Optimal Scheduling of Multiproduct Batch Processes for Various

4058

Ind. Eng. Chem. Res. 1996, 35, 4058-4066

PROCESS DESIGN AND CONTROL Optimal Scheduling of Multiproduct Batch Processes for Various Intermediate Storage Policies Minseok Kim,† Jae Hak Jung,‡ and In-Beum Lee*,† School of Environmental Engineering, Automation Research Center, Pohang University of Science and Technology, San 31, Hyoja-Dong, Pohang 790-784, Korea, and School of Chemical Engineering and Technology, Engineering College, Yeungnam University, Dae-Dong 214-1, Kyongsan 712-749, Korea

In this paper, we suggest optimal scheduling strategies for serial multiproduct batch processes to minimize the makespan as an objective function. The strategies are formulated as mixed-integer nonlinear programming (MINLP) for various storage policies under consideration of nonzero transfer and setup times. The major storage policies for multiproduct multiunit batch processes considered in this paper are unlimited intermediate storage (UIS), no intermediate storage (NIS), finite intermediate storage (FIS), zero wait (ZW), and mixed intermediate storage (MIS). The algorithms to calculate the process completion times are developed for each policy. Two examples are tested to evaluate the robustness of this strategy and reasonable computational times. Introduction As the life cycles of products are shorter and shorter, the requirement of producing high-value-added, lowvolume products is increased. Because batch or semicontinuous processes are suitable for these products, Chemical Process Industry (CPI) pays attention to multiproduct or multipurpose batch processes. In batch process operations, there are many interesting research fields to enhance productivity and operability. The multiproduct batch scheduling problem is one of them. Reklaitis (1982, 1991) and Ku et al. (1987) reviewed the scheduling problems for process operations. In this field, the main issues of study are the development of algorithms for the completion time and the optimal scheduling. Multiproduct serial batch scheduling problems have N (N > 1) products to be produced and M (M > 1) processing equipment (units) whose sequence is fixed for processing every product. The processing times for product i (i ) 1, 2, ..., N) at unit j (j ) 1, 2, ..., M) are usually given. The completion time algorithm is to calculate the completion time of each product i (i ) 1, 2, ..., n) on each unit j (j ) 1, 2, ..., M). The optimal scheduling is to find out the best sequence of production and the shortest total operation time for producing all products, the so-called makespan. Calculation of completion times and the optimal scheduling have been studied for a decade. Especially, serial multiproduct, multiunit batch processes have been studied for different operating types of intermediate storages. The different types of intermediate storage policies which have been frequently studied are unlimited intermediate storage (UIS), no intermediate storage (NIS), finite intermediate storage (FIS), zero wait (ZW), and mixed intermediate storage (MIS). To overcome the limitation of these storage policies, a shared storage system by Ku and Karini (1990) and CIS * To whom all correspondence should be addressed. Telephone: +82-562-279-2274. Fax: +82-562-279-2699. E-mail: [email protected]. † Pohang University of Science and Technology. ‡ Yeungnam University.

S0888-5885(96)00181-9 CCC: $12.00

policy as a flexible storage policy by Jung et al. (1996) were suggested. Besides the inclusion of various operating policies, completion time algorithms have been extended by considering nonzero transfer and setup times. However, some optimal scheduling has been developed by considering only processing times. Furthermore, whatever considers only processing times or transfer and setup times, the optimal scheduling for MIS policy has been rarely studied. A completion time algorithm under consideration of nonzero transfer and setup times was developed by Rajagopalan and Karimi (1989). They included them splendidly but did not consider sequence-dependent setup times for intermediate storage tanks. Optimal scheduling of FIS policy was also developed with consideration of only processing times by Ku and Karimi (1988). Although it was proposed, the optimal scheduling problems have not been researched for serial multiproduct, multiunit processes considering nonzero transfer and sequence-dependent setup times under several intermediate storage policies. In this study, contents are divided into three parts. The first part is concerned with developing completion time algorithms under UIS, NIS, FIS, ZW, and MIS policies considering nonzero transfer and sequencedependent setup times for units and storages. The second part explains the optimal scheduling with completion time algorithms by formulating the problems as mixed-integer nonlinear programming (MINLP) formulations. Finally, two examples are tested to evaluate the robustness and reasonability of this MINLP strategy by checking the computational times. Completion Time Algorithms Although Rajagopalan and Karimi (1989) developed the completion time algorithms for UIS, NIS, FIS, ZW, and MIS considering transfer and setup times, they assumed that the setup times for storages are negligible. However, the setup times for storages are as important as those for processing units when intermediate storages are introduced. So the setup times for storages should also be determined as sequence-dependent nonzero parameters. © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4059

the next unit (j + 1) for a product i in unit j is busy, the product should be held in unit j. A general Gantt chart of NIS policy for developing a completion time algorithm is shown in Figure 3. When we simply consider only processing times, Cij for NIS policy is calculated as Figure 1. UIS flowshop batch process.

Cij ) max [C(i-1)j, Ci(j-1), C(i-1)(j+1) - tij] + tij (3)

UIS Policy. The batch process plant operated under UIS policy has N - 1 storages between every pair of consecutive units. That means that every product i which is finished at unit j need not occupy unit j until ready to process product i on the next unit. If the next unit (unit j + 1) is busy, the product i can be transferred from unit j to any available storage at any time. Figure 1 shows a general schematic feature of the UIS flowshop batch process, while Figure 2 describes its Gantt chart for calculating completion times. In the simple case of considering only processing times, the completion time for product i at unit j under UIS policy is developed as

Cij ) max [C(i-1)j, Ci(j-1)] + tij

(1)

When transfer times and setup times are considered, eq 1 is changed to

where C0j, Ci0, ti0, t0j, and Ci(M+1) are zero. Equation 3 can be reduced from the condition of Ci(j-1) g C(i-1)j for j > 1 as follows:

Cij ) max [C(i-1)j, C(i-1)(j+1) - tij] + tij Cij ) max [Ci(j-1), C(i-1)(j+1) - tij] + tij

for j ) 1 for j > 1 (4)

If sequence-dependent setup and transfer times are considered, completion time for NIS flowshop is:

Cij ) max [Ci(j-1), C(i-1)j + S(i-1)ij + ai(j-1), C(i-1)(j+1) + S(i-1)i(j+1) - tij] + tij + aij (5) Equation 5 can also be reduced from Ci(j-1) g C(i-1)j + S(i-1)ij + ai(j-1) for j > 1.

for j ) 1: Cij ) max [Ci(j-1), C(i-1)j + S(i-1)ij + ai(j-1)] + tij + aij (2) The setup times for storages are not introduced in eq 2 because the storages are fully available and dedicated in the UIS flowshop process. We assumed that the transfer time of product i from unit j to unit j + 1 is equal to that from unit j to an available storage and that from the used storage to unit j + 1 as shown in Figure 2. NIS Policy. Under NIS policy, the intermediate storage is not used in the batch process plant. When

Figure 2. General Gantt chart of UIS policy.

Figure 3. General Gantt chart of NIS policy.

Cij ) max [C(i-1)j + S(i-1)ij + ai(j-1), C(i-1)(j+1) + S(i-1)i(j+1) - tij] + tij + aij for j ) 2, ..., M: Cij ) max [Ci(j-1), C(i-1)(j+1) + S(i-1)i(j+1) - tij] + tij + aij (6) FIS Policy. Under FIS policy, finite storage tanks are available between batch processing units. The number of them between unit j and unit j + 1, zj (j ) 1,

4060 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996

Figure 4. General Gantt chart of FIS policy.

..., M - 1), is fixed and given. A completion time algorithm for FIS flowshop considering only processing time is developed by Ku and Karimi (1988) as follows:

Cij ) max [C(i-1)j, Ci(j-1), C(i-zj-1)(j+1) - tij] + tij

(7)

where Cij ) 0, if i e 0 or j e 0 or j > M. In the case considering not only processing times but also transfer times and sequence-dependent setup times for units and storages, eq 7 should be changed as follows:

Cij ) max [Ci(j-1) + tij + aij, C(i-1)j + S(i-1)ij + ai(j-1) + tij + aij, C(i-zj-1)(j+1) + S(i-zj-1)(i-zi)(j+1) + (zj)(SC(i-zj)ij + a(i-zj)j) + aij] (8) The Gantt chart shown in Figure 4 may give an explanation of why eq 8 represents the recurrence relations of completion times for FIS policy. If all storage tanks and the next processing unit (unit j + 1) are busy, product i which has been finished on unit j must be held in unit j. To complete the processing of product i on unit j, one of the storages (zj) should be available, and for this, unit j + 1 must be ready to process product (i - zj) which has been stored in one of the storage tanks. So the setup time of the storage tank is dependent on the sequence of product (i - zj) and i, i.e., producing product i after product (i - zj). We assumed that the conditions of all storage tanks are equal. ZW Policy. In ZW policy, all products must be transferred immediately from unit j to the next unit (j + 1). It does not require any intermediate storage tanks. Jung et al. (1994) studied new completion time algorithms and the optimal scheduling with MINLP under consideration of nonzero transfer and setup times of units. Considering only the processing time, completion time for each product i at each unit j is

Cij ) Ci(j-1) + tij M

CiM ) max [C(i-1) +

∑ tik, C(i-1)1 +

k)1 M

∑

k)2

M

M

tik, ..., C(i-1)(M-1) +

∑

tik, C(i-1)M +

∑ tik]

k)M

k)(M-1)

(9) Cij ) C(i-1)M -

∑

tik

(10)

k)(j+1)

Equations 9 and 10 should be changed to eqs 11 and 12 for additional consideration of transfer and sequence-

dependent setup times.

Ci1 ) Ci0 + ti1 + ai1 Cij ) Ci(j-1) + tij + aij M

CiM ) max [(C(i-1)1 + S(i-1)i1 + M

(C(i-1)2 + S(i-1)i2 +

∑

M

tik +

k)1 M

∑ aik),

k)0

∑ tik + k)1 ∑ aik), (C(i-1)M +

k)2

M

S(i-1)iM +

M

∑ tik + ∑ k)M

aik)] (11)

k)(M-1)

M

Cij ) CiM -

∑

k)(j+1)

M

tik +

∑

aik

(12)

k)(j+1)

To show how eqs 11 and 12 are developed, a general Gantt chart for ZW policy is shown in Figure 5. MIS Policy. Generally, the intermediate storage tanks are used to enhance the efficiency of the productivity. As mentioned earlier, multiproduct batch process operations have used several patterns of intermediate storage. In UIS policy, an unlimited number (gN - 1) of storage may be installed between every two adjacent units, and in FIS mode, only finite storages may be held. In both ZW and NIS policy, there is no intermediate storage between every interunit, but ZW policy should be operated in a special manner. From this point of view, FIS policy includes UIS and NIS policies. If the number of storage tanks between every unit under FIS policy is N - 1 or zero, it becomes UIS or NIS policy, respectively. In MIS policy, every earlier-mentioned policy can be used between every unit as shown in Figure 6. The general feature of the MIS system can be explained as the FIS system with inserted ZW blocks. Its diagram is shown in Figure 7. To determine the completion times of the MIS system, we should define the boundary of FIS and ZW blocks. As shown in Figure 7, let unit P be the first unit of the ZW block and unit Q the last one on the condition of the unit number of 1 e P e Q e M. In Figure 8, a Gantt chart of MIS flowshop is shown for developing completion times and proposition 1 should be predefined. Proposition 1: The completion times for MIS policy of multiproduct batch processes are based on the FIS system with a ZW block which starts from unit P to unit Q. The completion times under consideration of nonzero transfer times, setup times of units, and setup times of


Figure 5. General Gantt chart of ZW policy.

Figure 6. Simple diagram of MIS plant.

Figure 7. General diagram of the MIS system.

storages are calculated as follows:

Cij ) max [Ci(i-1), C(i-1)j + S(i-1)ij + ai(i-1), C(i-zi-1)(i+1) + S(i-zi-1)(i-zj)(j+1) + F(zj)(SC(i-zj)ij + a(i-zj)j) - tij] + tij + aij (for j ) 1, 2, ..., P - 1 and Q + 1, Q + 2, ..., M) Q

CiP ) max [C′iQ -

Q

∑ tij - j)P ∑ aij, Ci(P-1)] + tiP + aiP i)P (for j ) P)

times has been studied in the mathematical formation, i.e., MILP formation, but nonzero transfer and setup times for units have to be considered for real operations. Recently completion time algorithm with transfer and setup times for various storage policies has been studied. As the number of products and units gets larger, the importance of transfer and setup times increases. Optimal scheduling method thus must consider the transfer and setup times. However, optimal scheduling under consideration of nonzero transfer and setup times has rarely been studied. Sequence-dependent setup times make the problem more complex. Because of such a variable as the setup time for unit which depends on the production sequence, the problem becomes a nonlinear form, and the optimal scheduling is formulated as a MINLP form. An objective function of optimal scheduling is to minimize CNM. For the formulation of the optimal scheduling, new binary variables are defined as follows:

Xij ) 1: if product i is in the position j in the sequence 0: otherwise Then, the constraints which mean a position in the sequence that must be assigned to only one product are expressed as follows: M

Cij ) Ci(j-1) + tij + aij (for j ) P + 1, ..., Q - 1) CiQ ) max [Ci(Q-1) + tiQ + aiQ, C(i-zQ-1)(Q+1) + S(i-zQ-1)(i-zQ)(Q+1) + F(zQ)(SC(i-zQ)ij + a(i-zQ)Q) + aiQ] (for j ) Q) (13) where Q

C′iQ ) max [(C(i-1)P + S(I-1)IP +

Q

∑tij + j)P-1 ∑ aij),

i)P Q

(C(i-1)(P+1) + S(i-1)(P+1) +

∑

i)P+1

Q

tij +

∑aij), ..., (C(i-1)Q +

j)P

Q

S(i-1)iQ +

Q

∑ tij + j)Q-1 ∑ aij)] i)Q

The Gantt chart shown in Figure 8 may give an explanation of why eq 13 represents the completion times for MIS policy. MINLP Formulations The optimal scheduling of multiproduct serial batch units which are assumed to have zero transfer and setup

Xij ) 1 ∑ j)1

(i ) 1, 2, ..., N)

(14)

An ordered product must be processed only once. So the constraints are constructed as follows: N

Xij ) 1 ∑ i)1

(j ) 1, 2, ..., M)

(15)

Equations 14 and 15 are the constraints that are commonly used in various intermediate storage policies. To solve the optimal scheduling of various storage policies, additional constraints which show the characteristics of each storage policy are needed. They are formulated from the completion time algorithms which are described in the previous section. UIS Policy. The remaining constraints for UIS policy under consideration of nonzero transfer and setup times which are formulated by eq 2 are as follows:

for unit 1 (j ) 1): N

Ci1 g C(i-1)1 +

N

∑ ∑ Slk1Xl(i-1)Xki + l)1 k)1 N

∑ (ak0 + ak1 + tk1)Xki

k)1

(16)

4062 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996

Figure 8. General Gantt chart of MIS policy.

for unit 2, ..., M (j ) 2, ..., M): N

Cij g Ci(j-1) +

∑ (akj + tkj)Xki

(17)

k)1 N

Ci1 g C(i-1)1 +

∑ (ak(j-1) + akj + tkj)Xki + k)1 N

for unit 1 (j ) 1):

N

∑ ∑ SlkjXl(i-1)Xki l)1 k)1

(18)

The quadratic term in eq 18 is the expression of sequence-dependent set-up times in the optimal scheduling problem. Due to this term, the problem is formulated as a MINLP form. Cij and Xij are 0 if i or j is equal to 0. With the objective function of min CNM, a formulation of UIS policy for the optimal scheduling is constructed with the constraints of eqs 14-18. NIS Policy. The constraints from eq 6 are as follows:

Ci1 g C(i-1)1 +

N

Ci1 g C(i-1)1 +

N

∑ (ak0 + ak1 + tk1)Xki k)1 N

Ci1 g C(i-1)2 + N

∑ ∑ Slk1Xl(i-1)Xki + l)1 k)1 ∑ (ak0 + ak1 + tk1)Xki

Ci1 g C(i-1)2 +

∑ (akj + tkj)Xki k)1

N

∑ ∑ l)1 k)1

N

(21)

(25)

(26)

N

∑

N

alkjXki + F(zi)(

k)1

N

Slk(j+1)Xl(i-1)Xki +


∑ ∑ Slk(j+1)Xl(i-zj-1)Xk(i-zj) + l)1 k)1

N

N

∑ ∑ SClkjXl(i-zj)Xki + l)1 k)1

∑ akjXki k)1

N

∑ alkjXk(i-zj))

(27)

k)1

(22)

for unit M (j ) M):

N

N

Cij g C1(M-1) +

1

N

Ci1 g C(i-zj-1)(i+1) +

N

N

∑ ak1Xk(i-z ))

k)1

(20)

for unit 2, ..., M - 1 (j ) 2, ..., M - 1): Cij g Ci(j-1) +

N

SClk1Xl(i-z1)Xki +

Cij g Ci(j-1) +

N

∑ ∑ Slk2Xl(i-1)Xki + k)1 ∑ ak1Xki l)1 k)1

Ci1 g C(I-1)(j-1) +

N

(19)

k)1 N

N

for unit 2, ..., M - 1 (j ) 2, ..., M - 1):

N

N

N

(24)

∑ ∑ Slk2Xl(i-zj-1)Xk(i-zj) + k)1 ∑ ak1Xki + l)1 k)1

F(zi)(

N

N

∑ ∑ Slk1Xl(i-1)Xki + l)1 k)1

∑ ∑ l)1 k)1

for unit 1 (j ) 1): N

policy for the optimal scheduling is made with the constraints of eqs 14, 15, and 19-23. FIS Policy. As mentioned above, setup times of storages (SCikj) should be considered for FIS policy additionally. It is shown in eq 8 in detail. Additional constraints for FIS policy are as follows:

∑ (akM + tkM)Xki k)1

(23)

Cij and Xij are 0 if i or j is equal to 0. With the objective function of min CNM, a formulation of NIS

Cij g C(i-1)j +

∑ (ak(j-1) + akj + tlkj)Xki +

k)1

N

N


(28)


for unit M:

for unit M (j ) M): N

CiM g Ci(M-1) +

N

∑ (akM + tkM)Xki k)1

(29)

CiM g C(i-1)M +

N

CiM g C(i-1)M +

N

∑ (aK(m-1) + aKm + tKm)XKI +

N

M

∑ ∑ l)1

k)1

N

N

∑ ∑ SlkMXl(i-1)Xki + l)1 k)1 alkXli +

k)(M-1)

M

∑ ∑ tlkXli l)1 k)M

(36)

N

∑ ∑ SLKmXl(i-1)Xki l)1 k)1

(30)

Cij and Xij are 0 if i or j is equal to 0. With the objective function of min CNM, a formulation of FIS policy for the optimal scheduling is formulated with the constraints of eqs 14, 15), and 24-30. UIS, NIS, and FIS policies have several similarities, and one of them is the completion time of the first product on each unit and the constraints for the first product. To reduce the number of constraints, equality constraints which are constructed from the condition of the first product may be useful. They can be applied to UIS, NIS, and FIS policies as follows:

for unit 1 (j ) 1):

and in the case of the second product (i ) 2), C(i-1)M’s in eq 33-36 become ClM, so they should be replaced by N

M

∑ ∑ l)1 k)0

N

M

∑ ∑ tlkXl1 l)1 k)1

alkXl1 +

MIS Policy. As mentioned earlier, setup times of storages should be considered for MIS policy additionally. It is shown in eq 13 in detail. Additional constraints for the scheduling of MIS policy under consideration of transfer and sequence-dependent setup times are as follows:

N

C11 )

∑ (ak0 + ak1 + tk1)Xk1

(31)

for unit 1 (j ) 1):

k)1

N

for unit 2, ..., M (j ) 2, ..., M):

Ci1 g C(i-1) +

N

C1j ) C1(j-1) +

N

∑ ∑ Slk1Xl(i-1)Xki +

k)1 k)1

∑ (akj + tkj)Xk1

(32)

N

∑ (ak0 + ak1 + tk1)Xki

k)1

k)1

ZW Policy. The optimal formulation of ZW policy under consideration of transfer and setup times had been studied by Jung et al. (1994). In this paper, we develop a MINLP formulation for ZW policy based on it. For the second to the Nth (final) products, the following are formulated:

for unit 1: CiM g C(i-1)M -

k)1 k)1 N

N

N

∑ ∑ SClkjXl(i-zj)Xki + k)1 ∑ akjXk(i-z1)) l)1 k)1

N

Slk1Xl(i-1)Xki +

M

∑ ∑ l)1 k)1

N

alkXli +

M

∑ ∑ tlkXli l)1 k)1

N

Cij g C(i-Zi-1)(j+1) + N

N

CiM g C(i-1)M -

∑ ∑ l)1 k)3


N

∑ ∑ Slk(j+1)Xl(i-z1)Xk(i-zj) + l)1 k)1 N

N

N

∑ akjXki + F(zj)(∑ ∑ SClkjXl(i-zj)Xki + k)1 ∑ akjXk(i-zj)) l)1 k)1

M

N

k)1

(tlk + alk)Xl(i-1) +

N

N

M

N

M

∑ ∑ Slk2Xl(i-1)Xki + ∑ ∑ alkXli + ∑ ∑ tlkXli l)1 k)1 l)1 k)1 l)1 k)2

Cij g C(i-1)j +

∑ (ak(j-1) + akj + tkj)Xki +

k)1

N

for unit (M - 1): N

CiM g C(i-1)M -

M

for unit P (j ) P):

∑ ∑ (tlk + alk)Xl(i-1) + l)1 k)M N

N

M

∑ ∑ Slk(M-1)Xl(i-1)Xki + ∑ ∑ l)1 k)1 l)1

N


(34)

N

(37)

N

Ci1 g Ci(j-1) +

for unit 2:

N

∑ ak0 +

k)1

Xki + F(zj)(

∑ ∑ (tlk + alk)Xl(i-1) + l)1 k)2 (33)

N

∑∑

N

Slk1Xl(i-z1)Xl(i-zj) +

M

N

∑ ∑ l)1 k)1

N

for unit 2, ..., P - 1 and Q + 1, ..., M - 1: N

N

N

Ci1 g C(i-zj-1)2 +

alkXli +

Cip g C′iQ -

k)(M-1)

N

tlkXli (35)

(tkj + akj)Xkj

j)(P+1)

k)(M-2) N M

∑ ∑ l)1

Q

∑ ∑ k)1

CiP g Ci(P-1) +

∑ (tkP + akP)Xki

k)1

(38)

4064 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 N

C′iQ g C(i-1)P +

Q

N

∑∑

tkjXki +

k)1 j)P

Table 1. Process Data for Example 1

Q

∑ ∑

akjXki +

k)1 i)P-1 N N

∑ ∑ SlkPXl(i-1)Xki l)1 k)1

N

C′iQ g C(i-1)(P+1) +

N

Q

∑ ∑

tkjXki +

k)1 j)(P+1) N

Q

∑ ∑ akjXki +

k)1 i)P N

∑ ∑ Slk(P+1)Xl(i-1)Xki l)1 k)1 N

C′iQ g C(i-1)Q +

Q

∑∑

N

tkjXki +

k)1 j)Q

Q

∑ ∑

akjXki +

k)1 i)(Q-1) N N

∑ ∑ SlkQXl(i-1)Xki l)1 k)1

(39)

Table 2. Process Data for Example 2

for unit P + 1, ..., Q - 1: N

∑ (tkj + akj)Xki k)1

Cij g Ci(j-1) +

(40)

for unit Q (j ) Q): N

∑ (tkQ + akQ)Xki

CiQ g C(i-1)Q +

k)1 N

CiQ g C(i-zQ-1)(Q+1) +

Q

∑ ∑ Slk(Q+1)Xl(i-zQ-1)Xk(i-zQ) + l)1 k)1 N

∑ akQXki + F(zQ)

k)1

for unit M (j ) M): N

CiM g CI(M-1) +

∑ (akM + tkM)Xki

(41)

k)1 N

CiM g C(i-1)M +

∑ (ak(M+1) + akM + tkM)Xki +

k)1

N

N

∑ ∑ SlkMXl(i-1)Xki l)1 k)1

(42)

N N The quadratic terms ∑l)1 SlkjXl(i-1)Xki in equations ∑k)1 are the expression of sequence-dependent setup times in the optimal scheduling problem. Due to this term, the problem is formulated as a MINLP form. Cij and Xij are 0 if i or j is equal to 0. With the objective function of min CNM, a formulation of MIS policy for the optimal scheduling is proposed with the constraints of eqs 14, 15, and 37-42.

Numerical Evaluations To evaluate the proposed completion time algorithms and the MINLP formulations, we solved two examples. The first one, as a small size problem (example 1), is a problem of producing four different products in a plant with four processing units. The other one, as a large size problem (example 2), is an example of producing eight products in the same type batch plant. The data

Figure 9. Process diagram for example problems.

used for the example problems are shown in Tables 1 and 2. They are solved for UIS, NIS, FIS, and ZW policies by MINLP formulations. In the case of the FIS system, it is assumed that one storage tank is introduced between units 3 and 4 (zl ) 0, z2 ) 0, and z3 ) 1). The process used for the example problems is shown in Figure 9. The same process is used for both small and large size example problems. It has four batch units and one storage tank between U3 and U4, but the stor-


Figure 10. Gantt chart for the result of example 1 in FIS policy. Table 3. Process Data for Example Problem of MIS Policy

Table 4. Results of Example 1 storage policy

optimal sequence

makespan (unit time)

CPU time (s)

UIS FIS NIS ZW

P1-P4-P3-P2 P1-P2-P4-P3 P1-P4-P2-P3 P1-P4-P2-P3

120 121 126 130

2.81 3.42 3.08 3.63

age tank is used only in FIS policy. The ZW block in the diagram is only used when MIS policy is considered. For MIS policy, the example problem consists of eight products and four units. The used process data are listed in Table 3. As shown in Figure 9, the process used for MIS policy is the same as before. It is assumed that one storage tank is introduced between unit 3 and 4 (z1 ) 0, z2 ) 0, and z3 ) 1). The process has a ZW block between U2 and U3.

Table 5. Results of Example 2 storage policy

optimal sequence

UIS FIS NIS ZW

P5-P7-P1-P2-P6-P8-P4-P3 P5-P7-P4-P1-P3-P6-P2-P8 P5-P7-P2-P4-P-P6-P8-P3 P5-P1-P6-P3-P7-P4-P2-P8

makespan CPU (unit time) time (s) 173 178 185 195

120.68 201.34 154.55 203.93

The GAMS ZOOM and DICOPT++ are used as the solvent for MILP and MINLP on a IBM RS/6000 Model 350 workstation, respectively. The results of example problems for UIS, NIS, FIS, and ZW are shown in Table 4 for example 1 and Table 5 for example 2. The results of the MIS example problem are shown in Table 6. For example 1, under FIS policy, the optimal production sequence is determined as P1-P2-P4-P3 and the makespan is calculated as 121 from the previously described completion time algorithm for FIS policy. From this result, a Gantt chart is constructed as shown in Figure 10. In this figure, (A) means the holding time for product P2 and U1. This holding time happens because the set-up processing of U2 for P2 is not finished at the end of processing P2 in U1. So P2 must be held in U1 until U2 is available. This status also happens when NIS policy is considered. It means that FIS policy includes NIS policy. (B) represents the holding time of product P2 in the storage tank. Because U4 is not available for P2 when the processing of P2 at U3 is finished, P2 should be transferred to the storage tank located between U3 and U4. P2 must be held in the storage tank until U4 is available for P2. Because P2 is transferred from U3 to the storage tank, U3 can be immediately available for the processing of P4. This effect happens in FIS policy. (C) means the setup time for the storage tank. During the setup time, the storage tank is prepared to hold P4 after holding P2. The setup times of general units are scheduled after the ending time of the previously processed product, but the setup time of the storage tank is scheduled before the beginning time of the next stored product. In this example problem, the setup times of the storage tanks do not affect the determination of the makespan. If the setup time to hold P4 after holding P2 is larger than this example, the start time of holding P4 can be delayed after the end of processing P4 at U3. Then, P4 must be held in U3 until the setup processing for the storage is finished. This will cause an increase of the makespan. So the setup times for storage tank are not negligible.

4066 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 6. Results of MIS Policy Problema

a

storage policy

transfer time set-up time

formula

optimal sequence

makespan

computational time (s)

MIS

zero nonzero

MILP MINLP

P7-P6-P1-P3-P5-P2-P4-P8 P5-P6-P1-P3-P7-P2-P4-P8

129 183

18.46 208.76

H/W: IBM RS 6000 Model 350 workstation. S/W: GAMS - ZOOM for MILP. GAMS - DICOPT++ for MINLP.

Conclusion The batch processes with multiunit and multiproduct can be operated more efficiently by solving the scheduling problems. In order to get the minimized makespan as an objective function for the scheduling problems of batch processes, we suggest MINLP formulations for each storage policy such as UIS, NIS, FIS, ZW, and MIS. The formulations are based on our developed completion time algorithms with nonzero transfer time, sequencedependent setup times in units, and the storage setup times. The developed algorithms are evaluated by two example problems. We obtained the optimal solutions successfully for various storage policies with nonzero transfer and setup times within reasonable computational times. These results show the suggested completion time algorithms and the MINLP formulations are generated effectively. Nomenclature Cij ) completion time of ith product in the sequence in batch unit j where the product is finished transfering out of unit j and filling in unit j + 1 F(zj) ) function defined as F ) 0 if zj ) 0 and F ) 1 if zj > 0 M ) number of batch units in the plant N ) number of products to be produced SCikj ) setup time required for product k after product i at the storage which is introduced between unit j and j +1 Sikj ) setup time required for product k after product i in batch unit j aij ) transfer time of product i out of batch unit j to batch unit j + 1

tij ) processing time of product i in batch unit j zj ) number of storage units between unit j and unit j + 1

Literature Cited Jung, J. H.; Lee, H.; Yang, D. R.; Lee, I.-B. Completion times and optimal scheduling for serial multi-product processes with transfer and setup times in zero-wait policy. Comput. Chem. Eng. 1994, 18, 537. Ku, H. M.; Karimi, I. A. Scheduling in serial multi-product batch processes with finite interstage storage: a mixed integer linear program formulation. Ind. Eng. Chem. Res. 1988, 27, 1840. Ku, H. M.; Karimi, I. A. Completion time algorithms for serial multi-product batch processes with shared storage. Comput. Chem. Eng. 1990, 14, 49. Ku, H. M.; Rajagopalan, D.; Karimi, I. A. Scheduling in batch process. Chem. Eng. Prog. 1987, Aug, 35. Jung, J. H.; Lee, H.-K.; Lee, I.-B. Completion times algorithm of multi-product batch processes for common intermediate storage policy (CIS) with nonzero transfer and set-up times. Comput. Chem. Eng. 1996, 20, 845. Rajagopalan, D.; Karimi, I. A. Completion times in serial mixedstorage multi-product processes with transfer and setup times. Comput. Chem. Eng. 1989, 13, 175. Reklaitis, G. V. Review of scheduling of process operations. AIChE Symp. Ser. 1982, 78, 119. Reklaitis, G. V. Perspectives on scheduling and planning of process operations. Proceedings of 4th International Symposium on PSE, Montebello, Canada, 1991.

Received for review March 27, 1996 Revised manuscript received August 7, 1996 Accepted August 7, 1996X IE9601817 X Abstract published in Advance ACS Abstracts, October 15, 1996.

Optimal Scheduling of Multiproduct Batch Processes for Various

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