Method and Scheduling for Optimal Work Practices

The Company

Northern England manufacturing company, Kart Co, manufacture metal goods primarily for the retail and food industry, such as shopping trolleys and baskets, roll carts, roll containers and garment rails. The company employs about 250 people mainly in their production facility. The production facility consists of three sections, metal products, roller castors and the branding unit.  

The metal section contains five stages of production:  cutting, bending, welding, plating and finishing. Where the unit operates for five days a week with a normal operating shift of 12 hours per day. The roller castor section has three stages of manufacture; cutting, bending and plating. This unit operates for five days a week with a normal operating shift of 8 hours per day. The branding section is where products are customised to suit particular customers, which only has one stage. Operating four days a week with a normal operating shift of 8 hours per day.

Kart Co has a plan for a four-week schedule of work. To maintain profits, improvements to the production facility need to be made. This has been further justified with a recent increase in customer complaints concerning late deliveries of orders. With the company not able to spend a substantial amount of money on external consultants or software. Improvements on the order of jobs in each section are investigated and discussed within the report.

Brief

Various methods to find the most optimal order of the relevant jobs have been used, the processes sort the jobs into an order which reduces the time taken for all the jobs to be completed. Simple methods such as sorting by smallest to largest have been used as well as more algorithm-based processes such as flow shop scheduling were used throughout the investigation. The methods suggested below improve the total time from the original version given, however it may not be the most optimal given the short amount of time to report back to you. If you require these, I am available to also investigate them for an additional charge, the additional work will be mentioned further into the report. The orders chosen have been based on reducing the make span or number of late jobs, with the largest reduction being the option chosen.

The recommendation chosen has been analysed against numerous other possibilities to ensure that the best is chosen out of those calculated.

Key Terms

In table 1, terms and definitions can be found of words used throughout this report.

Term	Description	Mathematics
Make Span	The latest completion time for all of the jobs	Cmax = the maximum completion of all the jobs C_max⁡=max⁡{Cii=1,n}
Mean Flow Time	The average time of a schedule that a job spends within a machine	F̅=1n∑i=1nFi
Lateness	How late a job is completed after its due date	Li=Ci-di Li=Completion of job – Due Date of job
Lower Bound	The theoretical lowest value of the make span	LB=Maxtotal processing time for each machine+Minsum of the machines at the left hand side of the bottleneck machine+Minsum of the machines at the right hand side of the bottleneck machine

Table 1: Terms and Definitions

Recommendations

It is recommended;

• That for the Metal facility the NEH Heuristic is used for finding the quickest time to complete the jobs. However, as this takes a long time the next best way to find the order is C+B+J & J+P+F if time is limited.
• That for the Casting facility the jobs are ordered by combining C+P & P+A.
• That for the Branding facility the jobs are ordered by Processing Time.
• Time for working from the metal facility is transferred to Castors and Branding, approximately at least 3 twelve-hour days to Castors and 1 twelve-hour day to Branding.

Heuristics and Information

Lower Bound

The lower bound is a theoretical value for the make span, which indicates the value the make span could be but, in most cases, this value can never be reached. The actual lowest make span should be equal or larger than this value but should be close to it. It is calculated by finding the total processing time of each machine, then find the maximum of those values (The Bottleneck Machine). Add the maximum value, to the minimum value from each machine on the left of the bottleneck machine and the minimum value from each machine on the right of the bottleneck machine.

General Sorting

For general sorting, the sort function in excel is all that is used. The jobs and machines are highlighted, then either sorting smallest to largest processing time or in case of the last machine largest to smallest as well. However, before sorting for the next machine the control data is re-entered so that the jobs are ordered just by one machine and not by one then another. 

Johnson’s Rule

Johnson's rule is a method of scheduling jobs in two machines. It reduces the amount of idle time between the machines, which helps to reduce the make span, finding the optimal sequence, its primary goal. This is done by listing the jobs and their times at each machine. Then select the job with the shortest activity time. If that activity time is for the first Machine, then schedule the job first. If that activity time is for the second machine, then the job is scheduled last. If the value is tied place the job at the top. Eliminate the shortest job from further consideration. Repeat, until the centre of the job schedule is reached, and all jobs are ordered.

Campbell, Dudek and Smith (CDS) Heuristic

The CDS heuristic is a multiple application of Johnson’s algorithm. It first generates (m - 1) sub- problems, where m is the number of stages. Johnson’s algorithm is then applied to each of the (m - 1) sub- problems to produce a make span. The sub-schedule that generates the best make span among all (m - 1) sub-schedules is used as the order for the entire problem. This means when there are more than 2 machines, adding machines in the facility together there by creates 2 machines, so Johnson’s Rule can be used. The new processing time is found by adding the processing times of the individual machines together to find the processing time of the new combined machine.

Nawaz-Enscore-Ham (NEH) Heuristic

When the current lowest make span is above 5% of the lower bound, Nawaz-Enscore-Ham (NEH) Heuristic can be used to potentially lower the make span even more. This is a constructive heuristic. It consists of sorting the n jobs in descending order of their total processing times, then taking the first two jobs and schedule them to minimise the partial make span as if there were only these two jobs. Then choose the smallest make span of the two possibilities, e.g. A+B or B+A, when chosen, insert the next job at the place, which minimises the next partial make span. E.g. If A+B had the smallest Make span previously when added the next job it would C+A+B, A+C+B and A+B+C, with the smallest make span being chosen and moving to the next job. The process is continued until all jobs have been included. With this heuristic if multiple routes have the same value for the make span both/all routes should be explored however, with the limited time you have given to investigate the first route in the list was picked if you would like all routes analysing I can investigate further for a fee.

Metals

It is recommended that jobs in the Metals section are ordered by NEH as 3 out of 5 times they produced the best result, however if the job order need to be found quickly the jobs should be ordered by C+B+J & J+P+F, as before NEH was performed on 3 of the Example extracts, C+B+J & J+P+F, gave the best make span.

Results

The summary of the key results of performing multiple scheduling techniques on the data extracts is shown in table 2.

	Make Span
Seed	Control	Ordered by Machine C	Ordered by Machine B	Ordered by Machine J	Ordered by Machine P	Ordered by Machine F	Machine F (Large to Small)	Ordered by  C & F	Ordered by C+B & P+F	Ordered by CBJ & JPF	Ordered by CBJP & BJPF	Ordered by CBJPF	NEH
9279	212	210	210	205	222	214	204	192	186	185	194	223	-
9284	197	190	204	213	200	209	198	192	187	179	194	209	-
9289	197	183	183	190	215	224	187	185	185	184	184	206	176
9294	191	196	205	205	205	202	199	184	181	181	183	197	175
9299	205	196	205	210	203 Table 2: Metals Seeds	204	196	184	184	183	185	205	179

Method

For metals, the 15 jobs are all taking around 3 weeks to complete which is in the 4-week scheduled plan 

The metal order 9279 and 9284 were both ordered by machine C+B+J & J+P+F being combined then sorted, as this has the smallest make span, this used the Campbell, Dudek and Smith (CDS) heuristic.

Order 9289, 9294 and 9299 also were analysed through CDS but when comparing to the lower bound they were 6%, 6% and 7% above the lower bound respectively, so Nawaz-Enscore-Ham (NEH) Heuristic.

Both processes are evaluating the total processing times (make span) and the smallest value of the make span is found. Whether it be the combining 2, 3, 4 or 5 machines to determine the order of all the jobs. Then ordering the jobs by the one or multiple machine/s processing times which in turn changes the make span.

Castors

It is recommended that jobs in the Castors section are ordered by C+P & P+A, as 5 out of 5 extracts analysed showed that ordering by C+P & P+A gave the lowest make span.

Results

The summary of the key results of performing multiple scheduling techniques on the data extracts is shown in table 3.

	Make Span
Seed	Control	Ordered by Machine C	Ordered by Machine P	Ordered by Machine A	Machine A (Large to Small)	Ordered by C & A	Ordered by C+P & P+A	Ordered by C+P+A
9279	164	175	186	186	162	154	150	175
9284	169	164	191	193	163	164	159	172
9289	186	180	180	207	186	180	177	187
9294	181	176	204	206	179	175	170	188
9299	173	171	197	199	173	170	163	181

Table 3: Castors Seeds

Method

For castors, the 15 jobs are all taking between 4 and 4.44 weeks, (around 0 to 3 days late) to complete, but were 3 times out of 5, over the 4-week scheduled plan

All casting orders were ordered by machine C+P & P+A being combined then sorted, as this has the smallest make span, this used the Campbell, Dudek and Smith (CDS) heuristic.

The process evaluates the processing times when completing CDS and the total processing times, make span when picking the most optimal process.

Branding

It is recommended that jobs in the Branding section are Processing Times as this creates the least number of late jobs as all the make spans are the same due there only being 1 machine, so another area would have to have been analysed when picking the best scheduling process.

Results

 

The summary of the key results of performing multiple scheduling techniques on the data extracts is shown in table 4.

 

	Number of Late Jobs
Seed	Control	Processing Time	Due date
9279	10	7	10
9284	9	8	11
9289	10	8	13
9294	7	8	11
9299	8	8	10

Table 4: Branding Seeds

Method

For castors, the 15 jobs are all taking between 3.9 and 4.2 weeks (around 0 to 1 day late) to complete. However, were 3 times out of 5, over the 4-week scheduled plan.

The castor seed 9294 was best when kept at the control order, as this produced the lowest number of late jobs. Though all other seeds when analysed had the lowest number of late jobs when ordered by processing time, reducing the number of late jobs by 0 to 3.

As the last seeds had two different orders resulting with the same number of late jobs, the total lateness was evaluated. With the value being 255 with the control and -11 when ordered by processing times, therefore processing time was picked as the best result as the lateness value was negative.

Further Recommendations

The recommendations of moving shifts from metals to castors and branding are built on the fact that all jobs within metals were completed in 3 weeks leaving 5, 12 hour working days that can be used complete other jobs within the factory.

Five 12-hour days is equivalent to 60 hours and with the maximum number of hours casting was late by was 17, equalling a maximum of 3 days late and branding being a maximum of 6 hours over the schedule (1 day). 4 working days from metals can be moved to castors and branding. 3 to castors and 1 to branding.

This may increase cost on operating the machines but in the long run will be cost effective as the number of complaints about late deliveries will be reduced, and in turn, may mean companies will use Kart Co in the future to produce their goods.

This will also still leave 1 day of production which could be moved to any facility or stay with metals if it is needed, giving extra time to ensure that the jobs are not delayed.

Further Work

To further improve the facilities and manufacturing processing time, more procedures and work can be investigated at a fee. Other techniques like NEH can be carried out on those seeds which were within 5% of the lower bound to see whether it can be reduced even further so it is closer to the lower bound.

A technique called branch and bound can be implemented, but this will take time as it will assess nearly every possibility but will find the most optimal solution.  

If Kart Co is willing to invest, exploring all possibilities could also be investigated but this would take a long time as with 15 jobs it would be 15! Possibilities (1,307,624,368,000) so this would need to be coded, to be able to work. This would mean a tool would be required to be created that the company would be able to use in the future.

Conclusion

To conclude, more working time is needed to be put into castors and branding as metals are being complete easily within the 4-week plan. Along with the recent increase in complaints about late deliveries this should be strongly considered, as it would help reduce the number of grievances.

The recommendation will also mean that no extra staff will have to be hired as staff can be transferred to other sections when metals are not in use, saving on expenses.

NEH, C+P & P+A and Processing Time should be used to order the jobs that will be scheduled in Metals, Castors and Branding, respectively.

The further work discussed earlier in the report should also be considered as this would also further reduce the time taken to produce the elements in each facility.