© Hindawi Publishing Corp. A SERIES-TYPE MULTISTAGE PRODUCTION SYSTEM WITH RANDOM DEMAND

We propose a mathematical model which considers the series-type product structure with n−1 predecessors. Our objective is to obtain the optimal production functions, in the planning horizon [0,T], based on the assumptions (1) that the cost of production unit is a linear function of production quantity in a time unit, (2) that sales of finished goods occur at the end of planning horizon, and (3) that product demand is a random variable. Then the phenomenon of optimal solution is discussed.

1. Introduction.Billington et al. [1] have described that there are four types of product structures in the multistage production systems: (1) series; (2) parallel; (3) assembly; (4) general.The simplest is the series-type product structure which illustrates a single product, produced in a series of steps.Zangwill [7] and Love [6] use a concave cost structure and present relatively efficient solution techniques.Whatever type of the multistage production systems, the assumption of the demand schedule for the end products has been established, and then one determines the lot sizes in each stage to minimize the total costs.On the contrary, the probabilistic demand is discussed in the inventory system.The typical one is the classical newsboy problem.It is a single-period, singleproduct inventory problem which considers the inventory size to be ordered for the sake of meeting random demand so as to maximize expected profit while balancing holding and shortage costs.There are many papers to discuss this problem in recent years.For example, Eppen [4] presented a multilocation newsboy problem with normal distribution of a location's demand, identical linear holding, and penalty cost functions.Chen and Lin [3] extend Eppen's model by considering the concave cost function with unspecified distribution of demand, then show that the Eppen's results are still true.
M. S. Chen and Y. C. Chen [2] have constructed a mathematical model for the newsboy problem with production and holding costs.The problem is, in the planning horizon [0,T ], how should the decision makers control the production rates to meet the random demand at the end of the period such that the expected profit is optimal?However, they assumed that the product structure is single-stage.Thus, we extend it by assuming that the product structure is series-type with n − 1 predecessors.Then how should the decision makers control the production rates at each stage to meet the random demand at the end of planning period such that the expected profit is optimal? 2. Notation and assumptions.For the sake of convenience, the following notation and assumptions are used in this paper: (i) [0,T ]: the available time interval for production and T is the selling time; (ii) v: the price unit of produced goods; (iii) h i : the holding unit cost of ith semifinished goods in a time unit, i = 1,...,n − 1, let h 0 = 0, and h n is the holding unit cost of finished goods in a time unit; (iv) b: the loss or treatment cost per unit of surplus goods.It occurs when the quantity of inventory on hand at time T is larger than the quantity of goods in demand; (v) p: the penalty cost per unit of lacking goods.It occurs when the quantity of inventory on hand is less than the quantity of goods in demand; (vi) S: the quantity of goods in demand at time T .Here, S is a random variable, its probability density function is f (s), and its cumulative distribution function is (vii) [t i ,T ]: the time interval during which the decision maker is actually engaged in the production of ith semifinished goods, and t i is the time to begin production of ith semifinished goods, where t i ≥ 0 and i = 1,...,n−1.
Let [t n ,T ] be the time interval during which the decision maker is actually engaged in the production of finished goods and let t n be the time to begin the production of finished goods, where t n ≥ 0; (viii) x i (t): the cumulative production of ith semifinished goods at time t, that is, the total production of semifinished goods in the time interval [t i ,t], where x i (t i ) = 0 and i = 1,...,n−1.Furthermore, x n (t) is the total production of finished goods in the time interval [t n ,t], where x n (t n ) = 0 and x n (T ) is the total inventory on hand at time T .When the decision makers make extra production plans in addition to the routine work, the cost will burden them because of the capital and the human resources.So, the production unit cost will increase as the production increases.Hence, in this paper, we assume that the production unit cost is an increasing function of production in the time unit.If the decision makers do not have production, they do not have to pay for the cost.Therefore, we have (ix) c i x i (t): the production unit cost of the ith semifinished goods at time t, where c i is a constant and i = 1, 2,...,n − 1.Furthermore, c n x n (t) is the production unit cost of finished goods at time t, where c n is a constant.
Thus, total production and holding costs equals Since the for-sale quantity of goods min{x n (T ), S} is a random variable, we have that (i) the expected revenue equals If our objective is the profit optimization, then the mathematical model is as follows: for all i = 1, 2,...,n, where x i (T ), t i are free for all i.Now, let (x * 1 (t), x * 2 (t),...,x * n (t)) be the optimal solution of (3.3) and consider two feasible solutions (3.4) Therefore, using the fact that the objective value of Similarly, the objective value of (x * ), then we have which implies that Combining (3.6) and (3.8), we have Then, Q must be one of the following three cases.
Case I.The sequence Q is an increasing sequence.
In this case, the optimal solution is shown in the next section.
Case II.The sequence Q is a strictly decreasing sequence.
Then by (3.9), we know that x * n−1 (t) = x * n (t).So, in this case, we let x n (t) = x n−1 (t) in (3.3), then the coefficients of x n−1 (t) and x n−1 (t) are h n −h n−1 and c n + c n−1 , respectively.
Hence, sequence Q becomes It is easy to see that if , by (3.9), we then have Continuing the process, we finally find that x * , where x n (T ), t n are free.In this case, problem (3.3) reduces to the single-stage problem which is discussed in [2].
Case III.The sequence Q is neither increasing nor strictly decreasing.
Step 0. Form the sequence Q In this situation, we recompute Q(i) and let Step 2. If Q(i) = " * " then i = i + 1; if i = n, then go to step 7; else redo step 2; else if Q(j) = " * , " then j = j + 1; if j = n + 1, then go to step 7; else redo step 2; go to step 3.
Step 3. If Q(i) > Q(j), then go to step 4 else go to step 6.
Step 6. i = j; j = j + 1; if i = n or j = n + 1, then go to step 7 else go to step 2. Step 7. Stop.Algorithm 3.1

Optimal solution.
To derive the optimal solution of Case I, note that (3.3) is not the standard form of calculus of variation, we first neglect the constraints x i (t) ≥ 0 for all i, and x i (t) ≥ x j (t) for 1 ≤ i < j ≤ n, and consider the following problem: t),..., xn (t)) be the optimal solution of (4.1).
It is easy to see that xi (t) − xj (t) ≥ 0 for 1 ≤ i < j ≤ n.For each i, since x i (t) > 0, x i (t) ≥ 0 if and only if x i (0) ≥ 0. It is easy to see that x i (0) ≥ 0, for all i if and only if x n (0) ≥ 0, and by (4.4) if and only if Let G be the function of xn (T ): It is easy to see that G ( xn (T )) < 0, so (4.
where the value of x * n (T ) is determined by the following equation: where the value of t * 1 is determined by the following equation: and the value of t * i is given by where the value of t * k+1 is determined by the following equation: then the value of t * i is given by where the value of x * n (T ) is determined by the following equation: where the value of t * n is determined by the following equation: (4.46)

Conclusion.
To analyze the characteristics of the optimal solution, the Q sequence should be considered first: Then there are some features of the optimal solution that are described as follows: (1) if Q is an increasing sequence, then the optimal solution is of Case 1.
In this situation, for Case 1.1, the optimal inventory size x * n (T ) will be greater than ((h n − h n−1 )/4c n )T 2 .For Case 1.2, the optimal inventory size x * n (T ) will be less than ((h 1 − h 0 )/4c 1 )T 2 .Otherwise, the optimal inventory size will be between them; (2) if Q is a strictly decreasing sequence, then the optimal solution is of Case 2. Furthermore, if (h i − h i−1 )/c i = (h i+1 − h i )/c i+1 is established, then the optimal solutions of Case 1 will reduce to that of Case 2, that is, the multistage problem will reduce to the single-stage problem like [2].Precisely speaking, the optimal production functions of all stages are the same; (3) there are two types of the optimal production plan for Cases I, II, and III: immediate production and postponed production.Briefly speaking, (h i − h i−1 )/c i , the ratio of the incremental holding cost to production cost per unit of ith semifinished goods, is an important factor which determines whether the optimal production functions are the same or not.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: xn(T ) 0 vsf (s)ds + ∞ xn(T ) vx n (T )f (s)ds; (ii) the expected cost of surplus goods equals xn(T ) 0 b(x n (T ) − s)f (s)ds; (iii) the expected cost of shortage goods equals ∞ xn(T ) p(s − x n (T ))f (s)ds.
and the coefficients of x * (t) and x * (t) are h n −h 0 and n i=1 c i , respectively.Then, problem (3.3) reduces to Max xn(T ) 0 vs − b x n (T ) − s f (s)ds