nano biochemical testing: Exact and heuristic algorithms

Computers & Operations Research 38 (2011) 942–953

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Computers & Operations Research journal homepage: www.elsevier.com/locate/caor

Scheduling large-scale micro/nano biochemical testing: Exact and heuristic algorithms Dong Tang a,, Udatta S. Palekar b,1 a b

Intel Corporation, 5000 West Chandler Boulevard, Chandler, AZ 85226, USA Department of Business Administration University of Illinois at Urbana-Champaign 4016 BIF 515 East Gregory Drive Champaign, IL 61820, USA

a r t i c l e i n f o

a b s t r a c t

Available online 14 October 2010

We consider a micro/nano fluidic toolbit that consists of a set of identical testing units, each of which contains a microchannel that has an array of equally spaced nanopores opened along it. In each microchannel, same equally spaced chemical liquid plugs shift back and forth under pneumatic pressure. Below each nanopore is a testing tube that accepts appropriate nanoscale chemical droplets from the microchannel above to perform biochemical tests. Each tube may require several different chemicals in sequence to get proper results. Liquid chemicals required in different tubes may be dropped simultaneously in a round if the liquid plug sequence in the microchannel above matches the chemical requirements in these tubes. The sizes of testing problems in terms of the numbers of tubes, liquid chemicals required in each tube and liquid plugs in the microchannel are large, efficient testing procedure requires careful ‘‘round’’ scheduling in order to shorten the testing time span. In this research, we model the biochemical test scheduling as the fixed plug sequence problem (FPSP), where the liquid plug layout in the microchannel is given. We show that the FPSP is NP-hard in general, and then develop both exact and heuristic algorithms. The computational performances of the proposed algorithms are provided and contrasted. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Micro/nano biochemical testing Scheduling Mathematical programming models Heuristic

1. Introduction The fears about another potential round of terrorist attacks by biological weapons have motivated researchers to find better ways to handle large-scale biochemical screening. Such efforts involve designing and implementing microfluidic devices, such as laboratory-on-a-chip devices. Besides deterring biological attacks, the technology of large-scale biochemical testing also facilitates rapid advances in gene discovery, genetic mapping and gene expression with broader applications ranging from infectious diseases, drug screening and cancer diagnostics to food quality and environmental evaluation. In this work, we consider a micro/nano fluidic toolbit, which is conceptually illustrated in Fig. 1, which contains a large set of identical testing units. Each testing unit has a microchannel with an array of equally spaced nanopores opened along it. In the microchannel, equally spaced chemical liquid plugs shift back and forth under pneumatic pressure. Below each nanopore is a testing tube that accepts appropriate nanoscale chemical droplets from the microchannel above to perform biochemical tests. Each tube may require several  Correspondence to: 2130 East Saltsage Drive, Phoenix, AZ 85048, USA. Tel.: + 1 480 552 6562; Cell: 1 217 721 5311. E-mail addresses: [email protected], [email protected] (D. Tang), [email protected] (U.S. Palekar). 1 Tel.: + 1 217 333 1665

0305-0548/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2010.10.006

different chemicals in sequence to get proper results. Liquid chemicals required in several different tubes may be dropped simultaneously in a round if the liquid plug sequence in the microchannel above matches the chemical requirements in these tubes. In each testing unit, chemical liquid plugs are introduced to the microchannel in such a way that the center distance between any two adjacent liquid plugs equals the distance between any two adjacent nanopores. The train of equal-length chemical liquid plugs in the microchannel shifts back and forth under pneumatic pressure. When the liquid plug train comes to a proper alignment with the underlying testing tubes, the corresponding nanopores are opened and the required nanoscale chemical droplets are deposited into the corresponding tubes to finish a round of biochemical tests. In each test round, the liquid plug train needs to be precisely sensed and positioned by sensor arrays arranged near the nanopores [1]. Since the liquid plugs in the microchannel are at a micro scale and the droplets are at a much smaller nano scale, we assume that the size of each liquid plug remains unchanged after each test round. Another concern is the potential liquid contamination due to the direct contact of different liquid plugs in the microchannel. In the micro scale, there is no turbulent mixing but only slow diffusions at the direct liquid-liquid interface at a very low Reynold number [2,3]. Ismagilov et al. [4] developed a technique that allows direct fluid–fluid contacts for combinatorial experiments and detection purposes. Every nano droplet comes out from the center of a liquid plug and therefore the impact of diffusion is negligible.

D. Tang, U.S. Palekar / Computers & Operations Research 38 (2011) 942–953

Pneumatic pressure Chemical liquid plugs

Testing unit

Pneumatic pressure

Micro gate

Testing unit

Testing unit

Testing unit

943

Supply liquid chemicals to testing unit

Connect to pneumatic gate

Testing unit details Liquid plug train moved by pneumatic pressure Microchannel

Liquid plug

Testing tube

Nanopore

Fig. 1. Concept of the physical structure of a micro/nano-fluidic toolbit.

The numbers of tubes, liquid chemicals required in each tube and liquid plugs in the microchannel are large, and for the system to run efficiently and to achieve high testing throughput requires careful scheduling. This research is devoted to exploring good scheduling methods to shorten the total biochemical testing time span. Each testing unit in the studied micro/nano fluidic toolbit can be considered independently, so we focus on a single testing unit in the following discussion. Assuming each testing task takes the same amount of time to complete, our objective of minimizing the testing time span is equivalent to scheduling the test sequence in such a way that the number of test rounds is minimized. In this work, we consider that the liquid plug layout in the microchannel is given, and so model the micro/nano biochemical test scheduling problem as the fixed plug sequence problem (FPSP). In Section 2, we establish the criteria for a scheduling solution to be feasible. Then the complexity of the FPSP is explored in Section 3. Mathematical programming models and exact algorithms are developed in Section 4. Several heuristics are provided and analyzed in Section 5. Solution qualities and performances of the proposed algorithms are computationally studied in Section 6. Section 7 concludes.

2. Notations and criteria for feasible scheduling Without loss of generality, we assume that the total number of testing tubes is n, each tube has m testing tasks and the microchannel contains q liquid plugs of p different chemicals. We index the tubes from left to right by 1, 2, 3, y, n, and the testing tasks in a tube by 1, 2, 3, y, m. A task with smaller index should be finished before a task with larger index in the same tube. For simplicity, we identify a testing task j in tube i as T(i, j) or simply (i, j), where 1rirn and 1rjrm. When it is clear in the context, we may use the required chemical or just the task index to represent a task for convenience. In order to facilitate the discussion, we may also graphically describe FPSPs. An example is shown in Fig. 2, in which liquid plugs are represented by rectangles and marked by chemicals, while testing tasks are represented by circles and marked by required chemicals. Note that, in order for a set of testing tasks to be scheduled in the same round, their layout in the testing tubes should match the layout of the liquid plugs of the required chemicals in the microchannel. For example, in Fig. 2, tasks T(1, 3), T(2, 3) and T(3, 1)

1

2

3 2 6

4

6

3

Task 1

5

Task 2

2

2

Task 3

Tube 2

Tube 3

4 a

5

b

7

7

c 1

Tube 1

c

Fig. 2. Graphical representation of FPSP.

require chemicals 1, 2, and 3, respectively. It is clear that the layout of these three tasks matches the layout of the liquid plugs of chemicals {1, 2, 3} in the microchannel. So tasks {T(1, 3), T(2, 3), T(3, 1)} can be completed together in a round. However, layout matching does not necessarily guarantee feasible scheduling since one round may conflict with another. For instance, in Fig. 2, rounds a, b and c indicate a conflict: round a needs to be done before round c since chemical 6 must be deposited before chemical 1 in tube 1; similarly round b needs to be done before round a and round c needs to be done before round b. So this example testing schedule is not feasible since it contains conflicting rounds. Round conflict can be easily detected as follows. Build a graph G in which vertices represent rounds, then compare every two vertices/rounds (denoted by r1 and r2) in G to see if one round needs to be done before the other. If r1 needs to be done before r2, an arc is connected from vertex r1 to vertex r2 and vice versa. Determining whether a schedule is conflict-free is equivalent to checking whether the digraph G contains a directed cycle. From basic graph theory, a digraph is acyclic if and only if its vertices can be topologically sorted. So we can use the topological sorting algorithm [5] to efficiently detect conflicts in a schedule.

3. Complexity of fixed plug sequence problem Given the liquid plug layout in the microchannel, it is clear that the FPSP with only one tube is trivial. When two tubes, namely 1 and 2, are involved, we build a bipartite graph G to represent these

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D. Tang, U.S. Palekar / Computers & Operations Research 38 (2011) 942–953

two tubes as shown in Fig. 3. In graph G, we connect tasks T(1, i) and T(2, j) by an arc if these two tasks can be done in a round, allowed by the liquid plug layout in the microchannel (this can be checked in O(q) steps). Denote by U the task set in tube 1 and by V the task set in tube 2. In graph G, two edges (i, j) and (iu,ju), where iAU, iu A U, jAV and ju A V, are said to cross each other if ðiiuÞ  ðjjuÞ r 0. It is seen that minimizing the number of test rounds is equivalent to finding the maximum number of non-crossing edges in graph G. This is a maximum non-cross matching problem in a bipartite graph, which can be solved in O(|U|  |V|), or O(m2), time [6]. However when three or more tubes are involved, we find the FPSP becomes intractable. The FPSP bears some properties that are similar to the unit-processing-time job shop problem: each testing tube is analogous to a job, and the tasks in a tube are analogous to a sequence of pre-ordered operations in the corresponding job. Each task takes the same amount of processing time, which without loss of generality; we can consider to be unit time. Job shop problems are known to be NP-hard. Even instances with 3 machines and unit processing times ðJ3jpij ¼ 1jCmax Þ are NP-hard [7]. When preemptions are allowed, instances with 3 machines and 3 jobs (J3jn ¼ 3jCmax and J3jn ¼ 3,pmtnjCmax ) are also NP-hard [8].

operations. By this means, a J3jn ¼ 3,pmtnjCmax problem can be polynomially reduced to a J3jn ¼ 3,pij ¼ 1jCmax problem, in which operations {1, 2, y, k} are in consecutive sequence within the job. &

Lemma 1. Job shop problem J3jn ¼ 3,pij ¼ 1jCmax is NP-hard.

4. Mathematical programming models and exact algorithms

Proof:. The proof follows a polynomial reduction from the NP-hard problem J3jn ¼ 3,pmtnjCmax [8] to J3jn ¼ 3,pij ¼ 1jCmax . In J3jn ¼ 3, pmtnjCmax , we break each operation into a set of consecutive operations with unit processing time. For example, an operation of k unit of processing time is broken into k consecutive operations {1, 2, y, k}, each of which has unit processing time. An operation of k-unit processing time in J3jn ¼ 3,pmtnjCmax is preemptive, which means that we can stop processing the job at operation i and later resume it at operation i+1 in the k broken-down consecutive

1

2

3

4

5

2

7

6

3

1

2

4

3

1

5

6

7

Fig. 3. Bipartite graph representation of a 2-tube FPSP.

1 (1, 1)

1

11

Based on lemma 1, we can show that the FPSP with 3 tubes and 3 liquid chemicals is NP-hard. Theorem 1. The FPSP with 3 tubes and 3 liquid chemicals is NP-hard. Readers are referred to Appendix A for proof. & Interestingly, we also find that even when each testing tube has only one task, the FPSP remains hard to solve. Theorem 2. The FPSP where each tube has only one task is NP-hard. Proof comes by a polynomial reduction from the minimum set cover problem. Readers are referred to Appendix B for details. & The proof of Theorem 2 brings bad news for solving the FPSP: the minimum set cover problem is inapproximable [9,10], so there exists no polynomial algorithm for the FPSP that can provide a constant approximation ratio.

This section considers solving the FPSP by exact algorithms based on mathematical programming models. Two different models are discussed and an exact solution is developed using the second model. 4.1. Disjunctive mathematical programming model Indeed, the FPSP can be represented elegantly by a disjunctive graph G(N, A, B). The nodes in N correspond to all testing tasks. A conjunctive arc in A, representing precedence relations between two tasks, is a directed arc from task (i, j) to task (i, j+ 1). Any two tasks in different tubes that cannot be processed in a round are connected by two disjunctive arcs going in opposite directions. The set of disjunctive arcs is denoted by B. In addition, we add a source S and a sink T, which are dummy nodes. The source S has conjunctive arcs emanating to the first tasks of all tubes, and the sink T has conjunctive arcs coming from the last tasks of all tubes. All arcs except those connected to S have length 1, while all arcs emanating from S have length 0. An example of the disjunctive graph representation is shown in Fig. 4, in which we use solid lines for conjunctive arcs and broken

(1, 2)

(1, 3)

1

0

S

0

(2, 1)

1

(2, 2)

1

(2, 3)

0

1

T

1 1 1 (3, 1)

(3, 2)

(3, 3)

11

Fig. 4. Disjunctive graph representation of FPSP.

D. Tang, U.S. Palekar / Computers & Operations Research 38 (2011) 942–953

lines for disjunctive arcs. A mathematical programming model based on the disjunctive graph can then be developed. We use tij to represent the starting time of task (i, j) in the scheduling. Since the processing time is unity for all tasks, there is no need to use a variable to represent the task processing time. Let Rmax denote the total testing time span or the number of rounds in the scheduling. The following mathematical programming model minimizes Rmax.

Minimize Rmax subject to: Rmax Z tij þ1,

8ði,jÞ A N

ð1Þ

tij þ 1 Z tij þ1,

8ði,jÞ-ði,j þ 1Þ A A

ð2Þ

tij Ztkl þ 1 or tkl Ztij þ 1,

8½ði,jÞ&ðk,lÞ A B

8ði,jÞ A N

ð3Þ ð4Þ

Constraint (1) states that the time span to finish all tasks is greater than the start time of the last task plus its processing time 1. Constraint (2) guarantees that the precedence requirements of the tasks in a same tube are satisfied. Constraint (3) indicates that any two tasks connected by a disjunctive arc pair cannot be done together. FPSP-1 is a nonlinear model. By introducing ( 1, if task ði,jÞ is done before taskðk,lÞ yði,jÞðk,lÞ ¼ 0, otherwise M  very large constant number we transform it into a mixed integer programming (MIP) model as follows.

Minimize Rmax subject to: Rmax Z tij þ1,

8ði,jÞ A N

ð5Þ

tij þ 1 Z tij þ1,

8ði,jÞ-ði,j þ 1Þ A A

ð6Þ

¼

1;

if task ði,jÞ is done in round r

0;

otherwise

We then develop the constraints to capture the facts that each round should be done with an appropriate liquid plug alignment and each task should be done in a round while observing the precedence relations. Constraint 1: Each task should be done in a round X yrði,jÞ ¼ 1, 8ði,jÞ r

Constraint 2: Each task should be done in a round with a proper liquid plug alignment X p yrði,jÞ r aði,jÞ  xrp , 8ði,jÞ, 8r p

When yrði,jÞ ¼ 1, e.g. task (i, j) is done in round r, correspondingly there should exist a liquid plug alignment that can be chosen for round r and also the task (i, j) can be done with this alignment. There may exist several such liquid plug alignments, but we choose only one. Constraint 3: A chosen round must be finished with only one liquid plug alignment X xrp r 1, 8r

ð11Þ

y1ði,jÞ ¼ 0,

ð7Þ

tij Ztkl þ 1ð1yðk,lÞði,jÞ Þ  M,

8½ði,jÞ&ðk,lÞA B

ð8Þ

8½ði,jÞ&ðk,lÞ A B

8ði,jÞ A N 8½ði,jÞ&ðk,lÞ A B

ru o r

ð10Þ

8½ði,jÞ&ðk,lÞA B

yði,jÞðk,lÞ þ yðk,lÞði,jÞ ¼ 1,

P Note that p xrp ¼ 0 means that round r is not included in the final solution. Constraint 4: In tube i, task j can only be done after all its predecessors have been finished X yrði,jÞ r yruði,j1Þ , 8ði,jÞ When yrði,jÞ ¼ 1, e.g. task (i, j) is done in round r, to satisfy this inequality we need yruði,j1Þ ¼ 1 for some round r0 or, which means that task (i, j  1) should be finished before round r. One can see that this constraint recursively enforces that task j can only be done after all its predecessors have been finished. However, we need to be cautious when r is equal to 1, which implies r0 o1. In this case the constraint is written as

tkl Z tij þ 1ð1yði,jÞðk,lÞ Þ  M,

yði,jÞðk,lÞ A f0,1g,

( yrði,jÞ

p

4.1.2. FPSP-2

tij Z0,

alignments exist. We denote ( 1; if test round r is done with liquid plug alignment p xrp ¼ 0; otherwise

Further define ( 1; if task ði,jÞcan be done with liquid plug alignment p apði,jÞ ¼ 0; otherwise

4.1.1. FPSP-1

tij Z0,

945

ð9Þ

One can see that a feasible scheduling of the FPSP corresponds to a selection of one arc from each disjunctive arc pair such that the resulting directed graph is acyclic. The number of rounds in the schedule is determined by the longest path from source S to sink T in the resulting disjunctive graph. So obtaining an optimal FPSP scheduling is equivalent to selecting disjunctive arcs so that the length of the longest path in the resulting disjunctive graph is minimized.

which means that any non-first task cannot be included in the first round. Similarly, one can see that any task with index bigger than r cannot be done in round r. Therefore yrði,jÞ ¼ 0,

Another mathematical programming model of the FPSP, which is similar to an assembly line balancing model, is discussed in this subsection. To establish this model, we first collect all possible ways that the liquid plugs in the microchannel can be aligned against the underneath testing tubes. In fact, only n + q 1

8j 4r

8i,

Our objective of minimizing the total number of rounds is captured by XX cr  xrp Minimize r

4.2. Assembly-line-balancing-like model

8j 41

8i,

p

in which, parameter cr serves as a penalty for including round r in the final solution. If cr is defined as cr Z M  cr1 where M is a sufficiently large positive number, minimizing the objective function will force the scheduler to finish all tasks in a minimum number of rounds. Formally, the model is written as

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D. Tang, U.S. Palekar / Computers & Operations Research 38 (2011) 942–953

4.3. Valid inequalities for FPSP-3

4.2.1. FPSP-3 XX cr  xrp Minimize r

subject to: X yrði,jÞ ¼ 1,

p

8ði,jÞ

ð12Þ

r

yrði,jÞ r

X p aði,jÞ  xrp ,

8ði,jÞ,

8r

ð13Þ

p

X xrp r 1,

8r

ð14Þ

p

yrði,jÞ r

X

yruði,j1Þ ,

8ði,jÞ

ð15Þ

ru o r

yrði,jÞ ¼ 0,

This subsection presents a few valid inequalities based on FPSP-3. We are only interested in the valid inequalities that define cutting planes of the linear programming (LP) relaxation polyhedron or even better-facets of the integer polyhedron. The cutting planes cut off fractional extreme points of the LP relaxation polyhedron while leaving the integer polyhedron untouched, thus tighten the LP bound and speed up the solution procedure. In the following discussion, we denote by r the upper bound of the round number obtained by a heuristic (see Section 5).

4.3.1. Clique inequalities The first family of valid inequalities proposed are clique inequalities defined on a graph G(V, E) with vertex set being the task set V ¼ fv : v ¼ ði,jÞg

8i,

xrp A f0,1g, yrði,jÞ A f0,1g,

8j 4r

8r,

ð16Þ

8p

8ði,jÞ,

ð17Þ 8r

ð18Þ

FPSP-3 is a mixed integer programming (MIP) model. The trivial upper bound of total round number is n  m. So when implementing this MIP model, we will not consider the constraints with r 4n  m. In fact, we can use the heuristics to be discussed in Section 5 to compute a better upper bound, which can help us eliminate a large number of unnecessary constraints and speed up the computation. Model FPSP-3 indicates that the FPSP is similar to the assembly line balancing problem [11], or more precisely, the simple assembly line balancing problem of type one (SALBP-1) [12], in which assembly tasks are assigned to workstations so that the number of stations is minimized while the task precedence relations are maintained. The similarities between the FPSP and the SALBP-1 lie in: i. Testing tasks in the FPSP can be viewed as assembly tasks in the SALBP-1. They all have precedence relations which must be observed. ii. A round in the FPSP can be viewed as a workstation in the SALBP-1, since they both finish a set of tasks. Also minimizing the number of rounds in the FPSP can be viewed as minimizing the number of workstations in the SALBP-1. However, the FPSP differs from the SALBP-1 in the following two ways: i. In the SALBP-1, a task can be put in any workstation. However, in the FPSP, a task can only be finished in a round with proper liquid plug alignment. ii. In the SALBP-1, it is feasible to put a task and its predecessors in the same workstation. However, in the FPSP, a task can never be done with its predecessors in a same round. If we relax the precedence constraints in the FPSP, the problem becomes very similar to the uncapacitated facility location problem [13–15], where a facility corresponds to a test round. If all facilities have the same cost, minimizing the number of facilities is similar to minimizing the number of test rounds. In the facility location problem, the group of customers a facility can serve is given. However, in the FPSP, the set of tasks a round can finish depends on the liquid plug alignment as well as the completed tasks in each tube, and is thus unknown.

and edge set being E ¼ fe : e ¼ ðv1 ,v2 Þ where tasks v1 and v2 cannot be done in a same roundg

A clique C in G(V, E) represents a set of tasks in which no two tasks can be finished together in a same round. This leads to the following inequality defined on such a clique: X yrði,jÞ r1, 8r ð19Þ ði,jÞ:v A C

which can be further strengthened as X X yrði,jÞ r xrp , 8r

ð20Þ

p

ði,jÞ:v A C

P When p xrp ¼ 0, e.g. the final solution does not contain round r, P it can be seen yrði,jÞ ¼ 0 for all (i, j). When p xrp ¼ 1, inequality (20) is

the same as inequality (19). Any two tasks in a same tube cannot be done together, so all tasks in a same tube form a clique. Therefore X yrði,jÞ r 1, 8r, 8i ð21Þ j

Similarly, valid inequality (21) can be further strengthened as X X yrði,jÞ r xrp , 8r, 8i ð22Þ p

j

Note that the clique inequality where the clique C consists of at most one task per testing tube is not a cutting plane. To see this, we use the fact that any two tasks from different testing tubes in C cannot be finished together with the same liquid plug alignment, P e.g. ap r 1,8p, and sum up constraint (13) over tasks in ði,jÞ A C ði,jÞ clique C X X X p X X p  X yrði,jÞ r aði,jÞ  xrp ¼ aði,jÞ  xrp r xrp ði,jÞ A C p

ði,jÞ A C

p

ði,jÞ A C

ð23Þ

p

which indicates that the clique inequality introduced by C can be obtained from constraint (13) and is thus redundant.

4.3.2. Two-task inequality X X yrði,jÞ þ yrði,j þ kÞ r1,

8r^ rr,

8i,

8j r m1, 8kA ½1,mj

r r r^

r Z r^

ð24Þ P

r r Z r^ yði,jÞ

P

r r r r^ yði,j þ kÞ

When ¼ 1, ¼ 0 since task (i, j +k) cannot be P P r r done before task (i, j). Also when r r r^ yði,j þ kÞ ¼ 1, r Z r^ yði,jÞ ¼ 0 because of the same reason. So inequality (24) is valid.

D. Tang, U.S. Palekar / Computers & Operations Research 38 (2011) 942–953

Step 4: Output clique C, which is the maximal weighted clique found.

4.3.3. Remaining task inequality XX X xrp  r  yrði,jÞ Z dði,jÞ, 8ði,jÞ r

p

ð25Þ

r

where d(i, j) is a parameter that represents the number of remaining tasks in testing tube i after task (i, j) has been completed. PP This inequality is valid since r p xrp represents the total number P of rounds in the final solution, and r r  yrði,jÞ is the round in which task (i, j) is completed. 4.3.4. Early round inequality X X xrp r xr1p , 8r A ½2,r p

947

4.4.2. End algorithm The separation procedures to find all non-clique valid inequalities are simple enumerations. But we will see later (Section 6) that all non-clique cuts together provide very limited improvement but require significantly longer CPU run time than the clique cuts.

5. Heuristic algorithms ð26Þ

p

P

It is easy to verify that this inequality is valid: if p xr1p is 1, it is P the same as constraint (14); if p xr1p is zero, e.g. the final solution P uses less than r  1 rounds, then p xrp should also be zero since the final solution should also use less than r rounds.

The exact algorithms presented in Section 4 can only solve small FPSPs to optimality. For practical applications, heuristics are needed to solve large size problems. We know that in the best case, every round finishes one task of each testing tube, and the number of rounds is m. In the worst case each task has to be done in an individual round, and the number of rounds is n  m. Thus the worstcase performance of any heuristic is bounded by r ¼(n  m)/m¼n.

4.4. Branch-and-cut procedure and separation problems 5.1. Greedy algorithms We adopt a standard branch-and-cut procedure to solve FPSP-3: (1) The LP relaxation is first solved and if the optimal solution is integer, then we are done; otherwise, (2) separation problems are solved to identify valid inequalities that cut off the extreme point of the optimal fractional solution. If no such valid inequalities can be found, we branch; otherwise, (3) the identified valid inequalities are added into the LP relaxation which is again solved to optimality. The procedure is repeated until we obtain an optimal integer solution or the problem is proven infeasible. Given an optimal fractional solution (xn, yn) of the LP relaxation, for the clique inequality, solving the separation problem is equivalent to determining a set of cliques where the clique inequalities are violated. To obtain such cliques, we solve the maximum weighted clique problem (MWCP) on G(V, E), where n the weight of vertex v or (i, j) is yr ði,jÞ , and find clique C such that the P P r r value of ði,jÞ A C  yði,jÞ is maximized. If ði,jÞ A C  yði,jÞ r 1, clearly no clique inequality is violated by the current fractional solution, otherwise the valid inequality introduced by clique Cn is the cutting plane that we are looking for. Since the traditional NP-hard maximum clique problem [16] is a special case of the MWCP where each vertex has unit weight, the MWCP is NP-hard in general. Nevertheless, we can use the following simple heuristic to solve the separation problem quickly. 4.4.1. MWCP heuristic Let w(v), e.g. yr ði,jÞ , be the weight of vertex v, e.g. (i, j), in graph G(V, E). Step 1: For each vertex v in graph G(V, E), compute its gain g(v), which is defined as X gðvÞ ¼ wðvÞ þ wðiÞ i A NðvÞ

where N(v) represents the vertices adjacent to v. Step 2: Find a vertex vn with the maximum gain and use it to initialize clique set C. Remove all vertices that are not adjacent to vn from G(V, E), and update both adjacency lists and gains of the remaining vertices. Step 3: If v is the latest vertex added to C, find a vertex u in the adjacency list of v that has the maximum gain and is not in C. If no such u exists, go to step 4, otherwise add u to C. Remove all vertices that are not in the adjacency list of u from G(V, E), again update both adjacency lists and gains of the remaining vertices. Repeat step 3.

It is clear that whenever the train of liquid plugs is aligned against the testing tubes, it is optimal to finish as many tasks as possible. Based on this observation, we can develop a greedy heuristic that takes the liquid plug train alignment that enables us to finish a maximal number of admissible tasks iteratively. The term ‘‘admissible’’ here refers to a task of which its predecessors have already been finished in earlier rounds. If more than one such liquid plug alignments exist, we randomly pick one. This heuristic is called online greedy algorithm, or ONG, since it only considers admissible tasks in each step. An offline greedy heuristic, or OFG in short, considers all tasks, not only admissible tasks, and can be algorithmically described as follows. 5.1.1. Offline greedy heuristic (OFG) Denote R the round set Step 1: Initialize R’+, and construct graph G(V, E) in such a way that each task is represented by a vertex, and an edge is established between two tasks that can potentially be done in a same round, allowed by the liquid plug layout in the microchannel. Note that a set of tasks that can be done in a same round form a clique in graph G(V, E). Step 2: Find test round rn that contains the maximum number of tasks. It is essentially to find the maximum clique in graph G(V, E). Since the maximum clique problem is NP-hard [16], we can use approximate algorithms [17,18] to solve it. Set R’R [ fr  g, and then remove all vertices in rn and edges that emanate from vertices in rn from graph G(V, E). Step 3: Resolve conflicts between R and edges in graph G(V, E) as follows. (i) Build a digraph in which rounds in R and edges in G(V, E) are represented by nodes; (ii) Establish a directed arc between any two nodes that have precedence relations; (iii) Run the topological sorting algorithm to detect directed cycles, i.e. conflicts; (iv) Break directed cycles, if any, by removing edges from G(V, E). Step 4: If the vertex set V of graph G(V, E) is empty, exit and output R; otherwise repeat steps 2 and 3.

5.1.2. End algorithm Though simple, the worst-case performances of these two greedy algorithms are not desirable. It can be shown that the

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worst-case approximate ratio r of these two heuristics approaches n. To see this, we consider an extreme case shown in Fig. 5 for the online greedy heuristic. By the online greedy algorithm, unluckily we may obtain a round schedule as {1, 1, y, 1, 2, 2, y, 2, 1, 3, 3, y 3, 1, y, 1, n, n, y, n}, which has totally m  n rounds. However, we easily observe that the optimum rounds scheduling is {1, 1, y, 1, (1, 2, 3, y, n), (1, 2, 3, y, n), y, (1, 2, 3, y, n)}, which contains n +m  1 rounds. The approximate ratio is r ¼(m  n)/(n+ m  1)-n, when m approaches infinity. For the offline greedy heuristic, we use extreme cases illustrated in Fig. 6. In Fig. 6, tasks scheduled in same rounds by the heuristic are linked by arcs. In (a), one can see that the optimal solution contains m rounds, but the heuristic provides a solution of 2m  1 rounds, the approximate ratio r approaches 2 as m approaches infinity. In (b) the optimal solution contains m rounds, the heuristic provides a solution of 3m  3 rounds, and the approximate ratio r approaches 3 as m approaches infinity. In (c) the optimal solution contains m rounds, the heuristic provides a solution of 4m  5 rounds, and the approximate ratio r approaches 4 as m approaches infinity. Similarly, if there are n tubes, we can construct an example in which the optimal solution contains m rounds, the heuristic P provides a solution of dm  n ni¼ 2 log2 ie rounds, and the approximate ratio r approaches n as m approaches infinity.

of non-conflict task pairs between the first two columns. We treat each connected task pair as a ‘‘compound’’ task. These compound tasks together with unconnected tasks form a ‘‘compound’’ column. Then the next tube/column is brought in and between the compound column and the new column, we connect a maximum number of nonconflict (compound) task pairs again. We repeat this procedure until all columns/tubes are added. By Ref. [6], in the worst case where no connection can be established in each step, solving columns 1 and 2 requires O(m2) time, solving (compound) columns 2 and 3 requires O(2m2) time, y, and solving (compound) columns n  1 and n requires O((n  1)m2) time. So the worst case run time of this heuristic is Oðm2 Þ þ Oð2m2 Þ þ . . . þOððn1Þm2 Þ which can be written as O(n2m2). When we construct a compound column, each (compound) task will be assigned an index. So the (compound) tasks need to be sorted in such way that the precedence constraints are met. This can be done by topologically sorting them in the directed acyclic precedence graph built on these (compound) tasks. Note that some (compound) tasks may have no precedence relations and so there exists multiple options to index them. In algorithm implementation, we sort them by local search that allows us to find a maximal number of connections in the next step. We can implement the column-wise algorithm in a way that new columns are added in sequence starting from the first tube (or the last tube), this implementation is called CW(I). Nevertheless, we can also implement the column-wise algorithm in a different way by picking the best two (compound) columns that gives us the maximum connections iteratively, that is, for every pair of adjacent (compound) columns we run the maximum non-cross matching algorithm and then pick the best pair that has the maximum connections. We call this implementation CW(II) in the remaining discussion. Notice that whenever the liquid plugs are aligned against the testing tubes, it is always optimal to finish as many tasks as possible. With this observation, the solution obtained by the column-wise heuristic can further be improved as follows: in sequence, for each round and liquid plug alignment suggested by the column-wise heuristic, finish as many admissible tasks as possible; if there is no admissible task for current round and alignment, e.g. all tasks suggested to be done in this round by the heuristic have already been finished in earlier rounds, we simply drop this round from the final solution.

5.2. Column-wise algorithm From the discussion in Section 3, we know that the FPSP with only two tubes can be converted to a maximum non-cross matching problem in a bipartite graph and then solved efficiently. Based on this, we devise a column-wise algorithm that brings a new tube into consideration in each step, that is, from left to right, we add tube/ column one by one. In the first step, we connect a maximum number

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5.3. Worst-case performance analysis of column-wise algorithm The essence of the column-wise algorithm is to establish as many connections between two adjacent columns as possible.

Fig. 5. Extreme case to evaluate the worst-case performance of the online greedy heuristic For the offline greedy heuristic, we use extreme cases illustrated by Fig. 6.

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Fig. 6. Extreme cases to evaluate the worst-case performance of the offline greedy heuristic.

D. Tang, U.S. Palekar / Computers & Operations Research 38 (2011) 942–953

sun3 Z maxf0,sn3 sun4 g Zsn3 sun4 . So we have

The term ‘‘connection’’ here refers to two tasks that are scheduled together in a same round, allowed by the given liquid plug layout in the microchannel. The number of connections directly determines the number of rounds in the final solution.

ðs1 su1 Þ þ ðs2 su2 su1 Þ þ ðs3 su3 su2 Þ þ::: þ ðsn3 sun3 sun4 Þ þ ðsn2 sun3 sn1 Þ r 0 By basic algebra, it can be verified that

Lemma 2. If a scheduling solution for the FPSP consists of K connections in total, then the solution contains m  n  K rounds.

Su ¼ su1 þ su2 þ . . . þ sun3 þsun2 þ sun1 Z su1 þ su2 þ . . . þsun3 s1 þ s2 þ . . . þsn1 S ¼ þ sn1 Z 2 2

Proof 1. . In the solution, some connections are further connected into chains. WLOG, we assume that there are o chains denoted by fC1 ,C2 , . . . ,Co g. Chain Ci, ir o, contains ai connections and ai +1 tasks. All tasks in a chain can be finished together in a same round. Each connection not in any chain introduces one round that accomplishes 2 tasks. Each task not in any connection needs an individual test round separately. Therefore, the total rounds r can be written as r ¼ m  n þ ðK

o X

ai Þ2  ðK

i¼1

o X

ai Þþ o

i¼1

o X

ðai þ 1Þ ¼ m  nK

Case 2: sn1 o sun2 . We have sun2 þ sun1 Zsun2 , su1 Z s1 , su2 Z maxf0,s2 su1 g Z s2 su1 , su3 Z maxf0,s3 su2 g Z s3 su2 , . . ., and sun2 Z maxf0,sn2 sun3 g Zsn2 sun3 . It follows that ðsu1 s1 Þ þ ðsu2 s2 þ su1 Þ þ ðsu3 s3 þsu2 Þ þ . . . þðsun2 sn2 þsun3 Þ þ ðsun2 sn1 Þ Z 0 By basic algebra, one can show that Su ¼ su1 þ su2 þ . . . þ sun3 þsun2 þ sun1 Z su1 þ su2 þ . . . þsun3 s1 þ s2 þ . . . þ sn1 S ¼ þ sun2 Z 2 2

&

i¼1

This finishes the proof.

Denoted by si the number of connections between columns i and i+ 1 in the optimal solution, and by sui the number of connections between columns i and i+ 1 in the solution obtained by the columnP wise heuristic. Let S ¼ n1 i ¼ 1 si be the total number of connections between all adjacent columns in the optimal solution, and P Su ¼ n1 i ¼ 1 sui be the total number of connections between all adjacent columns in the solution obtained by the column-wise heuristic.

Theorem 3. The approximate ratio r of the column-wise heuristic is bounded by r r ðm  nS =2Þ=ðm  nK  Þ, where Sn and Kn are the total number of connections between adjacent columns and the total number of all connections in the optimal solution, respectively.

Proof. WLOG, we assume the column-wise algorithm works from column 1 to column n in sequence. It is seen that su1 Zs1 , since the exact maximum non-cross matching algorithm can always obtain the maximum number of connections between columns 1 and 2. Between columns 2 and 3, we can obtain at least maxf0,s2 su1 g number of connections in case that all su1 connections between columns 1 and 2 cannot further be extended to include any task from column 3. So, su2 Z maxf0,s2 su1 g. Similarly, we can establish su3 Z maxf0,s3 su2 g, . . . ,sun1 Zmaxf0,sn1 sun2 g. It follows ( sn1 , if sn1 Z sun2 sun2 þsun1 Z sun2 þ maxf0,sn1 sun2 g ¼ sun2 , if sn1 o sun2

Proof. By Lemma 2, the number of rounds in the optimal solution is rOPT ¼ m  nK  . Let KALG denote the total number of connections in the solution obtained by the column-wise heuristic. The number of rounds obtained by the column-wise heuristic is thus rALG ¼ m  nKALG . Since KALG ZSu, by lemma 3, we have KALG Z S =2, which indicates that rALG r m  nS =2. So the approximate ratio is

r ¼ rALG =rOPT r ðm  nS =2Þ=ðm  nK  Þ: This finishes the proof.

Then, we consider the following two cases. Case 1: sn1 Z sun2 . In this case, it is seen that sn1 Z sun2 Zmaxf0,sn2 sun3 g Zsn2 sun3 , sun2 þ sun1 Zsn1 , su1 Zs1 , su2 Z maxf0,s2 su1 g Zs2 su1 , su3 Z maxf0,s3 su2 g Zs3 su2 , . . ., and

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The proposed exact algorithm and heuristics are implemented, and their computational performances are tested out on

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6. Computational studies

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The bound given by Lemma 3 is tight. With connections being represented by arcs, Fig. 7 shows an example in which Su ¼ S =2. Totally 7k+ 1 liquid plugs, 4k+ 1 chemicals, 2k+ 1 testing tubes and 4k+ 2 tasks are involved in this example.

Lemma 3. Su Z ðS =2Þ.

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Fig. 7. D represents dummy liquid plug. (a) represents the optimal solution which contains 2k connections between adjacent columns, i.e. Sn ¼2k; (b) represents the solution obtained by the column-wise heuristic. k connections are established between adjacent columns, i.e. S0 ¼ k. So Su ¼ S =2.

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a computer with a Pentium 4 1.82 GHz processor and 512 MB RAM. Test cases are randomly generated according to the following parameters: i. ii. iii. iv.

number number number number

of of of of

chemicals—p liquid plugs in the microchannel—q testing tubes—n testing tasks in each tube—m

with constraint q4p to guarantee that each chemical has at least one liquid plug in the microchannel. For each set of parameters, we generate 5 different test cases. Small FPSPs are formulated into FPSP-3 MIP models and then solved by ILOG CPLEX 9.1. Since the number of variables and constraints in the FPSP-3 heavily depend upon the pre-determined number of rounds, we adopt the heuristics presented in Section 5 to provide upper bounds of the round numbers. For slightly larger test cases that cannot be solved to optimality by MIP models, the LP lower bounds are provided by solving the corresponding LP relaxations. The direct LP relaxations from FPSP-3 provide poor lower bounds as Table 1 shows. The LP/MIP gaps,

Table 1 Optimal solutions and LP lower bounds for small test cases. Problems

p ¼ 5, q¼ 5, 1 2 3 4 5

Direct

Cutting plane group 1

Cutting plane group 2

OPT

Rounds

Rounds

Rounds

Rounds

CPU time (s)

n¼ 5, m¼5 11 14 10 15 10 15 9 14 10 13

0.31 0.32 0.32 0.28 0.34

14 15 15 14 13

0.79 0.81 0.76 0.74 0.78

16 15 17 15 15

3.81 4.22 4.46 4.03 3.87

15 16 17 17 16

11.41 11.82 11.03 11.58 11.65

18 19 20 19 18

311.32 267.13 424.51 403.33 398.22

35 32 41 36 33

831.09 782.44 927.51 935.13 892.74

p ¼ 5, q¼ 10, n¼ 10, m ¼5 1 12 15 2 10 16 3 13 17 4 11 16 5 14 16 p ¼ 5, q¼ 10, n¼ 10, m ¼10 1 15 34 2 13 32 3 15 40 4 15 34 5 12 33

CPU time (s)

– – – – –

Note: Direct-direct LP relaxations from FPSP-3; Cutting plane group 1-LP relaxations from FPSP-3 together with pre-determined clique inequalities; Cutting plane group 2-LP relaxations from FPSP-3 together with all pre-determined valid inequalities; OPT-optimal solutions.

which is defined as ðzIP zLP Þ=zIP , where ZIP and ZLP are the optimal solutions of the MIP and LP relaxation respectively, are around 25–50%. So the valid inequalities presented in Section 4.3 are added to tighten the LP relaxations. To evaluate the effects of these valid inequalities, we classify them into two groups of cutting planes. Cutting plane group 1 consists of only clique inequalities and cutting plane group 2 contains all valid inequalities. All cutting planes are pre-determined and then directly added to the root LP relaxation. The computational results shown in Table 1 indicate that the LP relaxations with valid inequalities provide much tighter lower bounds than the direct LP relaxations. The LP/MIP gaps are narrowed down to 10–15%. Cutting plane group 2 marginally improves the results from cutting plane group 1, although it takes a much longer time to run. So we only use clique inequalities in the exact algorithm. We run the proposed heuristics against two groups of test cases. Test case group 1 contains small test cases, from which optimal solutions or LP lower bounds can be obtained. If optimal solutions are not attainable, the LP lower bounds are computed as follows: a fractional solution is obtained by solving the direct LP relaxation of FPSP-3; then separation problems for clique inequalities are solved to generate a set of clique cuts that cut off this fractional solution; the new LP relaxations with these clique cuts are re-solved by the dual simplex algorithm to obtain a new fraction solution; this procedure is repeated until no further clique cut can be generated and then report the obtained objective value as the LP lower bound. The purpose of test case group 1 is to gauge the solution qualities of the heuristics. Test case group 2 consists of large test cases that are close to real-world biochemical testing problems. This group allows us to compare both solution qualities and computational efforts of the proposed heuristics. The computational results for test case groups 1 and 2 are presented in Tables 2 and 3, respectively. For each FPSP (p, q, n, m) in Table 2, we run 5 randomly generated test cases and report the average approximate ratios and CPU time of each heuristic. Approximate ratios are obtained by dividing the round numbers obtained from heuristics by the optimal round numbers or the LP lower bounds. Note that, when LP lower bounds are used, the obtained approximate ratios are larger than the actual values. From Tables 2 and 3, we observe that OFG does slightly better than ONG, and both greedy algorithms are significantly outperformed by the column-wise algorithms. CW(II) performs better than CW(I). However, due to the fact that CW(II) has to compare all 2-column results in each iteration, it takes much longer time to run with less than 5% improvement from CW(I). Based on the disjunctive graph model of Section 4.1, we also adapted and implemented the shifting bottleneck algorithm that is very successful in solving job shop problems [19]. Our computational study found it does not perform better than the

Table 2 Computational results from heuristics-test case group 1. Problems (p, q, n, m)

(5, (5, (5, (5, (5,

5, 5, 5) 10, 10, 5) 10, 10, 10) 15, 15, 5) 15, 15, 10)

ONG

OFG

CW(I)

CW(II)

Average ratio

Average CPU time (s)

Average ratio

Average CPU time (s)

Average ratio

Average CPU time (s)

Average ratio

Average CPU time (s)

1.345 1.451 1.673 1.442 1.669

0.009 0.025 0.037 0.042 0.087

1.281 1.365 1.600 1.390 1.598

0.017 0.041 0.040 0.049 0.106

1.027 1.043 1.195 1.096 1.185

0.015 0.022 0.035 0.035 0.049

1.013 1.033 1.178 1.086 1.148

0.023 0.034 0.043 0.046 0.094

Note 1: When implementing CW(I), we run the algorithm twice starting from two different sides of the tube array. In the first run we start from column 1 and add new columns from left to right one by one. In the second run we start from column n and add new columns from right to left one by one. The reported results are the best values obtained from the two runs. Note 2: The average approximate ratios in bold italic are obtained by dividing the round numbers obtained from heuristics by the optimal round numbers. Others are obtained by dividing the round numbers obtained from heuristics by the LP lower bounds.

D. Tang, U.S. Palekar / Computers & Operations Research 38 (2011) 942–953

951

Table 3 Computational results from heuristics-test case group 2 Problems

ONG Rounds

OFG CPU time (s)

Rounds

CW(I) CPU time (s)

Rounds

CW(II) CPU time (s)

Rounds

CPU time (s)

p ¼ 25, q ¼50, n¼ 50, m¼ 20 1 418 2 436 3 440 4 483 5 435

0.537 0.541 0.584 0.556 0.562

412 429 435 472 431

1.682 1.578 1.613 1.741 1.692

322 318 323 339 339

1.281 1.219 1.275 1.353 1.396

314 311 309 333 332

8.207 8.442 8.309 8.513 8.611

p ¼ 50, q¼ 100, n¼ 100, m ¼20 1 879 2 893 3 812 4 868 5 841

3.578 3.434 3.465 3.485 3.394

865 881 809 859 848

11.738 12.104 12.857 12.314 11.977

656 663 648 659 658

8.984 8.951 8.973 8.688 8.828

641 647 635 638 643

62.132 65.794 64.238 63.873 65.024

p ¼ 75, q ¼150, n¼ 150, m¼ 50 1 3170 2 3419 3 3399 4 3282 5 3483

26.141 27.796 28.875 27.562 28.203

3158 3398 3385 3276 3478

209.614 215.361 224.183 207.396 216.368

2337 2316 2317 2314 2332

154.684 131.388 132.529 131.513 132.232

2322 2314 2303 2301 2315

1013.871 1021.512 1015.139 1018.025 1014.465

Note: When implementing CW(I), we run the algorithm twice starting from two different sides of the tube array. In the first run we start from column 1 and add new columns from left to right one by one. In the second run we start from column n and add new columns from right to left one by one. The reported results are the best values obtained from the two runs.

greedy algorithms, but takes significant longer CPU time. The reason lies in that there are O(n2m2) number of disjunctive arc pairs in a randomly generated FPSP. In each iteration, the adapted shifting bottleneck algorithm will evaluate all disjunctive arc pairs in order to identify the bottleneck. In the local improvement stage, the tasks connected by those selected disjunctive arcs will be re-sequenced and the whole evaluation procedure will be executed again thereafter. Evaluating all disjunctive arc pairs only schedules two tasks in the FPSP, while in the job shop problem this evaluation will schedule all tasks performed on a specific machine. This may explain why the adapted shifting bottleneck algorithm does not perform well for our problem as compared to its application in the job shop problem.

7. Conclusions Efficient operation of micro/nano fluidic devices to perform largescale biochemical testing requires careful test-task scheduling. Given the liquid plug layout in the microchannel, we modeled this scheduling problem as the fixed plug sequence problem (FPSP) and then showed that FPSPs are NP-hard in general. Since the minimum set cover problem is polynomially reducible to a special case of the FPSP, this problem is inapproximable in terms of constant approximate ratio. Mathematical programming models and exact algorithms are developed to solve small FPSPs. We also provided several heuristics to solve large size FPSPs. The proposed heuristics are implemented and benchmarked against the exact solution as well as each other on a set of randomly generated test cases. Computational studies showed that the column-wise algorithm, which solves the FPSP tube by tube, wins out and can be used in real-world biochemical test scheduling.

Acknowledgement The authors wish to thank the editor and two anonymous referees for their helpful suggestions that greatly improved the presentation in this paper.

Appendix A

Proof of Theorem 1. According to Section 2, by the topological sorting algorithm, we can polynomially check whether a given scheduling solution is feasible and the total round number in the solution is less than k. So the decision version of FPSP belongs to P. We then reduce the NP-hard J3jn ¼ 3,pij ¼ 1jCmax to the FPSP having 3 tubes and 3 liquid chemicals. We treat each tube as a job, the sequenced tasks in each tube as ordered operations in the job, and each chemical as a machine. One can see that this treatment converts the FPSP to a job shop problem with unit processing time except that the machines are not independent: shifting one chemical liquid plug causes the shifting of all other liquid plugs in the microchannel, that is, the binding liquid plug train imposes an extra constraint on choosing chemicals/machines for tasks/ operations. In order to remove this constraint, we introduce a liquid plug layout including all permutations of these three chemicals (6 permutations in total), where each permutation is separated by a sufficiently large amount of dummy liquid plugs to ensure that no two adjacent permutations can be used simultaneously (it can be seen that 2 dummy liquid plugs between any two adjacent permutations are enough). Such a layout is illustrated by Fig. 8, where D represents a dummy liquid plug. We then argue that the FPSP shown in Fig. 8 is equivalent to J3jn ¼ 3,pij ¼ 1jCmax . No two permutations can be used simultaneously, so each liquid chemical can be used for no more than one task at a time; this is equivalent to the requirement that no machine can handle more than one operation at a time in the job shop problem. The microchannel contains all permutations of the three chemicals, which means that there is no ordering among these three chemicals and thus each chemical can be treated as an independent machine. Since each testing task in the FPSP takes the same amount of time (unit time), to finish, the objective of the FPSP to finish all tasks in the minimum number of rounds is equivalent to completing all operations in J3jn ¼ 3,pij ¼ 1jCmax within the minimum make-span. A solution with less than k rounds for this FPSP can be polynomially transformed into a solution for J3jn ¼ 3,pij ¼ 1jCmax with make-span less than k unit time. And a solution with

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D. Tang, U.S. Palekar / Computers & Operations Research 38 (2011) 942–953

Permutation 1 1

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Permutation 2 D

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Permutation 4

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Fig. 8. FPSP with 3 tubes and 3 liquid chemicals.

3

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D

7

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Fig. 10. Liquid cluster for subset {3, 4, 7}.

Chemical a (Machine a)

T(i, j)

…

…

Roundr (Time r)

…

Fig. 9. Gantt diagram representation.

make-span less than k unit time for J3jn ¼ 3,pij ¼ 1jCmax can also be polynomially transformed into a solution for this FPSP with less than k rounds. The transformation is below. (1) FPSP-J3jn ¼ 3,pij ¼ 1jCmax A Gantt diagram with horizontal axis representing rounds and the vertical axis representing chemicals can be drawn as shown in Fig. 9. If testing task T(i, j) is done in round r and requires chemical a (1ra r3), then a rectangle representing task T(i, j) is drawn at coordinate (r, a) in the Gantt diagram. Since each chemical can be used for at most one task in each round, there will be less than 1 rectangle at coordinate (r, a) in the Gantt diagram. Treating chemicals as machines, tubes as jobs and rounds as time buckets, we can see the Gantt diagram exactly represents the solution for the corresponding J3jn ¼ 3,pij ¼ 1jCmax . (2) J3jn ¼ 3,pij ¼ 1jCmax -FPSP Similarly, we can draw a Gantt diagram for the solution of J3jn ¼ 3,pij ¼ 1jCmax . Treating machines as chemicals, jobs as tubes and time buckets as test rounds, because the microchannel contains all permutations of the three chemicals, we can always find a permutation to finish all tasks in each round. So the Gantt diagram also represents the scheduling solution for the corresponding FPSP. It is seen that the reduction is polynomial since only 6 permutations of 3 liquid chemicals and 28 liquid plugs (dummy liquid plugs included) are involved. This finishes the proof. &

WLOG, we index elements in the finite set S of the minimum set cover problem by {1, 2, 3, y, k}, then construct a FPSP with k tubes and index the tubes from left to right in sequence by {1, 2, 3, y, k}. In the constructed FPSP, each tube has only one testing task and no two tasks require the same liquid chemical, which means that the tubes, tasks and chemicals are one-on-one mapping to each other and all can be conveniently indexed by {1, 2, 3, y, k} without introducing confusion. It is also seen that each element in set S corresponds to a tube, task or chemical, and each subset, denoted by H, in collection C corresponds to a group of tubes, tasks or chemicals. Then we represent each subset H in collection C of the set cover problem by a cluster of liquid plugs in the microchannel, and the layout of the liquid chemicals in this cluster is arranged in such a way that it can finish all the tasks in H within a same round. The chemical liquid plugs are spaced out accordingly by dummy liquid plugs. For example, if a subset H of collection C is {3, 4, 7}, then the layout of the corresponding cluster of liquid plugs in the microchannel is arranged as {3, 4, D, D, 7}, as shown in Fig. 10, where D represents a dummy liquid plug. Also liquid plug clusters are separated by dummy liquid plugs in such a way that no two clusters can be used to drop liquid chemicals simultaneously, which means that each cluster must be used independently. Now we can see that obtaining the minimum number of subsets in collection C to cover all elements in S is equivalent to obtaining the minimum number of clusters in the microchannel to finish all testing tasks. This is essentially to find the minimum number of rounds to finish all tasks in the FPSP. The conversion can be done in polynomial steps and only less than 2  |C|  |S| liquid plugs (dummy liquid plugs included) are needed in the microchannel. This finishes the proof. &

References Appendix B

Proof of Theorem 2. : We show that the NP-hard minimum set cover problem [16] is polynomially reducible to this FPSP. The statement of the minimum set cover problem is as follows. Instance: Collection C of subsets of a finite set S. Objective: A set cover for S, i.e., a sub-collection Cu D Csuch that every element in S belongs to at least one member of C0 , where the size of C0 is minimized.

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