INTRODUCTION

As we know, the COVID-19 pandemic has had a negative impact on UMKM, especially in the East Java region, including a decline in sales, difficulties in raw materials, capital problems to production process barriers. Various efforts to increase sales have been carried out both through mentoring and research, among others by trying to do online marketing (Imron & Nurdian, 2021), and tempe chips packaging designs (Jessica, at al. 2015).

To maintain its existence, especially during the COVID-19 pandemic, companies are required to be more observant in dealing with all existing problems and other problems that may arise. Analysis related to the composition of the number of production units, the efficiency of the production capital used, determining the selling price and optimizing other resources owned by the company need to be carried out carefully and thoroughly. Determining the composition of the number of production units in a company is the main thing that needs attention for planning, because this has a direct effect on the company's profit.

Determining the composition of the number of production units to maximize company profits by looking at the company's limited resources can be solved using a linear programming model. There are several ways to solve problems with a linear programming model, including the simplex method. The Simplex method is the solution of a linear program model whose solution is presented in the form of a graph before which calculations were carried out to find common ground on each axis. The general procedure is to transform a descriptive situation into a linear programming problem by determining the variables, constants, objective functions, and constraints so that the problem can be presented in graphical form and the solution interpreted (Suwirmayanti, 2017).

A company and all its resources such as energy, time, money and others, in order to achieve real growth requires professional management, with the aim of obtaining maximum profit while using minimal resources (production capital). This is nothing but one of the optimization problems in operations research (Chandra, 2015).

This research was conducted in the home industry "ENY", in Ngawi Regency, East Java, which daily produces Tempe Chips, which are served in 3 packages, namely small packages, medium packages and large packages. During the COVID-19 pandemic, this company continued to carry out the production process every day, but by reducing the production volume, namely 600 small packaging units, 80 medium packaging units and 250 large packaging units. At the time of observation, information was obtained that the production volume was fixed every day, and production costs were also relatively fixed. The composition of the number of units for each type of packaging of tempe chips produced, and the selling price of the product per unit, are determined on the basis of experience while running the business, so there is no optimal planning or analysis method.

Based on this condition, the researcher wants to help the owner of the company to make a plan about determining the composition of the number of units for each type of packaging of tempe chips that must be produced so that the company obtains maximum profit, while taking into account the limited resources available.

The purpose of this study is to solve the problems faced by the company, by discussing the determination of the number of small packaging units, medium packaging and large packaging that must be produced so that the profit is maximum, calculating the maximum profit value in one day, calculating the efficiency (decrease) in resource use. limited, namely production capital and production time.

Some of the limitations and assumptions in this study are the total number of units of the three types of packaging tempe chips produced is the largest equal to the total number of units produced before the study, the total production time is the largest equal to the total production time before the study, the production capital is at most equal to the capital production before the study, the results of calculating the maximum profit value will be achieved with the assumption that all products are sold out.

The results of this study are expected to provide benefits, input and suggestions, especially for company owners in making decisions to change the composition of many production units for each type of tempe chip packaging, in order to obtain maximum profit.

The problem faced by this company is the Linear Programming Problem (LPP). Because the problem consists of 3 decision variables, the simplex method in a linear program is the right method to solve the problems experienced by the company, by decomposition of the many product units produced based on the limited resources available, with the aim of obtaining maximum profit (Christian, 2013).

Linear programming is a mathematical method with linear characteristics to find an optimal solution by maximizing or minimizing the objective function against the set of constraints (Budiasih, 2013).

The simplex method is a mathematical procedure to find the optimal solution of a linear programming problem based on an iterative process (Aulia et al., 2013).

Linear programming is an optimization technique used to obtain the most optimal solution for a real world problem. There are simplex method. The simplex method provides two methods to solve linear programming problems, namely the graphical method and the a systematic way of examining the vertices of the feasible region to determine the optimal value of the objective function. The "linprog" function in Mat Lab can be used to solve linear programming problems (Velinov & Gicev, 2018). One method that can be used to solve LPP is the Simplex Algorithm in Linear Programs (Sukanta, 2016).

The following are several studies related to the application of the simplex method for the purpose of optimizing the completion of LPP. (Rumetna et al., 2020) conducted a study entitled ï¿½Optimizing Revenue Making Banners and Billboards Using the Simplex Method (Case Study: Shiau Printing Printing Business)ï¿½, from the analysis it is stated that Shiau Printing Printing can get a maximum profit of Rp. 15,000,000,- every month. (Fardiana, 2013) conducted a study entitled "Maximizing Profits at Martabak Doni Cake Shop with the Simplex Method", the results of data analysis are that the optimal combination of inputs obtained, will provide a maximum profit of Rp. 106,817,-. (Budiyanto, Mujiharjo, & Umroh, 2017) conducted a study with the title "Profit Maximization in the Roti Bunda Bakery Company Using the Simple Method" the results of the research were optimizing the use of resources, it would be able to increase profits from Rp. 1,663,914,- per week to Rp. 2,286,049,- per week. Akram, A. Sahari, and AI Jaya conducted a study entitled "Optimizing the Bread Production Method Using Branch And Bound" method Branch And Bound is solving technique that can be used not only on the LPP, but can be applied to a wide range of issues different. This method is used together with the simplex method (Akram, Sahari, & Jaya, 2016).

METHOD

Thinking Framework

Stages in the process of solving this research problem consist of:

1. Identification of research variables, profit functions, and limiting functions,

2. Compiling the problem into a mathematical model, in this case the form of linear program modeling,

3. Changing the mathematical model into a table simplex,

4. Perform an iterative process of solving with the algorithm in the simplex method, until the optimal table is obtained,

5. Presenting the optimal solution in mathematical language,

6. Interpreting the optimal solution of the research problem.

Data Collection Procedure

Collection was carried out by direct observation to the location and interviews, which were divided into two stages, as follows:

1. Preliminary observation, this stage was carried out to see firsthand the general condition of the company, and notify the research objectives to the company owner, so that the company understands the usefulness of the research results later, besides that at this preliminary stage the researcher can give a message to prepare data related to the data needed in research,

2. Data collection, at this stage interviews and clarification of the data that have been collected prepared by the owner, with the hope that detailed and in-depth information can be obtained, related to the analysis plan to be carried out.

Data Analysis Techniques

1. Describing the Data.

In this step, the transformation process is carried out from raw data from interviews, which are all materials and resources needed in the production process, into data that is ready for mathematical modeling.

2. The Identify Variables

making:

𝑋1 = Many units small packaging should be produced

𝑋2 = Many units medium packaging should be produced

𝑋3 = Many units large packaging should be produced

3. Constraints

total unit Production: 𝑎11𝑋1 + 𝑎12𝑋2 + 𝑎13𝑋3ï¿½ 𝑏 ≤1 (aij = 1, for every i and j)

Total cost: 𝑎21𝑋1 + 𝑎22𝑋2 + 𝑎23𝑋3ï¿½ï¿½ ≤2

Total time: 𝑎31𝑋1 + 𝑎32𝑋2 + 𝑎33𝑋3ï¿½ï¿½ ≤3

Functions Advantages 𝑍 = 𝑐1𝑋1 + 𝑐2𝑋2 + 𝑐3𝑋3

mathematical models of research problems in general:

Specifies 𝑋1, 𝑋2, and 𝑋3 with constraints:

𝑎11𝑋1 + 𝑎12𝑋2 + 𝑎13𝑋3ï¿½ï¿½ ≤1 (aij = 1, for every i and j)

𝑎21𝑋1 + 𝑎22𝑋2 + 𝑎23𝑋3ï¿½ï¿½ ≤2

𝑎31𝑋1 + 𝑎32𝑋2 + 𝑎33𝑋3ï¿½ï¿½ ≤3

so that the values 𝑋1, 𝑋2, and 𝑋3 obtained will maximize the profit function 𝑍 = 𝑐1𝑋1 + 𝑐2𝑋2 + 𝑐3𝑋3. Furthermore, this mathematical model will be solved by the simplex method.

To complete the LPP the simplex method in the case of maximizing, mathematical models of LPP expressed as linear as follows:

Determine ï¿½

Which satisfy the constraint: 𝐴_𝑚_ï¿½_𝑛𝑋_𝑛_ï¿½1 (≤, =)_ï¿½₁

Optimizing [𝑍1 ï¿½ 1 = 𝐶1 ï¿½𝑛 𝑛𝑋ï¿½1].

Here are the steps to resolve the case of LPP to maximize the objective function in the simplex method:

1. Turn all obstacles to the canonical form by adding a variable (variable) Slack's. The existing slack variables are added to the target function and given a coefficient of 0, Step 1 causes the matrix A to be of size mx (n+m) and contains an identity matrix of order m. Continue by compiling the initial simplex table,

2. Determine the key column, which is to determine the incoming variable to be the new base variable.

J column is the key column ↔(𝑍𝑗 - 𝐶𝑗)>0 smallest,

3. Determine the key lines that define variables which should be out long base being replaced by a new variable basis. I line is line ↔ key index i> 0, the smallest,

4. 𝑖𝑗𝑎 called a key element, doing surgery rows: row i i new = old / 𝑖𝑗𝑎

5. Perform row operations on the other lines so that the elements sekolom the key element to 0,

6. table ↔ were optimal for all j values(𝑍𝑗 - 𝐶𝑗)>0,

7. If the table is not optimal return to step 2. (Sriwasito, Surarso, & Sarwoko, 2011).

To avoid data analysis with a lengthy iteration process, this study used POM QM for Windows software to help solve research problems.

RESULTS AND DISCUSSION

Data Description

Following is a description of the research data, which has been processed and reduced from the raw data obtained from interviews and observations.

Table 1

Observation Data

NO	NEED FOR	SMALL PACKAGING		MEDIUM PACKAGING		LARGE PACKAGING
		MANY UNITS = 600		MANY UNITS = 80		MANY UNITS = 250
		VOLUME	PRICE	VOLUME	PRICE	VOLUME	PRICE
1	Soybeans	36 kg	306.000	3 kg	25.500	45 kg	382.500
2	Ragi Tempe	4 ounces	20.000	2 ounces	4.000	5 ounces	25.000
3	Cooking Oil	36 L	396.000	3 L	33.000	45 L	495.000
4	Flour	36 kg	288.000	3 kg	24.000	45 kg	360.000
5	Garlic	8 ounces	20.000	2 ounces	4.000	1 kg	25.000
6	Pecan	4 ounces	12.000	2 ounces	6.000	5 ounces	15.000
7	Salt	4 ounces	4.000	2 ounces	2.000	4 ounces	4.000
8	Seasonings	5 gr	5.000	4 gr	4.000	5 gr	5.000
9	Eggseggs	12	12.000	0.5 kg	12.000	15 eggs	15.000
10	Coriander	3 ounces	6.000	2 ounces	4.000	3 ounces	6.000
10	WAGES		250.000		100.000		250.000
11	PACKAGING FEE		200.000		90.000		200.000
TOTAL		Rp 1.519.000		Rp 308.500		Rp 1.782.500
CAPITAL/UNIT		Rp 2.550		Rp 3.900		Rp 7.150
SELLING PRICE/UNIT		Rp 5.000		Rp 6.500		Rp 12.500
TOTAL TIME		950 MINUTES		729 MINUTES		1.010 MINUTES

Table 2

Description of Data Per Unit of Production

No	Type Packaging	Many Units	Total Cost	Total Time	Cost per Unit	Selling price per Unit	Time per Unit	Total Selling	Profit per unit
1	Small	600	1.519.000	950	2.550	5000	1,58	3.000.000	2.450
2	Medium	80	308.500	729	3.900	6.500	9,11	520.000	2.600
3	Large	250	1.782.500	1.010	7.150	12.500	4,04	3.125.000	5.350
TOTAL		930	3.610.000	2.689	-	-	-	6.645.000	-

Table 7

Iteration

Cj	Basic Variables	Quantity	2450 X₁	2600 X₂	5350 X₃	0 slack 1	0 slack 2	0 slack 3
Iteration 1
0	slack 1	930	1	1	1	1	0	0
0	slack 2	3,610,000	2,550	3,900	7,150	0	1	0
0	slack 3	2,689	1.58	9.11	4.04	0	0	1
	zj	0	0	0	0	0	0	0
	cj-zj		2,450	2,600	5,350	0	0	0
Iteration 2
0	slack 1	425.1049	0.6434	0.4545	0	1	-0.0001	0
5350	X3	504.8951	0.3566	0.5455	1	0	0.0001	0
0	slack 3	649.2238	0.1392	6.9064	0	0	-0.0006	1
	zj	2,701,188.75	1908.04	2918.18	5350	0	.75	0
	cj-zj		541.958	-318.1818	0	0	-0.7483	0
Iteration 3
2450	X1	660.7609	1	0.7065	0	1.5543	-0.0002	0
5350	X3	269.2391	0	0.2935	1	-0.5543	0.0002	0
0	slack 3	557.2717	0	6.808	0	ï¿½ï¿½ -0.2163	-0.0005	1
	zj	3,059,293.5	2450	3301.09	5350	842.39	.63	0
	cj-zj		0	-701.087	0	ï¿½ -842.3913	-0.6304	0

CONCLUSION

To be able to achieve a maximum profit of 3,058,600 rupiah, the company must produce 661 units of small packaging and 269 large packaging and the company gets an additional profit of 23,600 rupiah per production period,

Under optimal conditions, the total production cost becomes 3,608,900 rupiah, there is an efficiency (decrease) of 1,100 rupiah.

If the company runs production according to optimal conditions, the time required is 557.86 minutes less than the production time before the study.

REFERENCES

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	X1	X2	X3		RHS	Equation form
Maximize	2450	2600	5350			Max 2450X1 + 2600X2 + 5350X3
Many units	1	1	1	<=	930	X1 + X2 + X3 <= 930
Production Cost/unit	2550	3900	7150	<=	3610000	2550X1 + 3900X2 + 7150X3 <= 3610000
Production Time/unit	1,58	9,11	4,04	<=	2689	1,58X1 + 9,11X2 + 4,04X3 <= 2689

	X1	X2	X3	ï¿½	RHS	Dual
Maximize	2450	2600	5350
Constraint 1	1	2600	1	<=	930	842.39
Constraint 2	2550	3900	7150	<=	3610000	.63
Constraint 3	1.58	9.11	4.04	<=	2689	0
Solution	660.76	0	269.24		3059294.0

	Initial Condition	Final Condition	Efficiency
Production	600 units small pack ï¿½ 80 medium pack 250 units big pack Number of units 930	661 units small pack 0 medium pack 269 units big pack Number of units 930	-
Prod. Cost (Rp.)	3.610.000	3.608.900	1.100
Time (Minutes)	2.689	2.131,14	557,86
Profit (Rp.)	3.035.000	3.058.600	23.600

Basis	X₁	X₂	X₃	S₁	S₂	S₃	H
S₁	1	1	1	1	0	0	930
S₂	2.550	3.900	7.150	0	1	0	3.610.000
S₃	1,58	9,11	4,04	0	0	1	2.689
Z	-2.450	-2.600	-5.350	0	0	0	0

Variable	Status	Value
X1	Basic	660.76
X2	NONBasic	0
X3	Basic	269.24
slack 1	NONBasic	0
slack 2	NONBasic	0
slack 3	Basic	557.27
Optimal Value (Z)		3059294.0