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A Fuzzy Goal Programming Model For Production Planning in Furniture Company
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Procedia Computer Science 00 (2018) 000–000
Procedia Computer Science 135 (2018) 544–552
3rd International Conference on Computer Science and Computational Intelligence 2018 3rd International Conference on Computer Science and Computational Intelligence 2018
A Fuzzy Goal Programming Model For Production Planning in Furniture Company A Fuzzy Goal Programming Model For Production Planning in Furniture Company Siti Komsiyaha, Meilianab, Hasegaf Ekaputera Centika a Siti Komsiyaha, Meilianab, Hasegaf Ekaputera Centika a
Mathematics Department, School of Computer Science, Bina Nusantara University, Jakarta, Indonesia 11480 b Computer Science Department, School of Computer Science, Bina Nusantara University, Jakarta, a Mathematics Department, School of Computer Science, Bina Nusantara University, Jakarta, Indonesia 11480 Indonesia 11480 b Computer Science Department, School of Computer Science, Bina Nusantara University, Jakarta, Indonesia 11480 a
Abstract In the industrial world, there is always a problem of supply chain and always be for a long time. Production planning is one of its Abstract stages. Almost all of companies want to make production process efficient and optimized with minimum expenses while still meet the market This paper presentofproduction planning problem in afurniture company with planning different is operational In thewith industrial world,demand. there is always a problem supply chain and always be for long time. Production one of its constraint, including capacity, and quantity of rawand materials. Thewith fuzzyminimum goal programming appliedstill to stages. Almost all ofproduction companiestime, wantwarehouse to make production process efficient optimized expenses while maximize cost production and minimize raw materials In this paper was conducted at the furniture meet with the the profit, marketminimize demand. the Thisproduction paper present planning problemcost. in furniture company with different operational company Arte Jaya which aims help company for decision making to production model by using fuzzy constraint,CV. including production time,towarehouse capacity, and quantity of regarding raw materials. The fuzzyplanning goal programming applied to goal programming. Method suitable forcost CV.and Arteminimize Jaya because the objective can bewas adapted to preferences of the maximize the profit,This minimize theis production raw materials cost.function In this paper conducted at the furniture company. TheArte result obtained decision problem about theplanning amountmodel of production is company CV. Jaya which application aims to helpprogram companyforforsolving decision makingmaking regarding to production by usingthat fuzzy exact and still meetThis with the market demand. TheArte application program that has function been created able totosimplify the of user's goal programming. Method is suitable for CV. Jaya because the objective can beisadapted preferences the performance CV. Arte Jaya. application program for solving decision making problem about the amount of production that is company. Theofresult obtained exact and still meet with the market demand. The application program that has been created is able to simplify the user's performance of CV. Arte Jaya. © 2018 The Authors. Published by Elsevier Ltd. © 2018 The Authors. by Elsevier Ltd. This is an open accessPublished article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection andAuthors. peer-review underby responsibility © 2018 The Published Elsevier Ltd.of the 3rd International Conference on Computer Science and Computational Selection and peer-review under responsibility of the 3rd International Conference on Computer Science and Computational Intelligence 2018. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Intelligence 2018. Selection and peer-review under responsibility of the 3rd International Conference on Computer Science and Computational Keywords: Supply chain, Decision making, Production planning, Fuzzy goal programming Intelligence 2018. Keywords: Supply chain, Decision making, Production planning, Fuzzy goal programming
1877-0509 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection peer-review under responsibility of the 3rdLtd. International Conference on Computer Science and Computational Intelligence 2018. 1877-0509and © 2018 The Authors. Published by Elsevier This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the 3rd International Conference on Computer Science and Computational Intelligence 2018. 1877-0509 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the 3rd International Conference on Computer Science and Computational Intelligence 2018. 10.1016/j.procs.2018.08.207
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1. Introduction In today’s era, business competition is no longer between the companies but does involve some production planning problem. The production planning problem involves the cost factor in which from these factors it will determine the income or benefits to be obtained from the company. Thus it need for good coordination between suppliers and companies to be able to satisfy the customer in terms of both quality and cost which when the supplier provides the raw material with good quality and as well as the cost, the manufacturing company will produce products of good quality and low production costs so the products can satisfy the consumers in terms of both quality and price. With the cost of raw materials can be moved up and down, a company must be smart in making decisions what products will be produced and what quantities which is exact but still meet the needs of the market. Taking into the cost factor allows the company to obtain the maximum benefit with minimum cost expenditures. Production planning is a complicated task that requires cooperation among multiple functional units in any organization, Therefore, new tools for production planning are required that consider these issues [5]. To achieve this, in this paper use fuzzy goal programming in decision making problems to determine production plans. Some researchers has already applied a fuzzy goal programming that is Gupta and Bhattacharjee [1] presents two weighted fuzzy goal programming methods to solve multiobjective goal programming problem , Li et al. [2] applied the fuzzy goal programming with multiple priorities via generalized varying-domain optimization method. Reddy et al. [3] use fuzzy goal programming approaches in production centers selection problem for a manufacturing firm of supply chain under production centers’ uncertainty and demand uncertainty. And Singh et al. [4] presents fuzzy goal programming approach to multiobjective linear plus linear fractional programming problem. A case study in this paper was conducted CV Arte Jaya which is a company engaged in the production of furniture. Until now CV. Arte Jaya using prediction of the amount of production but using that method often spending exceeded budget it should be. CV. Arte Jaya want to achieve a target or the goals that the company afloat and not lose even lose customers. Target which to be achieved by CV. Arte Jaya is the minimum target profit to be obtained, the maximum target expenditure of raw material costs, and the maximum target production expenses. Thus, in this paper use fuzzy goal programming to perform troubleshooting on the amount of production in order to obtain the optimal results and meet the targets or criteria of CV. Arte Jaya.. 2. Literature Review 2.1
Fuzzy Goal Programming (FGP)
The FGP problem formulated as [6] : Find 𝑥𝑥 ∗ Maximize λ To satisfy: 𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥) ≥ λ 𝐴𝐴𝐴𝐴 ≤ 𝑏𝑏, 𝑥𝑥 ≥ 0 where the fuzzy membership function each objective functions are : if 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝑓𝑓𝑖𝑖 then 1 , 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑈𝑈𝑖𝑖 −𝐹𝐹𝑖𝑖 (𝑥𝑥)
𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥) = {𝑈𝑈 −𝑓𝑓 𝑖𝑖
If 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≥ 𝑓𝑓𝑖𝑖 , then
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
, 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 < 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝑈𝑈𝑖𝑖
(3)
0 , 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≥ 𝑈𝑈𝑖𝑖
1 , 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≥ 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝐹𝐹𝑖𝑖 (𝑥𝑥)−𝐿𝐿𝑖𝑖
𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥) = {𝑓𝑓
(1) (2)
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 −𝐿𝐿 𝑖𝑖
, 𝐿𝐿𝑖𝑖 < 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
0 , 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝐿𝐿𝑖𝑖
(4)
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3
The membership function form of equation (3) and (4) can be draw as following Fig 1.
1
1
𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥)
0 0
a
𝐹𝐹𝑖𝑖 (𝑥𝑥)
𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥)
𝑈𝑈𝑖𝑖
0 0
a
𝐹𝐹𝑖𝑖 (𝑥𝑥)
𝐿𝐿𝑖𝑖
𝑓𝑓𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚
Fig 1. Membership function form
The greater the value of λ, the fuzzy membership value for each function of the objective function will be greater so that the solution obtained near the optimal value. 2.2
Fuzzy Goal Programming Model
From observations and interviews with the CV. Arte Jaya the following model is a model to be applied in the CV. Arte Jaya: Goal function: Max 𝜆𝜆 Constraint function: 𝑏𝑏𝑏𝑏𝑏𝑏 𝑞𝑞 ∑𝑛𝑛𝑖𝑖 𝑏𝑏𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∗ 𝑥𝑥𝑖𝑖 − ∑𝑛𝑛𝑖𝑖 𝑃𝑃𝑖𝑖 ∗ 𝑥𝑥𝑖𝑖 − ∑𝑛𝑛𝑖𝑖 ∑𝑚𝑚 ∗ 𝑥𝑥𝑖𝑖 − ∑𝑘𝑘 𝑓𝑓𝑘𝑘 ∗ 𝑔𝑔𝑘𝑘 − (𝑍𝑍 ∗ − 𝑍𝑍̅)𝜆𝜆 ≥ 𝑍𝑍̅ 𝑗𝑗 𝑏𝑏𝑖𝑖𝑖𝑖 𝑏𝑏𝑏𝑏𝑏𝑏 ∑𝑛𝑛𝑖𝑖 ∑𝑚𝑚 ∗ 𝑥𝑥𝑖𝑖 + (𝐵𝐵̅ − 𝐵𝐵 ∗ )𝜆𝜆 ≤ 𝐵𝐵̅ 𝑗𝑗 𝑏𝑏𝑖𝑖𝑖𝑖 𝑛𝑛 ∑𝑖𝑖 𝑃𝑃𝑖𝑖 ∗ 𝑥𝑥𝑖𝑖 + (𝑃𝑃̅ − 𝑃𝑃 ∗ )𝜆𝜆 ≤ 𝑃𝑃̅ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ≤ ∑𝑛𝑛𝑖𝑖 𝑡𝑡𝑖𝑖 ∗ 𝑥𝑥𝑖𝑖 ≤ 𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∑𝑛𝑛𝑖𝑖(𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖𝑖𝑖−1 ) ∗ 𝐿𝐿𝑖𝑖 ≤ 𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑚𝑚 ∑𝑛𝑛𝑖𝑖 ∑𝑚𝑚 𝑗𝑗 𝑟𝑟𝑖𝑖𝑖𝑖 ≤ ∑𝑗𝑗 𝑐𝑐𝑗𝑗 𝑛𝑛 𝑛𝑛 𝑚𝑚𝑚𝑚𝑚𝑚 ∑𝑖𝑖 𝑥𝑥𝑖𝑖 ≥ ∑𝑖𝑖 𝑥𝑥𝑖𝑖 ∑𝑛𝑛𝑖𝑖 𝑥𝑥𝑖𝑖 ≤ ∑𝑛𝑛𝑖𝑖 𝑥𝑥𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚
Where : Index symbol i : Index of types of product (1,2,3,…,n) j : Index of types of material (kaca, triplek 18 mm,…) k : Index of worker (1,2,3,…q) w : Index of time period (1,2,3,…) Parameter symbol 𝑥𝑥𝑖𝑖 : Quantity of product i : Selling price of product i 𝑏𝑏𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 : Production cost of product i 𝑃𝑃𝑖𝑖 𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑖𝑖𝑖𝑖 : Purchase cost of material j to produce product i : Cost of worker k 𝑓𝑓𝑘𝑘
(5) (6) (7) (8) (9) (10) (11) (12) (13)
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𝑔𝑔𝑘𝑘 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑡𝑡𝑖𝑖 𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥𝑖𝑖𝑖𝑖−1 𝐿𝐿𝑖𝑖 𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑖𝑖𝑖𝑖 𝑐𝑐𝑗𝑗 𝑥𝑥𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 𝑍𝑍 ∗ 𝑍𝑍̅ 𝐵𝐵∗ 𝐵𝐵̅ 𝑃𝑃 ∗ 𝑃𝑃̅
547
: Worker k : Time required to manufacture products i : Minimal capacity of production time : Maximal capacity of production time : Quantity of product i which has been produced from the previous period and remained in storage : Spacious storage of products i : Available storage capacity : Quantity of material j which is used to manufacture of product i : The capacity of raw materials j available in the warehouse : The minimum production quantity for product I which specified by the company : The maximum production quantity for product I which specified by the company : The Maximum profit which obtained by using linear programming : The Minimum profit value which want to be obtained by company : The minimum value of raw material which obtained by using linear programming : The Maximum value of raw material which determined by company : The minimum value of production cost which obtained by using linear programming. : The Maximum value of production cost which determined by company
3. Material and Method 3.1
Data Analysis
The data used in this paper are primary data or data taken directly from the CV. Arte Jaya. Data is collected by observation and direct interview with owner CV. Arte Jaya. Data being collected includes data products sold by CV. Arte Jaya, the data of raw materials, data on the use of raw materials for a product, the data of purchase of raw materials, the production time of a product, production cost data for a product, the data of sales price of a product, the operating hours of employees, labor costs, and warehouse area. 3.2
Fuzzy Goal Programming Methods
The data obtained from CV. Arte Jaya then be analyzed in which will build a model of mathematical equations that will be the base in solving optimization problems on the proper production quantity and in accordance with the criteria of CV. Arte Jaya using Fuzzy Goal Programming. Once the mathematical equation model has been constructed then the number of production optimization problems can be solved. The steps in Fuzzy Goal Programming is : a) Choose products that want to be optimized. b) Entering input values for each product like product selling prices, costs of production, the minimum amount of production, and the maximum amount of production. c) Input three levels of aspiration or goal function to be achieved. d) Go to the first calculation phase, seeking optimal value for the third factor that is maximize profits, minimize costs of raw materials, and minimize production costs. If there is no an optimal solution to the first calculation phase then go back to step b. e) Comparing the three level 3 aspirations with the optimal value goal factors. If all three levels of aspiration meets the requirements then phase calculation using Fuzzy Goal Programming can be done. If there is an aspiration level which does not meet the requirements then go back to step c. f) Establish a Fuzzy Goal Programming calculation model. g) Entry into the calculation phase Fuzzy Goal Programming. Fuzzy Goal Programming calculations model will yield a solution that will be displayed on the screen. When Fuzzy Goal Programming model not produce a solution then go back to step b. A flowchart for fuzzy goal programming model for production planning can be seen in Fig 2
Siti Komsiyah et al. / Procedia Computer 135 (2018) 544–552 Author name / Procedia Computer Science Science 00 (2018) 000–000
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c
5
Fig 2. Flowchart for fuzzy goal programming.
4. Simulation Result and Discussion The simulation show that the firm wants to produce products : student desk, cabinet, work table, bed divan, and wardrobe. The following data taken by the company: Table 1 Input Data for Simulation No
Product Name (x)
Selling Price (IDR)
Production Cost (IDR)
Min Production (Unit)
Max Production (Unit)
1
student desk
8.500.000
700.000
0
10
2
cabinet
10.000.000
800.000
5
7
3
work table bed divan wardrobe
5.000.000
600.000
0
10
8.000.000
750.000
2
10
36.000.000
2.000.000
3
8
4 5
6
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Suppose the company has a target for minimum profit to be obtained is IDR. 70.000.000, -, target for maximum cost of raw material to be spent is IDR. 150.000.000, - and targets for maximum production costs to be spent is IDR. 30.000.000, The results of data processing Goal function: 1).Maximize of Profit 8500 𝑥𝑥1 + 10000 𝑥𝑥2 + 5000 𝑥𝑥3 + 8000 𝑥𝑥4 + 36000 𝑥𝑥5 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5 +50000 ________ _____ − 3905 𝑥𝑥1 + 3860 𝑥𝑥2 + 2125 𝑥𝑥3 + 4080 𝑥𝑥4 + 20270 𝑥𝑥5 − 50000 2). Minimize of Material Cost 650 𝑥𝑥1 1020 𝑥𝑥1 + 1275 𝑥𝑥2 + 765 𝑥𝑥3 + 1020 𝑥𝑥4 + 3060 𝑥𝑥5 + 1170 𝑥𝑥5 775 x1 + 465 x2 + 310 x3 + 775 𝑥𝑥4 + 375 𝑥𝑥4 + 1000 𝑥𝑥5 500 𝑥𝑥4 + 3500 𝑥𝑥5 1200 𝑥𝑥1 + 500 𝑥𝑥4 + 5000 𝑥𝑥5 250 𝑥𝑥1 + 3600 𝑥𝑥2 + 1200 𝑥𝑥3 __ + 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5
3). Minimize of Production Cost 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5
Constraint function: 1). Production time => 24 ≤ 𝑥𝑥1 + 0.5 𝑥𝑥2 + 0.5 𝑥𝑥3 + 𝑥𝑥4 + 3.125 𝑥𝑥5 ≤ 25 2). Inventory =>0.78 𝑥𝑥1 + 1.5 𝑥𝑥2 + 1.2 𝑥𝑥3 + 2 𝑥𝑥4 + 1.5 𝑥𝑥5 ≤ 250 3). Materials Glass => 𝑥𝑥1 ≤ 100 triplex 18 mm => 4 𝑥𝑥1 + 5 𝑥𝑥2 + 3 𝑥𝑥3 + 4 𝑥𝑥4 + 12 𝑥𝑥5 ≤ 600 triplex 15 mm => 6 𝑥𝑥5 ≤ 300 triplex 12 mm => 5 𝑥𝑥1 + 3 𝑥𝑥2 + 2 𝑥𝑥3 + 5 𝑥𝑥4 ≤ 600 triplex 9 mm => 3 𝑥𝑥4 + 8 𝑥𝑥5 ≤ 300 melaminto => 12 𝑥𝑥1 + 5 𝑥𝑥4 + 35 𝑥𝑥5 ≤ 600 paint => 𝑥𝑥1 + 2 𝑥𝑥4 + 20 𝑥𝑥5 ≤ 300 HPL => 6 𝑥𝑥2 + 2 𝑥𝑥3 ≤ 300 4). Quantity of product student desk (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 5). Quantity of product cabinet (min and max) => 5 ≤ 𝑥𝑥2 ≤ 7 6). Quantity of product work table (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 7). Quantity of product bed divan (min and max) => 2 ≤ 𝑥𝑥2 ≤ 10 8). Quantity of product wardrobe (min and max) => 3 ≤ 𝑥𝑥5 ≤ 8 The optimum profit obtained using linear programming is: 𝑍𝑍 ∗ = 𝐼𝐼𝐼𝐼𝐼𝐼. 111.664.800, − The optimum material cost obtained using linear programming is: 𝐵𝐵∗ = 𝐼𝐼𝐼𝐼𝐼𝐼. 107.866.875, −
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The optimum production cost obtained using linear programming is: 𝑃𝑃 ∗ = 𝐼𝐼𝐼𝐼𝐼𝐼. 17.980.000, −
7
For aspiration levels of profit is acceptable because the value of 𝑍𝑍 ∗ ≥ 𝑍𝑍̅ which is 𝑅𝑅𝑅𝑅. 111.664.800 ≥ 𝑅𝑅𝑅𝑅. 70.000.000 , for aspiration levels of material costs may also be accepted as the value of 𝐵𝐵 ∗ ≤ 𝐵𝐵̅ is 𝑅𝑅𝑅𝑅. 107.866.875 ≤ 𝑅𝑅𝑅𝑅. 150.000.000, and for aspiration level of the production cost can also be accepted because of the value of 𝑃𝑃∗ ≤ 𝑃𝑃̅ is 𝑅𝑅𝑅𝑅. 17.980.000 ≤ 𝑅𝑅𝑅𝑅. 30.000.000. With all the criteria are met then the next step is to form a fuzzy goal programming models. The following is a fuzzy goal programming models: Goal function: Maximize 𝜆𝜆 Constraint function: 3905 𝑥𝑥1 + 3860 𝑥𝑥2 + 2125 𝑥𝑥3 + 4080 𝑥𝑥4 + 20270 𝑥𝑥5 − 50000 − (111664.8 − 70000) ≥ 70000 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5 + (150000 − 74230)𝜆𝜆 ≤ 150000 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5 + (30000 − 11500) 𝜆𝜆 ≤ 30000 Production time => 24 ≤ 𝑥𝑥1 + 0.5 𝑥𝑥2 + 0.5 𝑥𝑥3 + 𝑥𝑥4 + 3.125 𝑥𝑥5 ≤ 25 Inventory=>0.78 𝑥𝑥1 + 1.5 𝑥𝑥2 + 1.2 𝑥𝑥3 + 2 𝑥𝑥4 + 1.5 𝑥𝑥5 ≤ 250 Materials Glass => 𝑥𝑥1 ≤ 100 triplex 18 mm => 4 𝑥𝑥1 + 5 𝑥𝑥2 + 3 𝑥𝑥3 + 4 𝑥𝑥4 + 12 𝑥𝑥5 ≤ 600 triplex 15 mm => 6 𝑥𝑥5 ≤ 300 triplex 12 mm => 5 𝑥𝑥1 + 3 𝑥𝑥2 + 2 𝑥𝑥3 + 5 𝑥𝑥4 ≤ 600 triplex 9 mm => 3 𝑥𝑥4 + 8 𝑥𝑥5 ≤ 300 melaminto => 12 𝑥𝑥1 + 5 𝑥𝑥4 + 35 𝑥𝑥5 ≤ 600 paint => 𝑥𝑥1 + 2 𝑥𝑥4 + 20 𝑥𝑥5 ≤ 300 HPL => 6 𝑥𝑥2 + 2 𝑥𝑥3 ≤ 300 Quantity of product student desk (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 Quantity of product cabinet (min and max) => 5 ≤ 𝑥𝑥2 ≤ 7 Quantity of product work table (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 Quantity of product bed divan (min and max) => 2 ≤ 𝑥𝑥2 ≤ 10 Quantity of product wardrobe (min and max) => 3 ≤ 𝑥𝑥5 ≤ 8
Then simplify result becomes,
Goal function: Maximize of 𝜆𝜆 Constraint function: 3905 𝑥𝑥1 + 3860 𝑥𝑥2 + 2125 𝑥𝑥3 + 4080 𝑥𝑥4 + 20270 𝑥𝑥5 − 41664.8 𝜆𝜆 ≥ 120000 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5 + 75770 𝜆𝜆 ≤ 150000 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5 + 88500 ≤ 30000 Production time => 24 ≤ 𝑥𝑥1 + 0.5 𝑥𝑥2 + 0.5 𝑥𝑥3 + 𝑥𝑥4 + 3.125 𝑥𝑥5 ≤ 25 Inventory=>0.78 𝑥𝑥1 + 1.5 𝑥𝑥2 + 1.2 𝑥𝑥3 + 2 𝑥𝑥4 + 1.5 𝑥𝑥5 ≤ 250 Materials Glass => 𝑥𝑥1 ≤ 100 triplex 18 mm => 4 𝑥𝑥1 + 5 𝑥𝑥2 + 3 𝑥𝑥3 + 4 𝑥𝑥4 + 12 𝑥𝑥5 ≤ 600 triplex 15 mm => 6 𝑥𝑥5 ≤ 300 triplex 12 mm => 5 𝑥𝑥1 + 3 𝑥𝑥2 + 2 𝑥𝑥3 + 5 𝑥𝑥4 ≤ 600 triplex 9 mm => 3 𝑥𝑥4 + 8 𝑥𝑥5 ≤ 300 melaminto => 12 𝑥𝑥1 + 5 𝑥𝑥4 + 35 𝑥𝑥5 ≤ 600 paint => 𝑥𝑥1 + 2 𝑥𝑥4 + 20 𝑥𝑥5 ≤ 300 HPL => 6 𝑥𝑥2 + 2 𝑥𝑥3 ≤ 300 Quantity of product student desk (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 Quantity of product cabinet (min and max) => 5 ≤ 𝑥𝑥2 ≤ 7
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Quantity of product work table (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 Quantity of product bed divan (min dan max) => 2 ≤ 𝑥𝑥2 ≤ 10 Quantity of product wardrobe (min dan max) => 3 ≤ 𝑥𝑥5 ≤ 8
The following are the results of calculations using fuzzy goal programming: Student desk = 𝑥𝑥1 = 0 Cabinet = 𝑥𝑥2 = 5 Work table = 𝑥𝑥3 = 0 Bed divan = 𝑥𝑥4 = 3 Wardrobe = 𝑥𝑥5 = 6 with 𝜆𝜆 = 0.76667
Thus, the amount of production that is right and appropriate and meets the desired objective function of the company criteria is to produce 5 pieces kabinet, 3 pieces bed divan, and 6 pieces wardrobe with the profit to be gained is: = 3905 𝑥𝑥1 + 3860 𝑥𝑥2 + 2125 𝑥𝑥3 + 4080 𝑥𝑥4 + 20270 𝑥𝑥5 − 50000 = 3905 ∗ 0 + 3860 ∗ 5 + 2125 ∗ 0 + 4080 ∗ 3 + 20270 ∗ 6 − 50000 = 103160 => IDR. 103.160.000,00 Material costs to be spent to produce that is: = 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5 = 3895 ∗ 0 + 5340 ∗ 5 + 2275 ∗ 0 + 3170 ∗ 3 + 13730 ∗ 6 = 118590 => IDR. 118.590.000,00
Production costs to be spent to produce that is: = 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5 = 700 ∗ 0 + 800 ∗ 5 + 600 ∗ 0 + 750 ∗ 3 + 2000 ∗ 6 = 18250 => IDR. 18.250.000,00
The results using fuzzy goal programming, all the goal function desirable by the company are meet where the profit exceed the profit desired by the company that is IDR. 103.160.000,00 ≥ IDR. 70.000.000,00 , material costs incurred is not exceed the material cost desired by the company that is IDR.118.590.000,00 ≤ 𝐼𝐼𝐼𝐼𝐼𝐼. 150.000.000,00 , and production cost incurred is not exceed the production cost desired by the company IDR. 18.250.000,00 ≤ IDR. 30.000.000,00. If the product defect 1 or 2 pieces, the results were still meet with targets or criteria by CV. Arte Jaya.
In the production process there must be a defective product because it can arise from a variety of factors. Defective products will reduce the gains from company which led to the profit targets to be achieved can’t be realized. However, by using fuzzy goal programming companies can have the possibility for the target / goal determined by the company remains to be realized. Suppose that there are defective products on the kabinet as much as 2 pieces. The total cost of the production process of the kabinet is the total cost of materials for the production of kabinet + total production costs for the production of the kabinet, Total production process to produce kabinet (2 pieces) = 2 ∗ (5340) + 2 ∗ (800) = 12280 =>IDR. 12.280.000,00 Total profit becomes = IDR. 103.160.000,00 – IDR. 12.280.000,00 = IDR. 90.880.000,00 Total profit still exceeded the company's targets to be achieved, which is IDR. 70,000,000.00. Therefore, by using fuzzy goal programming defective product can still be handled. Simulation program display result to input product and display Result to Solution Result Using Fuzzy Goal Programming shown in Fig 3a and Fig 3b below :
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Fig 3a. Application Display to Input Products
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Fig 3b Application Display to Solution Result Using Fuzzy Goal Programming
5. Conclusion Based on the result analysis and discussion can be concluded that the fuzzy goal programming models can be applied to CV. Arte Jaya where the level of aspiration can be adaptable to the preferences of the CV. Arte Jaya. Furthermore, by using fuzzy goal programming, losses from defective products can still be handled where the profit is still meet with the criteria from CV. Arte Jaya. The difference of profit from the practice in the company and FGP model is obtained IDR 33.160.000,00 or saving about 32.14 %. And moreover the application program that has been created is very helpful for user performance of CV. Arte Jaya in calculating using fuzzy goal programming to determine the amount of furniture product that is exact and in accordance with the preferences of company in the production planning model. Acknowledgements We thank to Bina Nusantara University was supported this research and who provided insight and expertise that greatly assisted this paper References [1] Gupta, M., and Bhattacharjee, D. (2012). “ Two Weighted Fuzzy Goal Programming Methods to Solve Multiobjective Goal Programming Problem ”. Journal of Applied Mathematics (2012) [2] Li S., Y.Yang, C.Teng. (2004). “ Fuzzy Goal Programming With Multiple Priorities via Generalized Varying-Domain Optimization Method. IEEE Transactions on Fuzzy System ”. Journal Mathematics 12 : 597 – 605 [3] Reddy, B. C. M., Reddy, K. H., and Reddy, C. N. M. (2012), “ Production Centers Selection Problem for A Manufacturing Firm of Supply Chain Under Production Centers’ Uncertainty and Demand Uncertainty by using Fuzzy Goal Programming Approaches “. International Journal of Research in Mechanical Engineering & Technology 2 [4] Singh P., Kumar, S. T., Singh, R. K. (2011). “Fuzzy Goal Programming Approach to Multiobjective Linear Plus Linear Fractional Programming Problem”. Journal of Applied Mathematics [5] Bhargava, A.K, Singh, S.R, and Divya Bansal. (2015). “Fuzzy Goal Programming Techniques for Production Planning in Industry“. International Journal of Computer Applications Technology and Research 4 (2) : 92-96 [6] Tampinongkol, F. F., Rindengan, A. J., Latumakulita, L. A. (2015) “Aplikasi Fuzzy Goal Programming (Studi Kasus: UD. Sinar Sakti Manado)”. Jurnal Matematika 4 (2)
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Procedia Computer Science 00 (2018) 000–000
Procedia Computer Science 135 (2018) 544–552
3rd International Conference on Computer Science and Computational Intelligence 2018 3rd International Conference on Computer Science and Computational Intelligence 2018
A Fuzzy Goal Programming Model For Production Planning in Furniture Company A Fuzzy Goal Programming Model For Production Planning in Furniture Company Siti Komsiyaha, Meilianab, Hasegaf Ekaputera Centika a Siti Komsiyaha, Meilianab, Hasegaf Ekaputera Centika a
Mathematics Department, School of Computer Science, Bina Nusantara University, Jakarta, Indonesia 11480 b Computer Science Department, School of Computer Science, Bina Nusantara University, Jakarta, a Mathematics Department, School of Computer Science, Bina Nusantara University, Jakarta, Indonesia 11480 Indonesia 11480 b Computer Science Department, School of Computer Science, Bina Nusantara University, Jakarta, Indonesia 11480 a
Abstract In the industrial world, there is always a problem of supply chain and always be for a long time. Production planning is one of its Abstract stages. Almost all of companies want to make production process efficient and optimized with minimum expenses while still meet the market This paper presentofproduction planning problem in afurniture company with planning different is operational In thewith industrial world,demand. there is always a problem supply chain and always be for long time. Production one of its constraint, including capacity, and quantity of rawand materials. Thewith fuzzyminimum goal programming appliedstill to stages. Almost all ofproduction companiestime, wantwarehouse to make production process efficient optimized expenses while maximize cost production and minimize raw materials In this paper was conducted at the furniture meet with the the profit, marketminimize demand. the Thisproduction paper present planning problemcost. in furniture company with different operational company Arte Jaya which aims help company for decision making to production model by using fuzzy constraint,CV. including production time,towarehouse capacity, and quantity of regarding raw materials. The fuzzyplanning goal programming applied to goal programming. Method suitable forcost CV.and Arteminimize Jaya because the objective can bewas adapted to preferences of the maximize the profit,This minimize theis production raw materials cost.function In this paper conducted at the furniture company. TheArte result obtained decision problem about theplanning amountmodel of production is company CV. Jaya which application aims to helpprogram companyforforsolving decision makingmaking regarding to production by usingthat fuzzy exact and still meetThis with the market demand. TheArte application program that has function been created able totosimplify the of user's goal programming. Method is suitable for CV. Jaya because the objective can beisadapted preferences the performance CV. Arte Jaya. application program for solving decision making problem about the amount of production that is company. Theofresult obtained exact and still meet with the market demand. The application program that has been created is able to simplify the user's performance of CV. Arte Jaya. © 2018 The Authors. Published by Elsevier Ltd. © 2018 The Authors. by Elsevier Ltd. This is an open accessPublished article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection andAuthors. peer-review underby responsibility © 2018 The Published Elsevier Ltd.of the 3rd International Conference on Computer Science and Computational Selection and peer-review under responsibility of the 3rd International Conference on Computer Science and Computational Intelligence 2018. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Intelligence 2018. Selection and peer-review under responsibility of the 3rd International Conference on Computer Science and Computational Keywords: Supply chain, Decision making, Production planning, Fuzzy goal programming Intelligence 2018. Keywords: Supply chain, Decision making, Production planning, Fuzzy goal programming
1877-0509 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection peer-review under responsibility of the 3rdLtd. International Conference on Computer Science and Computational Intelligence 2018. 1877-0509and © 2018 The Authors. Published by Elsevier This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the 3rd International Conference on Computer Science and Computational Intelligence 2018. 1877-0509 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the 3rd International Conference on Computer Science and Computational Intelligence 2018. 10.1016/j.procs.2018.08.207
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1. Introduction In today’s era, business competition is no longer between the companies but does involve some production planning problem. The production planning problem involves the cost factor in which from these factors it will determine the income or benefits to be obtained from the company. Thus it need for good coordination between suppliers and companies to be able to satisfy the customer in terms of both quality and cost which when the supplier provides the raw material with good quality and as well as the cost, the manufacturing company will produce products of good quality and low production costs so the products can satisfy the consumers in terms of both quality and price. With the cost of raw materials can be moved up and down, a company must be smart in making decisions what products will be produced and what quantities which is exact but still meet the needs of the market. Taking into the cost factor allows the company to obtain the maximum benefit with minimum cost expenditures. Production planning is a complicated task that requires cooperation among multiple functional units in any organization, Therefore, new tools for production planning are required that consider these issues [5]. To achieve this, in this paper use fuzzy goal programming in decision making problems to determine production plans. Some researchers has already applied a fuzzy goal programming that is Gupta and Bhattacharjee [1] presents two weighted fuzzy goal programming methods to solve multiobjective goal programming problem , Li et al. [2] applied the fuzzy goal programming with multiple priorities via generalized varying-domain optimization method. Reddy et al. [3] use fuzzy goal programming approaches in production centers selection problem for a manufacturing firm of supply chain under production centers’ uncertainty and demand uncertainty. And Singh et al. [4] presents fuzzy goal programming approach to multiobjective linear plus linear fractional programming problem. A case study in this paper was conducted CV Arte Jaya which is a company engaged in the production of furniture. Until now CV. Arte Jaya using prediction of the amount of production but using that method often spending exceeded budget it should be. CV. Arte Jaya want to achieve a target or the goals that the company afloat and not lose even lose customers. Target which to be achieved by CV. Arte Jaya is the minimum target profit to be obtained, the maximum target expenditure of raw material costs, and the maximum target production expenses. Thus, in this paper use fuzzy goal programming to perform troubleshooting on the amount of production in order to obtain the optimal results and meet the targets or criteria of CV. Arte Jaya.. 2. Literature Review 2.1
Fuzzy Goal Programming (FGP)
The FGP problem formulated as [6] : Find 𝑥𝑥 ∗ Maximize λ To satisfy: 𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥) ≥ λ 𝐴𝐴𝐴𝐴 ≤ 𝑏𝑏, 𝑥𝑥 ≥ 0 where the fuzzy membership function each objective functions are : if 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝑓𝑓𝑖𝑖 then 1 , 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑈𝑈𝑖𝑖 −𝐹𝐹𝑖𝑖 (𝑥𝑥)
𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥) = {𝑈𝑈 −𝑓𝑓 𝑖𝑖
If 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≥ 𝑓𝑓𝑖𝑖 , then
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
, 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 < 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝑈𝑈𝑖𝑖
(3)
0 , 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≥ 𝑈𝑈𝑖𝑖
1 , 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≥ 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝐹𝐹𝑖𝑖 (𝑥𝑥)−𝐿𝐿𝑖𝑖
𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥) = {𝑓𝑓
(1) (2)
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 −𝐿𝐿 𝑖𝑖
, 𝐿𝐿𝑖𝑖 < 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
0 , 𝐹𝐹𝑖𝑖 (𝑥𝑥) ≤ 𝐿𝐿𝑖𝑖
(4)
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3
The membership function form of equation (3) and (4) can be draw as following Fig 1.
1
1
𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥)
0 0
a
𝐹𝐹𝑖𝑖 (𝑥𝑥)
𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝜇𝜇𝑓𝑓𝑓𝑓 (𝑥𝑥)
𝑈𝑈𝑖𝑖
0 0
a
𝐹𝐹𝑖𝑖 (𝑥𝑥)
𝐿𝐿𝑖𝑖
𝑓𝑓𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚
Fig 1. Membership function form
The greater the value of λ, the fuzzy membership value for each function of the objective function will be greater so that the solution obtained near the optimal value. 2.2
Fuzzy Goal Programming Model
From observations and interviews with the CV. Arte Jaya the following model is a model to be applied in the CV. Arte Jaya: Goal function: Max 𝜆𝜆 Constraint function: 𝑏𝑏𝑏𝑏𝑏𝑏 𝑞𝑞 ∑𝑛𝑛𝑖𝑖 𝑏𝑏𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∗ 𝑥𝑥𝑖𝑖 − ∑𝑛𝑛𝑖𝑖 𝑃𝑃𝑖𝑖 ∗ 𝑥𝑥𝑖𝑖 − ∑𝑛𝑛𝑖𝑖 ∑𝑚𝑚 ∗ 𝑥𝑥𝑖𝑖 − ∑𝑘𝑘 𝑓𝑓𝑘𝑘 ∗ 𝑔𝑔𝑘𝑘 − (𝑍𝑍 ∗ − 𝑍𝑍̅)𝜆𝜆 ≥ 𝑍𝑍̅ 𝑗𝑗 𝑏𝑏𝑖𝑖𝑖𝑖 𝑏𝑏𝑏𝑏𝑏𝑏 ∑𝑛𝑛𝑖𝑖 ∑𝑚𝑚 ∗ 𝑥𝑥𝑖𝑖 + (𝐵𝐵̅ − 𝐵𝐵 ∗ )𝜆𝜆 ≤ 𝐵𝐵̅ 𝑗𝑗 𝑏𝑏𝑖𝑖𝑖𝑖 𝑛𝑛 ∑𝑖𝑖 𝑃𝑃𝑖𝑖 ∗ 𝑥𝑥𝑖𝑖 + (𝑃𝑃̅ − 𝑃𝑃 ∗ )𝜆𝜆 ≤ 𝑃𝑃̅ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ≤ ∑𝑛𝑛𝑖𝑖 𝑡𝑡𝑖𝑖 ∗ 𝑥𝑥𝑖𝑖 ≤ 𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∑𝑛𝑛𝑖𝑖(𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖𝑖𝑖−1 ) ∗ 𝐿𝐿𝑖𝑖 ≤ 𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑚𝑚 ∑𝑛𝑛𝑖𝑖 ∑𝑚𝑚 𝑗𝑗 𝑟𝑟𝑖𝑖𝑖𝑖 ≤ ∑𝑗𝑗 𝑐𝑐𝑗𝑗 𝑛𝑛 𝑛𝑛 𝑚𝑚𝑚𝑚𝑚𝑚 ∑𝑖𝑖 𝑥𝑥𝑖𝑖 ≥ ∑𝑖𝑖 𝑥𝑥𝑖𝑖 ∑𝑛𝑛𝑖𝑖 𝑥𝑥𝑖𝑖 ≤ ∑𝑛𝑛𝑖𝑖 𝑥𝑥𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚
Where : Index symbol i : Index of types of product (1,2,3,…,n) j : Index of types of material (kaca, triplek 18 mm,…) k : Index of worker (1,2,3,…q) w : Index of time period (1,2,3,…) Parameter symbol 𝑥𝑥𝑖𝑖 : Quantity of product i : Selling price of product i 𝑏𝑏𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 : Production cost of product i 𝑃𝑃𝑖𝑖 𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑖𝑖𝑖𝑖 : Purchase cost of material j to produce product i : Cost of worker k 𝑓𝑓𝑘𝑘
(5) (6) (7) (8) (9) (10) (11) (12) (13)
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𝑔𝑔𝑘𝑘 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑡𝑡𝑖𝑖 𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥𝑖𝑖𝑖𝑖−1 𝐿𝐿𝑖𝑖 𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑖𝑖𝑖𝑖 𝑐𝑐𝑗𝑗 𝑥𝑥𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 𝑍𝑍 ∗ 𝑍𝑍̅ 𝐵𝐵∗ 𝐵𝐵̅ 𝑃𝑃 ∗ 𝑃𝑃̅
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: Worker k : Time required to manufacture products i : Minimal capacity of production time : Maximal capacity of production time : Quantity of product i which has been produced from the previous period and remained in storage : Spacious storage of products i : Available storage capacity : Quantity of material j which is used to manufacture of product i : The capacity of raw materials j available in the warehouse : The minimum production quantity for product I which specified by the company : The maximum production quantity for product I which specified by the company : The Maximum profit which obtained by using linear programming : The Minimum profit value which want to be obtained by company : The minimum value of raw material which obtained by using linear programming : The Maximum value of raw material which determined by company : The minimum value of production cost which obtained by using linear programming. : The Maximum value of production cost which determined by company
3. Material and Method 3.1
Data Analysis
The data used in this paper are primary data or data taken directly from the CV. Arte Jaya. Data is collected by observation and direct interview with owner CV. Arte Jaya. Data being collected includes data products sold by CV. Arte Jaya, the data of raw materials, data on the use of raw materials for a product, the data of purchase of raw materials, the production time of a product, production cost data for a product, the data of sales price of a product, the operating hours of employees, labor costs, and warehouse area. 3.2
Fuzzy Goal Programming Methods
The data obtained from CV. Arte Jaya then be analyzed in which will build a model of mathematical equations that will be the base in solving optimization problems on the proper production quantity and in accordance with the criteria of CV. Arte Jaya using Fuzzy Goal Programming. Once the mathematical equation model has been constructed then the number of production optimization problems can be solved. The steps in Fuzzy Goal Programming is : a) Choose products that want to be optimized. b) Entering input values for each product like product selling prices, costs of production, the minimum amount of production, and the maximum amount of production. c) Input three levels of aspiration or goal function to be achieved. d) Go to the first calculation phase, seeking optimal value for the third factor that is maximize profits, minimize costs of raw materials, and minimize production costs. If there is no an optimal solution to the first calculation phase then go back to step b. e) Comparing the three level 3 aspirations with the optimal value goal factors. If all three levels of aspiration meets the requirements then phase calculation using Fuzzy Goal Programming can be done. If there is an aspiration level which does not meet the requirements then go back to step c. f) Establish a Fuzzy Goal Programming calculation model. g) Entry into the calculation phase Fuzzy Goal Programming. Fuzzy Goal Programming calculations model will yield a solution that will be displayed on the screen. When Fuzzy Goal Programming model not produce a solution then go back to step b. A flowchart for fuzzy goal programming model for production planning can be seen in Fig 2
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c
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Fig 2. Flowchart for fuzzy goal programming.
4. Simulation Result and Discussion The simulation show that the firm wants to produce products : student desk, cabinet, work table, bed divan, and wardrobe. The following data taken by the company: Table 1 Input Data for Simulation No
Product Name (x)
Selling Price (IDR)
Production Cost (IDR)
Min Production (Unit)
Max Production (Unit)
1
student desk
8.500.000
700.000
0
10
2
cabinet
10.000.000
800.000
5
7
3
work table bed divan wardrobe
5.000.000
600.000
0
10
8.000.000
750.000
2
10
36.000.000
2.000.000
3
8
4 5
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Suppose the company has a target for minimum profit to be obtained is IDR. 70.000.000, -, target for maximum cost of raw material to be spent is IDR. 150.000.000, - and targets for maximum production costs to be spent is IDR. 30.000.000, The results of data processing Goal function: 1).Maximize of Profit 8500 𝑥𝑥1 + 10000 𝑥𝑥2 + 5000 𝑥𝑥3 + 8000 𝑥𝑥4 + 36000 𝑥𝑥5 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5 +50000 ________ _____ − 3905 𝑥𝑥1 + 3860 𝑥𝑥2 + 2125 𝑥𝑥3 + 4080 𝑥𝑥4 + 20270 𝑥𝑥5 − 50000 2). Minimize of Material Cost 650 𝑥𝑥1 1020 𝑥𝑥1 + 1275 𝑥𝑥2 + 765 𝑥𝑥3 + 1020 𝑥𝑥4 + 3060 𝑥𝑥5 + 1170 𝑥𝑥5 775 x1 + 465 x2 + 310 x3 + 775 𝑥𝑥4 + 375 𝑥𝑥4 + 1000 𝑥𝑥5 500 𝑥𝑥4 + 3500 𝑥𝑥5 1200 𝑥𝑥1 + 500 𝑥𝑥4 + 5000 𝑥𝑥5 250 𝑥𝑥1 + 3600 𝑥𝑥2 + 1200 𝑥𝑥3 __ + 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5
3). Minimize of Production Cost 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5
Constraint function: 1). Production time => 24 ≤ 𝑥𝑥1 + 0.5 𝑥𝑥2 + 0.5 𝑥𝑥3 + 𝑥𝑥4 + 3.125 𝑥𝑥5 ≤ 25 2). Inventory =>0.78 𝑥𝑥1 + 1.5 𝑥𝑥2 + 1.2 𝑥𝑥3 + 2 𝑥𝑥4 + 1.5 𝑥𝑥5 ≤ 250 3). Materials Glass => 𝑥𝑥1 ≤ 100 triplex 18 mm => 4 𝑥𝑥1 + 5 𝑥𝑥2 + 3 𝑥𝑥3 + 4 𝑥𝑥4 + 12 𝑥𝑥5 ≤ 600 triplex 15 mm => 6 𝑥𝑥5 ≤ 300 triplex 12 mm => 5 𝑥𝑥1 + 3 𝑥𝑥2 + 2 𝑥𝑥3 + 5 𝑥𝑥4 ≤ 600 triplex 9 mm => 3 𝑥𝑥4 + 8 𝑥𝑥5 ≤ 300 melaminto => 12 𝑥𝑥1 + 5 𝑥𝑥4 + 35 𝑥𝑥5 ≤ 600 paint => 𝑥𝑥1 + 2 𝑥𝑥4 + 20 𝑥𝑥5 ≤ 300 HPL => 6 𝑥𝑥2 + 2 𝑥𝑥3 ≤ 300 4). Quantity of product student desk (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 5). Quantity of product cabinet (min and max) => 5 ≤ 𝑥𝑥2 ≤ 7 6). Quantity of product work table (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 7). Quantity of product bed divan (min and max) => 2 ≤ 𝑥𝑥2 ≤ 10 8). Quantity of product wardrobe (min and max) => 3 ≤ 𝑥𝑥5 ≤ 8 The optimum profit obtained using linear programming is: 𝑍𝑍 ∗ = 𝐼𝐼𝐼𝐼𝐼𝐼. 111.664.800, − The optimum material cost obtained using linear programming is: 𝐵𝐵∗ = 𝐼𝐼𝐼𝐼𝐼𝐼. 107.866.875, −
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The optimum production cost obtained using linear programming is: 𝑃𝑃 ∗ = 𝐼𝐼𝐼𝐼𝐼𝐼. 17.980.000, −
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For aspiration levels of profit is acceptable because the value of 𝑍𝑍 ∗ ≥ 𝑍𝑍̅ which is 𝑅𝑅𝑅𝑅. 111.664.800 ≥ 𝑅𝑅𝑅𝑅. 70.000.000 , for aspiration levels of material costs may also be accepted as the value of 𝐵𝐵 ∗ ≤ 𝐵𝐵̅ is 𝑅𝑅𝑅𝑅. 107.866.875 ≤ 𝑅𝑅𝑅𝑅. 150.000.000, and for aspiration level of the production cost can also be accepted because of the value of 𝑃𝑃∗ ≤ 𝑃𝑃̅ is 𝑅𝑅𝑅𝑅. 17.980.000 ≤ 𝑅𝑅𝑅𝑅. 30.000.000. With all the criteria are met then the next step is to form a fuzzy goal programming models. The following is a fuzzy goal programming models: Goal function: Maximize 𝜆𝜆 Constraint function: 3905 𝑥𝑥1 + 3860 𝑥𝑥2 + 2125 𝑥𝑥3 + 4080 𝑥𝑥4 + 20270 𝑥𝑥5 − 50000 − (111664.8 − 70000) ≥ 70000 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5 + (150000 − 74230)𝜆𝜆 ≤ 150000 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5 + (30000 − 11500) 𝜆𝜆 ≤ 30000 Production time => 24 ≤ 𝑥𝑥1 + 0.5 𝑥𝑥2 + 0.5 𝑥𝑥3 + 𝑥𝑥4 + 3.125 𝑥𝑥5 ≤ 25 Inventory=>0.78 𝑥𝑥1 + 1.5 𝑥𝑥2 + 1.2 𝑥𝑥3 + 2 𝑥𝑥4 + 1.5 𝑥𝑥5 ≤ 250 Materials Glass => 𝑥𝑥1 ≤ 100 triplex 18 mm => 4 𝑥𝑥1 + 5 𝑥𝑥2 + 3 𝑥𝑥3 + 4 𝑥𝑥4 + 12 𝑥𝑥5 ≤ 600 triplex 15 mm => 6 𝑥𝑥5 ≤ 300 triplex 12 mm => 5 𝑥𝑥1 + 3 𝑥𝑥2 + 2 𝑥𝑥3 + 5 𝑥𝑥4 ≤ 600 triplex 9 mm => 3 𝑥𝑥4 + 8 𝑥𝑥5 ≤ 300 melaminto => 12 𝑥𝑥1 + 5 𝑥𝑥4 + 35 𝑥𝑥5 ≤ 600 paint => 𝑥𝑥1 + 2 𝑥𝑥4 + 20 𝑥𝑥5 ≤ 300 HPL => 6 𝑥𝑥2 + 2 𝑥𝑥3 ≤ 300 Quantity of product student desk (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 Quantity of product cabinet (min and max) => 5 ≤ 𝑥𝑥2 ≤ 7 Quantity of product work table (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 Quantity of product bed divan (min and max) => 2 ≤ 𝑥𝑥2 ≤ 10 Quantity of product wardrobe (min and max) => 3 ≤ 𝑥𝑥5 ≤ 8
Then simplify result becomes,
Goal function: Maximize of 𝜆𝜆 Constraint function: 3905 𝑥𝑥1 + 3860 𝑥𝑥2 + 2125 𝑥𝑥3 + 4080 𝑥𝑥4 + 20270 𝑥𝑥5 − 41664.8 𝜆𝜆 ≥ 120000 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5 + 75770 𝜆𝜆 ≤ 150000 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5 + 88500 ≤ 30000 Production time => 24 ≤ 𝑥𝑥1 + 0.5 𝑥𝑥2 + 0.5 𝑥𝑥3 + 𝑥𝑥4 + 3.125 𝑥𝑥5 ≤ 25 Inventory=>0.78 𝑥𝑥1 + 1.5 𝑥𝑥2 + 1.2 𝑥𝑥3 + 2 𝑥𝑥4 + 1.5 𝑥𝑥5 ≤ 250 Materials Glass => 𝑥𝑥1 ≤ 100 triplex 18 mm => 4 𝑥𝑥1 + 5 𝑥𝑥2 + 3 𝑥𝑥3 + 4 𝑥𝑥4 + 12 𝑥𝑥5 ≤ 600 triplex 15 mm => 6 𝑥𝑥5 ≤ 300 triplex 12 mm => 5 𝑥𝑥1 + 3 𝑥𝑥2 + 2 𝑥𝑥3 + 5 𝑥𝑥4 ≤ 600 triplex 9 mm => 3 𝑥𝑥4 + 8 𝑥𝑥5 ≤ 300 melaminto => 12 𝑥𝑥1 + 5 𝑥𝑥4 + 35 𝑥𝑥5 ≤ 600 paint => 𝑥𝑥1 + 2 𝑥𝑥4 + 20 𝑥𝑥5 ≤ 300 HPL => 6 𝑥𝑥2 + 2 𝑥𝑥3 ≤ 300 Quantity of product student desk (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 Quantity of product cabinet (min and max) => 5 ≤ 𝑥𝑥2 ≤ 7
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Quantity of product work table (min and max) => 0 ≤ 𝑥𝑥2 ≤ 10 Quantity of product bed divan (min dan max) => 2 ≤ 𝑥𝑥2 ≤ 10 Quantity of product wardrobe (min dan max) => 3 ≤ 𝑥𝑥5 ≤ 8
The following are the results of calculations using fuzzy goal programming: Student desk = 𝑥𝑥1 = 0 Cabinet = 𝑥𝑥2 = 5 Work table = 𝑥𝑥3 = 0 Bed divan = 𝑥𝑥4 = 3 Wardrobe = 𝑥𝑥5 = 6 with 𝜆𝜆 = 0.76667
Thus, the amount of production that is right and appropriate and meets the desired objective function of the company criteria is to produce 5 pieces kabinet, 3 pieces bed divan, and 6 pieces wardrobe with the profit to be gained is: = 3905 𝑥𝑥1 + 3860 𝑥𝑥2 + 2125 𝑥𝑥3 + 4080 𝑥𝑥4 + 20270 𝑥𝑥5 − 50000 = 3905 ∗ 0 + 3860 ∗ 5 + 2125 ∗ 0 + 4080 ∗ 3 + 20270 ∗ 6 − 50000 = 103160 => IDR. 103.160.000,00 Material costs to be spent to produce that is: = 3895 𝑥𝑥1 + 5340𝑥𝑥2 + 2275 𝑥𝑥3 + 3170 𝑥𝑥4 + 13730 𝑥𝑥5 = 3895 ∗ 0 + 5340 ∗ 5 + 2275 ∗ 0 + 3170 ∗ 3 + 13730 ∗ 6 = 118590 => IDR. 118.590.000,00
Production costs to be spent to produce that is: = 700 𝑥𝑥1 + 800 𝑥𝑥2 + 600 𝑥𝑥3 + 750 𝑥𝑥4 + 2000 𝑥𝑥5 = 700 ∗ 0 + 800 ∗ 5 + 600 ∗ 0 + 750 ∗ 3 + 2000 ∗ 6 = 18250 => IDR. 18.250.000,00
The results using fuzzy goal programming, all the goal function desirable by the company are meet where the profit exceed the profit desired by the company that is IDR. 103.160.000,00 ≥ IDR. 70.000.000,00 , material costs incurred is not exceed the material cost desired by the company that is IDR.118.590.000,00 ≤ 𝐼𝐼𝐼𝐼𝐼𝐼. 150.000.000,00 , and production cost incurred is not exceed the production cost desired by the company IDR. 18.250.000,00 ≤ IDR. 30.000.000,00. If the product defect 1 or 2 pieces, the results were still meet with targets or criteria by CV. Arte Jaya.
In the production process there must be a defective product because it can arise from a variety of factors. Defective products will reduce the gains from company which led to the profit targets to be achieved can’t be realized. However, by using fuzzy goal programming companies can have the possibility for the target / goal determined by the company remains to be realized. Suppose that there are defective products on the kabinet as much as 2 pieces. The total cost of the production process of the kabinet is the total cost of materials for the production of kabinet + total production costs for the production of the kabinet, Total production process to produce kabinet (2 pieces) = 2 ∗ (5340) + 2 ∗ (800) = 12280 =>IDR. 12.280.000,00 Total profit becomes = IDR. 103.160.000,00 – IDR. 12.280.000,00 = IDR. 90.880.000,00 Total profit still exceeded the company's targets to be achieved, which is IDR. 70,000,000.00. Therefore, by using fuzzy goal programming defective product can still be handled. Simulation program display result to input product and display Result to Solution Result Using Fuzzy Goal Programming shown in Fig 3a and Fig 3b below :
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Fig 3a. Application Display to Input Products
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Fig 3b Application Display to Solution Result Using Fuzzy Goal Programming
5. Conclusion Based on the result analysis and discussion can be concluded that the fuzzy goal programming models can be applied to CV. Arte Jaya where the level of aspiration can be adaptable to the preferences of the CV. Arte Jaya. Furthermore, by using fuzzy goal programming, losses from defective products can still be handled where the profit is still meet with the criteria from CV. Arte Jaya. The difference of profit from the practice in the company and FGP model is obtained IDR 33.160.000,00 or saving about 32.14 %. And moreover the application program that has been created is very helpful for user performance of CV. Arte Jaya in calculating using fuzzy goal programming to determine the amount of furniture product that is exact and in accordance with the preferences of company in the production planning model. Acknowledgements We thank to Bina Nusantara University was supported this research and who provided insight and expertise that greatly assisted this paper References [1] Gupta, M., and Bhattacharjee, D. (2012). “ Two Weighted Fuzzy Goal Programming Methods to Solve Multiobjective Goal Programming Problem ”. Journal of Applied Mathematics (2012) [2] Li S., Y.Yang, C.Teng. (2004). “ Fuzzy Goal Programming With Multiple Priorities via Generalized Varying-Domain Optimization Method. IEEE Transactions on Fuzzy System ”. Journal Mathematics 12 : 597 – 605 [3] Reddy, B. C. M., Reddy, K. H., and Reddy, C. N. M. (2012), “ Production Centers Selection Problem for A Manufacturing Firm of Supply Chain Under Production Centers’ Uncertainty and Demand Uncertainty by using Fuzzy Goal Programming Approaches “. International Journal of Research in Mechanical Engineering & Technology 2 [4] Singh P., Kumar, S. T., Singh, R. K. (2011). “Fuzzy Goal Programming Approach to Multiobjective Linear Plus Linear Fractional Programming Problem”. Journal of Applied Mathematics [5] Bhargava, A.K, Singh, S.R, and Divya Bansal. (2015). “Fuzzy Goal Programming Techniques for Production Planning in Industry“. International Journal of Computer Applications Technology and Research 4 (2) : 92-96 [6] Tampinongkol, F. F., Rindengan, A. J., Latumakulita, L. A. (2015) “Aplikasi Fuzzy Goal Programming (Studi Kasus: UD. Sinar Sakti Manado)”. Jurnal Matematika 4 (2)