Job-shop scheduling

NP-hardness

Since the traveling salesman problem is NP-hard, the job-shop problem with sequence-dependent setup is also NP-hard since the TSP is a special case of the JSP with a single job (the salesman in TSP) and the machines (the cities in TSP).

Problem representation

The disjunctive graph is one of the popular models used for describing the job-shop scheduling problem instances.

A mathematical statement of the problem can be made as follows:

Let ${\displaystyle M=\{M_1,M_2,\dots ,M_m\}}$ and ${\displaystyle J=\{J_1,J_2,\dots ,J_n\}}$ be two finite sets. On account of the industrial origins of the problem, the ${\displaystyle {\displaystyle M_i}}$ are called machines and the ${\displaystyle {\displaystyle J_j}}$ are called jobs.

Let ${\displaystyle {\displaystyle \mathcal{X}}}$ denote the set of all sequential assignments of jobs to machines, such that every job is done by every machine exactly once; elements ${\displaystyle x\in \mathcal{X}}$ may be written as ${\displaystyle n\times m}$ matrices, in which column ${\displaystyle {\displaystyle i}}$ lists the jobs that machine ${\displaystyle {\displaystyle M_i}}$ will do, in order. For example, the matrix

${\displaystyle x=\left(\begin{array}{cc}1& 2\\ 2& 3\\ 3& 1\end{array}\right)}$

means that machine ${\displaystyle {\displaystyle M_1}}$ will do the three jobs ${\displaystyle {\displaystyle J_1,J_2,J_3}}$ in the order ${\displaystyle {\displaystyle J_1,J_2,J_3}}$, while machine ${\displaystyle {\displaystyle M_2}}$ will do the jobs in the order ${\displaystyle {\displaystyle J_2,J_3,J_1}}$.

Suppose also that there is some cost function ${\displaystyle C:\mathcal{X}\to [0,+\mathrm{\infty }]}$. The cost function may be interpreted as a "total processing time", and may have some expression in terms of times ${\displaystyle C_{ij}:M\times J\to [0,+\mathrm{\infty }]}$, the cost/time for machine ${\displaystyle {\displaystyle M_i}}$ to do job ${\displaystyle {\displaystyle J_j}}$.

The job-shop problem is to find an assignment of jobs ${\displaystyle x\in \mathcal{X}}$ such that ${\displaystyle {\displaystyle C(x)}}$ is a minimum, that is, there is no ${\displaystyle y\in \mathcal{X}}$ such that ${\displaystyle {\displaystyle C(x)>C(y)}}$.

Scheduling efficiency

Scheduling efficiency can be defined for a schedule through the ratio of total machine idle time to the total processing time as below:

${\displaystyle C^{\prime }=1+\frac{\sum _i{l}_i}{\sum _{j,k}{p}_{jk}}=\frac{C.m}{\sum _{j,k}{p}_{jk}}}$

Where ${\displaystyle l_i}$ represents the idle time of machine i, C is the makespan, and m is the number of machines. This formulation normalizes the makespan by the number of machines and total processing time, allowing for the comparison of resource utilization across job-shop scheduling (JSP) instances of varying sizes.

The problem of infinite cost

One of the first problems that must be dealt with in the JSP is that many proposed solutions have infinite cost: i.e., there exists ${\displaystyle x_{\mathrm{\infty }}\in \mathcal{X}}$ such that ${\displaystyle C(x_{\mathrm{\infty }})=+\mathrm{\infty }}$. In fact, it is quite simple to concoct examples of such ${\displaystyle x_{\mathrm{\infty }}}$ by ensuring that two machines will deadlock, so that each waits for the output of the other's next step.

Major results

Graham introduced the List scheduling algorithm in 1966, which is (2 − 1/m)-competitive, where m is the number of machines. It was later proven to be the optimal online algorithm for two and three machines. The Coffman–Graham algorithm (1972) for uniform-length jobs is also optimal for two machines and (2 − 2/m)-competitive.

In 1992, Bartal, Fiat, Karloff, and Vohra presented a 1.986-competitive algorithm, followed by a 1.945-competitive algorithm by Karger, Philips, and Torng in 1994. That same year, Albers introduced a different 1.923-competitive algorithm. The best known result is by Fleischer and Wahl, achieving a 1.9201 competitive ratio.

Albers also established a lower bound of 1.852. Taillard instances play a key role in developing job-shop scheduling with a makespan objective.

In 1976, Garey proved that this problem is NP-complete for m > 2, meaning no optimal solution can be computed in polynomial time unless P=NP.

In 2011, Xin Chen et al. provided optimal algorithms for online scheduling on two related machines, improving previous results.

Offline makespan minimization

Atomic jobs

The simplest form of the offline makespan minimisation problem deals with atomic jobs, that is, jobs that are not subdivided into multiple operations. It is equivalent to packing a number of items of various different sizes into a fixed number of bins, such that the maximum bin size needed is as small as possible. (If instead the number of bins is to be minimised, and the bin size is fixed, the problem becomes a different problem, known as the bin packing problem.)

Dorit S. Hochbaum and David Shmoys presented a polynomial-time approximation scheme in 1987 that finds an approximate solution to the offline makespan minimisation problem with atomic jobs to any desired degree of accuracy.

Jobs consisting of multiple operations

The basic form of the problem of scheduling jobs with multiple (M) operations, over M machines, such that all of the first operations must be done on the first machine, all of the second operations on the second, etc., and a single job cannot be performed in parallel, is known as the flow-shop scheduling problem. Various algorithms exist, including genetic algorithms.

Johnson's algorithm

A heuristic algorithm by S. M. Johnson can be used to solve the case of a 2 machine N job problem when all jobs are to be processed in the same order. The steps of algorithm are as follows:

Job P_i has two operations, of duration P_i1, P_i2, to be done on Machine M1, M2 in that sequence.

• Step 1. List A = { 1, 2, …, N }, List L1 = {}, List L2 = {}.
• Step 2. From all available operation durations, pick the minimum.

If the minimum belongs to P_k1,

Remove K from list A; Add K to end of List L1.

If minimum belongs to P_k2,

Remove K from list A; Add K to beginning of List L2.

• Step 3. Repeat Step 2 until List A is empty.
• Step 4. Join List L1, List L2. This is the optimum sequence.

Johnson's method only works optimally for two machines. However, since it is optimal, and easy to compute, some researchers have tried to adopt it for M machines, (M > 2.)

The idea is as follows: Imagine that each job requires m operations in sequence, on M1, M2 … Mm. We combine the first m/2 machines into an (imaginary) Machining center, MC1, and the remaining Machines into a Machining Center MC2. Then the total processing time for a Job P on MC1 = sum (operation times on first m/2 machines), and processing time for Job P on MC2 = sum(operation times on last m/2 machines).

By doing so, we have reduced the m-Machine problem into a Two Machining center scheduling problem. We can solve this using Johnson's method.

Makespan prediction

Machine learning has been recently used to predict the optimal makespan of a JSP instance without actually producing the optimal schedule. Preliminary results show around 80% accuracy in classifying small randomly generated JSP instances by optimal scheduling efficiency using supervised learning.

Example

Here is an example of a job-shop scheduling problem formulated in AMPL as a mixed-integer programming problem with indicator constraints:

<span id="mwARs"></span><span class="kd" id="mwARw">param</span><span class="w" id="mwAR0"> </span><span class="nv" id="mwAR4">N_JOBS</span><span class="p" id="mwAR8">;</span> <span class="kd" id="mwASA">param</span><span class="w" id="mwASE"> </span><span class="nv" id="mwASI">N_MACHINES</span><span class="p" id="mwASM">;</span> <span class="kd" id="mwASQ">set</span><span class="w" id="mwASU"> </span><span class="nv" id="mwASY">JOBS</span><span class="w" id="mwASc"> </span><span class="kt" id="mwASg">ordered</span><span class="w" id="mwASk"> </span><span class="o" id="mwASo">=</span><span class="w" id="mwASs"> </span><span class="mi" id="mwASw">1</span><span class="o" id="mwAS0">..</span>N_JOBS<span class="p" id="mwAS4">;</span> <span class="kd" id="mwAS8">set</span><span class="w" id="mwATA"> </span><span class="nv" id="mwATE">MACHINES</span><span class="w" id="mwATI"> </span><span class="kt" id="mwATM">ordered</span><span class="w" id="mwATQ"> </span><span class="o" id="mwATU">=</span><span class="w" id="mwATY"> </span><span class="mi" id="mwATc">1</span><span class="o" id="mwATg">..</span>N_MACHINES<span class="p" id="mwATk">;</span> <span class="kd" id="mwATo">param</span><span class="w" id="mwATs"> </span><span class="nv" id="mwATw">ProcessingTime</span><span class="p" id="mwAT0">{</span>JOBS<span class="p" id="mwAT4">,</span><span class="w" id="mwAT8"> </span>MACHINES<span class="p" id="mwAUA">}</span><span class="w" id="mwAUE"> </span><span class="o" id="mwAUI">></span><span class="w" id="mwAUM"> </span><span class="mi" id="mwAUQ">0</span><span class="p" id="mwAUU">;</span> <span class="kd" id="mwAUY">param</span><span class="w" id="mwAUc"> </span><span class="nv" id="mwAUg">CumulativeTime</span><span class="p" id="mwAUk">{</span>i<span class="w" id="mwAUo"> </span><span class="kr" id="mwAUs">in</span><span class="w" id="mwAUw"> </span>JOBS<span class="p" id="mwAU0">,</span><span class="w" id="mwAU4"> </span>j<span class="w" id="mwAU8"> </span><span class="kr" id="mwAVA">in</span><span class="w" id="mwAVE"> </span>MACHINES<span class="p" id="mwAVI">}</span><span class="w" id="mwAVM"> </span><span class="o" id="mwAVQ">=</span> <span class="w" id="mwAVU"> </span><span class="kr" id="mwAVY">sum</span><span class="w" id="mwAVc"> </span><span class="p" id="mwAVg">{</span>jj<span class="w" id="mwAVk"> </span><span class="kr" id="mwAVo">in</span><span class="w" id="mwAVs"> </span>MACHINES<span class="p" id="mwAVw">:</span><span class="w" id="mwAV0"> </span><span class="nb" id="mwAV4">ord</span><span class="p" id="mwAV8">(</span>jj<span class="p" id="mwAWA">)</span><span class="w" id="mwAWE"> </span><span class="o" id="mwAWI"><=</span><span class="w" id="mwAWM"> </span><span class="nb" id="mwAWQ">ord</span><span class="p" id="mwAWU">(</span>j<span class="p" id="mwAWY">)}</span><span class="w" id="mwAWc"> </span>ProcessingTime<span class="p" id="mwAWg">[</span>i<span class="p" id="mwAWk">,</span>jj<span class="p" id="mwAWo">];</span> <span class="kd" id="mwAWs">param</span><span class="w" id="mwAWw"> </span><span class="nv" id="mwAW0">TimeOffset</span><span class="p" id="mwAW4">{</span>i1<span class="w" id="mwAW8"> </span><span class="kr" id="mwAXA">in</span><span class="w" id="mwAXE"> </span>JOBS<span class="p" id="mwAXI">,</span><span class="w" id="mwAXM"> </span>i2<span class="w" id="mwAXQ"> </span><span class="kr" id="mwAXU">in</span><span class="w" id="mwAXY"> </span>JOBS<span class="p" id="mwAXc">:</span><span class="w" id="mwAXg"> </span>i1<span class="w" id="mwAXk"> </span><span class="o" id="mwAXo"><></span><span class="w" id="mwAXs"> </span>i2<span class="p" id="mwAXw">}</span><span class="w" id="mwAX0"> </span><span class="o" id="mwAX4">=</span> <span class="w" id="mwAX8"> </span><span class="nb" id="mwAYA">max</span><span class="w" id="mwAYE"> </span><span class="p" id="mwAYI">{</span>j<span class="w" id="mwAYM"> </span><span class="kr" id="mwAYQ">in</span><span class="w" id="mwAYU"> </span>MACHINES<span class="p" id="mwAYY">}</span> <span class="w" id="mwAYc"> </span><span class="p" id="mwAYg">(</span>CumulativeTime<span class="p" id="mwAYk">[</span>i1<span class="p" id="mwAYo">,</span>j<span class="p" id="mwAYs">]</span><span class="w" id="mwAYw"> </span><span class="o" id="mwAY0">-</span><span class="w" id="mwAY4"> </span>CumulativeTime<span class="p" id="mwAY8">[</span>i2<span class="p" id="mwAZA">,</span>j<span class="p" id="mwAZE">]</span><span class="w" id="mwAZI"> </span><span class="o" id="mwAZM">+</span><span class="w" id="mwAZQ"> </span>ProcessingTime<span class="p" id="mwAZU">[</span>i2<span class="p" id="mwAZY">,</span>j<span class="p" id="mwAZc">]);</span> <span class="kd" id="mwAZg">var</span><span class="w" id="mwAZk"> </span><span class="nv" id="mwAZo">end</span><span class="w" id="mwAZs"> </span><span class="o" id="mwAZw">>=</span><span class="w" id="mwAZ0"> </span><span class="mi" id="mwAZ4">0</span><span class="p" id="mwAZ8">;</span> <span class="kd" id="mwAaA">var</span><span class="w" id="mwAaE"> </span><span class="nv" id="mwAaI">start</span><span class="p" id="mwAaM">{</span>JOBS<span class="p" id="mwAaQ">}</span><span class="w" id="mwAaU"> </span><span class="o" id="mwAaY">>=</span><span class="w" id="mwAac"> </span><span class="mi" id="mwAag">0</span><span class="p" id="mwAak">;</span> <span class="kd" id="mwAao">var</span><span class="w" id="mwAas"> </span><span class="nv" id="mwAaw">precedes</span><span class="p" id="mwAa0">{</span>i1<span class="w" id="mwAa4"> </span><span class="kr" id="mwAa8">in</span><span class="w" id="mwAbA"> </span>JOBS<span class="p" id="mwAbE">,</span><span class="w" id="mwAbI"> </span>i2<span class="w" id="mwAbM"> </span><span class="kr" id="mwAbQ">in</span><span class="w" id="mwAbU"> </span>JOBS<span class="p" id="mwAbY">:</span><span class="w" id="mwAbc"> </span><span class="nb" id="mwAbg">ord</span><span class="p" id="mwAbk">(</span>i1<span class="p" id="mwAbo">)</span><span class="w" id="mwAbs"> </span><span class="o" id="mwAbw"><</span><span class="w" id="mwAb0"> </span><span class="nb" id="mwAb4">ord</span><span class="p" id="mwAb8">(</span>i2<span class="p" id="mwAcA">)}</span><span class="w" id="mwAcE"> </span><span class="kt" id="mwAcI">binary</span><span class="p" id="mwAcM">;</span> <span class="kd" id="mwAcQ">minimize</span><span class="w" id="mwAcU"> </span><span class="nv" id="mwAcY">makespan</span><span class="p" id="mwAcc">:</span><span class="w" id="mwAcg"> </span><span class="kr" id="mwAck">end</span><span class="p" id="mwAco">;</span> <span class="kd" id="mwAcs">subj to</span><span class="w" id="mwAcw"> </span><span class="nv" id="mwAc0">makespan_def</span><span class="p" id="mwAc4">{</span>i<span class="w" id="mwAc8"> </span><span class="kr" id="mwAdA">in</span><span class="w" id="mwAdE"> </span>JOBS<span class="p" id="mwAdI">}:</span> <span class="w" id="mwAdM"> </span><span class="kr" id="mwAdQ">end</span><span class="w" id="mwAdU"> </span><span class="o" id="mwAdY">>=</span><span class="w" id="mwAdc"> </span>start<span class="p" id="mwAdg">[</span>i<span class="p" id="mwAdk">]</span><span class="w" id="mwAdo"> </span><span class="o" id="mwAds">+</span><span class="w" id="mwAdw"> </span><span class="kr" id="mwAd0">sum</span><span class="p" id="mwAd4">{</span>j<span class="w" id="mwAd8"> </span><span class="kr" id="mwAeA">in</span><span class="w" id="mwAeE"> </span>MACHINES<span class="p" id="mwAeI">}</span><span class="w" id="mwAeM"> </span>ProcessingTime<span class="p" id="mwAeQ">[</span>i<span class="p" id="mwAeU">,</span>j<span class="p" id="mwAeY">];</span> <span class="kd" id="mwAec">subj to</span><span class="w" id="mwAeg"> </span><span class="nv" id="mwAek">no12_conflict</span><span class="p" id="mwAeo">{</span>i1<span class="w" id="mwAes"> </span><span class="kr" id="mwAew">in</span><span class="w" id="mwAe0"> </span>JOBS<span class="p" id="mwAe4">,</span><span class="w" id="mwAe8"> </span>i2<span class="w" id="mwAfA"> </span><span class="kr" id="mwAfE">in</span><span class="w" id="mwAfI"> </span>JOBS<span class="p" id="mwAfM">:</span><span class="w" id="mwAfQ"> </span><span class="nb" id="mwAfU">ord</span><span class="p" id="mwAfY">(</span>i1<span class="p" id="mwAfc">)</span><span class="w" id="mwAfg"> </span><span class="o" id="mwAfk"><</span><span class="w" id="mwAfo"> </span><span class="nb" id="mwAfs">ord</span><span class="p" id="mwAfw">(</span>i2<span class="p" id="mwAf0">)}:</span> <span class="w" id="mwAf4"> </span>precedes<span class="p" id="mwAf8">[</span>i1<span class="p" id="mwAgA">,</span>i2<span class="p" id="mwAgE">]</span><span class="w" id="mwAgI"> </span><span class="o" id="mwAgM">==></span><span class="w" id="mwAgQ"> </span>start<span class="p" id="mwAgU">[</span>i2<span class="p" id="mwAgY">]</span><span class="w" id="mwAgc"> </span><span class="o" id="mwAgg">>=</span><span class="w" id="mwAgk"> </span>start<span class="p" id="mwAgo">[</span>i1<span class="p" id="mwAgs">]</span><span class="w" id="mwAgw"> </span><span class="o" id="mwAg0">+</span><span class="w" id="mwAg4"> </span>TimeOffset<span class="p" id="mwAg8">[</span>i1<span class="p" id="mwAhA">,</span>i2<span class="p" id="mwAhE">];</span> <span class="kd" id="mwAhI">subj to</span><span class="w" id="mwAhM"> </span><span class="nv" id="mwAhQ">no21_conflict</span><span class="p" id="mwAhU">{</span>i1<span class="w" id="mwAhY"> </span><span class="kr" id="mwAhc">in</span><span class="w" id="mwAhg"> </span>JOBS<span class="p" id="mwAhk">,</span><span class="w" id="mwAho"> </span>i2<span class="w" id="mwAhs"> </span><span class="kr" id="mwAhw">in</span><span class="w" id="mwAh0"> </span>JOBS<span class="p" id="mwAh4">:</span><span class="w" id="mwAh8"> </span><span class="nb" id="mwAiA">ord</span><span class="p" id="mwAiE">(</span>i1<span class="p" id="mwAiI">)</span><span class="w" id="mwAiM"> </span><span class="o" id="mwAiQ"><</span><span class="w" id="mwAiU"> </span><span class="nb" id="mwAiY">ord</span><span class="p" id="mwAic">(</span>i2<span class="p" id="mwAig">)}:</span> <span class="w" id="mwAik"> </span><span class="o" id="mwAio">!</span>precedes<span class="p" id="mwAis">[</span>i1<span class="p" id="mwAiw">,</span>i2<span class="p" id="mwAi0">]</span><span class="w" id="mwAi4"> </span><span class="o" id="mwAi8">==></span><span class="w" id="mwAjA"> </span>start<span class="p" id="mwAjE">[</span>i1<span class="p" id="mwAjI">]</span><span class="w" id="mwAjM"> </span><span class="o" id="mwAjQ">>=</span><span class="w" id="mwAjU"> </span>start<span class="p" id="mwAjY">[</span>i2<span class="p" id="mwAjc">]</span><span class="w" id="mwAjg"> </span><span class="o" id="mwAjk">+</span><span class="w" id="mwAjo"> </span>TimeOffset<span class="p" id="mwAjs">[</span>i2<span class="p" id="mwAjw">,</span>i1<span class="p" id="mwAj0">];</span> <span class="kr" id="mwAj4">data</span><span class="p" id="mwAj8">;</span> <span class="kd" id="mwAkA">param</span><span class="w" id="mwAkE"> </span><span class="nv" id="mwAkI">N_JOBS</span><span class="w" id="mwAkM"> </span><span class="p" id="mwAkQ">:</span><span class="o" id="mwAkU">=</span><span class="w" id="mwAkY"> </span><span class="mi" id="mwAkc">4</span><span class="p" id="mwAkg">;</span> <span class="kd" id="mwAkk">param</span><span class="w" id="mwAko"> </span><span class="nv" id="mwAks">N_MACHINES</span><span class="w" id="mwAkw"> </span><span class="p" id="mwAk0">:</span><span class="o" id="mwAk4">=</span><span class="w" id="mwAk8"> </span><span class="mi" id="mwAlA">4</span><span class="p" id="mwAlE">;</span> <span class="kd" id="mwAlI">param</span><span class="w" id="mwAlM"> </span><span class="nv" id="mwAlQ">ProcessingTime</span><span class="p" id="mwAlU">:</span> <span class="w" id="mwAlY"> </span><span class="mi" id="mwAlc">1</span><span class="w" id="mwAlg"> </span><span class="mi" id="mwAlk">2</span><span class="w" id="mwAlo"> </span><span class="mi" id="mwAls">3</span><span class="w" id="mwAlw"> </span><span class="mi" id="mwAl0">4</span><span class="w" id="mwAl4"> </span><span class="p" id="mwAl8">:</span><span class="o" id="mwAmA">=</span> <span class="mi" id="mwAmE">1</span><span class="w" id="mwAmI"> </span><span class="mi" id="mwAmM">5</span><span class="w" id="mwAmQ"> </span><span class="mi" id="mwAmU">4</span><span class="w" id="mwAmY"> </span><span class="mi" id="mwAmc">2</span><span class="w" id="mwAmg"> </span><span class="mi" id="mwAmk">1</span> <span class="mi" id="mwAmo">2</span><span class="w" id="mwAms"> </span><span class="mi" id="mwAmw">8</span><span class="w" id="mwAm0"> </span><span class="mi" id="mwAm4">3</span><span class="w" id="mwAm8"> </span><span class="mi" id="mwAnA">6</span><span class="w" id="mwAnE"> </span><span class="mi" id="mwAnI">2</span> <span class="mi" id="mwAnM">3</span><span class="w" id="mwAnQ"> </span><span class="mi" id="mwAnU">9</span><span class="w" id="mwAnY"> </span><span class="mi" id="mwAnc">7</span><span class="w" id="mwAng"> </span><span class="mi" id="mwAnk">2</span><span class="w" id="mwAno"> </span><span class="mi" id="mwAns">3</span> <span class="mi" id="mwAnw">4</span><span class="w" id="mwAn0"> </span><span class="mi" id="mwAn4">3</span><span class="w" id="mwAn8"> </span><span class="mi" id="mwAoA">1</span><span class="w" id="mwAoE"> </span><span class="mi" id="mwAoI">5</span><span class="w" id="mwAoM"> </span><span class="mi" id="mwAoQ">8</span><span class="p" id="mwAoU">;</span>

Related problems

• Flow-shop scheduling is a similar problem but without the constraint that each operation must be done on a specific machine (only the order constraint is kept).
• Open-shop scheduling is a similar problem but also without the order constraint.