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Ind. Eng. Chem. Res. 2006, 45, 6622-6625
Multiperiod Planning Strategies with Simultaneous Consideration of Demand Fluctuations and Capacity Expansion Jun-hyung Ryu* Samsung Electronics, San 16 Banwol-Ri Taean-Eup, Hwasung-City Gyeonggi-Do, 445-701 Korea
Considerable attention has been given to multiperiod planning in process communities in order to hedge against varying conditions. The negative impact of variations can be minimized by preparing production and inventory levels of the current and future operations. This paper aims to highlight how external conditions in the planning model are synchronized with internal process operations to obtain the maximum effect. Two strategies are discussed. One is to modify the external condition, mainly varying demands, suitable for the plant operation, and the other is to expose explicit constraints limiting capacity expansion based on actual expansion times in the model. Two numerical examples are presented to illustrate the potential of the proposed strategies with some discussion on applying these strategies in practice. Introduction
(i) Material balance
Multiperiod planning problems are concerned with how to operate processes over a relatively long horizon with an aim to maximizing their associated profits despite variations. A great deal of attention has been thereby given to multiperiod planning in the literature. From the literature, we can find that planning models are generally subject to various constraints such as (i) production capacities with or without capacity expansions, (ii) demands satisfaction, (iii) resource availabilities, (iv) inventory requirements, and (v) material balance. The production configuration may take the form of selecting particular units or processing types in response to varying external conditions, which are, in most cases, uncertain demands.1 Consequently, most planning models are mathematically formulated into a mixed-integer linear programming (MILP) problem. This can be explained by the fact that the major focus of the planning model involves the discrete decision, such as which unit to select or when to expand the current capacity by what amount, etc. Most previous studies are concerned with formulating the planning practice into mathematical models and computing their solutions. It surely is an important issue to compute the solution of the model. It is also worthwhile to review how external conditions are interpreted in the model to properly reflect actual practices. Particularly, it should be highlighted that there exist different time scales in planning between external demands and capacity expansion. Actual demands generally change rapidly in a matter of days, sometimes weeks. On the other hand, it is often much longer for manufacturers to respond to the variation, often in a matter of weeks or months. In this paper, we are going to focus on synchronizing such different time scales in order for the plant to respond to external conditions, particularly fast demand variations. Inventories can be used for the easy response, but it is not always sufficient to assume that there is always enough inventory to meet varying demands. 2. Conventional Multiperiod Production Planning Model of Chemical Processes A planning model is generally subject to the following constraints: * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: 82-31-208-6270. Fax: 82-31-208-6488.
∀ l, i, t
bliRlt + INVlit g Slit + INVlit+1
(1)
For product i, the capacity of plant l in time t, Rlt, is related with its current inventory INVlit and a future inventory. Slit denotes a demand which is allocated to plant l of product i at time t, and bli is the performance coefficient of product i in plant l. (ii) Capacity expansion
Rlt ) Rlt-1 + CElt ∀ l, t
(2)
The capacity of plant l at time t, Rlt, is increased from the one in the previous time period, Rlt-1 as much as the capacity expansion amount of CElt. (iii) Upper and lower bounds on inventory There exist physical limitations of inventory amount, which are expressed
INVLlit e INVlit e INVUlit ∀ l, i, t
(3)
(iv) Upper and lower bounds on capacity expansion quantity If the binary variable ylt is 1, the capacity expansion has an upper bound; otherwise, the capacity expansion amount is 0:
yltCELlt e CElt e yltCEUlt ∀ l, t
(4)
(v) Limited inventory of a plant The sum of all inventories at a plant is limited to the total inventory associated with the plant:
cilitINVlit e TotalINVlt ∑ i)1
∀ l, t
(5)
(vi) Objective function The objective function is to maximize its profit, which is the difference between sales revenue and the sum of inventory holding cost and capacity expansion cost.
max Profit ) prlitSlit -
∑ l,i,t
prIhdlitINVlit - ∑prEXPltylt - ∑CEltβl,t ∑ l,i,t l,i,t l,t
10.1021/ie060601k CCC: $33.50 © 2006 American Chemical Society Published on Web 08/16/2006
(6)
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Although the above model is based on the various planning practices, the solution of the model may provide results which are unrealistic to implement. The reason can be traced in two ways. One is the model is not complete in reflecting the practice, and the other is that we need to carefully modify the real practice into the model to get the realistic solution. As an illustration of being unrealistic, this paper is particularly interested in the repeated expansion of capacities. Capacity is one of the most important factors in the process planning. We can expand capacity directly by installing additional equipment or indirectly by improving performances of the equipment or associated activities through the effort of innovation activities. For both cases, it is true that it takes much longer for the capacity to be expanded than for the demand to change, because the time needed for new equipment to be commissioned during expansion is generally long. Therefore, we should coordinate the fast change demands with slow capacity expansion. The objective is to arrange capacity expansion in accordance with demand variation. In the next section, two methods are presented with the succeeding numerical illustrations. In terms of modeling, the issues this paper brings into attention may be thought of as trivial in academia. However, the difference in the time unit for demand management and the time unit for the manufacturing part may cause serious confusion in real businesses. A company generally consists of a manufacturing department and a sales and marketing department. Each focuses its own issues in their own working time schedule. The capacity planning is, thus, very important, because it simultaneously considers internal manufacturing with external sales operations. That is a motivation of this paper for the author to explicitly highlight its importance. 3. Novel Strategies The previous section pointed out that the frequent variations of the external condition, mainly in the form of demand change, should be properly incorporated with relatively slow internal actions of expanding capacities in order to construct practically implementable planning model. Considering the fact that the inventory can do a very limited role of buffer between process network entities, expanding the capacity is the only available resource. A single time scheme should then be introduced to synchronize the decision time unit between demand handling and capacity expansion, because the lead time of the demand change is generally much faster than that of expanding the capacity. As alternative strategies, this paper proposes the following two strategies. One is to aggregate the demand change periods. This is the approach most previous research using deterministic modeling is based on, for example, that by Sahinidis et al.1 In the planning model, demand, Slit in eq 1, needs to represent the aggregated demand. For instance, if the true demand changes every four month and a capacity expansion can be done over a year, the actual demand should be aggregated to give a single yearly demand within a planning model. The single yearly demand may be then the upper bound, lower bound, or average of the true demands. Whether to choose the upper or lower level is dependent on the companies' strategy and industry practices. For example, if the industry has the long total lead time, more buffer is generally preferred by the process network entities. They want to have more inventory. Then it would be preferred to have less conservative inventory. On the other hand, if they are under severe inventory holding cost with volume constraints, they want to minimize it as much as possible in a very
Table 1. Multiperiod Data for Case 1 demand (ton)
price ($/ton)
time
A
B
A
B
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
15 000 37 000 24 000 41 000 25 000 44 000 45 000 50 000 40 000 37 000
30 000 45 000 40 000 42 000 35 000 40 000 42 000 35 000 40 000 42 000
142 129 133 141 147 152 154 155 148 156
135 134 125 136 143 149 148 150 149 152
Table 2. Capacity Plan under Different Strategies in Case 1 capacity for each time period with different strategies time period
fixed capacity
aggregated demand
adding limiting constraints
no restriction on capacity expansion
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
72 102 72 102 72 102 72 102 72 102 72 102 72 102 72 102 72 102 72 102
56 962 56 962 64 251 64 251 69 451 69 451 73 951 73 951 73 951 73 951
46 835 46 835 53 256 53 256 59 756 59 756 68 256 68 256 68 256 68 256
35 250 44 750 50 632 56 132 56 132 63 136 63 136 63 136 63 136 63 136
profit ($)
87.08M
88.15M
89.63M
89.85M
aggressive manner. Other parameters such as price can also be defined in a similar way. The resulting planning model can represent the average status from the original practice. One negative impact is that the resulting planning and inventory levels are also in the aggregated form. The detailed levels should be computed again from the resulting same repetitive values. The second is to add explicit new constraints in the model. A new constraint limiting the number of times the capacity is expanded in response to the changing demand can be added in the original planning model. For the case where demand can change every week but an expansion requires at least one month or four weeks, an explicit constraint which limits the expansion number to once per every four weeks may be added with the following form, N
∑j yt+j e 1
∀t
(7)
where N denotes the necessary number of demand time periods for an event of capacity expansion. 4. Case Studies Two numerical case problems are solved using the proposed methods to illustrate the proposing strategies in this section. 4.1. Case 1: Two Products and Ten Time Periods. Consider a company which makes two products A and B. The time horizon of the company is 10 time periods: the planning department computes their planning decisions for 10 time periods. Demands over these periods vary as in Table 1 in terms of quantity and price. Because of the plant characteristics, it takes at least 2 time periods to expand the manufacturing capacity. A plan should be prepared covering the next 10 time periods with respect to varying demands; see Table 2.
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The above proposed approaches can be introduced as follows: (1) Three operation planning policies for the operation of the plant can be suggested to meet the demand. The initial design of the plant is computed, and the plant is operated based on the fixed capacity. This is the multiperiod design,2 which has fixed capacity during the operation. (2) The original 10-time period problem is transformed into a 5-time period problem. Each two time periods are aggregated to allow capacity to be modified. Since 2 time periods are required to expand the capacity, time 1 and time 2 are grouped as new time 1, time 3 and time 4 are grouped as new time 2, and so on. The new demand can be a simple sum during a respective 2 time periods. (3) The third policy involves an explicit constraint that restricts the number of capacity expansions in the given time periods. In this example, since one expansion for every 2 time periods is assumed to be possible, the following constraints are added in the generic planning model:
yt-1 + yt e 1 t ) t2...t10
(8)
At last, the generic planning problem is solved to compare the result with no restriction on capacity expansion constraints. As can be seen in the result, the model without considering the synchronizing issue provides the highest profit: it can expand its capacity whenever necessary. That is to say, operation based on the low capacity with gradual increase is the best operation strategy. The point this case aims to address is that the difference between ab unconditional increase in capacity and the limited capacity increase does exist and we have to make a realistic choice. 4.2. Case 2: Five Products and Twenty Time Periods. Consider a company that manufactures five products. Their planning horizon is 20 weeks, and at least 2 time periods is requested to expand their manufacturing capacity by technology innovation parts or by purchasing and installing additional equipment. In response to demand trajectory of those products by the marketing department, their capacity planning should be made by the planning department. As in Case 1, the proposed approach provided the result as summarized in Table 3. From the result in Table 3, we can confirm that the model without any consideration on the capacity expansion time provides the highest profit. This may be the best solution in terms of profit maximization. However, in terms of execution in practice, it represents the upper-bound value that the planning model can provide, which may not be possible in practice because of the physical expansion time. 4.3. Remarks. It is worthwhile to mention some issues regarding the proposed strategies and the results on the case studies. It may not be clear as to why the proposed method is superior to other existing methods. Moreover, two cases are solved with different strategies, in which the proposed method does not offer better results in terms of maximizing profit. However, the major proposition of this paper is to add more detailed constraints in the original planning model in order to obtain more practically suitable solution. This may not be the solution with a higher profit but is one that is more meaningful in terms of practical implementation. In other words, the proposed methodology provides realistic multiperiod planning strategies that are suitable for actual implementation Second, the issue of whether the demand is deterministic or stochastic also needs to be addressed. It is assumed in this work that the demand is known. To forecast demands without actually knowing demands is a very difficult task. Regarding this, we
Table 3. Result on Case 2 with Twenty Time Periods for Five Products capacity for each time period with different strategies time period
no restriction
demand aggregation
adding constraints
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20
6 074 6 074 7 074 7 074 7 074 8 239 8 239 8 239 8 239 9 739 10 739 11 739 12 739 13 739 15 239 16 739 18 339 19 939 21 639 23 339
31 402 31 402 32 402 32 402 34 102 34 102 35 102 35 102 36 602 36 602 37 602 37 602 38 602 38 602 40 102 40 102 41 702 41 702 43 402 43 402
11 164 11 164 11 164 12 164 12 164 13 864 13 864 14 864 14 864 16 364 16 364 17 364 17 364 18 364 18 364 19 864 19 864 21 464 21 464 23 164
profit ($)
205M
147M
200M
can point out that it is less difficult to forecast parameters in the aggregated format over long-term periods than ones with short horizons. In practice, it may be very difficult to forecast exactly the demand for tomorrow but it would be less uncertain to forecast the expected demands over the next week. Thus, we may have relatively reliable information on the varying parameters. Third, it was found out from the two case studies that, between the two strategies, adding new constraints has some advantage over the other. As can be seen in the results of the examples, the profit according to adding constraints is higher than that of the other strategy. It also has the advantage of providing the more detailed timing of when to expand the capacity. This can be explained by the fact that the time constraints are more divided in detail than the aggregated time. However, there may be cases in which the demand aggregation is preferred over the adding-constraint strategy. Fourth, if the capacity expansion time is long, such as >4 time periods, the difference between no constraints and the two proposed strategies is increased. The distinction between speeds in external variation and internal operations is the dominant issue to raise their efficiency. The computational time is not included here, because recent technological improvement has significantly reduced the computational expense and the problem size handled in this paper is solved within reasonably short times. This paper does not provide any new mathematical formulation. From the literature review, it is already stressed that mathematical modeling and computational issues have been widely addressed and have significantly improved. However, this paper aims to highlight that the actual implementation methodology of the developed mathematical model should be also improved. Oh and Karimi3 have introduced a similar constraint that accounts for the time needed for capacity expansion. For instance, eq 7, which corresponds to industry practice, should be compared with works that considered construction lead time. However they did not explicitly highlight the importance of the inherent issue of the distinctive time frames in the planning problem. Although this paper does not provide any new mathematics but rather provides a new
Ind. Eng. Chem. Res., Vol. 45, No. 19, 2006 6625
interpretation of old mathematics, its inherent importance in practice should be properly deserved.
between supply chain entities may also be an important future issue, because it definitely reduces the distribution of redundancies in a supply chain decision framework.
5. Discussion As business environments become complex and rapidly changing, the industry is compelled to be more competitive. The industry should consider various issues in and out of their business environment to eliminate the unnoticed redundancy and to be, thereby, more effective. Synchronizing external condition handling with internal operations is, thus, an important issue to raise the effectiveness of companies. Two novel multiperiod planning strategies have been suggested in this paper to unite the different time scales and have been demonstrated with two case studies. Regarding eliminating the redundancy, academia and industries have been recently interested in supply chains and managing them, which is called supply chain management (SCM). Synchronizing the speed of response
Literature Cited (1) Sahinidis, N. V.; Grossmann, I. E.; Fornari, R.; Chathathi, M. Optimization model for long range planning in the chemical industry. Comput. Chem. Eng. 1989, 13, 1049. (2) Sahinidis, N. V.; Grossmann, I. E. Multiperiod investment model for processing networks with dedicated and flexible plants. Ind. Eng. Chem. Res. 1991, 30, 1165. (3) Oh, H.-C.; Karimi, I. A. Regulatory Factors and Capacity-Expansion Planning in Global Chemical Supply Chains. Ind. Eng. Chem. Res. 2004, 43, 3364.
ReceiVed for reView May 16, 2006 ReVised manuscript receiVed July 25, 2006 Accepted August 7, 2006 IE060601K
