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Joint determination of target value and production run for a process with multiple markets
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Int. J. Production Economics 96 (2005) 201–212 www.elsevier.com/locate/dsw
Joint determination of target value and production run for a process with multiple markets Moncer A. Harigaa,, M.A. Al-Fawzanb a
Industrial Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia b King Abdulaziz City for Science and Technology, P.O. Box 6086, Riyadh 11442, Saudi Arabia Received 10 June 2003; accepted 26 April 2004 Available online 25 June 2004
Abstract Setting the target value (process mean) for an industrial process when its outputs are sold to quality-dependent customers is an important and challenging decision for quality and manufacturing managers. The determination of an appropriate process mean does not only affect the output rate of conforming units but also affects other manufacturing decisions such as finished product and raw material lot sizing policies. In this paper, we address the integrated targetinginventory problem to determine simultaneously the optimal production cycle time and the target value that maximizes the total expected profit per unit of time. Depending on its quality characteristic, the output of the manufacturing process may be sold to different customers from the same market but with different quality requirements. The output rejected by original customers may be stored for some time and then sold to other customers in the same market at a reduced price. We built the developed mathematical model in Microsoft Excel and used SOLVER to search for the optimal production run and process mean. We also carried out a sensitivity analysis to study the effects of some problem parameters on the quality and inventory decisions. r 2004 Elsevier B.V. All rights reserved. Keywords: Process mean; Production run; Multiple markets
1. Introduction Setting the target value (process mean) for an industrial process when its outputs are sold to qualitydependent customers is an important and challenging decision for quality and manufacturing managers. This problem is often referred to as the targeting problem (Hunter and Kartha, 1977) and has received much interest from both academicians and practitioners in recent years. Several variants of the targeting Corresponding author. Fax: +966-1-467-6652.
E-mail addresses: [email protected] (M.A. Hariga), [email protected] (M.A. Al-Fawzan). 0925-5273/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2004.04.008
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problem are encountered in the quality control literature. Originally, the targeting problem is concerned with the setting of the process mean that maximizes the net income per unit produced. It is assumed that the quality characteristic, which can be a measure of the raw material used such as weight, volume, or concentration, is a random variable with an adjustable mean, m, and a constant standard deviation, S. The process output is classified as conforming when its quality characteristic is larger or equal to a lower specification limit, L. Earlier works related to the targeting problem can be attributed to Springer (1951) and Bettes (1962). Other researchers have addressed the targeting problem under different assumptions including Bisgaard et al. (1984), Golhar (1987), Golhar and Pollock (1988), Arcelus and Rahim (1990, 1994), Melloy (1991), and Duffuaa and Siddiqui (2003) among others. Different alternatives of disposing of unacceptable units are observed in practice depending on the industrial process and the market. In some cases, the out-of-specification units are discarded, reworked for later sale, or sold in a secondary market at a lower price. In other cases, rejected units are held for some time and then sold to other customers in the primary market. To the best of the author’s knowledge, Hunter and Kartha (1977) were among the first to consider the targeting problem with multiple markets. They presented a profit maximization model where rejected products are sold in a secondary market at a fixed price. Carlsson (1984) applied Hunter and Kartha’s model with some modifications in the cost and income components of the profit function to the steel beam industry. Tang (1990) considered a situation in which the produced items are sorted into two grades according to predetermined specifications and sold at two different prices. The objective of his model is to determine the optimal grading specifications that maximize the expected per-item profit. In a recent paper, Shao et al. (2000) considered the targeting problem in the context of the steel galvanization industry. Within this industry, different customers from the same market of the galvanized steel sheets may require different specifications in terms of zinc coating weight. Therefore, the out-of-specification output may be stored for some time and then sold in the same primary market to customers with lower coating weight requirement for a slightly lower price instead of selling it in a secondary market. The problem for the manufacturer is then to determine the optimal process mean by maximizing the expected total profit that takes into account the costs associated with holding rejected products for later sale. We also observed a similar problem in a local Saudi industrial fiber company producing a polyester product called ‘‘Partially Oriented Yarn’’, which is generally known as ‘‘POY’’ in the commercial and market parlance and has been referred to as POY. The polyester is used in the production of textured yarns and flat-drawn yarns for weaving, knitting, and for various textile end uses. The different grades of the POY, which are differentiated by their deniers (difference between external and internal radii of the filament) and number of filaments, are sold to different classes of customers. For example, the POY 200 series is made in wide range from 50 deniers to 530 deniers and 36 filaments to 72 filaments to meet the requirement of different market segments. Table 1 shows the classification of the different types of POY 200 series according to the deniers and number of filaments. Table 1 Properties of the different types of POY 200 series POY type
Deniers
No. of filament
201 203 211 213 221 223 231
150 150 100 100 76 76 50
36 72 36 72 36 56 36
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Practically speaking, the optimal setting of the process mean does not only affect the output rate of conforming units but also affects other manufacturing decisions such as finished product and raw material lot sizing policies. In the above reviewed literature, inventory and production issues were not addressed in the targeting problem. Recently, Roan et al. (1997, 2000) proposed an integrated production and targeting model. The objective of their model is to minimize the total costs for finished product and raw material with respect to process mean and to procurement decisions. They considered two cases depending on the order frequency of the raw material within the production cycle. Gong et al. (1998) extended the work of Roan et al. by allowing quantity discounts in the raw material acquisition cost. In this paper, we propose an integrated two-echelon inventory-targeting model for a single finished product that can be sold to three types of customers depending on the level of its quality characteristic. As in Shao et al. (2000), we assume that only one major raw material affects the quality performance of the finished product. The problem is then to determine the optimal process mean, production lot size for the finished product and ordering lot size for the raw material that maximize the expected total profit per unit of time. The remainder of the paper is organized as follows. The problem statement and model formulation are presented in the next section. In the third section, the model is illustrated with a numerical example and a sensitivity analysis is carried out to study the effects of some problem parameters on the quality and inventory decisions. Finally, the last section concludes the paper.
2. Problem statement and formulation A finished product with a constant demand rate D per unit of time is produced in a manufacturing facility having a production rate r per unit of time. The quality characteristic, x, is a measure of the raw material used in the manufacturing of the finished product such as weight or volume. It is assumed that the quality characteristic is a random variable following a normal distribution with an adjustable mean m and a constant standard deviation S. A unit of the process output is classified as a first-class unit, which can be sold in the primary market at a price p1 , if its x value is larger than or equal to the customer’s specification limit L1 . On the other hand, if the quality characteristic of the process output is smaller than the customer’s specification L1 but larger than the plant tolerance specification limit L2 (L2 is close to L1 ), then the finished product belongs to the second-class units and may be stored for a random period of time y and then sold to another customer in the primary market at price p2 . Finally, a manufactured unit of the finished product is classified as a third-class unit to be sold in a secondary market at a price p3 if its quality characteristic is smaller than L2 . A summary of the main notation used in the model formulation is reported in Table 2. For a given process mean m, the yield rate for each class item is Z 1 L1 m l1 ðmÞ ¼ r f ðxÞ dx ¼ r 1 F for first-class units; s L1 Z
f ðxÞ dx ¼ rF
L1 m L2 m F s s
f ðxÞ dx ¼ rF
L2 m s
L1
l2 ðmÞ ¼ r L2
for second-class units;
and Z
L2
l3 ðmÞ ¼ r 1
for third-class units;
where f ðxÞ is the probability density function of the quality characteristic, x, and FðÞ is the standard normal distribution function. It is assumed that the expected total number of units produced for first-class units is larger than the quantity demanded, D, so that the situation of shortages cannot occur in any
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Table 2 List of symbols used in the model formulation D r S L1 L2 pj y my m T lj ðmÞ cj ðmÞ H j ðT; mÞ H 0 ðT; mÞ Pj ðT; mÞ AðT; mÞ GðT; mÞ EPUTðT; mÞ
Demand rate per unit of time Production rate per unit of time Standard deviation of the quality characteristic Customer’s specification limit Plant tolerance specification limit Unit price of the jth class item (j ¼ 1; 2; 3) Random period of storage for second class items Mean of the random period of storage for second class items Process mean Production cycle time Yield rate of the jth class item (j ¼ 1; 2; 3) Expected unit production cost for the jth class item (j ¼ 1; 2; 3) Expected holding cost per cycle for the jth class item (j ¼ 1; 2; 3) Expected holding cost per cycle for the raw material Expected production cost per cycle for the jth class item (j ¼ 1; 2; 3) Expected material acquisition cost per cycle Expected material giveaway cost per cycle Expected profit per unit of time
production cycle. Therefore, l1 ðmÞ must be larger than D to ensure that the process output is large enough to meet the demand for first-class units. It is also assumed that the direct unit production cost for the finished product is a linear function of the quality characteristic (e.g. Shao et al., 2000; and Bisgaard et al., 1984): gðxÞ ¼ b þ cf x; where b is a fixed production cost, cf is the cost of obtaining a specific quality characteristic for one unit of the finished product. Therefore, the expected unit production cost for each of the three classes of the finished product is given by Class 1 finished product: xXL1 Z 1 c1 ðmÞ ¼ ðcf x þ bÞf ðxÞ dx L1
L1 m xf ðxÞ dx þ bF s L1 L1 m L1 m ¼ ðcf m þ bÞF þ scf f ; s s Z
1
¼ cf
(1)
¼ 1 FðÞ is the standard normal complementary distribution function and fðÞ is the standard where FðÞ normal density function. Class 2 finished product: L2 pxoL1 Z L1 c2 ðmÞ ¼ ðcf x þ bÞf ðxÞ dx L2
L1 m L2 m ¼ cf xfðxÞ dx þ b F F s s L1 L1 m L2 m L2 m L1 m ¼ ½cf m þ b F F þ scf f f : s s s s Z
L2
(2)
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Class 3 finished product: xoL2 Z
L2
c3 ðmÞ ¼
ðcf x þ bÞf ðxÞ dx 1
L2 m xf ðxÞ dx þ bF s 1 L2 m L2 m ¼ ½cf m þ b F scf f : s s Z
L2
¼ cf
(3)
Finished Product
Letting i be the inventory holding rate for the finished product and raw material, then the unit inventory holding cost per unit of time for the jth class of the finished product is icj ðmÞ and for the raw material is icm , where cm is the unit acquisition cost of the raw material. At the beginning of the production cycle of length T, the required quantity of the raw material is received, the manufacturing process is setup at a cost S f and the production of the finished product starts at that moment. During the inventory uptime period of the production cycle ½0; t , where t is the production time, the yield rate for the jth class is Lj and the stock for raw material is depleted at rate ðrmÞ. The inventory variations over time for the three classes of units and the raw material are shown in Fig. 1. It can be observed that the inventory line for the second-class items becomes stable after time t since the quantity produced for this class will be stored for a random period of time y before being sold in the primary market. The length of the period depends on how fast the company can find customers from the primary market who are willing to purchase the second-class items with a quality characteristic smaller than L1 but larger than L2 . Using this figure, the inventory holding costs per cycle for each class of the finished product and the raw material are as follows.
λ1 − D
y
λ2 λ3
time T
Raw Material
t
rµ
time Fig. 1. Inventory variation for the three classes of the finished product and raw material.
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Class 1 finished product: H 1 ðT; mÞ ¼ ic1 ðmÞðl1 ðmÞ DÞt
T T 2 D l1 ðmÞ D DT 2 ¼ ic1 ðmÞ ¼ ic1 ðmÞrðmÞ ; 2 2 l1 ðmÞ 2
ð4Þ
where rðmÞ ¼ 1 D=l1 ðmÞ and t ¼ DT=l1 ðmÞ by the no-shortages assumption. Class 2 finished product: t2 l2 ðmÞ DT H 2 ðT; mÞ ¼ ic2 ðmÞ l2 ðmÞ þ l2 ðmÞtmy ¼ ic2 ðmÞ DT þ my ; l1 ðmÞ 2l1 ðmÞ 2
ð5Þ
where my is the mean of the random variable representing the storage period for the second class of the finished product before its sale. Class 3 finished product: H 3 ðT; mÞ ¼ ic3 ðmÞl3 ðmÞ
D2 T 2 l3 ðmÞ D2 T 2 : ¼ ic3 ðmÞ 2 2 l1 ðmÞ 2 2l1 ðmÞ
ð6Þ
Raw material: H 0 ðT; mÞ ¼ icm rm
t2 m D2 T 2 ¼ icm r 2 : 2 l1 ðmÞ 2
ð7Þ
The production cost per cycle for the jth class of the finished product is given by Pj ðm; TÞ ¼ cj ðmÞlj ðmÞt ¼ cj ðmÞ
lj ðmÞ DT; l1 ðmÞ
j ¼ 1; 2; 3:
ð8Þ
The cost components relative to the raw material are the material ordering cost S m , the material acquisition cost and the material giveaway cost. The material acquisition is Aðm; TÞ ¼ cm rmt ¼ cm rm
DT : l1 ðmÞ
ð9Þ
The material giveaway cost is the cost of excess quality for the first class of the finished product (Hunter and Kartha, 1977) and is given by Z 1 DT L1 m L1 m L1 m F Gðm; TÞ ¼ cm rt s f ðx L1 Þf ðxÞ dx ¼ cm r : ð10Þ l1 ðmÞ s s s L1 The revenue from sales of each class of the finished product is pj lj ðmÞt ¼ pj
lj ðmÞ DT; l1 ðmÞ
j ¼ 1; 2; 3:
ð11Þ
Using the developed costs components and sales revenue, the expected profit per unit of time as function of the process mean and the production cycle time can be expressed mathematically as 3 X lj ðmÞ cm rsD L1 m L1 m L1 m rm EPUTðm; TÞ ¼ D f D ½pj cj ðmÞ F cm l1 ðmÞ s s s l1 ðmÞ l1 ðmÞ j¼1 " # l2 ðmÞ Sf þ Sm i Dc2 ðmÞl2 ðmÞ Dc3 ðmÞl3 ðmÞ cm rmD ic2 ðmÞ m D DT c1 ðmÞrðmÞ þ þ 2 þ : (12) l1 ðmÞ y 2 T l21 ðmÞ l21 ðmÞ l1 ðmÞ From the last mathematical expression, it is simple to show that the expected profit per unit of time is concave in T for a given value of the process mean. The optimal production cycle time as function of the
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35000.000 Expected profit
30000.000 25000.000 20000.000 15000.000 10000.000 5000.000 30 9. 31 0 2. 31 0 5. 31 0 8. 32 0 1. 32 0 4. 32 0 7. 33 0 0. 33 0 3. 33 0 6. 33 0 9. 34 0 2. 34 0 5. 0
0.000
Process mean
Fig. 2. Variation of the expected profit per unit of time as function of the process mean.
process mean is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Sf þ Sm u
: T ðmÞ ¼ u2 t Dc3 ðmÞl3 ðmÞ cm rmD 2 ðmÞ iD c1ðmÞrðmÞ þ Dc2lðmÞl þ þ 2 ðmÞ l2 ðmÞ l2 ðmÞ 1
1
ð13Þ
1
Substituting Eq. (13) into Eq. (12), the expected profit per unit of time becomes a function of the single variable m and takes the following mathematical form: 3 X lj ðmÞ cm rsD L1 m L1 m L1 m f ½pj cj ðmÞ F EPUTðmÞ ¼ D l1 ðmÞ s s s l1 ðmÞ j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # u u Dc2 ðmÞl2 ðmÞ Dc3 ðmÞl3 ðmÞ cm rmD t þ þ 2 2iDðS f þ S m Þ c1 ðmÞrðmÞ þ l21 ðmÞ l21 ðmÞ l1 ðmÞ cm
rm l2 ðmÞ D ic2 ðmÞ m D: l1 ðmÞ l1 ðmÞ y
(14)
The single variable optimization problem to solve is then Maximize subject to
EPUTðmÞ l1 ðmÞXD; mX0:
We built the above single variable optimization problem in a Microsoft Excel spreadsheet and used SOLVER to search for the optimal process mean and production cycle time. Experience with numerical solutions to the optimization problem revealed that the expected profit per unit of time is concave in the process mean (see Fig. 2) which indicates the obtained solution is a global optimal one.
3. Numerical example and analysis In this section, we illustrate the developed optimization model with a numerical example and then carry out a sensitivity analysis to study the effects of some problem parameters on the expected profit and the decision variables.
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3.1. Illustrative example Consider a product that requires a major raw material for its production. The manufacturing process used to produce this product has a production rate of 1000 units per year. Any unit of the finished product that contains more than or equal to 310 mg of the raw material can be sold directly in the primary market at a price of $250. The annual demand for conforming units is constant at a known rate of 500 units. If the raw material content is less than 310 mg but larger than 305 mg, the finished product is kept in stock for an average period of 1 year and then sold at a price of $150 per unit. On the other hand, the output of the manufacturing process is considered non-conforming and will be sold in a secondary market when the raw material content is less than 305 mg. Given the variation in the production process, the raw material quantity used in the manufacturing of the finished product follows a normal distribution with an adjustable process mean and a constant standard deviation of 8 mg. The setup cost of the production process is $150, the fixed production cost is $100 per unit, and the variable production cost is $0.1/mg. The raw material is procured from an outside source at a cost of $0.3/mg and it costs the company $80 each time an order is placed to this source. The company is using an annual inventory holding rate of 2%. Using Excel Solver, we found that the optimal solution is to set the process mean at 312.07 mg and to run the production process every 0.54 year. The length of the production period is 0.45. The resulting maximum yearly profit is $29330.22. As shown in Fig. 2, the obtained solution for the process mean is indeed global optimal since the expected profit per unit of time is concave in m. 3.2. Sensitivity analysis We now present the results of the sensitivity analysis that we performed to assess the effects of some input parameters on the model output values such as the expected profit, process mean, cycle time, and yield rate for each of the three classes of the finished product. We carried the experiment by varying the standard deviation, customer specification value, plant tolerance value, demand rate, discount rates for second- and third-class items, and mean storage period for second-class items. All the remaining problem parameters are kept as in the illustrative example. 3.2.1. Effect of the process standard deviation In Table 3, we report the results of the experimental analysis when we varied the process standard deviation. The table shows the obvious result that the target value should be set away above the customer’s specification limit for large process variation. Because of the increase in the process mean for large standard deviation, the yield rate for first-class items will also increase. The increase of the production cycle time is also a result of the chain effects of the increase in the process mean. Another expected result is that the yield rate for third-class items sold at the secondary market increases as the process variation increases. We can also observe from the same table that as the process standard deviation increases, the expected profit Table 3 Effect of process variation s
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
3 5 8 12 20
310.1115 310.4395 312.0736 314.7253 320.0909
0.451 0.484 0.542 0.572 0.587
514.82 535.05 602.26 653.12 693.06
440.97 326.64 209.45 138.63 81.68
44.21 183.31 188.29 208.84 225.26
39990.90 36568.37 29330.22 22998.54 15770.93
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Table 4 Effect of customer’s specification limit L1
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
308 310 312 314 316
311.5616 312.0737 313.0219 314.3927 316.1847
0.585 0.542 0.499 0.466 0.449
671.91 602.26 550.82 519.97 509.21
122.03 209.45 291.18 360.24 409.75
206.05 188.29 157.99 120.18 81.04
23442.32 29330.22 33661.72 36094.02 36841.55
Table 5 Effect of plant tolerance’s limit L2
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
301 303 305 307 309
310.2889 310.9782 312.0737 313.5947 315.7144
0.464 0.498 0.542 0.584 0.617
514.40 548.66 602.26 673.41 762.48
362.80 292.03 209.45 121.72 36.87
122.80 151.32 188.29 204.87 200.64
37420.74 34285.36 29330.22 22918.45 15844.36
decreases. This is mainly due to the increase in material requirement and sales at the secondary market for large standard deviation. 3.2.2. Effect of the customer’s specification limit Table 4 displays the effect of the customer’s specification limit on the model’s outputs. The results of the analysis show that the process mean tends to converge to the customer’s specification limit as the later increases. Therefore, the yield rate for first-class items will decrease with an increase in L1 . As expected, a reverse effect is observed for the yield rate of the second-class items. Given that the yield rate L1 is decreasing, one also expects that the production cycle time will decrease. Finally, when the customer’s specification limit increases, the expected profit per unit of time increases because of the increase in the quantity of the finished product sold at the primary market to both original and non-original customers (note that L3 is decreasing with L1 ). 3.2.3. Effect of the plant’s tolerance limit From Table 5, we note that as the plant’s tolerance limit increases, the process mean increases and, consequently, the yield rate for first-class items increases. As a result, it is also observed that the production cycle time is increasing with L2 . As expected, the yield rate for items sold at the primary market to nonoriginal customers decreases as L2 increases. Finally, for the given pricing of the three classes of the finished product, the expected profit per unit of time decreases with an increase in the plant’s tolerance limit due to the increase in the sales at the secondary market. 3.2.4. Effect of the annual demand rate We can observe from Table 6 that the process mean and the yield rate for first-class items increase as the demand increases. One would expect this result since for large demand rates, the process mean should be set large enough to meet this demand. Note also that for small demand rates, we observe a slight increase in the process mean. When the demand rate gets closer to the production rate (D is larger than 700), the
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Table 6 Effect of the annual demand rate D
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
300 400 500 600 700 800 900
312.0575 312.0669 312.0736 312.0787 314.1952 316.7330 320.2524
0.780 0.638 0.542 0.472 0.479 0.485 0.487
601.48 601.94 602.26 602.50 700.00 800.00 900
209.68 209.54 209.45 209.37 174.80 128.76 71.71
188.84 188.52 188.29 188.12 125.20 71.24 28.29
17518.51 23422.50 29330.22 35240.19 38588.56 36584.97 28822.13
Table 7 Effect of the discount rate for second-class items 1 p2 =p1 (%)
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
20 30 40 50
310.9556 311.4903 312.0736 312.7136
0.500 0.521 0.542 0.562
547.54 573.89 602.26 632.77
224.16 217.51 209.45 199.76
228.30 208.60 188.29 167.47
38801.86 33873.46 29330.22 25182.78
constraint of the optimization problem becomes binding and the optimal process mean is given by m ¼ L1 SF1 ð1 D=rÞ. Furthermore, as the demand rate increases, the production cycle time first decreases and then takes almost the same value starting from D ¼ 700. In fact, when the yield rate is nearly equal to the demand rate, a low inventory for first-class items is carried out and the production cycle time becomes the time needed to deplete the raw material lot size (see Fig. 1). Finally, the expected profit per unit time first increases, then starts to decrease when the demand rate becomes closer to the production rate. This finding can be justified by the fact that the production capacity is not large enough to satisfy large values of the demand rate. 3.2.5. Effect of the discount rate for second-class items The discount rate for the units of the finished product sold at the primary market to non-original customers is given by ð1 p2 =p1 Þ%. Table 7 gives the results of the sensitivity analysis for selected values of this discount rate. From this table, we note a slight increase in the process mean as the discount rate increases. This can be explained by the fact that an increase in the discount rate should be accompanied by an increase in the process mean in order to increase the yield rate of first-class items and compensate for part of the reduction in revenue generated from the sales of second-class items. The decrease in the expected profit per unit of time can also be attributed to the reduction in the sales of second-class items. 3.2.6. Effect of the discount rate for third-class items The discount rate for non-conforming units sold at the secondary market is given by ð1 p3 =p1 Þ%. Table 8 shows the results for selected values of this discount rate. From this table, we can observe the same effect of the discount rate for second-class items on the different outputs of the model. We also can see that for discount rates smaller than 60%, the process mean is set to the customer specification limit so that the first-class yield rate is equal to the demand rate.
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Table 8 Effect of the discount rate for third class items 1 p3 =p1 (%)
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
50 60 70 80 84 90
310.0000 310.0000 310.6130 311.6831 312.0736 312.6529
0.459 0.459 0.486 0.528 0.542 0.599
500.00 500.00 530.54 583.32 602.26 628.63
234.01 234.01 228.00 214.93 209.45 201.13
265.99 265.99 241.46 201.75 188.29 170.24
48978.19 42328.56 35935.20 30974.49 29330.22 27147.64
Table 9 Effect of the mean storage period for second class items my
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
0.5 0.75 1 1.25 1.5
312.0638 312.0687 312.0736 302.0785 312.0834
0.541 0.542 0.542 0.542 0.542
601.79 602.02 602.26 602.50 602.73
209.59 209.52 209.45 209.38 209.30
188.62 188.46 188.29 188.13 187.96
29377.89 29354.04 29330.22 29306.42 29282.65
3.2.7. Effect of the mean storage period for second-class items Table 9 shows the effect of the mean storage period for second-class items on the model output values. It is clear from this table that the mean storage period my did not have significant effects on all the model outputs. This result is quite expected since this input parameter affects only the second class of the finished product through the expression of the holding cost (see expression (5)).
4. Conclusions In this paper, the classical targeting problem was integrated with a two-echelon inventory problem to jointly determine the optimal production cycle time and process mean. Depending on its quality characteristic, the output of the production process is sold to two types of customers in the primary market or sold in a secondary market. The developed mathematical model is solved in Excel using SOLVER. We also performed a sensitivity analysis to study the effects of some problem parameters on the model’s outputs. The developed model assumed that conforming units immediately become available to meet the demand from original customers of the primary market. A possible extension to this model is the case where the output of conforming units cannot be used until an entire batch is produced.
References Arcelus, F.J., Rahim, M.A., 1990. Optimal process levels for the joint control of variables and attributes. European Journal of Operational Research 45, 224–230. Arcelus, F.J., Rahim, M.A., 1994. Simultaneous economic selection of a variables and an attribute target mean. Journal of Quality Technology 26, 125–133.
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Bettes, D.C., 1962. Finding an optimum target value in relation to a fixed lower limit and an arbitrary upper limit. Applied Statistics 11, 202–210. Bisgaard, S., Hunter, W.G., Pallesen, L., 1984. Economic selection of quality of manufactured product. Technometrics 26, 9–18. Carlsson, O., 1984. Determining the most profitable process level for a production process under different sales conditions. Journal of Quality Technology 16, 44–49. Duffuaa, S.O., Siddiqui, A.W., 2003. Process targeting with multi-class screening and measurement error. International Journal of Production Research 41, 1373–1392. Golhar, D.Y., 1987. Determination of the best mean contents for a canning problem. Journal of Quality Technology 19, 82–84. Golhar, D.Y., Pollock, S.M., 1988. Determination of the optimal mean and the upper limit for a canning problem. Journal of Quality Technology 20, 188–192. Gong, L., Roan, J., Tang, K., 1998. Process mean determination with quantity discounts in raw material cost. Decision Sciences 29, 271–302. Hunter, W.G., Kartha, C.P., 1977. Determining the most profitable target value for a production process. Journal of Quality Technology 9, 176–181. Melloy, B.J., 1991. Determining the optimal process mean and screening limits for packages subject to compliance testing. Journal of Quality Technology 23, 318–323. Roan, J., Gong, L., Tang, K., 1997. Process mean determination under constant material supply. European Journal of Operational Research 99, 353–365. Roan, J., Gong, L., Tang, K., 2000. Joint determination of process mean, production run size and material order quantity for a container-filling process. International Journal of Production Economics 63, 303–317. Shao, Y.E., Fowler, J.W., Runger, G.C., 2000. Determining the optimal target for a process with multiple markets and variable holding costs. International Journal of Production Economics 65, 229–242. Springer, C.H., 1951. A method of determining the most economic position of a process mean. Industrial Quality Control 8, 36–39. Tang, K., 1990. Design of product grading procedures. Decision Sciences 21, 434–445.
Int. J. Production Economics 96 (2005) 201–212 www.elsevier.com/locate/dsw
Joint determination of target value and production run for a process with multiple markets Moncer A. Harigaa,, M.A. Al-Fawzanb a
Industrial Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia b King Abdulaziz City for Science and Technology, P.O. Box 6086, Riyadh 11442, Saudi Arabia Received 10 June 2003; accepted 26 April 2004 Available online 25 June 2004
Abstract Setting the target value (process mean) for an industrial process when its outputs are sold to quality-dependent customers is an important and challenging decision for quality and manufacturing managers. The determination of an appropriate process mean does not only affect the output rate of conforming units but also affects other manufacturing decisions such as finished product and raw material lot sizing policies. In this paper, we address the integrated targetinginventory problem to determine simultaneously the optimal production cycle time and the target value that maximizes the total expected profit per unit of time. Depending on its quality characteristic, the output of the manufacturing process may be sold to different customers from the same market but with different quality requirements. The output rejected by original customers may be stored for some time and then sold to other customers in the same market at a reduced price. We built the developed mathematical model in Microsoft Excel and used SOLVER to search for the optimal production run and process mean. We also carried out a sensitivity analysis to study the effects of some problem parameters on the quality and inventory decisions. r 2004 Elsevier B.V. All rights reserved. Keywords: Process mean; Production run; Multiple markets
1. Introduction Setting the target value (process mean) for an industrial process when its outputs are sold to qualitydependent customers is an important and challenging decision for quality and manufacturing managers. This problem is often referred to as the targeting problem (Hunter and Kartha, 1977) and has received much interest from both academicians and practitioners in recent years. Several variants of the targeting Corresponding author. Fax: +966-1-467-6652.
E-mail addresses: [email protected] (M.A. Hariga), [email protected] (M.A. Al-Fawzan). 0925-5273/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2004.04.008
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problem are encountered in the quality control literature. Originally, the targeting problem is concerned with the setting of the process mean that maximizes the net income per unit produced. It is assumed that the quality characteristic, which can be a measure of the raw material used such as weight, volume, or concentration, is a random variable with an adjustable mean, m, and a constant standard deviation, S. The process output is classified as conforming when its quality characteristic is larger or equal to a lower specification limit, L. Earlier works related to the targeting problem can be attributed to Springer (1951) and Bettes (1962). Other researchers have addressed the targeting problem under different assumptions including Bisgaard et al. (1984), Golhar (1987), Golhar and Pollock (1988), Arcelus and Rahim (1990, 1994), Melloy (1991), and Duffuaa and Siddiqui (2003) among others. Different alternatives of disposing of unacceptable units are observed in practice depending on the industrial process and the market. In some cases, the out-of-specification units are discarded, reworked for later sale, or sold in a secondary market at a lower price. In other cases, rejected units are held for some time and then sold to other customers in the primary market. To the best of the author’s knowledge, Hunter and Kartha (1977) were among the first to consider the targeting problem with multiple markets. They presented a profit maximization model where rejected products are sold in a secondary market at a fixed price. Carlsson (1984) applied Hunter and Kartha’s model with some modifications in the cost and income components of the profit function to the steel beam industry. Tang (1990) considered a situation in which the produced items are sorted into two grades according to predetermined specifications and sold at two different prices. The objective of his model is to determine the optimal grading specifications that maximize the expected per-item profit. In a recent paper, Shao et al. (2000) considered the targeting problem in the context of the steel galvanization industry. Within this industry, different customers from the same market of the galvanized steel sheets may require different specifications in terms of zinc coating weight. Therefore, the out-of-specification output may be stored for some time and then sold in the same primary market to customers with lower coating weight requirement for a slightly lower price instead of selling it in a secondary market. The problem for the manufacturer is then to determine the optimal process mean by maximizing the expected total profit that takes into account the costs associated with holding rejected products for later sale. We also observed a similar problem in a local Saudi industrial fiber company producing a polyester product called ‘‘Partially Oriented Yarn’’, which is generally known as ‘‘POY’’ in the commercial and market parlance and has been referred to as POY. The polyester is used in the production of textured yarns and flat-drawn yarns for weaving, knitting, and for various textile end uses. The different grades of the POY, which are differentiated by their deniers (difference between external and internal radii of the filament) and number of filaments, are sold to different classes of customers. For example, the POY 200 series is made in wide range from 50 deniers to 530 deniers and 36 filaments to 72 filaments to meet the requirement of different market segments. Table 1 shows the classification of the different types of POY 200 series according to the deniers and number of filaments. Table 1 Properties of the different types of POY 200 series POY type
Deniers
No. of filament
201 203 211 213 221 223 231
150 150 100 100 76 76 50
36 72 36 72 36 56 36
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Practically speaking, the optimal setting of the process mean does not only affect the output rate of conforming units but also affects other manufacturing decisions such as finished product and raw material lot sizing policies. In the above reviewed literature, inventory and production issues were not addressed in the targeting problem. Recently, Roan et al. (1997, 2000) proposed an integrated production and targeting model. The objective of their model is to minimize the total costs for finished product and raw material with respect to process mean and to procurement decisions. They considered two cases depending on the order frequency of the raw material within the production cycle. Gong et al. (1998) extended the work of Roan et al. by allowing quantity discounts in the raw material acquisition cost. In this paper, we propose an integrated two-echelon inventory-targeting model for a single finished product that can be sold to three types of customers depending on the level of its quality characteristic. As in Shao et al. (2000), we assume that only one major raw material affects the quality performance of the finished product. The problem is then to determine the optimal process mean, production lot size for the finished product and ordering lot size for the raw material that maximize the expected total profit per unit of time. The remainder of the paper is organized as follows. The problem statement and model formulation are presented in the next section. In the third section, the model is illustrated with a numerical example and a sensitivity analysis is carried out to study the effects of some problem parameters on the quality and inventory decisions. Finally, the last section concludes the paper.
2. Problem statement and formulation A finished product with a constant demand rate D per unit of time is produced in a manufacturing facility having a production rate r per unit of time. The quality characteristic, x, is a measure of the raw material used in the manufacturing of the finished product such as weight or volume. It is assumed that the quality characteristic is a random variable following a normal distribution with an adjustable mean m and a constant standard deviation S. A unit of the process output is classified as a first-class unit, which can be sold in the primary market at a price p1 , if its x value is larger than or equal to the customer’s specification limit L1 . On the other hand, if the quality characteristic of the process output is smaller than the customer’s specification L1 but larger than the plant tolerance specification limit L2 (L2 is close to L1 ), then the finished product belongs to the second-class units and may be stored for a random period of time y and then sold to another customer in the primary market at price p2 . Finally, a manufactured unit of the finished product is classified as a third-class unit to be sold in a secondary market at a price p3 if its quality characteristic is smaller than L2 . A summary of the main notation used in the model formulation is reported in Table 2. For a given process mean m, the yield rate for each class item is Z 1 L1 m l1 ðmÞ ¼ r f ðxÞ dx ¼ r 1 F for first-class units; s L1 Z
f ðxÞ dx ¼ rF
L1 m L2 m F s s
f ðxÞ dx ¼ rF
L2 m s
L1
l2 ðmÞ ¼ r L2
for second-class units;
and Z
L2
l3 ðmÞ ¼ r 1
for third-class units;
where f ðxÞ is the probability density function of the quality characteristic, x, and FðÞ is the standard normal distribution function. It is assumed that the expected total number of units produced for first-class units is larger than the quantity demanded, D, so that the situation of shortages cannot occur in any
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Table 2 List of symbols used in the model formulation D r S L1 L2 pj y my m T lj ðmÞ cj ðmÞ H j ðT; mÞ H 0 ðT; mÞ Pj ðT; mÞ AðT; mÞ GðT; mÞ EPUTðT; mÞ
Demand rate per unit of time Production rate per unit of time Standard deviation of the quality characteristic Customer’s specification limit Plant tolerance specification limit Unit price of the jth class item (j ¼ 1; 2; 3) Random period of storage for second class items Mean of the random period of storage for second class items Process mean Production cycle time Yield rate of the jth class item (j ¼ 1; 2; 3) Expected unit production cost for the jth class item (j ¼ 1; 2; 3) Expected holding cost per cycle for the jth class item (j ¼ 1; 2; 3) Expected holding cost per cycle for the raw material Expected production cost per cycle for the jth class item (j ¼ 1; 2; 3) Expected material acquisition cost per cycle Expected material giveaway cost per cycle Expected profit per unit of time
production cycle. Therefore, l1 ðmÞ must be larger than D to ensure that the process output is large enough to meet the demand for first-class units. It is also assumed that the direct unit production cost for the finished product is a linear function of the quality characteristic (e.g. Shao et al., 2000; and Bisgaard et al., 1984): gðxÞ ¼ b þ cf x; where b is a fixed production cost, cf is the cost of obtaining a specific quality characteristic for one unit of the finished product. Therefore, the expected unit production cost for each of the three classes of the finished product is given by Class 1 finished product: xXL1 Z 1 c1 ðmÞ ¼ ðcf x þ bÞf ðxÞ dx L1
L1 m xf ðxÞ dx þ bF s L1 L1 m L1 m ¼ ðcf m þ bÞF þ scf f ; s s Z
1
¼ cf
(1)
¼ 1 FðÞ is the standard normal complementary distribution function and fðÞ is the standard where FðÞ normal density function. Class 2 finished product: L2 pxoL1 Z L1 c2 ðmÞ ¼ ðcf x þ bÞf ðxÞ dx L2
L1 m L2 m ¼ cf xfðxÞ dx þ b F F s s L1 L1 m L2 m L2 m L1 m ¼ ½cf m þ b F F þ scf f f : s s s s Z
L2
(2)
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Class 3 finished product: xoL2 Z
L2
c3 ðmÞ ¼
ðcf x þ bÞf ðxÞ dx 1
L2 m xf ðxÞ dx þ bF s 1 L2 m L2 m ¼ ½cf m þ b F scf f : s s Z
L2
¼ cf
(3)
Finished Product
Letting i be the inventory holding rate for the finished product and raw material, then the unit inventory holding cost per unit of time for the jth class of the finished product is icj ðmÞ and for the raw material is icm , where cm is the unit acquisition cost of the raw material. At the beginning of the production cycle of length T, the required quantity of the raw material is received, the manufacturing process is setup at a cost S f and the production of the finished product starts at that moment. During the inventory uptime period of the production cycle ½0; t , where t is the production time, the yield rate for the jth class is Lj and the stock for raw material is depleted at rate ðrmÞ. The inventory variations over time for the three classes of units and the raw material are shown in Fig. 1. It can be observed that the inventory line for the second-class items becomes stable after time t since the quantity produced for this class will be stored for a random period of time y before being sold in the primary market. The length of the period depends on how fast the company can find customers from the primary market who are willing to purchase the second-class items with a quality characteristic smaller than L1 but larger than L2 . Using this figure, the inventory holding costs per cycle for each class of the finished product and the raw material are as follows.
λ1 − D
y
λ2 λ3
time T
Raw Material
t
rµ
time Fig. 1. Inventory variation for the three classes of the finished product and raw material.
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Class 1 finished product: H 1 ðT; mÞ ¼ ic1 ðmÞðl1 ðmÞ DÞt
T T 2 D l1 ðmÞ D DT 2 ¼ ic1 ðmÞ ¼ ic1 ðmÞrðmÞ ; 2 2 l1 ðmÞ 2
ð4Þ
where rðmÞ ¼ 1 D=l1 ðmÞ and t ¼ DT=l1 ðmÞ by the no-shortages assumption. Class 2 finished product: t2 l2 ðmÞ DT H 2 ðT; mÞ ¼ ic2 ðmÞ l2 ðmÞ þ l2 ðmÞtmy ¼ ic2 ðmÞ DT þ my ; l1 ðmÞ 2l1 ðmÞ 2
ð5Þ
where my is the mean of the random variable representing the storage period for the second class of the finished product before its sale. Class 3 finished product: H 3 ðT; mÞ ¼ ic3 ðmÞl3 ðmÞ
D2 T 2 l3 ðmÞ D2 T 2 : ¼ ic3 ðmÞ 2 2 l1 ðmÞ 2 2l1 ðmÞ
ð6Þ
Raw material: H 0 ðT; mÞ ¼ icm rm
t2 m D2 T 2 ¼ icm r 2 : 2 l1 ðmÞ 2
ð7Þ
The production cost per cycle for the jth class of the finished product is given by Pj ðm; TÞ ¼ cj ðmÞlj ðmÞt ¼ cj ðmÞ
lj ðmÞ DT; l1 ðmÞ
j ¼ 1; 2; 3:
ð8Þ
The cost components relative to the raw material are the material ordering cost S m , the material acquisition cost and the material giveaway cost. The material acquisition is Aðm; TÞ ¼ cm rmt ¼ cm rm
DT : l1 ðmÞ
ð9Þ
The material giveaway cost is the cost of excess quality for the first class of the finished product (Hunter and Kartha, 1977) and is given by Z 1 DT L1 m L1 m L1 m F Gðm; TÞ ¼ cm rt s f ðx L1 Þf ðxÞ dx ¼ cm r : ð10Þ l1 ðmÞ s s s L1 The revenue from sales of each class of the finished product is pj lj ðmÞt ¼ pj
lj ðmÞ DT; l1 ðmÞ
j ¼ 1; 2; 3:
ð11Þ
Using the developed costs components and sales revenue, the expected profit per unit of time as function of the process mean and the production cycle time can be expressed mathematically as 3 X lj ðmÞ cm rsD L1 m L1 m L1 m rm EPUTðm; TÞ ¼ D f D ½pj cj ðmÞ F cm l1 ðmÞ s s s l1 ðmÞ l1 ðmÞ j¼1 " # l2 ðmÞ Sf þ Sm i Dc2 ðmÞl2 ðmÞ Dc3 ðmÞl3 ðmÞ cm rmD ic2 ðmÞ m D DT c1 ðmÞrðmÞ þ þ 2 þ : (12) l1 ðmÞ y 2 T l21 ðmÞ l21 ðmÞ l1 ðmÞ From the last mathematical expression, it is simple to show that the expected profit per unit of time is concave in T for a given value of the process mean. The optimal production cycle time as function of the
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35000.000 Expected profit
30000.000 25000.000 20000.000 15000.000 10000.000 5000.000 30 9. 31 0 2. 31 0 5. 31 0 8. 32 0 1. 32 0 4. 32 0 7. 33 0 0. 33 0 3. 33 0 6. 33 0 9. 34 0 2. 34 0 5. 0
0.000
Process mean
Fig. 2. Variation of the expected profit per unit of time as function of the process mean.
process mean is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Sf þ Sm u
: T ðmÞ ¼ u2 t Dc3 ðmÞl3 ðmÞ cm rmD 2 ðmÞ iD c1ðmÞrðmÞ þ Dc2lðmÞl þ þ 2 ðmÞ l2 ðmÞ l2 ðmÞ 1
1
ð13Þ
1
Substituting Eq. (13) into Eq. (12), the expected profit per unit of time becomes a function of the single variable m and takes the following mathematical form: 3 X lj ðmÞ cm rsD L1 m L1 m L1 m f ½pj cj ðmÞ F EPUTðmÞ ¼ D l1 ðmÞ s s s l1 ðmÞ j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # u u Dc2 ðmÞl2 ðmÞ Dc3 ðmÞl3 ðmÞ cm rmD t þ þ 2 2iDðS f þ S m Þ c1 ðmÞrðmÞ þ l21 ðmÞ l21 ðmÞ l1 ðmÞ cm
rm l2 ðmÞ D ic2 ðmÞ m D: l1 ðmÞ l1 ðmÞ y
(14)
The single variable optimization problem to solve is then Maximize subject to
EPUTðmÞ l1 ðmÞXD; mX0:
We built the above single variable optimization problem in a Microsoft Excel spreadsheet and used SOLVER to search for the optimal process mean and production cycle time. Experience with numerical solutions to the optimization problem revealed that the expected profit per unit of time is concave in the process mean (see Fig. 2) which indicates the obtained solution is a global optimal one.
3. Numerical example and analysis In this section, we illustrate the developed optimization model with a numerical example and then carry out a sensitivity analysis to study the effects of some problem parameters on the expected profit and the decision variables.
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3.1. Illustrative example Consider a product that requires a major raw material for its production. The manufacturing process used to produce this product has a production rate of 1000 units per year. Any unit of the finished product that contains more than or equal to 310 mg of the raw material can be sold directly in the primary market at a price of $250. The annual demand for conforming units is constant at a known rate of 500 units. If the raw material content is less than 310 mg but larger than 305 mg, the finished product is kept in stock for an average period of 1 year and then sold at a price of $150 per unit. On the other hand, the output of the manufacturing process is considered non-conforming and will be sold in a secondary market when the raw material content is less than 305 mg. Given the variation in the production process, the raw material quantity used in the manufacturing of the finished product follows a normal distribution with an adjustable process mean and a constant standard deviation of 8 mg. The setup cost of the production process is $150, the fixed production cost is $100 per unit, and the variable production cost is $0.1/mg. The raw material is procured from an outside source at a cost of $0.3/mg and it costs the company $80 each time an order is placed to this source. The company is using an annual inventory holding rate of 2%. Using Excel Solver, we found that the optimal solution is to set the process mean at 312.07 mg and to run the production process every 0.54 year. The length of the production period is 0.45. The resulting maximum yearly profit is $29330.22. As shown in Fig. 2, the obtained solution for the process mean is indeed global optimal since the expected profit per unit of time is concave in m. 3.2. Sensitivity analysis We now present the results of the sensitivity analysis that we performed to assess the effects of some input parameters on the model output values such as the expected profit, process mean, cycle time, and yield rate for each of the three classes of the finished product. We carried the experiment by varying the standard deviation, customer specification value, plant tolerance value, demand rate, discount rates for second- and third-class items, and mean storage period for second-class items. All the remaining problem parameters are kept as in the illustrative example. 3.2.1. Effect of the process standard deviation In Table 3, we report the results of the experimental analysis when we varied the process standard deviation. The table shows the obvious result that the target value should be set away above the customer’s specification limit for large process variation. Because of the increase in the process mean for large standard deviation, the yield rate for first-class items will also increase. The increase of the production cycle time is also a result of the chain effects of the increase in the process mean. Another expected result is that the yield rate for third-class items sold at the secondary market increases as the process variation increases. We can also observe from the same table that as the process standard deviation increases, the expected profit Table 3 Effect of process variation s
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
3 5 8 12 20
310.1115 310.4395 312.0736 314.7253 320.0909
0.451 0.484 0.542 0.572 0.587
514.82 535.05 602.26 653.12 693.06
440.97 326.64 209.45 138.63 81.68
44.21 183.31 188.29 208.84 225.26
39990.90 36568.37 29330.22 22998.54 15770.93
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Table 4 Effect of customer’s specification limit L1
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
308 310 312 314 316
311.5616 312.0737 313.0219 314.3927 316.1847
0.585 0.542 0.499 0.466 0.449
671.91 602.26 550.82 519.97 509.21
122.03 209.45 291.18 360.24 409.75
206.05 188.29 157.99 120.18 81.04
23442.32 29330.22 33661.72 36094.02 36841.55
Table 5 Effect of plant tolerance’s limit L2
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
301 303 305 307 309
310.2889 310.9782 312.0737 313.5947 315.7144
0.464 0.498 0.542 0.584 0.617
514.40 548.66 602.26 673.41 762.48
362.80 292.03 209.45 121.72 36.87
122.80 151.32 188.29 204.87 200.64
37420.74 34285.36 29330.22 22918.45 15844.36
decreases. This is mainly due to the increase in material requirement and sales at the secondary market for large standard deviation. 3.2.2. Effect of the customer’s specification limit Table 4 displays the effect of the customer’s specification limit on the model’s outputs. The results of the analysis show that the process mean tends to converge to the customer’s specification limit as the later increases. Therefore, the yield rate for first-class items will decrease with an increase in L1 . As expected, a reverse effect is observed for the yield rate of the second-class items. Given that the yield rate L1 is decreasing, one also expects that the production cycle time will decrease. Finally, when the customer’s specification limit increases, the expected profit per unit of time increases because of the increase in the quantity of the finished product sold at the primary market to both original and non-original customers (note that L3 is decreasing with L1 ). 3.2.3. Effect of the plant’s tolerance limit From Table 5, we note that as the plant’s tolerance limit increases, the process mean increases and, consequently, the yield rate for first-class items increases. As a result, it is also observed that the production cycle time is increasing with L2 . As expected, the yield rate for items sold at the primary market to nonoriginal customers decreases as L2 increases. Finally, for the given pricing of the three classes of the finished product, the expected profit per unit of time decreases with an increase in the plant’s tolerance limit due to the increase in the sales at the secondary market. 3.2.4. Effect of the annual demand rate We can observe from Table 6 that the process mean and the yield rate for first-class items increase as the demand increases. One would expect this result since for large demand rates, the process mean should be set large enough to meet this demand. Note also that for small demand rates, we observe a slight increase in the process mean. When the demand rate gets closer to the production rate (D is larger than 700), the
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Table 6 Effect of the annual demand rate D
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
300 400 500 600 700 800 900
312.0575 312.0669 312.0736 312.0787 314.1952 316.7330 320.2524
0.780 0.638 0.542 0.472 0.479 0.485 0.487
601.48 601.94 602.26 602.50 700.00 800.00 900
209.68 209.54 209.45 209.37 174.80 128.76 71.71
188.84 188.52 188.29 188.12 125.20 71.24 28.29
17518.51 23422.50 29330.22 35240.19 38588.56 36584.97 28822.13
Table 7 Effect of the discount rate for second-class items 1 p2 =p1 (%)
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
20 30 40 50
310.9556 311.4903 312.0736 312.7136
0.500 0.521 0.542 0.562
547.54 573.89 602.26 632.77
224.16 217.51 209.45 199.76
228.30 208.60 188.29 167.47
38801.86 33873.46 29330.22 25182.78
constraint of the optimization problem becomes binding and the optimal process mean is given by m ¼ L1 SF1 ð1 D=rÞ. Furthermore, as the demand rate increases, the production cycle time first decreases and then takes almost the same value starting from D ¼ 700. In fact, when the yield rate is nearly equal to the demand rate, a low inventory for first-class items is carried out and the production cycle time becomes the time needed to deplete the raw material lot size (see Fig. 1). Finally, the expected profit per unit time first increases, then starts to decrease when the demand rate becomes closer to the production rate. This finding can be justified by the fact that the production capacity is not large enough to satisfy large values of the demand rate. 3.2.5. Effect of the discount rate for second-class items The discount rate for the units of the finished product sold at the primary market to non-original customers is given by ð1 p2 =p1 Þ%. Table 7 gives the results of the sensitivity analysis for selected values of this discount rate. From this table, we note a slight increase in the process mean as the discount rate increases. This can be explained by the fact that an increase in the discount rate should be accompanied by an increase in the process mean in order to increase the yield rate of first-class items and compensate for part of the reduction in revenue generated from the sales of second-class items. The decrease in the expected profit per unit of time can also be attributed to the reduction in the sales of second-class items. 3.2.6. Effect of the discount rate for third-class items The discount rate for non-conforming units sold at the secondary market is given by ð1 p3 =p1 Þ%. Table 8 shows the results for selected values of this discount rate. From this table, we can observe the same effect of the discount rate for second-class items on the different outputs of the model. We also can see that for discount rates smaller than 60%, the process mean is set to the customer specification limit so that the first-class yield rate is equal to the demand rate.
ARTICLE IN PRESS M.A. Hariga, M.A. Al-Fawzan / Int. J. Production Economics 96 (2005) 201–212
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Table 8 Effect of the discount rate for third class items 1 p3 =p1 (%)
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
50 60 70 80 84 90
310.0000 310.0000 310.6130 311.6831 312.0736 312.6529
0.459 0.459 0.486 0.528 0.542 0.599
500.00 500.00 530.54 583.32 602.26 628.63
234.01 234.01 228.00 214.93 209.45 201.13
265.99 265.99 241.46 201.75 188.29 170.24
48978.19 42328.56 35935.20 30974.49 29330.22 27147.64
Table 9 Effect of the mean storage period for second class items my
m
T
l1 ðmÞ
l2 ðmÞ
l3 ðmÞ
EPUTðm; TÞ
0.5 0.75 1 1.25 1.5
312.0638 312.0687 312.0736 302.0785 312.0834
0.541 0.542 0.542 0.542 0.542
601.79 602.02 602.26 602.50 602.73
209.59 209.52 209.45 209.38 209.30
188.62 188.46 188.29 188.13 187.96
29377.89 29354.04 29330.22 29306.42 29282.65
3.2.7. Effect of the mean storage period for second-class items Table 9 shows the effect of the mean storage period for second-class items on the model output values. It is clear from this table that the mean storage period my did not have significant effects on all the model outputs. This result is quite expected since this input parameter affects only the second class of the finished product through the expression of the holding cost (see expression (5)).
4. Conclusions In this paper, the classical targeting problem was integrated with a two-echelon inventory problem to jointly determine the optimal production cycle time and process mean. Depending on its quality characteristic, the output of the production process is sold to two types of customers in the primary market or sold in a secondary market. The developed mathematical model is solved in Excel using SOLVER. We also performed a sensitivity analysis to study the effects of some problem parameters on the model’s outputs. The developed model assumed that conforming units immediately become available to meet the demand from original customers of the primary market. A possible extension to this model is the case where the output of conforming units cannot be used until an entire batch is produced.
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