Strategic production planning of an automated manufacturing system through market forecasting Kuei-Shu Hsu Department of Automation Engineering Kao Yuan Institute of Technology Lu-Chu Hsiang, Kaohsiung Taiwan 821 R.O.C. E-mail :
[email protected] Abstract The attention of the maximum-profit production scheduling and control grows up in automated manufacturing with considerations of market demand and product price. This paper not only constructs a mathematical model under deterministic market demand and product sales price, but also implements the Lagrange Method to optimize the production profit of an automated production system. Additionally, the step-by-step algorithm in achieving the optimum production schedule of the system through the forecasted probabilistic market is also prepared. To exemplify the applicability of this study, the numerical simulation under fluctuating market situations is furthermore introduced with V ISUAL B ASIC. Through this study, the product scheduling, production cost estimating, and even the order negotiating for an automated manufacturing system under deterministic or probabilistic market can then be approached. This paper not only contributes the applicable production scheduling strategy for an automated manufacturing system, but also provides the valuable conception in maximizing the production profit to the modern manufacturing industry. Keywords : Production scheduling, lagrange method, market demand, operation cost, fixture cost.
1.
Introduction
For many years, the decision making process at the shop-floor level has been headed by the implicit idea to optimize the use of machines. Production control has been addressed from various machine states [1] ——————————– Journal of Information & Optimization Sciences Vol. 25 (2004), No. 3, pp. 461–472 c Taru Publications
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to real-time control problem [2]. As automation and flexibility have become ingredients in the recipe for success in global manufacturing, the control of an automated manufacturing system is thus the main objective of manufacturing management. Kogan and Levner [3] have mentioned several mathematical models to control a production operation; nevertheless, they were basically constructed to minimize the maximal completion time of all operations. Levner et al. [4] have also introduced the buffer capacity between operations into a small-scale manufacturing cell to optimize the productivity. However, their complicated and hard solving control model is only verified to a particular case. Production control is often modeled as optimization problem for constructing the maximum profit, so the most concerned problem challenging the manufacturing industry. Whilst the marginal cost of production is a linear increasing function of the production rate, the operational cost of a manufacturing system is in direct proportion to the square of the production rate [5]. This denotes that the higher production rate results higher operational cost, such as machine depreciation and tooling cost. Besides, the machines won’t have operational cost while idle [6]. This is because the consumption of input materials does not exist, and electricity fees of idle machines are relatively small when compared with those of the whole system. But, the maintenance cost for the machines has to be taken into consideration as the fixture cost [7]. Mosheiov [8] has stated that the promised due-date is to be determined during sales negotiations with the customers in many practical scheduling environments, and meeting due-date is clearly one of management’s primary objectives. Kingsman et al. [9] have designated that supplying products in response to a customer order in competition with other companies is the major problem confronting manufacturing. Soroush [10] also mentioned that meeting the production deadline is the most desirable objective of management. Therefore, meeting the production deadline of an order is critical to production projects. Efficient utilization of the modern automated systems is heavily dependent on proper scheduling of jobs. The interest in the minimumcost operation control grows up in modern manufacturing systems with the necessity of being more and more flexible to match the order quantity and production deadline. There is an economic need to control an auto-
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mated manufacturing system for a deterministic quantity under deadline constraint. Nevertheless, the market demand varies [11] and the product sales price fluctuates [12] seasonally for long-term manufacturing. Therefore, the production scheduling of a long-term manufacturing system for future uncertainty is also necessary to be addressed through the forecasted data. The mathematical model proposed in this study surely provides the practical solutions to the technique, and contributes the significant approaches to schedule and control the production of an automated manufacturing system. 2.
Assumptions and notations
2.1
Assumptions
Several conditions are assumed throughout the paper. described as follows :
They are
1. The manufacturing process is a continuous operation without breakdown. 2. There allows only one part being manufactured in the system at one time. 3. No delay or scrapping of parts occurs during production. 4. The marginal operation cost of the automated manufacturing system is a linear function of the production rate, and the operational cost is also directly proportional to the square of the production rate [5]. 5. All finished products are held in the manufacturing system until the whole production quantity is done, and the whole quantity is shipped to customers exactly at the deadline. 6. The production deadline is within its feasible range that will not affect the quality requirement of the products. 2.2
Parameters and notations The parameters and notations are listed and described as follows : B : maximum number of forecasting time periods. c : marginal operation constant when operating the automated manufacturing system, where c is a constant.
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c f : fixture cost of the automated manufacturing system per unit time. ch : holding cost of unit product per unit time. p : constant sales price per unit product. p(t) : forecasted sales price per unit manufactured product at time period t . Q : constant production quantity of an order. Q(t) : forecasted demand quantity at time period t . T : production deadline, which is given by the customer or job schedule. ¯ T : production time interval, which can be selected as one week, one month or one season. t : forecasting time period, where t = 1, 2, 3, . . . . c : marginal operation cost when operating tp
the
automated
manufacturing system per unit time at the production rate
1 . tp
c : operational cost when operating the automated manufacturing t2p 1 system per unit time at t e production rate . tp 2.3
Decision variables
t p : production time per unit part in the automated manufacturing system. ¯t queuing time of the production project before manufacturing. 3.
Mathematical modelling
3.1
Model formulation
When considering the constant order quantity and the constant product sales price, Qt p + t¯ has to be denoted in matching the production deadline T . And, Qt p describes the actual production period. c 1 Therefore, × × Qt p and c f ( Qt p + t¯) represent the operational cost tp tp and the fixture cost of the manufacturing system during production Z Qt p t dt states the product holding cost during respectively. Besides, c h tp 0
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production. With all the description above, the maximum profit objective function of a production project for an automated manufacturing system under constant situations is constructed as follow. Z Qt p c 1 t ¯ max pQ − × × Qt p − ch + dt − c f ( Qt p + t) . tp tp tp 0 Rearrange the above objective function, the mathematical model and its constraint are then formulated and shown as follows: " # 2t c Q cQ p h maximize pQ − − − c f ( Qt p + t¯) tp 2 subject to Qt + t¯ = T . p
3.2
Optimal solution
Let (t∗p , t¯∗ ) be the optimal solution of the model, and set L be the Lagrange Function. Applying the Lagrange Method, it is then defined as cQ ch Q2 t p ( T − t¯) ¯ L = pQ − − − c f ( Qt p + t) + λ Q − . tp 2 tp Here, λ is the Lagrange Multiplier. Since the partial differentials of L with respect to t p , t¯ and λ are zero, we then have 0 =
cQ c Q2 ∂L λ ( T − t¯) = 2 − h − cf Q + ∂t p 2 tp t2p
(1)
0 =
∂L λ = −c f + ¯ ∂t tp
(2)
0 =
∂L ( T − t¯) = Q− . ∂λ tp
(3)
With Eq. (2) and Eq. (3), we have
λ = cf tp
(4)
T − t¯ = Q. tp
(5)
and
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Multiplying Eq. (4) to Eq. (5), it is obtained that
λ ( T − t¯) = cf Q . t2p
(6)
Introducing Eq. (6) into Eq. (1), the optimal production time is found as s 2c ∗ . (7) tp = ch Q Substitute Eq. (7) into Eq. (3), the optimal queuing time is thus derived as s 2cQ t¯∗ = T − . (8) ch 3.3
Feasibility discussion
From Eq. (8), it is necessary to discuss the feasibility of the three possible situations. They are discussed as follows. r 2cQ 1. When T > , this explains that t¯∗ > 0 . There must exist an ch optimal queuing time before manufacturing. This optimal queuing time t¯∗ can always be scheduled for the machine maintenance or small production projects to promote the of the r r efficient time utilization 2c 2cQ and t¯∗ = T − . system. The optimal solution is t ∗p = ch Q ch r 2cQ , this shows that t¯∗ < 0 . There doesn’t exist any 2. When T < ch optimal queuing time before manufacturing. The optimal solution is T t∗p = and t¯∗ = 0 Q r 2cQ , this denotes that t¯∗ = 0 . The optimal solution is 3. When T = c h r 2c ∗ and t¯∗ = 0 . tp = ch Q 4.
Step-by-step algorithm
When considering the uncertain market, the demand of production quantity and the product sales price are surely time-variant. To approach
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the optimal production scheduling of the uncertain future, a computerized tool in monitoring the market change through the forecasted market demand and product sales price is also necessary to be constructed. With the optimal solution of the mathematical model and the feasibility discussion above, the concept of the flow chart is then described as follows. p(t), Q(t), c, c f , ch , and the selected scheduling period T¯ should be given before the following algorithm. Initialize t = 1 and T = T¯ . Step 1. Find p = p(t) and Q = Q(t) r 2cQ Step 2. If T < , go to Step 5. ch Otherwise, go to Step 3. Step 3. Compute the optimal solution t∗p
=
s
2c ch Q
and
t¯∗ = T −
s
2cQ . ch
Then, find the maximum profit of production. L = pQ −
cQ ch Q2 t p − − c f ( Qt p + t¯) . tp 2
Step 4. Plot (t, L∗ ), (t, t∗p ) and (t, t¯) . Go to Step 7. Step 5. Compute the optimal solution t∗p =
T Q
and
t¯∗ = 0.
Then, find the maximum profit of production. L = pQ −
cQ ch Q2 t p − − c f ( Qt p + t¯) . tp 2
Step 6. Plot (t, L∗ ), (t, t∗p ) and (t, t¯) . Go to Step 7.
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Step 7. If t ≥ B , stop the program. Otherwise, set t = t + 1 and return to Step 1.
5.
Numerical illustration
To demonstrate the extensive versatility of this study, a numerical case with the year-round forecasts of the demand quantity and product sales price is introduced. Here, a theoretical approximation of the seasonal product sales price [13] shown in Figure 1 is selected for simulation. And, the discrete capacity step change on the probabilistic demand quantity [14] shown in Figure 2 is also suggested for the study.
Figure 1
Theoretical approximation of seasonal product sales price
Figure 2 Theoretical approximation of step-wise market demand
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The simulating time interval is selected to be a week; that is, T¯ = 2400 minutes. All other relevant data compiled are transformed into SI units as well as US dollars. They are listed as follows : p(t) = 35 − sin(0.115t), Q(t) = 150 + 100 sin(0.001t), dollar − minute c = 1.5614 , c f = 2.1528(dollar/minute), part dollar ch = 0.00011 , T¯ = 2400 minutes. minute − part With the step-by-step algorithm described, the computerized decision tool written in V ISUAL B ASIC is then constructed, and the simulated results are presented as Figure 3, 4 and 5. It is noted that the system maintenance or small production projects can be scheduled into the optimum queuing time shown in Figure 3. With the optimal production time in Figure 4, operations engineers can always modify the production parameters of the automated manufacturing system; therefore, the production time for each time interval is capable to be adjusted to satisfactorily match the optimal production time t ∗p . Besides, the optimum production profit can be estimated with Figure 5; therefore, the budget control and even the contract negotiation can then distinctly be further approached with this study.
Figure 3
Optimum queuing time before production
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Figure 4
Optimum production time per unit product
Figure 5
Optimum production profit
From the simulated results, the optimal solution of the proposed model surly shows good agreement in its applicability and adaptability. The algorithm in achieving the optimal solution of the proposed model provides exact values indicating the optimum queuing time and production time to be followed by the manufacturing system.
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Summary and remarks
Automation and flexibility have become ingredients in the recipe for success in global manufacturing. The interest of the maximum-profit production control grows up in the automated manufacturing systems. The operational cost and fixture cost of the manufacturing system as well as the product holding cost are necessitated to be considered simultaneously in maximizing the production profit for a deterministic as well as the probabilistic market under deadline constraint. This is a hardsolving and complicated issue. However, the problem becomes concretely solvable through our proposed model. The optimal solution of operation control is prepared using the Lagrange Method in this paper. With the optimal controlling time from the proposed model, operations engineers can always modify the production period of the automated manufacturing system to satisfactorily match the optimal production time t ∗p ; therefore, the maximum production profit is thus accomplished. Besides, the optimal queuing time, t¯∗ , can also be scheduled for system maintenance or small production projects to promote the efficient time utilization of the system. Thus, the production scheduling, production profit estimating, and even the order negotiating for an automated manufacturing system can further be achieved through this study. Future researches on the modeling of multi-order operation control with various deadlines for an automated manufacturing system are fully encouraged. In sum, the mathematical model proposed in this study surely provides the better and practical scheme for production control in maximizing the production profit of an automated manufacturing system under a deterministic as well as the probabilistic market with deadline constraint. Acknowledgements. This work of the paper is supported by National Science Council, Taiwan, R.O.C. under project NSC-93-2622-E-244-003CC3. References [1] A. Sharifnia (1988), Production control of a manufacturing system with multiple machine states, IEEE Transactions on Automatic Control, Vol. 33 (7), pp. 620–625. [2] G. Cohen (1998), Neural networks implementations to control realtime manufacturing systems, Computer Integrated Manufacturing Systems, Vol. 11 (4), pp. 243–251.
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[3] K. Kogan and E. Levner (1998), A polynomial algorithm for scheduling small-scale manufacturing cells served by multiple robots, Computers Operations Research, Vol. 25 (1), pp. 53–62. [4] E. Levner, K. Kogan and I. Levin (1995), Scheduling a two-machine robotic cell : a solvable case, Annals of Operations Research, Vol. 57, pp. 217–232. [5] M. Kamien and N. Schwartz (1991), Dynamic Optimization, Elsevier Science Publishing Co. Inc., New York. [6] T. S. Lan and C. H. Lan (2000), On a comparison of objective function in optimal design of flexible production line with the unreliable machines, Journal of Information & Optimization Sciences, Vol. 21 (2), pp. 221–231. [7] M. S. Chen and C. H. Lan (2001), The maximal profit flow model in designing multiple-production-line system with obtainable resource capacity, International Journal of Production Economics, Vol. 70 (2), pp. 176–184. [8] G. Mosheiov (2001), A common due-date assignment problem on parallel identical machines, Computers & Operations Research, Vol. 28, pp. 719–732. [9] B. Kingsman, L. Hendry, A. Mercer and A. Souza (1996), Responding to customer enquiries in make-to-order companies : problem and solutions, International Journal of Production Economics, Vol. 46-47, pp. 219–231. [10] H. Soroush (1999), Sequencing and due-date determination in the stochastic single machine problem with earliness and tardiness costs, European Journal of Operational Research, Vol. 113, pp. 450–468. [11] B. Kang, H. Kim and C. Han (1996), A demand-based model for forecasting innovation diffusion, Computer Industrial Engineering, Vol. 30 (3), pp. 487-499. [12] F. J. Shilpi (1996), Estimating the level of protection : the implications of seasonal price fluctuations, World Development, Vol. 24 (5), pp. 929937. [13] L. C. Hendry and B. G. Kingsman (1993), Customer enquiry management : part of a hierarchical system to control lead times in make-to-order companies, Journal of Operational Research Society, Vol. 44 (1), pp. 61–70. [14] J. Olhager, M. Rudberg and J. Wikner (2001), Long-term capacity management: linking the perspectives from manufacturing strategy and sales and operations planning, International Journal of Production Economics, Vol. 69, pp. 215–225.
Received September, 2003
