Optimal Design of Batch-Storage Network under Sporadic Operating

Article pubs.acs.org/IECR

Optimal Design of Batch-Storage Network under Sporadic Operating Time Loss Gyeongbeom Yi,† Bomsock Lee,‡ and Euy Soo Lee*,§ †

Department of Chemical Engineering, Pukyong National University, 100 San Yongdang-Dong Nam-Gu, Busan 608-739, South Korea ‡ Department of Chemical Engineering, Kyung Hee University, 1732 Dugyoungeaero Gyhung-Gu Yonginsi Kyunggi 446-701, South Korea § Department of Chemical and Biological Engineering, Dongguk University, 26 3-Ga Pil-Dong Jung-Gu, Seoul 100-715, South Korea. ABSTRACT: The purpose of this study was to find an analytic solution for the optimal capacity of a batch-storage network that meets the demand for finished products under sporadic operating time losses. Batch processes are prone to random but infrequent operating time losses. Two common remedies for such failures are the duplication of the process and increases in the process and storage capacity, which are both very costly. A model that minimizes the total cost, which consists of setup and inventory holding costs as well as the capital costs of constructing processes and storage units, has been developed by using the framework of a batch-storage network with the flows that are susceptible to sporadic operating time losses. A graphical analysis for the estimation of the upper and lower bounds of the flows under sporadic shutdowns is used in this approach. The advantage of our model is that it provides a set of simple analytic solutions in spite of its realistic description of the material flows between processes and storage units. loss,1,2 which include the case of sporadic shutdown as a subset. The material flow patterns of the two cases are shown in Figure 1. The solution obtained in frequent shutdown may not be tight one for the case of sporadic shutdown. Therefore, a new optimization model is needed for batch-storage networks with sporadic shutdown. Sporadic shutdowns are very common in batch chemical processing. A certain class of production shutdowns, caused by planned preventive maintenance such as the inspection or replacement of rapidly aging parts and periodic switching of the operation mode as well as seasonal production, commonly occur in periodical and sometimes sporadic patterns. Processes are also susceptible to infrequent random processing failures as well as intentional shutdowns. Random processing failures usually originate from equipment damage, operator mistakes, process troubles, irregular or insufficient feedstock deliveries, or sudden sales fluctuations in the product market. Such failures generally require emergency corrective maintenance and therefore result in sporadic shutdowns. Sporadic shutdowns produce sporadic losses in operation and business time and increase maintenance costs. In spite of such operation or business time losses, the demand of customers for the final products must still be satisfied. In order to meet customer demand during shutdown or failure periods, the duplication of processes and/or increase in storage capacity are unavoidable. However, process duplication and storage capacity increases are

1. INTRODUCTION Batch processes are susceptible to unexpected operating time losses because of equipment malfunctions and/or operator mistakes. Every operating cycle can sustain a random operating time loss, which in this article is called a frequent shutdown. Previous research has treated this problem by introducing the concept of availability, which is defined as minimum operating time without failure time divided by average operating time with failure time.1,2 An analytical solution for the optimal cycle time has previously been found for batch processing that undergoes frequent shutdown, that is, when there are random variations in cycle time. In this study, we focus on random operating time losses that occur repeatedly after some (uncertain) number of normal operating cycles. We describe this type of operating failure as sporadic shutdown. Figure 1

Figure 1. Types of operating time loss. Special Issue: PSE-2012

shows the difference between the two flow types: (a) frequent shutdown and (b) sporadic shutdown. The previous solutions for frequent shutdown were developed under the assumption that every operating cycle sustains a random operating time © 2013 American Chemical Society

Received: Revised: Accepted: Published: 7116

September 27, 2012 January 18, 2013 January 24, 2013 January 24, 2013 dx.doi.org/10.1021/ie302404g | Ind. Eng. Chem. Res. 2013, 52, 7116−7126

Industrial & Engineering Chemistry Research

Article

included into the batch-storage network model.13 Optimal lot sizes and startup times are calculated with analytic equations; the optimal average flow rates through the network can then be computed by solving a separable concave cost minimization network flow problem. A special network structure was identified that has an analytic solution for the optimal average flow rates. A cash inventory can be installed in the batchstorage network, with financial transactions and cash flows then represented by using the PSW model.14 For example, the setup cost is withdrawn from the cash inventory when the processing was incurred. Purely financial activities, such as temporary investments in marketable securities, can be incorporated into the model. Cash and material inventories are maintained to prevent shortages. The production processes considered were originally confined to batch processes that fabricate the same products periodically. However, other types of production processes do need to be considered for chemical industries. One important example of such a chemical process is a multitasking semicontinuous process that operates continuously in a production mode that can be varied without turning it off. The types of processes that can now be described by the batch-storage network model include multitasking semicontinuous processes15 and multisite transportation processes.16 The PSW flows through the network are susceptible to uncertainties, which can be characterized by using the joint random variations in lot size and cycle time. A novel graphical method has been developed for displaying the upper, lower, and average values of the random PSW flows, which enables analytic optimal lot sizes to be obtained for stochastic optimization problems.1,2 Uncertainties in process operations can result in off-specification waste materials, and additional processes and flows for the treatment of these waste materials can be included in the network. A previous study with this approach of financial transactions and cash flows14 assumed that only one currency and one currency storage were required. In multinational corporations, multiple currencies and currency storages are involved in financial transactions and currency flows. Thus, the batch-storage network structure was enlarged to include multiple currency storages.17 Incoming currency flows arise from final product sales, and all costs represented by related currencies are included as outgoing currency flows. The currencies are permitted to move between the currency storages, and the exchange rates between currencies are included as conversion factors. Transfer prices, corporate income taxes, and customs duties can also be accounted for in the optimization model. The objective function of the optimization calculates the opportunity costs of the annualized capital investments and currency/material inventories minus the benefit to stockholders in the numeraire currency, and must be minimized. A notable result is that the optimal lot sizes are typically 20% smaller when corporate income tax is taken into consideration than when it is not considered.17 The PSW model is based on the periodicity of operations. Many nonperiodic operations that occur in the real world are not treated by the current PSW model. The PSW model has previously been modified to yield a multiperiod model with a time horizon that is divided into multiple time buckets to treat slowly time-varying operations.18,19 The great advantage of the PSW model of batch-storage networks is that analytic lot sizing equations are available for sophisticated network structures. A simple analytic solution can facilitate rapid and appropriate investment decisions for preliminary supply chain design problems confronted with diverse economic situations. Analytic

no longer easy: the design and construction costs of chemical processes with respect to their total capacities are increasing substantially. Nowadays, even storage facilities are increasingly expensive because of rising land values, heightened environmental concerns, and severe quality control requirements. Moreover, modern plant design concepts such as JIT (just-in-time) encourage businesses to reduce storage capacity. Therefore, the storage capacity in conjunction with process capacity should be selected carefully to consider the operational characteristics such as frequent or sporadic shutdowns. Failure prone processing is a long pursued research subject in the chemical engineering. The effect of equipment failure can be mitigated by online scheduling,3 reactive scheduling,4 and combined maintenance and production scheduling.5−8 However, the problem of dealing with uncertainty is not simply a topic of academic research; it is a current and significant problem in the real world. With a view to creating a practical field technique to estimate process reliability, a method based on failure modes and effects analysis (FMEA) was introduced.9 This technique is composed of a unique definition of uncertainty and a set of simple equations. The fact that this technique effectively treats uncertainty in real problems without depending on advanced mathematical formalisms and with negligible computation suggests that it could represent a new research direction. A pioneering attempt has been conducted for the design of intermediate storage to connect batch input and continuous output where the batch size is subject to uncertainty.10 In the present study, we introduce a novel optimization model resulting in simple analytical solutions with negligible computational burden. In this study, a complex supply chain network is modeled as a batch-storage network that includes most supply chain components, for example, raw material purchasing, production, and finished product demand. The production process units in the network are all connected to inventory units. Each production process consumes materials from feedstock inventory units and supplies materials to product inventory units according to a given fixed feedstock recipe and product yield. The inventory units in the network are all connected to process units: some that supply the material to the connected inventory unit, and some that consume materials from the connected inventory unit. Each inventory unit stores one type of material. If there is no material flow between a process unit and an inventory unit, the corresponding feedstock recipe or product yield is zero. Raw material purchasing and finished product demand are considered to be special process units and are connected to the corresponding inventory units. There is no discrimination between upstream and downstream stages, so multistage reverse material flows can be accounted for naturally as part of the network. The material flow pattern between a process unit and an inventory unit is described with a periodic square wave (PSW) model that resembles the flow patterns of economic production quantity (EPQ) models. The PSW model has been successfully used to obtain analytic solutions for various systems; it was first introduced to optimize the lot sizes of processes and the volumes of inventory units in a single stage parallel system composed of multiple feedstock inventory units, multiple production processes, and multiple product inventory units.11 The plant structure in the model was then expanded to a multistage batch-storage network without reverse material flows, with multiple parallel inventory units and processes arranged sequentially.12 Reverse material flows have since been 7117

dx.doi.org/10.1021/ie302404g | Ind. Eng. Chem. Res. 2013, 52, 7116−7126


Article

the start-up time of the feedstock stream and the start-up time of the product stream,

solutions also provide good insights into optimality; such insights are nontrivial when numerical solutions must be computed. For example, the optimal lot size is proportional to the square roots of the setup cost and the average flow rate. In this study, we modified the previous PSW model so as to account for sporadic shutdowns. We would not use an ordinary probabilistic analysis of random failures; instead, we will use a judicious graphical method to find the upper and lower bounds of the flows under sporadic shutdowns. In particular, we focus on obtaining a compact set of analytical solutions for systems with minimal shutdown information. The key point of our analysis is to find upper/lower bounds and the average of the inventory level: the upper bound of the inventory level is used to compute the storage size, the lower bound of the inventory level is used for the no-depletion constraint of the optimization model, and the average inventory level is used to compute the inventory holding cost. The upper/lower bounds and the average of the inventory level can be calculated if those of each flow connected to the storage unit are known. The upper/lower bounds of each flow are computed from the graphical analysis of two extreme flow cases. The overall optimization formulation procedures resemble those of the previous study1,2 except in the determination of the upper and lower bounds. For simplicity, we would not consider transportation processes and currency flows with costs such as those of transportation setup cost, transfer price, corporate income tax, customs duty, interest rate, or exchange rate. The approach of this study can easily be extended to cover such aspects of multinational supply chain networks by following those of previous studies.18,19

ti′ = ′ti + Δti( ·)

(1)

where Δti(·) is a function of arbitrary variables except startup times. The above definitions of the variables and parameters hold not only for batch production flows but also for raw material purchasing and finished product shipping flows. For convenience of presentation, variables without superscripts or subscripts, B, ω, γ, d, and x are used here to represent the batch size, cycle time, batch frequency, shutdown duration, and storage operation time fraction, respectively, of raw material purchase, production, and finished product demand. We now define the long cycle time ω̃ and the long batch size B̃ , where ω̃ and B̃ represent ω̃ jk, ω̃ i, ω̃ jm, and B̃ jk, B̃ i, B̃ jm, respectively. The exact definitions of ω̃ and B̃ are given below. Let Di be the average material flow rate for process i, which is the long batch size B̃ i divided by the long cycle time ω̃ i. The average material flows of raw material purchases from suppliers and of finished product shipping to consumers are denoted by Djk and Djm, respectively, where Djk = B̃ jk/ω̃ jk and Djm = B̃ jm/ω̃ jm. The overall material balance with respect to storage results in the following relationships: |I |

|K (j)|

|I |

|M(j)|

∑ gi jDi + ∑ Dkj = ∑ fi j Di + ∑ i=1

k=1

i=1

Dmj

m=1

(2)

Figure 2 shows typical flows with sporadic shutdown F(t). Sporadic shutdowns have random characteristics. The random

2. DEFINITIONS OF PARAMETERS AND VARIABLES We use the definitions of parameters and variables proposed by previous work.1,2 A chemical plant that converts raw materials into final products through multiple physicochemical processing steps is composed of a set of storage units (J) and a set of batch processes (I). Each process requires multiple feedstock materials with fixed compositions (f ji) and produces multiple products with fixed product yields (gji). Note that the storage index j is a superscript and that the process index i is a subscript. Storage is associated with four types of material movement: purchases from suppliers (k ∈ K(j)), shipping to consumers (m ∈ M(j)), feeds to processes, and production from processes. Each material flow from process to storage (or from storage to process) is expressed by using the periodic square wave model with sporadic shutdown, as shown in Figure 1b. Each process generates a batch of product during every cycle time ωi and after γi cycles, a shutdown of duration di will occur repeatedly, where γi and di are random variables. The cycle time of a production unit is composed of the feedstock feeding time (′xiωi), the processing time ([1 − x′i − ′xi]ωi), and the product discharging time(x′i ωi), where 0 ≤ x′i , ′xi ≤ 1 are the storage operation time fractions. Note that a back prime on a variable indicates that the variable represents the feeding flow to a process and a prime on a variable indicates that the variable represents the discharging flow from a process. Processing is initiated at the start-up time ′ti (or ti′). Therefore, the material flow representation of the PSW model with sporadic shutdown is composed of six variables: the batch size Bi, the cycle time ωi, the storage operation time fraction ′xi (or x′i ), the start-up time ′ti (or t′i ), the shutdown duration di, and the batch frequency γi (≥2). The following expression describes the timing relationship between

Figure 2. Definition of uncertainty in sporadic shutdown.

properties of sporadic shutdowns are characterized in our approach with two random variables γ(l) and d(l), as shown in Figure 2b, where the subscript (l) represents the sequence of occurrence. It is not necessary to know the exact distribution functions of γ(l) and d(l). We assume that γ(l) and d(l) have symmetrical distribution functions with γ ̲ ≤ γ(I) ≤ γ ̿ and d ̲ ≤ d(I) ≤ d̿, i.e., that the maximum and minimum values of the random variables are already known. The mean values of γ(l) and d(l) are 0.5(γ ̲ + γ)̿ and 0.5(d ̲ + d̿), respectively. Note that γ ̲ , γ,̿ and 0.5(γ ̲ + γ)̿ all have integer values. For 0.5(γ ̲ + γ)̿ to be an integer, both γ ̲ and γ ̿ must be odd or even numbers. Suppose that γ(l) and d(l) have identical independent distribution functions with respect to (l). For given convergence limits 0 < ε1, ε2 ≪ 1 and confidence levels 0 < δ1, δ2 ≪ 1, the weak law of large numbers says that there exists an integer η such that P{|(1/η)∑ηl=1γ(l) − η 0.5(γ ̲ + γ)| ̿ < ε1} ≥ 1 − δ1 and P{|(1/η)∑l=1d(l) − 0.5(d ̲ + d̿)| < 7118



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ε2} ≥ 1 − δ2. From Chebycheff’s inequality, η ≥ (Var(γ(l))/ δ1ε12) and η ≥ (Var(d(l))/δ2ε22), that is, η = max{int[Var(γ(l))/ δ1ε12], int[Var(d(l))/δ1ε12]} + 1 if the least integer is chosen.1 The term η is known as the occurrence number and should be an even number so that 0.5η has an integer value. The time interval during which η number of shutdowns occur is defined as the long cycle time ω̃ . The sample means of γ(l) and d(l) converge to their mean values over a long cycle time according to the weak law of large numbers, so the total number of batches and the total shutdown duration during a long cycle time are 0.5(γ ̲ + γ)η ̿ and 0.5(d ̲ + d̿)η, respectively. Thus, ω̃ = 0.5η⌊(γ ̿ + γ ̲ )ω + d̿ + d ̲ ⌋ and B̃ = 0.5η(γ ̿ + γ ̲ )B. Here, the long cycle time means the shortest period within which all random effects diminish with a given confidence level. We note that a flow with sporadic shutdowns has a constant average flow rate D = {[(γ ̿ + γ ̲ )B]/[(γ ̿ + γ ̲ )ω + d̿ + d ̲ ]} over the long cycle time. This expression means that in spite of the randomness of the shutdowns, the total quantity processed during a long cycle time is a constant. In order to specify the formulation of the optimization, we need the upper/lower bounds and the average inventory level of the storage units under sporadic shutdowns. The upper bound of the inventory level is used to compute the storage size, the lower bound of the inventory level is used in the optimization constraint that ensures the inventory level is always nonnegative, and the average inventory level is used to compute the inventory holding cost. There are two extreme cases of a flow with sporadic shutdowns: (a) the upper bound case and (b) the lower bound case, as shown in Figure 3. In the upper bound

Figure 4. Cumulative flow functions for two extreme cases.

D[t + (1 − x)ω + d + Ω+ + Ξ] ≡

∫0

∫0

F (t ) d t

t

≥ ≥

t

∫0

F (t ) d t t

F (t ) d t

≡ D[t − d − Ξ]

(3)

where Ω ≡ [(γ ̲ d̿ − γd̿ ̲ )/(γ ̿ + γ ̲ )], Ξ ≡ 0.5{[η(d̿γ ̿ − d ̲ γ ̲ )]/(γ ̿ + γ ̲ )], and Ω+ ≡ max{0, Ω}. The upper bound of the inventory level, V j , is computed by adding the upper bounds of all incoming flow integrals and subtracting the lower bounds of all outgoing flow integrals from the initial inventory. The lower bound of the inventory level, V j , is computed by adding the lower bounds of all incoming flow integrals and subtracting the upper bounds of all outgoing flow integrals from the initial inventory. The incoming flows are raw material purchase flows and product discharging flows from production processes. The outgoing flows are feed flows to production processes and finished product demand flows. We can find the upper and lower bounds of the inventory level by using eq 3. |K (j)|

V j = V j(0) +

Figure 3. Two extreme cases of sporadic shutdown.

|K (j)|

∑ (1 − xkj)Dkjωkj − ∑ Dkjtkj k=1

|K (j)|

case, there are 0.5η occurrences of minimum shutdown duration d ̲ with maximum batch frequency γ ̿ and 0.5η occurrences of maximum shutdown duration d̿ with minimum batch frequency γ ̲ within repeated long cycle times. In the lower bound case, there are 0.5η occurrences of maximum shutdown duration d̿ with minimum batch frequency γ ̲ and 0.5η occurrences of minimum shutdown duration d ̲ with maximum batch frequency γ ̿ within repeated long cycle times. Figure 4 shows the cumulative flow functions of the two cases. The dotted lines are the upper and lower bounds of the two extreme cases. The three large arrows indicate contacting points. All integrals of flows with sporadic shutdowns ∫ t0F(t) dt exist between the dotted lines, whose equations are given in Figure 4:

+

∑ Dkjd ̲ kj + ∑ (Ωkj)+ Dkj + ∑ ΞkjDkj k=1 |I |

+

k=1

∑ (1 − xi′)gi jDiωi − ∑ gi jDi[′ti + Δti(. )] i=1 |I | i=1 |I |

∑ m=1

7119

|I |

i=1 |M(j)|

∑ fi j Did ̲ i + ∑ Ξifi j Di + ∑ i=1 |M(j)|

+

|I |

∑ gi jDid ̲ i + ∑ Ω+i gi jDi + ∑ Ξigi jDi + ∑ fi j Di′ti i=1 |I |

+

k=1

|I |

i=1 |I |

+

k=1 |K (j)|

|K (j)|

i=1

ΞmjDmj

m=1

i=1 |M(j)|

Dmjtmj +

∑ m=1

Dmjd ̲ mj

(4)


Industrial & Engineering Chemistry Research |K (j)|

V i = V j(0) −

|K (j)|

|K (j)|

k=1

k=1

k=1

|I | i=1

i=1

i=1

|I |

|M(j)|

∑ Ω+i fi j Di − ∑ Ξifi j Di − ∑ i=1 |M(j)|

Dmjtmj −

∑ m=1 |M(j)|

−

|I |

∑ (1 − ′x i)fi j Diωi + ∑ fi j Di′ti − ∑ fi j Did ̲ i

i=1 |M(j)|

+

i=1

|I |

i=1 |I |

−

|I |

∑ gi jDi[′ti + Δti(. )] − ∑ gi jDid ̲ i − ∑ Ξigi jDi i=1 |I |

−

capital cost per capacity of the purchasing facility for raw material j, ai ($/(y L)) is the annual capital cost per capacity of process i, and bj ($/(y L)) is the annual capital cost per capacity of storage unit j. We then attempt to minimize the objective function for the design of the batch-storage network, which is the annualized expectation of the total cost, and consists of the setup cost of the processes, the inventory holding cost of the storage units, and the capital cost of the processes and storage units for a given minimum/maximum shutdown duration, minimum/maximum batch frequency, and occurrence number over a long cycle time for each process:

∑ Dkjtkj − ∑ Dkjd ̲ kj − ∑ ΞkjDkj

|I |

−

Article

∑ m=1

(1 − xmin)Dmjωmj

m=1 |M(j)|

Dmjd ̲ mj −

∑

|J | |K (j)|

(Ωmj)+ Dmj

|I |

+ (5)

m=1

t

F(t ) dt = D[t + 0.5(1 − x)ω + 0.5Ω+]

|K (j)|

∑

k=1 |K (j)|

+ 0.5

(1 − xkj) j j Dk ωk − 2 |I |

∑ (Ωkj)+ Dkj + ∑ k=1

i=1

−

(6)

−

∑ Dkjtkj

∑ ∑

(1 − 2

xmj)

Dmjωmj

m=1 |M(j)|

− 0.5

∑ m=1

(Ωmj)+ Dmj

+

k=1

(8)

Akj Dkj Ψkj

j

−

dk̿ + d ̲ kj (γk̿ j + γ ̲ j ) k

Bkj =

,

di̿ + d ̲ i Ai , − Di Ψi (γi ̿ + γ ̲ )

B=

Ai Di Ψi

Akj Dkj Ψkj

(9)

(10)

where ⎛ Hj ⎞ Ψkj = ⎜ + b j⎟(1 − xkj) + akj ⎝2 ⎠

(11)

|J | ⎛ Hj ⎞ Ψi = ai + (1 − ′x i) ∑ ⎜ + b j ⎟f i j ⎠ j=1 ⎝ 2

|I |

i=1 |M(j)|

∑ [H jV j + b jV j ]

i

∑ fi j Di′ti − 0.5 ∑ Ω+i fi j Di ∑

=

ωi =

i=1

i=1 |M(j)|

−

ωkj

|K (j)|

|I |

ωĩ

V j are obtained from extensions of eqs 7 and 4. The constraints on optimization are that there must be no depletion of any storage unit, i. e., 0 ≤ V j , where V j is given by eq 5. Table 1 summarizes the optimization problem for sporadic shutdowns. The solution procedure for the Kuhn−Tucker conditions is given in the Appendix. The optimal cycle times are

∑ gi jDi[′ti + Δti(. )] + 0.5 ∑ Ω+i gi jDi (1 − ′x i) j fi Diωi + 2

⎢⎣

⎤ + aiBi ⎥ ⎥⎦

where B = D{⌊(γ ̿ + γ ̲ )ω + d̿ + d ̲ ⌋/(γ ̿ + γ ̲ )}. Note that V j and

|I |

i=1 |I |

i

j=1

(1 − xi′) j gi Diωi 2

|I |

⎡ 0.5η (γ ̿ + γ ̲ )Ai i i

⎤ + akjBkj ⎥ ⎥ ⎦

|J |

+

The average inventory level of a storage unit is computed by adding the average of all incoming flow integrals and subtracting the average of all outgoing flow integrals from the initial inventory, where the average of the flow integral is given by eq 6. V j = V j(0) +

ω̃ kj

⎣

∑⎢ i=1

The average inventory level is highly dependent on the random properties of failures. The exact value of the average inventory level cannot be obtained without defining the probability distribution function of all random variables, which is a nontrivial task. In this study, we took an intuitive approach to this task. The average of ∫ t0F(t) dt was selected as the line equidistant from the upper and lower bounds.1

∫0

⎡ 0.5η j(γ j + γ ̲ j )A j k k̿ k k

∑ ∑ ⎢⎢ j=1 k=1

m=1

ΞmjDmj

∑

TC =

i=1

|J | ⎛ Hj ⎞ + (1 − xi′) ∑ ⎜ + b j ⎟g i j ⎠ j=1 ⎝ 2

Dmjtmj

m=1

(12)

ωjk,

Note that, because of the conditions ωi ≥ 0, ≥ [(d̿jk + d ̲ jk)/(γjk̿ + γ ̲ jk)] and (Ai/ΨiDi)1/2 ≥ [(d̿i + d ̲ i)/ (γi̿ + γ ̲ i)]. The objective function in eq 8 is convex with respect to ωjk and ωi, so the bounds nearest to eqs 9 and 10, respectively, can be optimum points if eqs 9 and 10 are out of bounds of variables. If there are no shutdowns, that is, d̿jk + d ̲ jk = d̿i + d ̲ i = 0, the optimal cycle times return to those of the case of no shutdowns5 which can be solved independently. Note also that the occurrence number η does not influence the optimal cycle times. The optimal batch sizes for the case of sporadic shutdowns are the same as those for the case of no shutdown. The optimal objective value is

(7)

3. OPTIMIZATION MODEL The purchasing setup cost of a raw material j is denoted by Ajk $/order, and the setup cost of process i is denoted by Ai $/batch. Note that these setup costs include shutdown and failure maintenance costs. The annual inventory holding cost of storage j is denoted by Hj $/(L y). We assume that the capital cost is proportional to the process capacity in order to obtain an analytical solution. Suppose that ajk ($/(y L)) is the annual 7120

(Ajk/ΨjkDjk)1/2


Industrial & Engineering Chemistry Research |J | |K (j)|

*TC(Dkj , Di) = 2 ∑

j=1 k=1 j

|J | |K (j)|

∑∑

−

Dmjωmj(1

∑

−

xmj)

+

j

(Ψkj − akj)(dk̿ + d ̲ kj)Dkj γk̿ j

+ γ̲

γi ̿ + γ ̲

i=1

∑ m=1 |J |

+

⎛

+

+

∑

⎛

(Ωmj)+ Dmj)

(13)

j

(14)

The optimal startup times are derived from the equality of the constraint 0 ≤ V j , where V j is given by eq 5. |I |

∑ Dkjtkj + ∑ (gi j − fi j )Di′ti k=1

i=1 |M(j)|

= V j(0) −

∑ m=1

|M(j)|

−

∑ m=1 |I |

−

∑ (f i j i=1 |I |

−

−

∑ ΞkjDkj − ∑ (1 − ′x i)fi j Diωi k=1

i=1

+ gi j)Did ̲ i − j

j

+ gi )Di −

∑ gi jDiΔti(. ) i=1 |K (j)|

∑ k=1

|M(j)|

Dkjd ̲ kj

−

∑

ΞmjDmj

m=1

|M(j)|

∑ Ω+i fi j Di − ∑ i=1

|I |

|I |

∑ Ξ i (f i i=1 |I |

Dmjtmj

m=1 |K (j)|

Dmjd ̲ mj −

∑

m=1

(Ωmj)+ Dmj

(Ωmj)+ Dmj

m=1

(16)

(17)

4. PLANT DESIGN EXAMPLES Consider a plant that produces three finished products from four raw materials, as shown in Figure 5. Figure 5 shows most of the input data for computation. According to the above procedure, the average material flow rates for all the storage and process units should be calculated first. We used the values of the average flow rates obtained in the previous study13 for comparison. We assumed that the mean shutdown durations are 20% of the total operating time (availability = 0.8) for all flows. The minimum shutdown durations were assumed to constitute 10% of the total operating time. The maximum shutdown duration was assumed to constitute 30% of the total operating time. The mean batch frequencies were assumed to be 5, the minimum batch frequencies were assumed to be 3, and the maximum batch frequencies were assumed to be 7. Then, by using the lot sizing and storage sizing equations obtained in this and previous studies,13,1 the optimal batch sizes, optimal cycle times, and optimal storage sizes were

|M(j)|

(1 − xmin)Dmjωmj +

∑

The last term of eq 17 indicates that the storage size increases linearly with the occurrence number and with the gap between the maximum and the minimum in the shutdown duration times the batch frequency. In the derivation of the above optimal solutions, the average flow rates Djk and Di are assumed to be constant; however, they usually do vary. The optimal average flow rates can be calculated by solving another optimization problem with an objective function given by eq 13 and a constraint given by eq 2; this is the second-level optimization problem and is summarized in Table 2. The global optimality of this decomposition of the original optimization problem is proved in Appendix A of the previous paper.13 The nonlinear objective function in eq 13 is a separable concave function and can be linearized in piecewise manner by using the specially ordered sets (SOS) formulation.14 The overall computation procedure is the same as that used in the previous study.13 First, the optimal average flow rates are obtained by solving the second-level problem in Table 2. Then, the analytical solutions for the cycle times, batch sizes, storage sizes, and startup times can be calculated by using eqs 9, 10, and 16.

j

⎞ (Ψi − ai)(di̿ + d ̲ i)Di + b j⎟Ω+i fi j Di − γi ̿ + γ ̲ ⎠ j=1 ⎝ 2 i

|K (j)|

+ gi j)Di +

+ 2(fi j + gi j)Di(Ξi + d ̲ i)

d ̲ mj)Dmj)

|J | ⎛ Hj ⎞ 2 Ai ΨiDi + 2 ∑ ⎜ + b j⎟(Ξi + d ̲ i)(fi j + gi j)Di ⎠ j=1 ⎝ 2 |J |

k=1

[(1 − ′x i)fi j + (1 − xi′)gi j]Diωi + Ω+i (fi j + gi j)Di

The contribution of process i to the objective values is

∑ ⎜H

∑ Ω+i (fi j

∑ (Ωkj)+ Dkj

The contribution of process i to storage j is

+ gi j)Di

m=1

+

k=1

|M(j)|

i=1

|M(j)|

+

∑ Dkj(Ξkj + d ̲ kj) |K (j)|

j

i=1 |I |

i

⎞ |I | + b j⎟(∑ Ω+i fi j Di ⎠ i=1 j=1 ⎝ 2

∑ ⎜H

Dmj(Ξmj + d ̲ mj) + 2

+ 2 ∑ (fi + gi )Di(Ξi + d ̲ i) +

i=1 |M(j)|

+

(1 − xmj)Dmjωmj

m=1 |K (j)|

j

|I |

(Ξmj

∑ m=1 |I |

k

⎛ Hj ⎞ |K (j)| + 2∑⎜ + b j⎟( ∑ (Ξkj + d ̲ kj)Dkj ⎠ k=1 j=1 ⎝ 2

∑ (Ξi + d ̲ i)(fi j

|M(j)|

∑ (1 − xkj)Dkjωkj + ∑

+2

j

|J |

+

+ (1 − xi′)gi j]Diωi

k=1 |M(j)|

m=1

(Ψi − ai)(di̿ + d ̲ i)Di

∑

∑ [(1 − ′x i)fi j i=1 |K (j)|

|M(j)|

j=1 k=1 |I |

Vj =

i=1

⎛H ⎞ + ∑⎜ + b j⎟ ⎠ j=1 ⎝ 2 −

|I |

|I |

Akj ΨkjDkj + 2 ∑ Ai ΨiDi

∑

|J |

Article

(15)

The optimal storage sizes are derived from eqs 4 and 13 and are given by 7121



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Table 1. Optimization Problem and Solution of Kuhn−Tucker Conditions for Sporadic Shutdown objective function

⎡ 0.5η j(γ j + γ ̲ j )A j k k̿ k k

|J | |K (j)|

∑ ∑ ⎢⎢

TC =

ω̃ kj

⎣

j=1 k=1

⎤ + akjBkj ⎥ + ⎥ ⎦

⎡ 0.5η (γ ̿ + γ ̲ )Ai i i

|I |

∑⎢

i

⎢⎣

i=1

ωĩ

⎤ + aiBi ⎥ ⎥⎦

|J |

+

∑ [H jV j + b jV j ] j=1

design variables constraints

ωkj ,

ωi , tkj , ′ti ,

|I |

Dkj , Di

|K (j)|

|M(j)|

|I |

∑ gi jDi + ∑ Dkj = ∑ fi j Di + ∑ i=1

k=1

solution of Kuhn−Tucker conditions with fixed Djk, Di

j

Akj

ωkj =

Dkj Ψkj

|K (j)|

Dmj , V j ≥ 0

m=1

i=1

−

dk̿ + d ̲ kj (γk̿ j

+ γ̲ )

di̿ + d ̲ i Ai − (γi ̿ + γ ̲ ) Di Ψi

ωi =

,

j

i

k

|I |

∑ Dkjtkj + ∑ (gi j − fi j )Di′ti k=1

i=1 |M(j)|

= V j(0) −

∑

|M(j)|


m=1 |I |

−

∑ (1 − ′xi)fi j Diωi − ∑ (fi j i=1 |K (j)|

∑ Ξi(fi j

+ gi j)Di −

i=1 |M(j)|

−

∑

|M(j)|

Dmjtmj −

∑

|K (j)|

Dmjd ̲ mj −

m=1

m=1 |I |

+ gi j)Did ̲ i −

∑ gi jDiΔti(·)

|I |

i=1 |I |

−

∑

|I |

ΞmjDmj −

m=1

k=1

k=1

i=1

|M(j)|

∑ Dkjd ̲ kj − ∑

∑ ΞkjDkj

∑ Ω+i fi j Di i=1

(Ωmj)+ Dmj

m=1 |I |

|J | |K (j)|

*TC(Dkj , Di) = 2 ∑


∑

i=1

j=1 k=1

⎛ Hj ⎞ |M(j)| + ∑⎜ + b j⎟ ∑ Dmjωmj(1 − xmj) ⎠ m=1 j=1 ⎝ 2 |J |

|J | ⎛ Hj ⎞ |K (j)| + 2∑⎜ + b j⎟( ∑ (Ξkj + d ̲ kj)Dkj ⎠ k=1 j=1 ⎝ 2 |I |

+

i=1 |J |

+

|M(j)|

∑ (Ξi + d ̲ i)(fi j ⎛

∑∑ j=1 k=1 |I |

−

∑

∑ i=1

(Ξmj + d ̲ mj)Dmj)

m=1 |M(j)|

⎞ |I | + b j⎟(∑ Ω+i fi j Di + ⎠ i=1 j=1 ⎝ 2

∑ ⎜H

j

|J | |K (j)|

−

+ gi j)Di +

∑

(Ωmj)+ Dmj)

m=1

j

(Ψkj − akj)(dk̿ + d ̲ kj)Dkj γk̿ j + γ ̲ j

k

(Ψi − ai)(di̿ + d ̲ i)Di γi ̿ + γ ̲

i

This result indicates that uncertainty in operating time significantly increases storage capacity. This storage requirement arises partly because the constraint 0 ≤ V j in the case of sporadic shutdowns demands a worst-case scenario approach and pessimistic solutions are obtained. The optimal storage size difference between frequent and sporadic shutdown mainly originates from a modeling difference: shutdown duration is assumed to be proportional to cycle time (d = [1 − α]ηω̅ ) in frequent shutdown which comes from the constant availability assumption, whereas shutdown duration is independent of cycle time in sporadic shutdown. In another words, the impact of uncertainty is absorbed by both batch and storage size for frequent shutdown whereas the impact of uncertainty is absorbed by only storage size for sporadic shutdown. These simulation results indicate that the proper selection of the

calculated. The results are summarized in Tables 3, 4, and 5, respectively. The cases compared are (i) no shutdowns,13 (ii) frequent shutdowns (all batch cycles undergo shutdowns with the specified availability1), and (iii) sporadic shutdowns. The objective values computed were (i) $2,285,557/y, (ii) $3,543,574/y, and (iii) $3,278,786/y. Note that cases ii and iii have the same average shutdown durations with availability = 0.8. Note that the optimal batch sizes of the no shutdown and sporadic shutdown cases are the same. The optimal batch size for the case of frequent shutdowns is smaller than those of the other cases. The optimal cycle time for the case of frequent shutdowns is smaller than those of the other cases, whereas the optimal cycle time of the case of no shutdowns is larger than those of the other cases. The optimal storage size for the case of sporadic shutdowns is much larger than those of other cases. 7122



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Table 2. Second Level Optimization Problem for Sporadic Shutdown objective function

|I |

|J | |K (j)|

*TC(Dkj , Di) = 2 ∑


∑

i=1

j=1 k=1

⎛

|J |

+

∑ ⎜H j=1

j

⎝2

j

∑∑

γk̿ j + γ ̲ j k

j=1 k=1 |I |

−

Dmjωmj(1 − xmj)

∑ m=1

(Ψkj − akj)(dk̿ + d ̲ kj)Dkj

|J | |K (j)|

−

⎞ + b j⎟ ⎠

|M(j)|

(Ψi − ai)(di̿ + d ̲ i)Di

∑

γi ̿ + γ ̲

i=1

i

⎞ |K (j)| ⎛H + 2∑⎜ + b j⎟( ∑ (Ξkj + d ̲ kj)Dkj ⎠ k=1 j=1 ⎝ 2 |J |

j

|I |

+

i=1 |J |

+ design variables constraints

|M(j)|

∑ (Ξi + d ̲ i)(fi j ⎛

+ gi j)Di +

∑

⎞ |I | + b j⎟(∑ Ω+i fi j Di + ⎠ i=1 j=1 ⎝ 2

∑ ⎜H

(Ξmj + d ̲ mj)Dmj)

m=1 |M(j)|

j

∑

(Ωmj)+ Dmj)

m=1

Dkj , Di |I |

|K (j)|

|I |

|M(j)|

∑ gi jDi + ∑ Dkj = ∑ fi j Di + ∑ i=1

k=1

i=1

Dmj

m=1

Figure 5. Example plant designinput data.

optimization model is very important in design problems with uncertain shutdowns.

shutdowns. This approach is useful for the optimal design of multiproduct, multistage production and inventory systems with uncertainties in operating time. The proposed modified uncertain PSW model with shutdowns can be used to represent the material flows between processes and storage units when infrequent operating time losses can arise. This approach minimizes the objective function for plant design, which

5. CONCLUSION This article discusses the determination of the optimal sizes of the batch processes and storage units interconnected in a network structure when the processes undergo sporadic 7123



Article

Table 3. Optimal Batch Process Sizes (L) process I1 I2 I3 I4 I5 I6

no shutdown

a

frequent shutdown

112.93 335.74 238.54 136.31 170.66 64.41

b

Table 5. Optimal Storage Sizes (L) sporadic shutdown

86.78 260.17 119.04 68.02 84.35 31.65

c

storage

112.93 335.74 238.54 136.31 170.66 64.41

no shutdowna

J1 J2 J3 J4 J5 J6 J7 J8 J9

Equation 12 and Bi = Diωi in previous work.5 bEquation 29 in previous work.1 cEquation 10.

a

frequent shutdownb

483.28 704.19 563.47 404.57 840.11 241.06 337.36 170.66 64.41

sporadic shutdownc

3067.24 4762.72 4121.95 3009.23 10616.96 3264.03 3501.69 1754.4 658.24

5219.46 7605.29 6085.47 4369.38 9073.14 2603.46 3643.49 1843.17 695.64

a c

Table 4. Optimal Cycle Times (h) process I1 I2 I3 I4 I5 I6

no shutdowna

frequent shutdownb

4.1 6.91 6.4 11.94 8.9 23.51

sporadic shutdownc

2.52 4.28 2.55 4.77 3.52 9.24

Analytical equations are very useful for quick size estimates and fault diagnosis. Our simulation results demonstrated that the selection of the optimization model is very important in design problems with uncertain operating time losses. The method used in this study is applicable to multitasking semicontinuous processes as well as to batch processes.14 This new version of the PSW model enhances the analysis of the optimal design and operation of failure prone processing networks.

3.28 5.53 5.12 9.55 7.12 18.81

a c

Equation 12 in previous work.5 bEquation 29 in previous work.1 Equation 10.

■

accounts for the sum of the process setup costs, the inventory holding costs, and the capital costs of constructing the process and storage units. The constraints on optimization include the stipulation that there must be no depletion of any material inventory in storage and that the finished product demand must be met. The solution of the Kuhn−Tucker conditions of optimality provides analytical lot and storage sizing equations.

|J | |K (j)|

L(ωkj , ωi , tkj , ′ti) =

∑∑ j=1 k=1

Equation 16 in previous work.5 bEquation 34 in previous work.1 Equation 14.

APPENDIX: KUHN−TUCKER SOLUTION TO THE OPTIMIZATION PROBLEM IN STOCHASTIC ANALYSIS FOR SPORADIC SHUTDOWN The Lagrangian for the optimization problem to minimize eq 13 subject to 0 ≤ V j , where V j is given from eq 5 with respect to ωjk, ωi, tjk, ′ti is

j ⎡ 0.5η j(γ j + γ j )A j [(γk̿ j + γ ̲ j )ωkj + dk̿ + d ̲ kj] ⎤⎥ ̲k k k k̿ ⎢ k + akjDkj j ⎢ ⎥ γk̿ j + γ ̲ j ) ( ω ̃ k ⎢⎣ ⎥⎦ k

⎡ 0.5η (γ + γ ̲ )A [(γi ̿ + γ ̲ )ωi + di̿ + d ̲ i] ⎤ i i i̿ i i ⎢ ⎥+ + aiDi +∑ ⎢ ⎥ ωĩ (γi ̿ + γ ̲ ) i=1 ⎣ ⎦ i |I |

|J |

|J | |K (j)|

−

−

|I |

|I |

∑ (H j + b j) ∑ gi jDiΔti i=1

|K (j)|

|I |

j

|K (j)|

|K (j)|

|I |

|I |

|I |

∑ λ j [V j(0) − ∑ Dkjtkj − ∑ Dkjd ̲ kj − ∑ ΞkjDkj − ∑ gi jDi[′ti + Δti(. )] − ∑ gi jDid ̲ i − ∑ Ξigi jDi k=1

|I |

k=1

|I |

k=1

i=1

|I |

|I |

i=1

|I |

∑ m=1

i=1

|M(j)|

Dmjtmj −

∑ m=1

i=1

i=1 |M(j)|

|M(j)|

Dmjd ̲ mj −

∑

(Ωmj)+ Dmj −

m=1

∑ m=1

7124

ΞmjDmj]

i=1

i=1

|M(j)|

∑ (1 − ′x i)fi j Diωi + ∑ fi j Di′ti − ∑ fi j Did ̲ i − ∑ Ω+i fi j Di − ∑ Ξifi j Di − ∑ i=1 |M(j)|

+

⎞ + b j⎟(1 − xkj)Dkjωkj ⎝2 ⎠ j

⎞ + b j⎟ ∑ (1 − xi′)gi jDiωi + constants ⎠ i=1 j=1 ⎝ 2

2

j=1

−

− gi j)Di′ti −

j=1

⎛

|J |

j

|J |

−

|J |

∑ ∑ H (1 − ′x i)fi j Diωi + ∑ ⎜ H j=1 i=1

j=1 k=1

j=1 i=1

j=1 k=1

⎛

∑ ∑ ⎜H

|I |

∑ ∑ (H j + b j)Dkjtkj + ∑ ∑ (H j + b j)(fi j |J |

|J | |K (j)|

(1 − xmin)Dmjωmj

m=1

(A1)



Article

λ j = Hj + b j

where λ j is the Lagrange multiplier and the constants are |J |

|J |

j |M(j)|

∑ [H j + b j]V j(0) − ∑ H ∑

constants =

j=1

+

∑ (H j + b j) ∑

+

j=1

|K (j)|

j=1

k=1

|J |

|K (j)|

|J |

m=1

i=1

*E-mail: [email protected]. Tel.: +822-2260-3796. Fax: +822-2260-8898. Notes

|M(j)|

+ gi j)Di + b j

i=1

∑

The authors declare no competing financial interest.

ΞmjDmj

■ ■

m=1

ACKNOWLEDGMENTS This work was supported by the Pukyong National University Research Fund in 2011(C-D-2011-0197).

⎞ |K(j)| ⎛ Hj ⎞ |I | + b j⎟ ∑ (Ωkj)+ Dkj + ⎜ + b j⎟ ∑ Ω+i gi jDi ⎠ k=1 ⎝2 ⎠ i=1 j=1 ⎝ 2 |J |

∑ ⎜H

j

|I |

|M(j)|

− 0.5H j ∑ Ω+i fi j Di − 0.5H j i=1

(Ωmj)+ Dmj

∑ m=1

Kuhn−Tucker conditions give

i = batch process index k = raw material vendor index (l) = The sequence of failure occurrence m = finished product customer index

(A2)

[0.5ηkj(γk̿ j + γ ̲ j )]2 Akj ∂L k =− j ∂ωkj (0.5ηkj[(γk̿ j + γ ̲ j )ωkj + dk̿ + d ̲ kj])2

Superscript

j = storage index

k

Normal Letters

⎡⎛ H j ⎤ ⎞ + ⎢⎜ + b j⎟(1 − xkj) + akj ⎥Dkj = 0 ⎠ ⎣⎝ 2 ⎦ |J |

ajk = annualized capital cost of raw material purchasing facility, $/(L y), parameter ai = annualized capital cost of unit i, $/(L y), parameter bj = annualized capital cost of storage facility, $/(L y), parameter Ajk = ordering cost of feedstock materials, $/order, parameter Ai = ordering cost of noncontinuous units, $/order, parameter B = either of Bjk, Bi, and Bjm, variable or parameter Bjk = raw material order size, L/order, variable Bi = batch unit size, L/order, variable Bjm = final product delivery size, L/order, parameter B̃ jk = long order size, L/order, variable B̃ i = long batch size, L/order, variable B̃ jm = long final product delivery size, L/order, parameter d = either of djk, di, and djm, parameter d(l) = random operating time loss, y, parameter djk = periodic or sporadic shutdown duration for raw material purchasing process, y, parameter di = periodic or sporadic shutdown duration for process i, y, parameter djm = periodic or sporadic shutdown duration for finished product demand, y, parameter d̿ = either of d̿jk, d̿i, and d̿jm, parameter d̿jk = maximum operating time loss for raw material purchase, y, parameter d̿i = maximum operating time loss for process i, y, parameter d̿jm = maximum operating time loss for finished product demand, y, parameter d ̲ = either of d ̲ jk, d ̲ i and d ̲ jm, parameter d ̲ jk = minimum operating time loss for raw material purchase, y, parameter d ̲ i = minimum operating time loss for process i, y, parameter d ̲ jm = minimum operating time loss for finished product demand, y, parameter D = either of Djk, Djm, and Di, variable

(A3)

|J |

∑ (H

j

j

j

j

+ b )(fi − gi )Di −

∑ λ j (f i j

j=1

− gi j)Di = 0

j=1

(A4)

[0.5ηi(γi ̿ + γ ̲ )]2 Ai ∂L i =− + aiDi ∂ωi (0.5ηi[(γi ̿ + γ ̲ )ωi + di̿ + d ̲ i])2 i

|J |

−

⎞ + b j⎟(1 − xi′)gi jDi ⎠ j=1 ⎝ 2 |J |

j

2

|J |

−

|J |

⎡

∑ λ j ⎢(1 − ′x i)fij j=1

⎢⎣

|K (j)|

λ j [V j(0) −

+ gi jDi

|K (j)|

k=1

k=1

|I | i=1

i=1

|I |

|I |

|I |

∑ (1 − ′xi)fi j Diωi + ∑ fi j Di′ti − ∑ fi j Did ̲ i − ∑ Ω+i fi j Di i=1

|M(j)|

∑ Ξifi j Di − ∑ i=1 |M(j)|

−

|I |

∑ gi jDi[′ti + Δti(. )] − ∑ gi jDid ̲ i − ∑ Ξigi jDi i=1 |I |

−

(A5)

|K (j)|

k=1

i=1 |I |

−

∂Δti ⎤ ⎥Di = 0 ∂ ωi ⎥⎦

∑ Dkjtkj − ∑ Dkjd ̲ kj − ∑ ΞkjDkj

|I |

−

j

∂Δti ∂ ωi

∑ (H j + b j)gijDi j=1

+

⎛

∑ H (1 − ′x i)fij Di + ∑ ⎜ H j=1

∑ m=1

∑

i=1 |M(j)|


m=1 |M(j)|

Dmjd ̲ mj −

∑

i=1

Dmjtmj

m=1 |M(j)|

(Ωmj)+ Dmj −

m=1

NOMENCLATURE

Subscript

∂L = −(H j + b j)Dkj + λ j Dkj = 0 ∂tkj

∂L = ∂′ti

AUTHOR INFORMATION

Corresponding Author

+ gi j)Did ̲ i

|I |

∑ b j ∑ ΞkjDkj + b j ∑ Ξi(fi j ⎛

Dmjd ̲ mj

|I |

j=1

k=1

■

|M(j)|

∑ bj ∑

∑ b j ∑ Dkjd ̲ kj + ∑ b j ∑ (fi j

j=1

+

|J |

Dmj[tmj] +

m=1

|J |

Solving eqs A3 and A5 with eq A7 gives eqs 9 and 10 in the main text. Solving eq A6 gives eq 15 in the main text.

(1 − xmj)Dmjωmj

m=1

|M(j)|

j=1

+

2

j=1 |J |

(A7)

∑ m=1

ΞmjDmj] = 0

(A6)

Solving eqs A2 and A4 gives 7125



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ω̃ jm = long cycle time of finished product demand, y, parameter Ω = [(γ ̲ d̿ − γd̿ ̲ )/(γ ̿ + γ ̲ )] Ξ = 0.5{[η(d̿γ ̿ − d ̲ γ ̲ )]/(γ ̿ + γ ̲ )], y, parameter Ψi = aggregated cost defined by eq 11, parameter Ψjk = aggregated cost defined by eq 12, parameter

Djk = average material flow of raw material supply, L/y, variable Djm = average material flow of customer demand, L/y, parameter Di = average material flow through process i, L/y, variable f ji = feedstock composition of unit i, parameter F(t) = a deterministic material flow, L/y, variable F(t) = a material flow with random failures, L/y, variable gji = product yield of unit i, parameter Hj = annual inventory holding costs, $/(L y), parameter I = batch process set J = storage set K(j) = raw material supplier set for storage j M(j) = consumer set for storage j tjm = start-up time of customer demand, y, parameter ′ti = start-up time of feedstock feeding to process i, y, variable t′i = start-up time of product discharging from process i, y, variable tjk = start-up time of raw material purchasing, y, variable

Special Functions

■

|X| = number of elements in set X Ω+ = max{0, Ω} Var(·) = variance

REFERENCES

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V j = upper bound of inventory level, L, variable V j = lower bound of inventory level, L, variable Vj(t) = inventory level, L, variable Vj(0) = initial inventory level, L, parameter V j = time averaged inventory level, L, variable xjk = storage operation time fraction of purchasing raw materials, parameter ′xi = storage operation time fraction of feeding to process i, parameter x′i = storage operation time fraction of discharging from process i, parameter xjm = storage operation time fraction of finished product demand, parameter Greek Letters

γ = either of γjk, γi, and γi, parameter γ(l) = random batch frequency, parameter γjk = batch frequency between shutdowns for raw material purchase, parameter γi = batch frequency between shutdowns for process i, parameter γjm = batch frequency between shutdowns for finished product demand, parameter γ ̿ = either of γjk̿ , γi̿ , and γjm̿ , parameter γjk̿ = maximum batch frequency for raw material purchase, parameter γi̿ = maximum batch frequency for process i, parameter γmj̿ = maximum batch frequency for finished product demand, parameter γ ̲ = either of γ ̲ jk, γ ̲ i, and γ ̲ jm, parameter γ ̲ jk = minimum batch frequency for raw material purchase, parameter γ ̲ i = minimum batch frequency for process i, parameter γ ̲ mj = minimum batch frequency for finished product demand, parameter ω = either of ωjm, ωjk, and ωi, y, variable ωjm = cycle time of finished product demand, y, variable ωjk = cycle time of raw material purchasing, y, variable ωi = cycle time of process unit, y, variable ω̃ = either of ω̃ i, ω̃ jk and ω̃ jm, variable or parameter ω̃ i = long cycle time of process unit, y, variable ω̃ jk = long cycle time of raw material purchasing, y, variable 7126


Optimal Design of Batch-Storage Network under Sporadic Operating

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