Optimization Of Production Through Linear Programming: An Essay.

1. Discuss and formulate the scenarios relevant for answering different questions
2. Demonstrate the interpretation of computer output to answer different questions 

The use of information technology has affected the business process in all aspect. But use of information technology in Supply chain management, has totally chained the face of doing business, all the jargons like E-Commerce, EDI, ERP, Barcode, QR code, 3PL, all are only possible after the application of computer in supply chain management. Here we will see that how optimization can be done in limited resources with the help of computer application (Szpilko, 2017).

As given in the question, the first work to John smith is the analyses the data as given in question, the normal demand of Mech. wire is around 4276 unit, and average output is around 2400 unit/month, Therefore it clear that, we have to arrange all the resources to maximize the profit from the given output. The main difference in output and demand is due to the reason, that, the plant’s machine utilization is around 63%. By analysis and calculation we try to find, how we can optimize the available machine and resources in give condition. We will solve this problem with the help of excel solver in Microsoft Excel 2016 (Štefan Kudlá?, 2017). 

The given situation, is suitable for linear programming, and this will be done by using excel solver, because, if we see the data we can analyze, LP is only the tools which can be used for this situation, The summary of data is as given below,

1 Next Month Order
Product	Units ordered
W0075C	1,400
W0033C	250
W0005X	1,510
W0007X	1,116

2 Standard Cost
Product	Material	Labour	Overhead	Selling Price
W0075C	$33.00	$9.90	$23.10	$100.00
W0033C	$25.00	$7.50	$17.50	$80.00
W0005X	$35.00	$10.50	$24.50	$130.00
W0007X	$75.00	$11.25	$64.75	$175.00

4 Plant Capacity
Drawing	Extrusion	Winding	Packaging
4,000	4,200	2,000	2,300

The above plant capacity data is full capacity data, but in question it is given that, the average machine utilization is 63% and 5% production is going to rework from winding, in this condition the utilized plant capacity is given as follows. 

6 Capacity utilised(Actual)
2,520	2,646	1,197	1,449

And finally the bill of labor is given as

5	Bill of labour (hours/unit)
Product	Drawing	Extrusion	Winding	Packaging
W0075C	1	1	1	1
W0033C	2	1	3	0
W0005X	0	4	0	3
W0007X	1	1	0	2

From the above situation we will formulate the linear programming which is as follows,

The actual profit /unit of wire can be taken as

Actual profit = Sell Price – (Material + labor + overhead)

The profit calculated for four different products is as follows,

W0075C	W0033C	W0005X	W0007X
$34.00	$30.00	$60.00	$25.00

Suppose, product W0075C is denoted as X₁, similarly other product, W0033C, W0005X, W0007X is X₂, X₃, and X₄ respectively, then our total profit will be calculated as

34x X₁ + 30x X₂ + 60x X₃ + 25x X₄ = Z, this will be objective which I have to maximize according to the above data.

The constraint can be given as

For plant capacity,

X₁ + 2X₂ + X₄   2520 …………(i)

X₁ + X₂ + 4X₃ + X₄   2646 …..... (ii)

Formulating the Linear Programming

X₁ + 3X₂   1197 ……………..…(iii)

X₁ + 3X₃ + 2X₄  1449 …………..(iv)

For demand capacity

X₁  1400 …………………..(v)

X₂  250 ……………………(vi)

X₃  1510…………….……..(vii)

X₄  1116, ……………..(viii)

Two more constrains is there because of commitment done by Vivian Napoli.

X₁  150 ………………(ix)

X₄  600 ………………(x)

Now to calculate the maximum capacity from the given data, we have to run the excel solver and put all the data as given in excel sheet.

Product	W0075C	W0033C	W0005X	W0007X
Units	249	250	0	600
Profit/Unit	$34.00	$30.00	$60.00	$25.00	$30,966.00
Available constraints
W0075C orders	1	0	0	0	249.00	<=	1400
W0033C orders	0	1	0	0	250.00	<=	250
W0005X orders	0	0	1	0	0.00	<=	1510
W0007X orders	0	0	0	1	600.00	<=	1116
Drawing time	1	2	0	1	1349.00	<=	2520
Extrusion time	1	1	4	1	1099.00	<=	2646
Winding time	1	3	0	0	999.00	<=	1197
Packaging time	1	0	3	2	1449.00	<=	1449
Minimum W0075C	1	0	0	0	249.00	>=	150
Minimum W0007X	0	0	0	1	600.00	>=	600

The cell given in green is calculated maximum profit for given condition. The answer report and sensitivity analysis is given in excel sheet.

If we will see the utilisation of different section we will observe that, the % utilisation of different section i.e. Drawing, Extrusion, Winding and Packaging, we see that, it is around 53%, 41%, 83% and 100% for packaging, in this condition, it is clear that, almost half of the manpower in drawing section and extrusion section is unused, If by any means If we shift the manpower to winding, and packaging section, we can increase the output,

Suppose by shifting the manpower, we have increased the rated capacity of winding and packaging, in this condition we must put the maximum value of winding and packaging.

After running the solver but putting the value 2000 and 2300 for winding and packaging, the result is as follows

Product	W0075C	W0033C	W0005X	W0007X
Units	1100	250	0	600
Profit/Unit	$34.00	$30.00	$60.00	$25.00	$59,900.00
Available constraints
W0075C orders	1	0	0	0	1100.00	<=	1400
W0033C orders	0	1	0	0	250.00	<=	250
W0005X orders	0	0	1	0	0.00	<=	1510
W0007X orders	0	0	0	1	600.00	<=	1116
Drawing time	1	2	0	1	2200.00	<=	2520
Extrusion time	1	1	4	1	1950.00	<=	2646
Winding time	1	3	0	0	1850.00	<=	2000
Packaging time	1	0	3	2	2300.00	<=	2300
Minimum W0075C	1	0	0	0	1100.00	>=	150
Minimum W0007X	0	0	0	1	600.00	>=	600

From the above solution it is clear that, if we shift the worker bay any means from drawing and extrusion department, the profit can be increase in such a way that, we can fulfil the required condition and earn maximum profit as profit $ 59,900.

The solution optimised solution suggest that, the product W0005X is not produced to maximise the profit, even profit margin for W0005X is highest, but it is also taking resources highest. The commitment done by Vivian Napoli can be easily fulfilled with the above condition,

Therefore, main problem is here is low utilisation of machine, and the mains constraints are packaging time, it is still clear from the above table is that if by any means we increase the packaging limit we can produce more with the given constraints, even if we increase the packaging by 3000, the almost all resources will be utilised and profit will be around $ 77000,

The best recommendation for john smith is that, it should stop producing W0005X, because resources are greatly utilised and any how increase the capacity of packaging department, so that maximum profit can be done without any further investment. The other recommendation is that the number of rejection should be reduced to, because this 5% will directly add to the profit margin.

Running Excel Solver

If we want to perform the sensitivity analysis for this problem, we have to develop table to identify the related information obtained from the sensitivity analysis, but sensitivity is not the report which can be presented directly to the meeting, we must change into lucrative form so that it can be presented.

The Sensitivity analysis of the above problem is given as follows,

Microsoft Excel 14.0 Sensitivity Report
Worksheet: [806856.xlsx]Sheet1
Report Created: 30-09-2018 04:08:42
Variable Cells
			Final	Reduced	Objective	Allowable	Allowable
	Cell	Name	Value	Cost	Coefficient	Increase	Decrease
	$J$4	Units W0075C	1100	0	34	1E+30	14
	$K$4	Units W0033C	250	0	30	1E+30	30
	$L$4	Units W0005X	0	-42	60	42	1E+30
	$M$4	Units W0007X	600	0	25	43	1E+30
Constraints
			Final	Shadow	Constraint	Allowable	Allowable
	Cell	Name	Value	Price	R.H. Side	Increase	Decrease
	$N$10	W0075C orders	1100	0	1400	1E+30	300
	$N$11	W0033C orders	250	30	250	50	250
	$N$12	W0005X orders	0	0	1510	1E+30	1510
	$N$13	W0007X orders	600	0	1116	1E+30	516
	$N$14	Drawing time	2200	0	2520	1E+30	320
	$N$15	Extrusion time	1950	0	2646	1E+30	696
	$N$16	Winding time	1850	0	2000	1E+30	150
	$N$17	Packaging time	2300	34	2300	150	950
	$N$18	Minimum W0075C	1100	0	150	950	1E+30
	$N$19	Minimum W0007X	600	-43	600	475	75

As per the report given above, we can analyses the situation; we can compare the production report with actual order report and can see that how much I have fulfilled the demand. In this way we can also show the way for management that which product is more necessary to produce. 

Product	Units ordered	Order Produced	Difference
W0075C	1,400	1100	300
W0033C	250	250	0
W0005X	1,510	0	1510
W0007X	1,116	600	516

The total product produce is 1950 against the order 4276, in this production; we have fulfilled the demand of product W0033C. Additionally I have fulfilled the demand of Vivian Napoli. The production of W0005X is not taken into consideration by solver, even it is the product of maximum margin, but in terms of resources, this product is grabbing too many resources, therefore for maximum profit, the product W0005X is stopped.

Further looking into the Sensitivity analysis we can see that, we have to put the actual order and find the differences the cost is gone up by $42 and profit go by $ 102, but removing it cost is decrease by $42 and we get maximum profit after removing the W0005X.

Another aspect we can analyse is that, the utilisation of resources in each department.

Department	Given Cap.	Resources consumed	Unutilised
Drawing	4000	2200	1800
Extrusion	4200	1950	2250
Winding	2000	1850	150
Packaging	2300	2300	0

From the above table we can see that, the as per capacity of plant, the resources unitised are almost 50% for drawing and extrusion, and resources for winding is around 95%, but resources for packaging is fully utilised. Therefore, packaging is the bottleneck for the operation, we must exploit and subordinate the packaging section, so that, further resources can be utilized. For fully utilising the other resources, we must increase the packaging resource up to 3000 hr.

The limit report is also providing the same thing

Microsoft Excel 14.0 Limits Report
Worksheet: [806856.xlsx]Sheet1
Report Created: 30-09-2018 04:08:42
		Objective
	Cell	Name	Value
	$N$5	Profit/Unit	? 59,900.00
		Variable			Lower	Objective		Upper	Objective
	Cell	Name	Value		Limit	Result		Limit	Result
	$J$4	Units W0075C	1100		150	27600		1100	59900
	$K$4	Units W0033C	250		0	52400		250	59900
	$L$4	Units W0005X	0		0	59900		0	59900
	$M$4	Units W0007X	600		600	59900		600	59900

The binding of constraints can also be visible in this section

Cell	Name	Cell Value	Formula	Status	Slack
$N$10	W0075C orders	1100.00	$N$10<=$P$10	Not Binding	300
$N$11	W0033C orders	250.00	$N$11<=$P$11	Binding	0
$N$12	W0005X orders	0.00	$N$12<=$P$12	Not Binding	1510
$N$13	W0007X orders	600.00	$N$13<=$P$13	Not Binding	516
$N$14	Drawing time	2200.00	$N$14<=$P$14	Not Binding	320
$N$15	Extrusion time	1950.00	$N$15<=$P$15	Not Binding	696
$N$16	Winding time	1850.00	$N$16<=$P$16	Not Binding	150
$N$17	Packaging time	2300.00	$N$17<=$P$17	Binding	0
$N$18	Minimum W0075C	1100.00	$N$18>=$P$18	Not Binding	950.00
$N$19	Minimum W0007X	600.00	$N$19>=$P$19	Binding	0.00

Here we can clearly see that the binding of data comes in W0033C order and packaging time. The order is as per demand, but packaging time is something which we can control. 

As we have seen in the table and Sensitivity report above, it clear that, the drawing department is underutilized, in this condition we should not send the temporary labor to drawing department, rather we should send it to packaging department, In fact not need of temporary worker, here, we can utilize own worker of the company to manage the packaging department, One important information from the above reports is that, if we increase packaging hour by 1, out inherited profit will rise by $34, this packaging hour can go up to, 2500 hour without any additional resources. But if we further want to increase the profitability, we have to add some winding hour also, this will give leverage to increase the profit.

Sensitivity Analysis

Conclusion

The use of excel solver becomes very common for purpose of optimization problem. The main reason behind this is easy to use Solver tools given in Microsoft excel. The use of complex mathematical concept is history, when linear programming being solved by Simplex method or any other manual method, in 1950s when linear programing first used in US military, to find the low cost diet with highest nutrition value by Jerry Cornfield, it took 2 years to find the optimized value, but today with the use of computer we can dot in two hours or less.

This is the report which can be presented by John Smith to the management. There are various other possible permutation and combination of data by which we can analyses the result obtained from solver. If manpower is given for department, we can reschedule the entire problem in better optimization using this solver.

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