that is proportional to the raw material used in production. Note that a - 1 is the relative value (cost) added on the raw material during the production process. It can be verified that, for given /x, the expected cost of yielding a conforming item is
( b + aclx)/p. Let h a be the cost of holding a monetary unit of the raw material for a unit time. Assume that the costs of holding a monetary unit of raw material and finished product are the same. Then, the cost of holding a conforming item for a unit time is
H = (ha/p) ( actx + b). Let q be the production run size, which is the number of items (including both conforming and
R a w material
I I
I
Time
q r
__Pq D
(b) 0
Time
Fig. 1. (a) I n v e n t o r y level o f f i n i s h e d product; (b) i n v e n t o r y l e v e l o f r a w material.
nonconforming items) produced in a production run. The inventory level as a function of time is described in part (a) of Fig. 1. Assume that a production run begins at time 0. The finished product inventory increases at a rate of r . p - D items per unit time until q items are produced. At time q/r, the production run is complete, and, then, the inventory decreases at a rate of D items per unit time until time qp/D when the inventory level reaches 0 and the second production run starts. Let S be the production setup cost. Since the total number of setups required per unit time is D/qp, the total setup cost is SD/qp. It can be verified that the average inventory level for the finished product is
½q/r(rp-D). As a result, the total holding cost for finished products is
H½q/r( rp - D). per unit time. Furthermore, because the expected cost of yielding a conforming item is (b + actz)/p, the per-unit-time direct production cost is
D(b + a c t z ) / p . Therefore, the finished product-related cost, which is the sum of the production cost, the setup cost and the inventory holding cost, is given by FPC(/x, q) -
D( b + act~ ) P
DS
+ --
Pq
+ H ~ r ( r p - D), where H = ( h J p ) ( a c ~ + b).
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356
The expected amount of the raw material required to produce one conforming item is Ix/p: It is assumed that the raw material arrives at a constant rate /3 per unit time, regardless of whether the production process is in operation or is idle. Assume all the raw material received by the producer is used in production. The inventory level of the raw material as a function of time is shown in Fig. lb. The requirement for the raw material is r/x per unit time during production, which should be higher than the raw material supply rate /3. Since q is the production run size, we assume that the initial raw material inventory level at time 0 is (q/r)(rlx-/3), which is sufficient to meet the requirement in the first production run. When production starts, the raw material inventory decreases at a rate of ( - r/z + / 3 ). At time q/r, the raw material inventory is depleted and the first production run is finished. Then, the process is idle, and the raw material inventory level increases at a rate of /3 until the second production run starts at time pq/D. At the start of a production run, the inventory level / 3 ( p q / D - q/r) should be same as the level at time 0; that is, (Pq q) q(r/x-/3)=/3r --D--r " For any given process mean /x, therefore, the proper supply rate should be a function of /x; i.e., /3 = Dix/p. On the other hand, /x corresponding to given /3 is determined by /x = p/3/D. In this paper, the material-related cost is expressed as a function of /3 instead of po. Let C(/3 ) denote the raw material cost when the supply rate is /3. The cost of holding a conforming item for a unit time can, therefore, be rewritten as H = h,(
+ b/p).
Since the supply rate is constant, the setup cost associated with the raw material is not considered. The total holding cost per unit time for the raw material is M C ( / 3 , q) =
h,C~r(rlx - / 3 )
=hlC~---~-q ( r p - D ) , zrp which can be rewritten as MC(/3,
C( /3)q q) = h , - - ( r p - D). 2rD
The finished product-related cost function can be expressed as a function of 13 and q as follows: F P C ( / 3 , q) =
Db DS aC( /3 ) + - - + - P Pq + H~r ( r p - D).
Three cases, in terms of raw material quantity discounts, are considered. Let c i > ci+ 1, and 0 ~
c ( / 3 ) =/3c0, where c o is the material unit cost. Case 2. Incremental quantity discounts; that is,
C(/3) = C(/3i) ~- (/3 -/3i) ci, for /3 ~ [/3i, /3i+l), i = 0 , 1 , 2 . . . . . k - 1. Case 3. All-units quantity discounts; that is,
c ( / 3 ) =/3c,, for /3 ~ [/3i, /3i+1), i = 0 , 1, 2 . . . . . k - 1. Let TC l(/3, q) and TC2(/3, q) denote the total cost functions for cases 1 and 2, respectively. Because purchasing excess material is not economical for cases 1 and 2, these two total cost functions are given by T C , ( / 3 , q) = F P C ( / 3 , q) + M C ( / 3 , q)
Db
=
+--+--
DS
P Pq ( aC(/3) + b ) q ( r p - D ) +hi ~ ~r +h l
C(- /3)q (r p - D ) 2rD
for i = 1
2.
(1) For case 3, the material purchasing cost C(/3) under the all-units quantity discounts is a discontinuous function of /3, and it is possible that the producer orders a larger-than-needed amount of the raw material and disposes of the excess raw material to take advantage of price differentials. Let /3u denote the actual material use rate, where /3u ~/3- Depending on whether excess raw material is ordered, there-
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J. Roan et a l . / European Journal of Operational Research 99 (1997) 353-365
fore, we further consider the following two cases under all-units quantity discounts. Case 3.1: No excess raw material ordered In this case, the raw material use rate is equal to the raw material supply ,rate; i.e., /3~ =/3. In other words, no excess raw material is ordered. It can be easily verified that the total cost, denoted by TC31(/3, q), is also given by (1). Case 3.2: Excess raw material (disposed) In this case, /3 is larger than /3u, and the excess material is disposed of at a cost of c d per unit material. If the supply rate is /3i, then the total cost function is
TC32(/3u, q) Db = a/3iC i -lr-
cient solution algorithms are proposed to find the optimal process mean, production run size, and material supply rate. 3.1. Analytical properties In this subsection, we study the properties of three situations: no discounts, incremental quantity discounts, and all-units quantity discounts. For given /3, the optimal production run sizes for the no-discount, incremental discount and all-units discount without disposal are found by solving OTC/(/3, q ) / O q = O. We obtain the following result. Result 1. For cases 1, 2, and 3.1, the optimal production run size associated with given /3 is
DS
+ m
•
Pq
P
q +hi
D
+ -p
~ r (rp - D )
/3uq . + h l c i - ~ - r ~ ( rp - D) + ( / 3 i - / 3 u ) ( C i + ca)"
(2) Note that ( ]3 i -- /3u)(Ci "~ Cd) = D E ( / 3 i ,
~
2rDS
hl(rp/D_l)[Db+p(~+l)C(/3)] "
=
In cases 3.2, it can be verified that TC32(/3u, q) is a convex function of q when /3u (or /x) is fixed. Consequently, the minimum point of the total cost for given /3u (or /x) can be found by finding the solution to 0TC32(/3u, q ) / O q = O.
q)
is the disposal cost per unit time. It is assumed that excess raw material is disposed when it is received, so that no holding cost is associated with the excess raw material. Therefore, the process mean /x is P/3u/D. The optimal solution to the model is obtained by minimizing the per-unit time expected total cost. Since it is assumed that all the demand must be satisfied, the optimal solution also minimizes the expected total cost of producing a conforming item.
The following result can be obtained. Result 2. For case 3.1, the optimal production run size associated with given /3u is , q
~ =
2rDS hl(rp/D-
1)[Db+p(ol+
1)ci/3u]
"
3. Optimal solution
Note that, for all-units discounts, using Result 1, TC3t(/3, q * ) becomes a function of /3. As a result, a search procedure can be used to find the solution /3 * that minimizes TC31(/3, q * ). However, for unit cost c i, the solution may not be feasible because /3 * may not be in the range [/3i' /3i+ 1)" If /3 * is in [/3i, /3i+ 1), we say /3 * is admissible.
In this section, we first derive several important analytical properties for the optimal solutions for the three cases defined in the last section; that is, no discounts, incremental discounts, and all-units quantity discounts policies. Based on these results, effi-
Result 3. Consider all-units discounts. If/3 *, which minimizes TC3~( ~, q), is admissible for given unit cost c i, then this solution has a lower total cost than all the solutions associated with the higher unit costs.
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J. Roan et al./ European Journal of Operational Research 99 (1997) 353-365
Proof. We let TC31(/3, q [ ci) and TC32(/3, q ] ci) be the total costs when c i is used in evaluating total cost functions, Eqs. (1) and (2), respectively. For cj > % we know that
WE31(/3, q[cj) > TC31(/3, q lci) and TC3e(/3, q lcj) > TC3~ (/3,
q* [ Ci) ~
TC31(/3, q l Ci)
for all /3 :/:/3 * and q ¢ q *. On the other hand, for anY /3 ~ / 3 i ' TC32(/3, q l cj) > TC32 (/3, q lci) and TC32(/3, q lci) >1 TC3~ (/3, q[ci). As a result, TC31(/3*, q * I ci) < TC3x (/3, q[cj) and
TC3x(/3 *,
q * [c,) ~ TC3, (/3, q[ci) < TC32(/3, q lcj).
Similar to case 1, the given /3 can be obtained process mean is given by search method is used material supply rate.
optimal production size for by using Result 1, and the p/3/D. The golden-section to search for the optimal
Case 3: All-units quantity discounts
q[ci).
Let q * be the production run size corresponding to /3 *. Since 13 * is admissible for ci,
TC31(/3 *,
Case 2: Incremental quantity discounts
[]
3.2. Solution algorithm The analytical results presented in the last section provide the basis for developing efficient solution procedures to find the optimal solutions for the three cases. The solution procedures for the three cases are given separately as follows. Note that the golden-section search is used for all the three cases below.
Case 1: No discounts Because the raw material supply rate is a function of the process mean and Result 1 gives the optimal production run size corresponding to a given process mean, the total cost becomes a function of the process mean. The golden-section search is used to locate the optimal process mean in the range [L, L + 4o-] and the search is terminated when the width of the interval of uncertainty is less than or equal to 0.0001.
The solution procedure is a search method based on the unit material cost c r Specifically, it starts with the lowest c k and case 3.1. The relationship between the material supply rate and the material unit cost is ignored first. The solution that minimizes TC31(/3, q lcg) is feasible and admissible if its resulting /3 is inside its cost (price) range. The procedure continues until the first feasible /3 is found. Let c' be the unit price associated with the first feasible solution (i.e., /3 * is admissible). If c' is the lowest unit cost, then the solution is optimal. Otherwise, based on Result 3, we can exclude all the solutions associated with the unit cost higher than c'. Therefore, we examine case 3.2 for all the unit costs lower than c', and compare the results with the first feasible solution (for case 3.1). The one with the lowest total cost is the optimal solution. The procedure is summarized by the following two steps.
Step 1. Starting with the lowest price, search the value of /3 * that minimizes TC31(/3, q [ Cg). /3 * is admissible if /3 * ~ [/3i, /3i+ l)- This process continues until an admissible /3 * is found. Step 2. If /3 * corresponding to the lowest unit material cost is admissible, then the solution is optimal. Otherwise, find the solutions for case 3.2 associated with the unit costs lower than cj. Comparing the solution obtained in Step 1 and those in this step for case 3.2, the one with the lowest total cost is the optimal solution. A F O R T R A N program has been written for implementing the solution procedures on an IBM 3090600E computer. The running time for solving each of the problems considered in this paper is only a few seconds.
4. An example In this section, an example is used to illustrate the solution procedures given in the last section. The
J. Roan et al./ European Journal of Operational Research 99 (1997) 353-365
example will also be used in the sensitivity analysis in Section 5. Consider a product that requires at least 8 mgs of main content in each item. Any item that is less than 8 mgs is considered nonconforming and is scrapped without salvage value. Because of the variation in the production process, the content of an item produced by the process follows a normal distribution, with an adjustable process mean and a constant standard deviation of 1.2 mgs. Assume that the product demand rate and the production rate are 5000 items and 7500 items per unit time, respectively. The setup cost per production run is $150, the fixed production cost is $0.16 per item, and c~ is 4. The raw material is constantly supplied by a vendor. Furthermore, the cost for holding $1 of inventory (finished product or raw material) is $0.03 per unit time. The material cost for the no-discounts policy is $0.1 per mg and the material costs under the incremental quantity discounts policy are as follows: $0.1
for each of the first 30 000 mgs,
$0.098
for each of the next 30 000 mgs,
$0.095
for each mg in excess of 60 000 mgs.
c=
The all-units quantity discounts are as follows: Order quantity
Price per mg ($)
1-29 999 30 0 0 0 - 5 9 999 _> 60 000
0.100 0.098 0.095
The raw material disposal cost to the producer is ca = $ - 0 . 0 4 5 / m g . The results are given as follows. Case 1: N o discounts The optimal process mean is found to be 9.915 mgs. The corresponding raw material supply rate and the production run size are 52 474.94 mgs and 5927 items, respectively. The total cost is $22 104.50. The process conforming rate is 94.48%. Case 2: Incremental quantity discounts The optimal process mean is found to be 9.916 mgs and the corresponding process conforming rate
359
is 94.48%. The raw material supply rate is /3 * = 52475.0 mgs and its corresponding raw material purchasing cost is C(/3 * ) = $5202.54. The optimal production run size is q * = 5952 items and the total cost is $21 924.31. Case 3: All-units quantity discounts The optimal raw material supply rate and use rate found are 60000 and 52475.53 mgs, respectively. Based on the raw material use rate, the optimal process mean is 9.919 mgs. The following two steps were used to find the optimal supply rate and production run size associated with the optimal process mean: Step 1. When c i = $0.098, the optimal /3 * =/3u = 52 475.07 mgs, which is within the range [30 000, 60 000). TC31 = $21 682.39, and c' = 0.098. Step 2. c i = $0.095 is the only unit cost that is smaller than c'. When c i = 0.095, the lowest total costs TC32 = $21 425.18, and its corresponding /3u = 52 475.53 mgs, which is less than /3 = 60 000. Comparing this result with the result in Step 1, we found that the lowest total cost is $21 425.18.
Therefore, the optimal solution is to use the supply rate 60000 mgs and dispose of 7524.47 mgs. The total disposal cost per unit time is $376.29. The resulting process mean is 9.919 mgs, and the process conforming rate is 94.51%. The optimal production run size is 6071 items.
5. Sensitivity analysis In this section, a sensitivity analysis is performed to study the effects of the model parameters on the optimal solutions of all three cases: 1) no discounts, 2) incremental quantity discounts, and 3) all-units quantity discounts. The model parameters included in the study are the demand rate D, the production rate r, the process standard deviation o-, the valueadded factor a , and the production setup cost S. For the all-units quantity discounts policy, the unit disposal cost, c d, is also studied. The sensitivity analy-
J. Roan et al./ European Journal of Operational Research 99 (1997) 353-365
360
ses are based on the example given in the last section. Per-unit costs are used to report the results in this section. They are per-item finished product-related cost:
q)/D,
PFPC = F P C ( / 3 ,
per-item material-holding cost: PMC = M C ( / 3 ,
q)/D,
per-item total cost: PTC = T C ( / 3 ,
q)/D,
and per-item disposal cost: PDC = D C ( / 3 ,
q)/D.
5.1. Effect of demand rate To study the effects of demand rate, we use the results for selected values of D ranging from 1500 to 7000 per unit time. As shown in Table 1, when D increases, the optimal production run size for cases 1 and 2 increases. This result makes intuitive sense. When the demand rate is very low relative to the production rate (process capacity), no clear patterns were observed on the optimal process mean. w h e n the demand rate is moderate or very high, both the
optimal process mean and the material supply rate increase steadily as the demand rate increases. When the demand rate is very close to the production rate, because of a low accumulation rate of finished products, the production run size in the first two cases becomes very large and sensitive to the change of D. The results of case 3 are very similar when the demand rate is lower than 5000. When the demand rate is larger than 5000, however, a different approach should be taken under the all-unit discounts policy. The reason is that both a larger demand and a higher process mean push the material use rate higher. When the material use rate is high enough, it is attractive to use the material supply rate at the next discount-applicable level. As a result, a larger material supply rate, /3 * = 60 000, is used to take advantage of quantity discounts, and excess materials are disposed. We also notice that a larger process mean is also used, which is expected because the material cost is lower. A noticeable difference in case 3 (compared with cases 1 and 2) is that, when the demand rate is very high, the optimal production run size is not affected by changes in the demand rate. The reason is that excess materials are available and the process yield rate is high; consequently, accumulation rate of finished products remains high. It is expected, however, that when the demand rate is
Table 1 Effect of demand rate D
Case 1 /z*
q*
/3'
/z*
q*
/3*
u*
q*
/3u
/3*
1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000
9.921 9.923 9.919 9.920 9.918 9.919 9.916 9.915 9.912 9.909 9.903 9.801
1 982 2399 2825 3 278 3 781 4356 5 049 5 927 7 136 9019 12854 1 609350
15 743 20990 26238 31 485 36733 41 980 47228 52475 57722 62969 68 216 73 510
9.921 9.923 9.919 9.920 9.918 9.918 9.917 9.916 9.912 9.911 9.902 9.801
1982 2399 2825 3280 3787 4368 5066 5952 7169 9067 12957 1623034
15 743 20990 26238 31 485 36733 41 980 47 228 52475 57 722 62969 68 216 73 510
9.921 9.922 9.919 9.921 9.918 9.917 9.917 9.919 9.918 9.918 9.918 9.918
1982 2399 2825 3310 3818 4400 5098 6071 7304 7304 7304 7304
15 743 20990 26238 31 485 36732 41 980 47228 52 476 57 723 57723 57 723 57 723
15 743 20990 26238 31 485 36732 41 980 47228 60000 60000 60000 60000 60000
D : p~* : q *: /3 * : /3u:
Case 2
Product demand rate. Optimal process mean. Optimal production lot size. Optimal raw material supply rate. Material use rate.
Case 3
J. Roan et al./ European Journal of'Operational Research 99 (1997) 353-365
361
Table 2 Effect of production rate r
Case 1
6000 6500 7 000 7 500 8000 8 500 9000 9 500 10000
Case 2
Case 3
/x*
q*
fl*
/x*
q*
/3*
/x*
q*
flu
/3*
9.905 9.910 9.914 9.915 9.917 9.917 9.917 9.919 9.919
9410 7473 6514 5927 5526 5233 5009 4829 4684
52474 52 474 52475 52475 52475 52475 52475 52476 52476
9.904 9.910 9.913 9.916 9.918 9.918 9.919 9.921 9.921
9453 7504 6541 5951 5548 5054 5028 4848 4703
52474 52474 52475 52475 52475 52 475 52475 52 476 52476
9.907 9.913 9.918 9.919 9.921 9.922 9.923 9.925 9.925
9636 7654 6671 6071 5661 5360 5130 4946 4798
52 474 52475 52475 52476 52476 52476 52 476 52476 52476
60000 60000 60000 60000 60000 60000 60000 60000 60000
r : Production rate. /z* : Optimal process mean. q * : Optimal production lot size. 13 * : Optimal raw material supply rate. /3u: Material use rate.
very close to the production rate, the optimal production run size will become sensitive to changes in the demand rate.
5.2. Effect of production rate Table 2 contains the results under selected values of r ranging from 6000 to 10000. The results of the three cases are very similar. It is clear that a large production rate results in a smaller q* and a larger
/x*. A larger /z* implies a higher production yield rate and a higher per-item holding cost. In order to avoid carrying a high level of finished product inventory, therefore, the manufacturer should lower the production run size, so as to reduce the cost of holding finished product inventory. Note that /3 * and flu remain very stable as r increases. This implies that, as production rate (capacity) increases, the manufacturer should adjust internal production planning rather than change the vendor's raw material supply rate.
Table 3 Effect of process variation o-
0.01 0.10 0.50 1.00 2.00 3.00 4.00 5.00 6.00 o- : /z* : q* : /3 * : flu:
Case 1
Case 2
]
]Case 3
/x*
q*
/3'
/~*
q*
/3*/
/x
q
flu
/3
8.030 8.266 9.005 9.689 10.640 11.203 11.410 11.140 10.584
6006 5 946 5 870 5 893 '6 167 6750 7 906 11302 1 435 473
40204 41 492 46048 50760 58 682 65 349 71 043 75 782 79 383
8.030 8.266 9.006 9.688 10.643 11.206 11.415 11.144 10.584
6021 5 962 5 890 5 918 6 192 6789 7958 11382 1449436
40204 41 492 46048 50760 58 692 65 349 71044 75 783 79783
8.030 8.266 9.005 9.692 10.648 11.211 11.414 11.144 10.584
6065 6004 5 928 6037 6 311 6907 8097 11 546 1471 628
40204 41 492 46 048 50761 58 683 65 350 71044 75 783 79383
40204 41 492 46048 60000 60000 65 350 71044 75 783 79383
Process standard deviation. Optimal process mean. Optimal production lot size. Optimal raw material supply rate. Material use rate.
J. Roan et al. / European Journal of Operational Research 99 (1997) 353-365
362
5.3. Effect of process standard deviation
5.4. Effect of value-added factor
As discussed, a smaller process standard deviation implies a better performance of the process and a smaller total cost of operating the process. Table 3 shows the results associated with selected values of O'.
The results of the three cases are very similar. As o- increases, the material supply rate generally increases. Two exceptions are ~r = 1.0 and 2.0 for case 3, in which excess materials are ordered to take advantage of quantity discounts. When tr is smaller than 0.5, a larger o- results in a larger process mean. As discussed, a larger mean implies a higher per-item inventory holding cost. The production run size decreases, therefore, in order to reduce a high inventory holding cost. Note that although the process mean increases as ~r increases, the process yield rate actually decreases. As a result, when tr is between 1.0 and 4.0, a larger production run size is used mainly because the process yield rate decreases very significantly. When o- is larger than 5.0, however, in order to avoid the higher per-item production cost and holding cost, the process mean is set at a lower level wherein the process yield rate is very close to the demand, resulting in a very large production run size.
A larger a implies a higher average cost of producing an item and also a higher average holding cost. To reduce these costs, a reasonable approach is to use a lower process mean. This result is observed in Table 4. The changes in the process mean are, however, quite small. Consequently, the process yield rate does not decrease very significantly. Furthermore, to avoid a high inventory holding cost caused by a larger a , the production run size decreases as a increases. From Table 4, we can also see that, when a is 3 and 4, the process mean in case 3 is much higher than those in cases 1 and 2. The purpose of the increased process mean is to raise the material supply rate to 60 000 for a lower per-unit material cost ($0.095/mg). The saving from the lower material unit cost is more than the cost of disposing of excess material and the cost incurred because of higher process mean. The material supply rates in cases 1 and 2 are relatively very stable with respect to the changes in a. In case 3, the material use rate is also very stable as a increases, although a shift occurs in the material supply rate at a = 3.0. From the above discussion, we can conclude that the production run size is the decision variable most affected by a change in a.
Table 4 Effect of value-added factor a
Case 1 /z*
q*
/3"
/z*
q*
/3'
p~*
q*
B~
/3*
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
9.981 9.937 9.923 9.915 9.911 9.908 9.906 9.905 9.903 9.903
9037 7538 6591 5927 5430 5040 4723 4459 4236 4041
52498 52479 52476 52475 52474 52 474 52474 52474 52474 52474
9.983 9.939 9.925 9.916 9.911 9.908 9.906 9.904 9.904 9.902
9070 7566 6615 5952 5453 5061 4743 4479 4253 4056
52499 52480 52476 52475 52474 52474 52474 52474 52474 52474
9.983 9.938 9.930 9.919 9.912 9.909 9.907 9.906 9.903 9.903
9119 7610 6745 6071 5566 5166 4842 4572 4344 4145
52 499 52 479 52477 52476 52475 52474 52 474 52474 52474 52 474
52499 52479 60000 60000 60000 60000 60000 60000 60000 60000
a : /x* : q *: /3 * : /3u:
Case 2
Value-added factor. Optimal process mean. Optimal production lot size. Optimal raw material supply rate. Material use rate.
Case 3
J. Roan et al. / European Journal of Operational Research 99 (1997) 353-365
363
Table 5 Effect of production setup cost S
Case 1
50 100 150 200 250 300 350 400 450 500 550
Case 2
Case 3
/x*
q*
/3"
/x*
q*
/3*
/x*
q*
/3u
/3*
9.919 9.918 9.915 9.915 9.911 9.910 9.910 9.908 9.907 9.906 9.905
3419 4837 5927 6845 7660 8392 9064 9 694 10284 10843 11 375
52475 52475 52475 52475 52474 52474 52474 52 474 52474 52474 52 474
9.921 9.918 9.916 9.914 9.912 9.910 9.910 9.908 9.908 9.905 9.905
3432 4857 5952 6875 7689 8427 9101 9 734 10326 10890 11 422
52476 52475 52475 52475 52475 52474 52474 52 474 52474 52474 52 474
9.924 9.921 9.919 9.918 9.915 9.914 9.912 9.910 9.910 9.910 9.907
3502 4955 6071 7013 7845 8595 9287 9 934 10538 11 106 11 656
52476 52476 52476 52475 52475 52475 52475 52 474 52474 52474 52 474
60000 60000 60000 60000 60000 60000 60000 60 000 60000 60000 60 000
s : Production setup cost. /x* : Optimal process mean. q * : Optimal production lot size. /3 * : Optimal raw material supply rate. /3u: Material use rate.
5.5. Effect o f production setup cost
duction setup cost increases. Note that a large prod u c t i o n r u n s i z e r e s u l t s in h o l d i n g a l a r g e r i n v e n t o r y ,
It is e x p e c t e d t h a t a l a r g e p r o d u c t i o n s e t u p c o s t l e a d s to a l a r g e p r o d u c t i o n r u n size in o r d e r to reduce the frequency of production setup. The results in T a b l e 5 s h o w t h a t this r e l a t i o n s h i p is t r u e f o r all
and that a smaller process mean, on the other hand, reduces per-item inventory cost and process yield rate. S i n c e t h e d e c r e a s e s in t h e p r o c e s s m e a n s are
t h e t h r e e c a s e s . It is i n t e r e s t i n g , h o w e v e r , t o o b s e r v e
s m a l l , t h e p r o c e s s y i e l d rate r e m a i n s q u i t e s t a b l e . F u r t h e r m o r e , t h e m a t e r i a l s u p p l y a n d u s e r a t e are
that t h e o p t i m a l p r o c e s s m e a n d e c r e a s e s as t h e p r o -
also
very
stable
when
the
production
setup
Table 6 Effect of unit disposal cost
Ca
P.*
q*
/3u
13 *
PFPC
PMC
PDC
PTC
-
9.918 9.919 9.919 9.919 9.916 9.916 9.916 9.916 9.916 9.916
6073 6071 6071 6071 5984 5984 5984 5984 5984 5984
52 475 52 476 52 476 52 476 52 475 52 475 52 475 52 475 52 475 52 475
60 000 60 000 60 000 60 000 52 475 52 475 52 475 52 475 52 475 52 475
4.2047 4.2047 4.2047 4.2047 4.3313 4.3313 4.3313 4.3313 4.3313 4.3313
0.005 l 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051
0.0226 0.0527 0.0828 0.1129 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
4.2323 4.2624 4.2925 4.3226 4.3364 4.3364 4.3364 4.3364 4.3364 4.3364
0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08 0.10
co: Unit disposal cost. /x* :Optimal process mean. q * : Optimal production lot size. /3 ".'Optimal raw material supply rate. /3u: Material use rate.
PFPC: PMC: PDC: PTC:
Per-item Per-item Per-item Per-item
finished product-related cost. material holding cost. raw material disposal cost. total cost.
cost
364
J. Roan et a l . / European Journal of Operational Research 99 (1997) 353-365
increases. Consequently, it can be concluded that only the production run size is significantly affected by the production setup cost.
5.6. Effect of unit disposal cost Table 6 gives the effects of some selected c a values ranging from $ - 0.08 to $0.1 for the all-units quantity discounts. When c o increases, the material use rate remains very stable. When c d is small (less than or equal to $ - 0.02), it is economical to order excess material to receive larger quantity discounts. When c a is larger than $ - 0.02, this practice is not attractive. Both the process mean and production run size are at a slightly higher level when excess material is ordered. When c d is larger than $ - 0.02, the solution remains unchanged because no excess material is ordered.
The model structure presented in this paper provides a useful framework for future research on several interesting issues related to this classical problem in quality control. In particular, the following several extensions are possible. First, the deterioration of raw material a n d / o r finished product can be considered. This factor will become very interesting and important when the deterioration rates of raw material and finished product are different. Second, the deterioration of the production process can be incorporated into the model. The third extension is to consider a multiple-level filling process in which several raw materials are added in different stages. This issue could become very complicated, however, when the product conformance is jointly determined by the amount of several raw materials. These issues have been included in the authors' future research plans.
Acknowledgments 6. Summary In this paper, it is assumed that the raw material is supplied at a constant rate from outside vendors. Three cases in terms of quantity discounts in the raw material cost are introduced into a two-echelon model, which incorporates the issues associated with production run size and raw material procurement policy into the classical process mean problem for a single-product production process. The performance variable has a lower specification limit, and the items that do not meet the limit are scrapped with no salvage value. The production cost of an item is a linear function of the amount of raw material used in producing the item. Three cases (no discount, incremental quantity discounts, and all-unit quantity discounts) are used in the raw material cost. The option of ordering an excess amount of raw material is considered. Efficient solution procedures have been developed for a joint determination of process mean, production setup and raw material ordering policies for minimizing the total cost incurred because of production, inventory holding, raw material procurement, and excess material disposing, if there is any. A sensitivity analysis reveals the effect of the model parameters on the optimal solutions under three cases.
We wish to thank the anonymous referees for many helpful suggestions that improved this paper.
References [1] Bettes, D.C., "Finding an optimal target value in relation to a fixed lower limit and an arbitrary upper limit", Applied Statistics 11 (1962) 202-210. [2] Bisgaard, S., Hunter, W.G., and Pallesen, L., "Economic selection of quality of manufactured product", Technometrics 26 (1984) 9-18. [3] Carlsson, O., "Determining the most profitable process level for a production process under different sales conditions", Journal of Quality Technology 16 (1984) 44-49. [4] Golhar, D.Y., "Determination of the best mean contents for a 'canning problem'", Journal of Quality Technology 19 (1987) 82-84. [5] Golhar, D.Y., and Pollock, S.M., "Determination of the optimal process mean and the upper limit for a canning problem", Journal of Quality TechnoTo-gy 20 (1988) 188192. [6] Golhar, D.Y., and Pollock, S.M., "Cost saving due to variance reduction in canning process", liE Transactions 24/1 (1992) 89-92. [7] Gong, L., Roan, J., and Tang, K., "Process mean determination with quantity discounts in raw material cost", Working Paper, Department of Quantitative Business Analysis, Louisiana State University, Baton Rouge, LA, 1994.
J. Roan et al./ European Journal of Operational Research 99 (1997) 353-365 [8] Hammer, M., and Champy, J., Reengineering the Corporation: A Manifesto for Business Revolution, Harper Business, New York, 1993. [9] Hunter, W.G., and Kartha, C.P., "Determining the most profitable target value for a production process", Journal of Quality Technology 9 (1977) 176-181. [10] Roan, J., Gong, L., and Tang, K., " A model for joint determination of process mean, production run size and material order quantity", Working Paper, Department of
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Quantitative Business Analysis, Louisiana State University, Baton Rouge, LA, 1994. Ill] Springer, C.H., " A method of determining the most economic position of a process mean", Industrial Quality Control 8 (1951) 36-39. [12] Tang, K., and Tang, J., "Design of screening procedures: A review", Journal of Quality Technology 26/3 (1994) 209226.
European Journal of OperationalResearch 99 (1997) 353-365
Theory and Methodology
Process mean determination under constant raw material supply J i n s h y a n g R o a n a, L i n g u o G o n g
b, K w e i T a n g b,. ,l
a Department of Business Administration, Soochow University, Taipei, Taiwan, ROC b Department oflnformation Systems and Decision Sciences, Louisiana State University, Baton Rouge, LA 70 803, USA
Received September 1994; revised October 1995
Abstract Setting the mean (target value) for a production process is an important decision for a producer when material cost is a significant portion of production cost. Because the process mean determines the process conforming rate, it affects other production decisions, including, in particular, pro.duction setup and raw material procurement policies. In this paper, we consider the situation in which the product of interest is assumed to have a lower specification limit, and the items that do not conform to the specification limit are scrapped with no salvage value. The production cost of an item is a linear function of the amount of the raw material used in producing the item, and the supply rate of the raw material is finite and constant. Furthermore, it is assumed that quantity discounts are available in the raw material cost and that the discounts are determined by the supply rate. Two types of discounts are considered in this paper: incremental quantity discounts and all-unit quantity discounts. A two-echelon model is formulated for a single-product production process to incorporate the issues associated with production setup and raw material procurement into the classical process mean problem. Efficient solution algorithms are developed for finding the optimal solutions of the model. © 1997 Elsevier Science B.V. Keywords: Production/operations management;Inventorytheory; Statistics
1. Introduction Setting the process mean (target value) of a production process is a classical problem in quality control. A typical situation considered in the problem is as follows. The product of interest has a performance variable with a lower specification limit, and the raw material requirement for producing the product is an increasing function of the performance variable. An item that meets this specification is
i Supported in part by National Science Foundation grant no. DDM 8857557. * Corresponding author.
classified as conforming and is sold at a fixed regular price, while the nonconforming item is sold at a reduced price, reprocessed, or scrapped, depending on the production and marketing environments. The product is produced by a production process with an adjustable mean and a constant variance. If the process mean is set too high, the probability of producing nonconforming items becomes small. The manufacturer saves the cost raised by nonconforming items at the expense of material cost. (The cost of extra material is sometimes referred to as 'give-away' COSt.) Consequently, a process mean should be determined based on a balance between production cost and economic consequences associated with conforming items and nonconforming items. Note that
0377-2217/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved SSDI 0377-2217(95)00334-7
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J. Roan et al./ European Journal of Operational Research 99 (1997) 353-365
the typical performance variable under study are weight, volume, count, or concentration. The problem is first addressed by Springer [11]. He considered a production situation where upper and lower specification limits are both presented and the performance variable follows a gamma distribution. Bettes [1] studied the problem of finding optimal values for both process mean and upper limit. All the nonconforming (lower than the lower limit or higher than the upper limit) items are reprocessed at a fixed cost. Hunter and Kartha [9] considered a product with a lower specification limit and discussed the situation where nonconforming items can be sold at a (constant) reduced price. Bisgaard, Hunter and Pallesen [2] argued that Hunter and Kartha's assumption concerning the selling price of the nonconforming items is not realistic and, therefore, modified the model by assuming that the selling price of the nonconforming items is a linear function of the performance variable. Carlsson [3] followed Hunter and Kartha's model except for assuming that the net income function is a piecewise linear function of the performance variable. Golhar [4] assumed that only the regular market (fixed selling price) is available for the conforming items and that the underfilled items are emptied and reprocessed at a fixed expense (reprocessing cost). Golhar and Pollock [5] extended Golhar's [4] model to include an artificial upper limit so that nonconforming items as well as the items larger than the upper limit are reprocessed. Golhar and Pollock [6] also studied the cost benefit of reducing the variance of the process. A detailed review of these and related studies was given by Tang and Tang [ 12]. The selection of process mean determines the process conforming rate and, thus, the production yield rate. As a result, other production decisions, such as production setup and raw material procurement policies, are also dependent on the process mean setting. Roan, Gong and Tang [10] incorporated the issues associated with production setup and raw material procurement into the process mean problem. It is assumed that the product of interest requires one major raw material, which is purchased from outside vendors at a fixed unit cost. The production cost of an item is a linear function of the amount of the raw material used in producing the item. A two-echelon model is formulated for jointly
determining the process mean, production lot size and raw material procurement policy in a singleproduct production process. Gong, Roan and Tang [7] extended the model by Roan et al. to consider quantity discounts in raw material cost. In this paper, we base on the model proposed by Roan et al. [10] to consider a situation wherein a constant supply rate of raw material is offered by a reliable source. This approach of receiving raw materials from a vendor is known as 'continuous replenishment', which has drawn much attention recently [8]. Basically, this approach is beneficial to both the manufacturer and the supplier: the manufacturer can practically eliminate the raw material inventory and avoid out-of-stock situations, and the supplier can better plan production and inventory of the raw material. A smooth and cooperative manufacturersupplier relation is essential for using this method. In addition, we also consider the situation in which quantity discounts, incremental and all-units quantity discounts, for raw material are offered by vendors. The organization of this paper is as follows. In the next section, assumptions are given and models are formulated for three cases in quantity discounts: no quantity discounts, incremental discounts, and allunits quantity discounts. In Section 3, analytical properties associated with the optimal solution are derived and solution procedures are proposed. In Section 4, numerical examples are given to illustrate the solution procedures, and in Section 5 a sensitivity analysis is carried out to compare the three cases in terms of quantity discounts. Finally, a brief summary of the results and a discussion of possible future extensions are given in the last section.
2. Model assumption and formulation Consider a product with a constant demand rate of D items per unit time. A production process with a production rate of r items per unit time is used to satisfy the demand. Let X denote the performance variable of interest, which is a measure of the raw material used in the production, such as weight and volume. Let L be the lower specification limit of X, so that an item is conforming if its X value is larger than or equal to L. Assume that the production process is stable and X follows a probability density
J. Roan et al. / European Journal of Operational Research 99 (1997) 353-365
function, f i x ) , with an adjustable mean /x and a constant variance cr 2. For given /x, the conforming rate of the production process is
355
Inventory q(rp-D) t _
_ _
(a) ~
p= f~f( x) dx. Assume that nonconforming items are scrapped with no salvage value. Consequently, for given /x, the yield rate of the production process is r- p items per unit time. It is assumed that all the demand will be satisfied in such a way that the expected total number of conforming items produced is equal to the total demand and no backlog is allowed. Note that use of the expected conforming items is reasonable, especially in high-speed production, because the production output can be treated as approximately constant. Otherwise, it may require considering safety stock or using other inventory models. Note that r. p has to be greater than or equal to D to ensure that the production capacity is large enough to meet the demand. Suppose the unit cost of the raw material is c; thus, cX is the material cost required for producing an item of the finished product. We further assume that the direct cost of producing an item is a linear function of the item's material cost:
g( X ) = b + acX, where b is the fixed production cost, and a is a constant larger than or equal to 1. This cost function implies that the production cost consists of a fixed cost and a variable cost that is proportional to the raw material used in production. Note that a - 1 is the relative value (cost) added on the raw material during the production process. It can be verified that, for given /x, the expected cost of yielding a conforming item is
( b + aclx)/p. Let h a be the cost of holding a monetary unit of the raw material for a unit time. Assume that the costs of holding a monetary unit of raw material and finished product are the same. Then, the cost of holding a conforming item for a unit time is
H = (ha/p) ( actx + b). Let q be the production run size, which is the number of items (including both conforming and
R a w material
I I
I
Time
q r
__Pq D
(b) 0
Time
Fig. 1. (a) I n v e n t o r y level o f f i n i s h e d product; (b) i n v e n t o r y l e v e l o f r a w material.
nonconforming items) produced in a production run. The inventory level as a function of time is described in part (a) of Fig. 1. Assume that a production run begins at time 0. The finished product inventory increases at a rate of r . p - D items per unit time until q items are produced. At time q/r, the production run is complete, and, then, the inventory decreases at a rate of D items per unit time until time qp/D when the inventory level reaches 0 and the second production run starts. Let S be the production setup cost. Since the total number of setups required per unit time is D/qp, the total setup cost is SD/qp. It can be verified that the average inventory level for the finished product is
½q/r(rp-D). As a result, the total holding cost for finished products is
H½q/r( rp - D). per unit time. Furthermore, because the expected cost of yielding a conforming item is (b + actz)/p, the per-unit-time direct production cost is
D(b + a c t z ) / p . Therefore, the finished product-related cost, which is the sum of the production cost, the setup cost and the inventory holding cost, is given by FPC(/x, q) -
D( b + act~ ) P
DS
+ --
Pq
+ H ~ r ( r p - D), where H = ( h J p ) ( a c ~ + b).
J. Roan et al. / European Journal of Operational Research 99 (1997) 353-365
356
The expected amount of the raw material required to produce one conforming item is Ix/p: It is assumed that the raw material arrives at a constant rate /3 per unit time, regardless of whether the production process is in operation or is idle. Assume all the raw material received by the producer is used in production. The inventory level of the raw material as a function of time is shown in Fig. lb. The requirement for the raw material is r/x per unit time during production, which should be higher than the raw material supply rate /3. Since q is the production run size, we assume that the initial raw material inventory level at time 0 is (q/r)(rlx-/3), which is sufficient to meet the requirement in the first production run. When production starts, the raw material inventory decreases at a rate of ( - r/z + / 3 ). At time q/r, the raw material inventory is depleted and the first production run is finished. Then, the process is idle, and the raw material inventory level increases at a rate of /3 until the second production run starts at time pq/D. At the start of a production run, the inventory level / 3 ( p q / D - q/r) should be same as the level at time 0; that is, (Pq q) q(r/x-/3)=/3r --D--r " For any given process mean /x, therefore, the proper supply rate should be a function of /x; i.e., /3 = Dix/p. On the other hand, /x corresponding to given /3 is determined by /x = p/3/D. In this paper, the material-related cost is expressed as a function of /3 instead of po. Let C(/3 ) denote the raw material cost when the supply rate is /3. The cost of holding a conforming item for a unit time can, therefore, be rewritten as H = h,(
+ b/p).
Since the supply rate is constant, the setup cost associated with the raw material is not considered. The total holding cost per unit time for the raw material is M C ( / 3 , q) =
h,C~r(rlx - / 3 )
=hlC~---~-q ( r p - D ) , zrp which can be rewritten as MC(/3,
C( /3)q q) = h , - - ( r p - D). 2rD
The finished product-related cost function can be expressed as a function of 13 and q as follows: F P C ( / 3 , q) =
Db DS aC( /3 ) + - - + - P Pq + H~r ( r p - D).
Three cases, in terms of raw material quantity discounts, are considered. Let c i > ci+ 1, and 0 ~
c ( / 3 ) =/3c0, where c o is the material unit cost. Case 2. Incremental quantity discounts; that is,
C(/3) = C(/3i) ~- (/3 -/3i) ci, for /3 ~ [/3i, /3i+l), i = 0 , 1 , 2 . . . . . k - 1. Case 3. All-units quantity discounts; that is,
c ( / 3 ) =/3c,, for /3 ~ [/3i, /3i+1), i = 0 , 1, 2 . . . . . k - 1. Let TC l(/3, q) and TC2(/3, q) denote the total cost functions for cases 1 and 2, respectively. Because purchasing excess material is not economical for cases 1 and 2, these two total cost functions are given by T C , ( / 3 , q) = F P C ( / 3 , q) + M C ( / 3 , q)
Db
=
+--+--
DS
P Pq ( aC(/3) + b ) q ( r p - D ) +hi ~ ~r +h l
C(- /3)q (r p - D ) 2rD
for i = 1
2.
(1) For case 3, the material purchasing cost C(/3) under the all-units quantity discounts is a discontinuous function of /3, and it is possible that the producer orders a larger-than-needed amount of the raw material and disposes of the excess raw material to take advantage of price differentials. Let /3u denote the actual material use rate, where /3u ~/3- Depending on whether excess raw material is ordered, there-
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J. Roan et a l . / European Journal of Operational Research 99 (1997) 353-365
fore, we further consider the following two cases under all-units quantity discounts. Case 3.1: No excess raw material ordered In this case, the raw material use rate is equal to the raw material supply ,rate; i.e., /3~ =/3. In other words, no excess raw material is ordered. It can be easily verified that the total cost, denoted by TC31(/3, q), is also given by (1). Case 3.2: Excess raw material (disposed) In this case, /3 is larger than /3u, and the excess material is disposed of at a cost of c d per unit material. If the supply rate is /3i, then the total cost function is
TC32(/3u, q) Db = a/3iC i -lr-
cient solution algorithms are proposed to find the optimal process mean, production run size, and material supply rate. 3.1. Analytical properties In this subsection, we study the properties of three situations: no discounts, incremental quantity discounts, and all-units quantity discounts. For given /3, the optimal production run sizes for the no-discount, incremental discount and all-units discount without disposal are found by solving OTC/(/3, q ) / O q = O. We obtain the following result. Result 1. For cases 1, 2, and 3.1, the optimal production run size associated with given /3 is
DS
+ m
•
Pq
P
q +hi
D
+ -p
~ r (rp - D )
/3uq . + h l c i - ~ - r ~ ( rp - D) + ( / 3 i - / 3 u ) ( C i + ca)"
(2) Note that ( ]3 i -- /3u)(Ci "~ Cd) = D E ( / 3 i ,
~
2rDS
hl(rp/D_l)[Db+p(~+l)C(/3)] "
=
In cases 3.2, it can be verified that TC32(/3u, q) is a convex function of q when /3u (or /x) is fixed. Consequently, the minimum point of the total cost for given /3u (or /x) can be found by finding the solution to 0TC32(/3u, q ) / O q = O.
q)
is the disposal cost per unit time. It is assumed that excess raw material is disposed when it is received, so that no holding cost is associated with the excess raw material. Therefore, the process mean /x is P/3u/D. The optimal solution to the model is obtained by minimizing the per-unit time expected total cost. Since it is assumed that all the demand must be satisfied, the optimal solution also minimizes the expected total cost of producing a conforming item.
The following result can be obtained. Result 2. For case 3.1, the optimal production run size associated with given /3u is , q
~ =
2rDS hl(rp/D-
1)[Db+p(ol+
1)ci/3u]
"
3. Optimal solution
Note that, for all-units discounts, using Result 1, TC3t(/3, q * ) becomes a function of /3. As a result, a search procedure can be used to find the solution /3 * that minimizes TC31(/3, q * ). However, for unit cost c i, the solution may not be feasible because /3 * may not be in the range [/3i' /3i+ 1)" If /3 * is in [/3i, /3i+ 1), we say /3 * is admissible.
In this section, we first derive several important analytical properties for the optimal solutions for the three cases defined in the last section; that is, no discounts, incremental discounts, and all-units quantity discounts policies. Based on these results, effi-
Result 3. Consider all-units discounts. If/3 *, which minimizes TC3~( ~, q), is admissible for given unit cost c i, then this solution has a lower total cost than all the solutions associated with the higher unit costs.
358
J. Roan et al./ European Journal of Operational Research 99 (1997) 353-365
Proof. We let TC31(/3, q [ ci) and TC32(/3, q ] ci) be the total costs when c i is used in evaluating total cost functions, Eqs. (1) and (2), respectively. For cj > % we know that
WE31(/3, q[cj) > TC31(/3, q lci) and TC3e(/3, q lcj) > TC3~ (/3,
q* [ Ci) ~
TC31(/3, q l Ci)
for all /3 :/:/3 * and q ¢ q *. On the other hand, for anY /3 ~ / 3 i ' TC32(/3, q l cj) > TC32 (/3, q lci) and TC32(/3, q lci) >1 TC3~ (/3, q[ci). As a result, TC31(/3*, q * I ci) < TC3x (/3, q[cj) and
TC3x(/3 *,
q * [c,) ~ TC3, (/3, q[ci) < TC32(/3, q lcj).
Similar to case 1, the given /3 can be obtained process mean is given by search method is used material supply rate.
optimal production size for by using Result 1, and the p/3/D. The golden-section to search for the optimal
Case 3: All-units quantity discounts
q[ci).
Let q * be the production run size corresponding to /3 *. Since 13 * is admissible for ci,
TC31(/3 *,
Case 2: Incremental quantity discounts
[]
3.2. Solution algorithm The analytical results presented in the last section provide the basis for developing efficient solution procedures to find the optimal solutions for the three cases. The solution procedures for the three cases are given separately as follows. Note that the golden-section search is used for all the three cases below.
Case 1: No discounts Because the raw material supply rate is a function of the process mean and Result 1 gives the optimal production run size corresponding to a given process mean, the total cost becomes a function of the process mean. The golden-section search is used to locate the optimal process mean in the range [L, L + 4o-] and the search is terminated when the width of the interval of uncertainty is less than or equal to 0.0001.
The solution procedure is a search method based on the unit material cost c r Specifically, it starts with the lowest c k and case 3.1. The relationship between the material supply rate and the material unit cost is ignored first. The solution that minimizes TC31(/3, q lcg) is feasible and admissible if its resulting /3 is inside its cost (price) range. The procedure continues until the first feasible /3 is found. Let c' be the unit price associated with the first feasible solution (i.e., /3 * is admissible). If c' is the lowest unit cost, then the solution is optimal. Otherwise, based on Result 3, we can exclude all the solutions associated with the unit cost higher than c'. Therefore, we examine case 3.2 for all the unit costs lower than c', and compare the results with the first feasible solution (for case 3.1). The one with the lowest total cost is the optimal solution. The procedure is summarized by the following two steps.
Step 1. Starting with the lowest price, search the value of /3 * that minimizes TC31(/3, q [ Cg). /3 * is admissible if /3 * ~ [/3i, /3i+ l)- This process continues until an admissible /3 * is found. Step 2. If /3 * corresponding to the lowest unit material cost is admissible, then the solution is optimal. Otherwise, find the solutions for case 3.2 associated with the unit costs lower than cj. Comparing the solution obtained in Step 1 and those in this step for case 3.2, the one with the lowest total cost is the optimal solution. A F O R T R A N program has been written for implementing the solution procedures on an IBM 3090600E computer. The running time for solving each of the problems considered in this paper is only a few seconds.
4. An example In this section, an example is used to illustrate the solution procedures given in the last section. The
J. Roan et al./ European Journal of Operational Research 99 (1997) 353-365
example will also be used in the sensitivity analysis in Section 5. Consider a product that requires at least 8 mgs of main content in each item. Any item that is less than 8 mgs is considered nonconforming and is scrapped without salvage value. Because of the variation in the production process, the content of an item produced by the process follows a normal distribution, with an adjustable process mean and a constant standard deviation of 1.2 mgs. Assume that the product demand rate and the production rate are 5000 items and 7500 items per unit time, respectively. The setup cost per production run is $150, the fixed production cost is $0.16 per item, and c~ is 4. The raw material is constantly supplied by a vendor. Furthermore, the cost for holding $1 of inventory (finished product or raw material) is $0.03 per unit time. The material cost for the no-discounts policy is $0.1 per mg and the material costs under the incremental quantity discounts policy are as follows: $0.1
for each of the first 30 000 mgs,
$0.098
for each of the next 30 000 mgs,
$0.095
for each mg in excess of 60 000 mgs.
c=
The all-units quantity discounts are as follows: Order quantity
Price per mg ($)
1-29 999 30 0 0 0 - 5 9 999 _> 60 000
0.100 0.098 0.095
The raw material disposal cost to the producer is ca = $ - 0 . 0 4 5 / m g . The results are given as follows. Case 1: N o discounts The optimal process mean is found to be 9.915 mgs. The corresponding raw material supply rate and the production run size are 52 474.94 mgs and 5927 items, respectively. The total cost is $22 104.50. The process conforming rate is 94.48%. Case 2: Incremental quantity discounts The optimal process mean is found to be 9.916 mgs and the corresponding process conforming rate
359
is 94.48%. The raw material supply rate is /3 * = 52475.0 mgs and its corresponding raw material purchasing cost is C(/3 * ) = $5202.54. The optimal production run size is q * = 5952 items and the total cost is $21 924.31. Case 3: All-units quantity discounts The optimal raw material supply rate and use rate found are 60000 and 52475.53 mgs, respectively. Based on the raw material use rate, the optimal process mean is 9.919 mgs. The following two steps were used to find the optimal supply rate and production run size associated with the optimal process mean: Step 1. When c i = $0.098, the optimal /3 * =/3u = 52 475.07 mgs, which is within the range [30 000, 60 000). TC31 = $21 682.39, and c' = 0.098. Step 2. c i = $0.095 is the only unit cost that is smaller than c'. When c i = 0.095, the lowest total costs TC32 = $21 425.18, and its corresponding /3u = 52 475.53 mgs, which is less than /3 = 60 000. Comparing this result with the result in Step 1, we found that the lowest total cost is $21 425.18.
Therefore, the optimal solution is to use the supply rate 60000 mgs and dispose of 7524.47 mgs. The total disposal cost per unit time is $376.29. The resulting process mean is 9.919 mgs, and the process conforming rate is 94.51%. The optimal production run size is 6071 items.
5. Sensitivity analysis In this section, a sensitivity analysis is performed to study the effects of the model parameters on the optimal solutions of all three cases: 1) no discounts, 2) incremental quantity discounts, and 3) all-units quantity discounts. The model parameters included in the study are the demand rate D, the production rate r, the process standard deviation o-, the valueadded factor a , and the production setup cost S. For the all-units quantity discounts policy, the unit disposal cost, c d, is also studied. The sensitivity analy-
J. Roan et al./ European Journal of Operational Research 99 (1997) 353-365
360
ses are based on the example given in the last section. Per-unit costs are used to report the results in this section. They are per-item finished product-related cost:
q)/D,
PFPC = F P C ( / 3 ,
per-item material-holding cost: PMC = M C ( / 3 ,
q)/D,
per-item total cost: PTC = T C ( / 3 ,
q)/D,
and per-item disposal cost: PDC = D C ( / 3 ,
q)/D.
5.1. Effect of demand rate To study the effects of demand rate, we use the results for selected values of D ranging from 1500 to 7000 per unit time. As shown in Table 1, when D increases, the optimal production run size for cases 1 and 2 increases. This result makes intuitive sense. When the demand rate is very low relative to the production rate (process capacity), no clear patterns were observed on the optimal process mean. w h e n the demand rate is moderate or very high, both the
optimal process mean and the material supply rate increase steadily as the demand rate increases. When the demand rate is very close to the production rate, because of a low accumulation rate of finished products, the production run size in the first two cases becomes very large and sensitive to the change of D. The results of case 3 are very similar when the demand rate is lower than 5000. When the demand rate is larger than 5000, however, a different approach should be taken under the all-unit discounts policy. The reason is that both a larger demand and a higher process mean push the material use rate higher. When the material use rate is high enough, it is attractive to use the material supply rate at the next discount-applicable level. As a result, a larger material supply rate, /3 * = 60 000, is used to take advantage of quantity discounts, and excess materials are disposed. We also notice that a larger process mean is also used, which is expected because the material cost is lower. A noticeable difference in case 3 (compared with cases 1 and 2) is that, when the demand rate is very high, the optimal production run size is not affected by changes in the demand rate. The reason is that excess materials are available and the process yield rate is high; consequently, accumulation rate of finished products remains high. It is expected, however, that when the demand rate is
Table 1 Effect of demand rate D
Case 1 /z*
q*
/3'
/z*
q*
/3*
u*
q*
/3u
/3*
1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000
9.921 9.923 9.919 9.920 9.918 9.919 9.916 9.915 9.912 9.909 9.903 9.801
1 982 2399 2825 3 278 3 781 4356 5 049 5 927 7 136 9019 12854 1 609350
15 743 20990 26238 31 485 36733 41 980 47228 52475 57722 62969 68 216 73 510
9.921 9.923 9.919 9.920 9.918 9.918 9.917 9.916 9.912 9.911 9.902 9.801
1982 2399 2825 3280 3787 4368 5066 5952 7169 9067 12957 1623034
15 743 20990 26238 31 485 36733 41 980 47 228 52475 57 722 62969 68 216 73 510
9.921 9.922 9.919 9.921 9.918 9.917 9.917 9.919 9.918 9.918 9.918 9.918
1982 2399 2825 3310 3818 4400 5098 6071 7304 7304 7304 7304
15 743 20990 26238 31 485 36732 41 980 47228 52 476 57 723 57723 57 723 57 723
15 743 20990 26238 31 485 36732 41 980 47228 60000 60000 60000 60000 60000
D : p~* : q *: /3 * : /3u:
Case 2
Product demand rate. Optimal process mean. Optimal production lot size. Optimal raw material supply rate. Material use rate.
Case 3
J. Roan et al./ European Journal of'Operational Research 99 (1997) 353-365
361
Table 2 Effect of production rate r
Case 1
6000 6500 7 000 7 500 8000 8 500 9000 9 500 10000
Case 2
Case 3
/x*
q*
fl*
/x*
q*
/3*
/x*
q*
flu
/3*
9.905 9.910 9.914 9.915 9.917 9.917 9.917 9.919 9.919
9410 7473 6514 5927 5526 5233 5009 4829 4684
52474 52 474 52475 52475 52475 52475 52475 52476 52476
9.904 9.910 9.913 9.916 9.918 9.918 9.919 9.921 9.921
9453 7504 6541 5951 5548 5054 5028 4848 4703
52474 52474 52475 52475 52475 52 475 52475 52 476 52476
9.907 9.913 9.918 9.919 9.921 9.922 9.923 9.925 9.925
9636 7654 6671 6071 5661 5360 5130 4946 4798
52 474 52475 52475 52476 52476 52476 52 476 52476 52476
60000 60000 60000 60000 60000 60000 60000 60000 60000
r : Production rate. /z* : Optimal process mean. q * : Optimal production lot size. 13 * : Optimal raw material supply rate. /3u: Material use rate.
very close to the production rate, the optimal production run size will become sensitive to changes in the demand rate.
5.2. Effect of production rate Table 2 contains the results under selected values of r ranging from 6000 to 10000. The results of the three cases are very similar. It is clear that a large production rate results in a smaller q* and a larger
/x*. A larger /z* implies a higher production yield rate and a higher per-item holding cost. In order to avoid carrying a high level of finished product inventory, therefore, the manufacturer should lower the production run size, so as to reduce the cost of holding finished product inventory. Note that /3 * and flu remain very stable as r increases. This implies that, as production rate (capacity) increases, the manufacturer should adjust internal production planning rather than change the vendor's raw material supply rate.
Table 3 Effect of process variation o-
0.01 0.10 0.50 1.00 2.00 3.00 4.00 5.00 6.00 o- : /z* : q* : /3 * : flu:
Case 1
Case 2
]
]Case 3
/x*
q*
/3'
/~*
q*
/3*/
/x
q
flu
/3
8.030 8.266 9.005 9.689 10.640 11.203 11.410 11.140 10.584
6006 5 946 5 870 5 893 '6 167 6750 7 906 11302 1 435 473
40204 41 492 46048 50760 58 682 65 349 71 043 75 782 79 383
8.030 8.266 9.006 9.688 10.643 11.206 11.415 11.144 10.584
6021 5 962 5 890 5 918 6 192 6789 7958 11382 1449436
40204 41 492 46048 50760 58 692 65 349 71044 75 783 79783
8.030 8.266 9.005 9.692 10.648 11.211 11.414 11.144 10.584
6065 6004 5 928 6037 6 311 6907 8097 11 546 1471 628
40204 41 492 46 048 50761 58 683 65 350 71044 75 783 79383
40204 41 492 46048 60000 60000 65 350 71044 75 783 79383
Process standard deviation. Optimal process mean. Optimal production lot size. Optimal raw material supply rate. Material use rate.
J. Roan et al. / European Journal of Operational Research 99 (1997) 353-365
362
5.3. Effect of process standard deviation
5.4. Effect of value-added factor
As discussed, a smaller process standard deviation implies a better performance of the process and a smaller total cost of operating the process. Table 3 shows the results associated with selected values of O'.
The results of the three cases are very similar. As o- increases, the material supply rate generally increases. Two exceptions are ~r = 1.0 and 2.0 for case 3, in which excess materials are ordered to take advantage of quantity discounts. When tr is smaller than 0.5, a larger o- results in a larger process mean. As discussed, a larger mean implies a higher per-item inventory holding cost. The production run size decreases, therefore, in order to reduce a high inventory holding cost. Note that although the process mean increases as ~r increases, the process yield rate actually decreases. As a result, when tr is between 1.0 and 4.0, a larger production run size is used mainly because the process yield rate decreases very significantly. When o- is larger than 5.0, however, in order to avoid the higher per-item production cost and holding cost, the process mean is set at a lower level wherein the process yield rate is very close to the demand, resulting in a very large production run size.
A larger a implies a higher average cost of producing an item and also a higher average holding cost. To reduce these costs, a reasonable approach is to use a lower process mean. This result is observed in Table 4. The changes in the process mean are, however, quite small. Consequently, the process yield rate does not decrease very significantly. Furthermore, to avoid a high inventory holding cost caused by a larger a , the production run size decreases as a increases. From Table 4, we can also see that, when a is 3 and 4, the process mean in case 3 is much higher than those in cases 1 and 2. The purpose of the increased process mean is to raise the material supply rate to 60 000 for a lower per-unit material cost ($0.095/mg). The saving from the lower material unit cost is more than the cost of disposing of excess material and the cost incurred because of higher process mean. The material supply rates in cases 1 and 2 are relatively very stable with respect to the changes in a. In case 3, the material use rate is also very stable as a increases, although a shift occurs in the material supply rate at a = 3.0. From the above discussion, we can conclude that the production run size is the decision variable most affected by a change in a.
Table 4 Effect of value-added factor a
Case 1 /z*
q*
/3"
/z*
q*
/3'
p~*
q*
B~
/3*
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
9.981 9.937 9.923 9.915 9.911 9.908 9.906 9.905 9.903 9.903
9037 7538 6591 5927 5430 5040 4723 4459 4236 4041
52498 52479 52476 52475 52474 52 474 52474 52474 52474 52474
9.983 9.939 9.925 9.916 9.911 9.908 9.906 9.904 9.904 9.902
9070 7566 6615 5952 5453 5061 4743 4479 4253 4056
52499 52480 52476 52475 52474 52474 52474 52474 52474 52474
9.983 9.938 9.930 9.919 9.912 9.909 9.907 9.906 9.903 9.903
9119 7610 6745 6071 5566 5166 4842 4572 4344 4145
52 499 52 479 52477 52476 52475 52474 52 474 52474 52474 52 474
52499 52479 60000 60000 60000 60000 60000 60000 60000 60000
a : /x* : q *: /3 * : /3u:
Case 2
Value-added factor. Optimal process mean. Optimal production lot size. Optimal raw material supply rate. Material use rate.
Case 3
J. Roan et al. / European Journal of Operational Research 99 (1997) 353-365
363
Table 5 Effect of production setup cost S
Case 1
50 100 150 200 250 300 350 400 450 500 550
Case 2
Case 3
/x*
q*
/3"
/x*
q*
/3*
/x*
q*
/3u
/3*
9.919 9.918 9.915 9.915 9.911 9.910 9.910 9.908 9.907 9.906 9.905
3419 4837 5927 6845 7660 8392 9064 9 694 10284 10843 11 375
52475 52475 52475 52475 52474 52474 52474 52 474 52474 52474 52 474
9.921 9.918 9.916 9.914 9.912 9.910 9.910 9.908 9.908 9.905 9.905
3432 4857 5952 6875 7689 8427 9101 9 734 10326 10890 11 422
52476 52475 52475 52475 52475 52474 52474 52 474 52474 52474 52 474
9.924 9.921 9.919 9.918 9.915 9.914 9.912 9.910 9.910 9.910 9.907
3502 4955 6071 7013 7845 8595 9287 9 934 10538 11 106 11 656
52476 52476 52476 52475 52475 52475 52475 52 474 52474 52474 52 474
60000 60000 60000 60000 60000 60000 60000 60 000 60000 60000 60 000
s : Production setup cost. /x* : Optimal process mean. q * : Optimal production lot size. /3 * : Optimal raw material supply rate. /3u: Material use rate.
5.5. Effect o f production setup cost
duction setup cost increases. Note that a large prod u c t i o n r u n s i z e r e s u l t s in h o l d i n g a l a r g e r i n v e n t o r y ,
It is e x p e c t e d t h a t a l a r g e p r o d u c t i o n s e t u p c o s t l e a d s to a l a r g e p r o d u c t i o n r u n size in o r d e r to reduce the frequency of production setup. The results in T a b l e 5 s h o w t h a t this r e l a t i o n s h i p is t r u e f o r all
and that a smaller process mean, on the other hand, reduces per-item inventory cost and process yield rate. S i n c e t h e d e c r e a s e s in t h e p r o c e s s m e a n s are
t h e t h r e e c a s e s . It is i n t e r e s t i n g , h o w e v e r , t o o b s e r v e
s m a l l , t h e p r o c e s s y i e l d rate r e m a i n s q u i t e s t a b l e . F u r t h e r m o r e , t h e m a t e r i a l s u p p l y a n d u s e r a t e are
that t h e o p t i m a l p r o c e s s m e a n d e c r e a s e s as t h e p r o -
also
very
stable
when
the
production
setup
Table 6 Effect of unit disposal cost
Ca
P.*
q*
/3u
13 *
PFPC
PMC
PDC
PTC
-
9.918 9.919 9.919 9.919 9.916 9.916 9.916 9.916 9.916 9.916
6073 6071 6071 6071 5984 5984 5984 5984 5984 5984
52 475 52 476 52 476 52 476 52 475 52 475 52 475 52 475 52 475 52 475
60 000 60 000 60 000 60 000 52 475 52 475 52 475 52 475 52 475 52 475
4.2047 4.2047 4.2047 4.2047 4.3313 4.3313 4.3313 4.3313 4.3313 4.3313
0.005 l 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051
0.0226 0.0527 0.0828 0.1129 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
4.2323 4.2624 4.2925 4.3226 4.3364 4.3364 4.3364 4.3364 4.3364 4.3364
0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08 0.10
co: Unit disposal cost. /x* :Optimal process mean. q * : Optimal production lot size. /3 ".'Optimal raw material supply rate. /3u: Material use rate.
PFPC: PMC: PDC: PTC:
Per-item Per-item Per-item Per-item
finished product-related cost. material holding cost. raw material disposal cost. total cost.
cost
364
J. Roan et a l . / European Journal of Operational Research 99 (1997) 353-365
increases. Consequently, it can be concluded that only the production run size is significantly affected by the production setup cost.
5.6. Effect of unit disposal cost Table 6 gives the effects of some selected c a values ranging from $ - 0.08 to $0.1 for the all-units quantity discounts. When c o increases, the material use rate remains very stable. When c d is small (less than or equal to $ - 0.02), it is economical to order excess material to receive larger quantity discounts. When c a is larger than $ - 0.02, this practice is not attractive. Both the process mean and production run size are at a slightly higher level when excess material is ordered. When c d is larger than $ - 0.02, the solution remains unchanged because no excess material is ordered.
The model structure presented in this paper provides a useful framework for future research on several interesting issues related to this classical problem in quality control. In particular, the following several extensions are possible. First, the deterioration of raw material a n d / o r finished product can be considered. This factor will become very interesting and important when the deterioration rates of raw material and finished product are different. Second, the deterioration of the production process can be incorporated into the model. The third extension is to consider a multiple-level filling process in which several raw materials are added in different stages. This issue could become very complicated, however, when the product conformance is jointly determined by the amount of several raw materials. These issues have been included in the authors' future research plans.
Acknowledgments 6. Summary In this paper, it is assumed that the raw material is supplied at a constant rate from outside vendors. Three cases in terms of quantity discounts in the raw material cost are introduced into a two-echelon model, which incorporates the issues associated with production run size and raw material procurement policy into the classical process mean problem for a single-product production process. The performance variable has a lower specification limit, and the items that do not meet the limit are scrapped with no salvage value. The production cost of an item is a linear function of the amount of raw material used in producing the item. Three cases (no discount, incremental quantity discounts, and all-unit quantity discounts) are used in the raw material cost. The option of ordering an excess amount of raw material is considered. Efficient solution procedures have been developed for a joint determination of process mean, production setup and raw material ordering policies for minimizing the total cost incurred because of production, inventory holding, raw material procurement, and excess material disposing, if there is any. A sensitivity analysis reveals the effect of the model parameters on the optimal solutions under three cases.
We wish to thank the anonymous referees for many helpful suggestions that improved this paper.
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