Berikut merupakan contoh kasus penyelesaian linear programming menggunakan aplikasi lips.
Case 1: Mukidi Tiles, Ltd.
Paimo and Surtini spent several summers during their college years working at archaeological sites in the Kasongan, Bantul, Yogyakarta. While at those digs, they learned how to make ceramic tiles from local artisans. After college they made use of their college experiences to start a tile manufacturing firm called Mukidi Tiles, Ltd. They opened their plant in Wonogiri, where they would have convenient access to a special clay they intend to use to make a clay derivative for their tiles. Their manufacturing operation consists of a few relatively simple but precarious steps, including molding the tiles, baking, and glazing.
Paimo and Surtini plan to produce two basic types of tile for use in home bathrooms, kitchens, sunrooms, and laundry rooms. The two types of tile are a larger, singlecolored tile and a smaller, patterned tile. In the manufacturing process, the color or pattern is added before a tile is glazed. Either a single color is sprayed over the top of a baked set of tiles or a stenciled pattern is sprayed on the top of a baked set of tiles.
The tiles are produced in batches of 100. The first step is to pour the clay derivative into specially constructed molds. It takes 18 minutes to mold a batch of 100 larger tiles and 15 minutes to prepare a mold for a batch of 100 smaller tiles. The company has 60 hours available each week for molding. After the tiles are molded, they are baked in a kiln: 0.27 hour for a batch of 100 larger tiles and 0.58 hour for a batch of 100 smaller tiles. The company has 105 hours available each week for baking. After baking, the tiles are either colored or patterned and glazed. This process takes 0.16 hour for a batch of 100 larger tiles and 0.20 hour for a batch of 100 smaller tiles. Forty hours are available each week for the glazing process. Each batch of 100 large tiles requires 32.8 pounds of the clay derivative to produce, whereas each batch of smaller tiles requires 20 pounds. The company has 6,000 pounds of the clay derivative available each week.
Mukidi Tiles earns a profit of $190 for each batch of 100 of the larger tiles and $240 for each batch of 100 smaller patterned tiles. Surtini and Paimo want to know how many batches of each type of tile to produce each week to maximize profit. In addition, they have some questions about resource usage they would like answered.
- Formulate a linear programming model for Mukidi Tiles, Ltd., and determine the mix of tiles it should manufacture each week.
- Transform the model into standard form.
- Solve the linear programming model graphically.
- Determine the resources left over and not used at the optimal solution point.
- Determine the sensitivity ranges for the objective function coefficients and constraint quantity values by using the graphical solution of the model.
- For artistic reasons, Paimo and Surtini prefer to produce the smaller, patterned tiles. They also believe that in the long run, the smaller tiles will be a more successful product. What must the profit be for the smaller tiles in order for the company to produce only the smaller tiles?
- Solve the linear programming model by using the computer and verify the sensitivity ranges computed in (E)
H. Paimo believes it may be able to reduce the time required for molding to 16 minutes for a batch of larger tiles and 12 minutes for a batch of smaller tiles. How will this affect the solution?
I. The company that provides Mukidi Tile Ltd. with clay has indicated that it can deliver an additional 100 pounds each week. Should Mukidi Tile Ltd, agree to this offer?
J. Paimo is considering adding capacity to one of its kilns to provide 20 additional glazing hours per week, at a cost of $90,000. Should it make the investment?
K. The kiln for glazing had to be shut down for 3 hours, reducing the available kiln hours from 40 to 37. What effect will this have on the solution?
Case 2: Ayu Ting-ting Food Booth
Ayu Ting-ting Sumodilogo is a senior at ITS, and she’s investigating different ways to finance her final year at school. She is considering leasing a food booth outside the ITS stadium at home football games. ITS sells out every home game, and Ayu Ting-ting knows, from attending the games herself, that everyone eats a lot of food. She has to pay $1,000 per game for a booth, and the booths are not very large. Vendors can sell either food or drinks on ITS property, but not both. Only the ITS athletic department concession stands can sell both inside the stadium. She thinks slices of cheese pizza, hot dogs, and barbecue sandwiches are the most popular food items among fans and so these are the items she would sell.
Most food items are sold during the hour before the game starts and during half time; thus it will not be possible for Ayu Ting-ting to prepare the food while she is selling it. She must prepare the food ahead of time and then store it in a warming oven. For $600 she can lease a warming oven for the six-game home season. The oven has 16 shelves, and each shelf is 3 feet by 4 feet. She plans to fill the oven with the three food items before the game and then again before half time.
Ayu Ting-ting has negotiated with a local pizza delivery company to deliver 14-inch cheese pizzas twice each game—2 hours before the game and right after the opening kickoff. Each pizza will cost her $6 and will include 8 slices. She estimates it will cost her $0.45 for each hot dog and $0.90 for each barbecue sandwich if she makes the barbecue herself the night before. She measured a hot dog and found it takes up about 16 square inches of space, whereas a barbecue sandwich takes up about 25 square inches. She plans to sell a slice of pizza and a hot dog for $1.50 apiece and a barbecue sandwich for $2.25. She has $1,500 in cash available to purchase and prepare the food items for the first home game; remaining five games she will purchase her ingredients with money she has made from the previous game.
Ayu Ting-ting has talked to some students and vendors who have sold food at previous football games at ITS as well as at other universities. From this she has discovered that she can expect to sell at least as many slices of pizza as hot dogs and barbecue sandwiches combined. She also anticipates that she will probably sell at least twice as many hot dogs as barbecue sandwiches. She believes that she will sell everything she can stock and develop a customer base for the season if she follows these general guidelines for demand.
If Ayu Ting-ting clears at least $1,000 in profit for each game after paying all her expenses, she believes it will be worth leasing the booth.
A. Formulate and solve a linear programming model for Ayu Ting-ting that will help you advise her if she should lease the booth.
B. If Ayu Ting-ting were to borrow some more money from a friend before the first game to purchase more ingredients, could she increase her profit? If so, how much should she borrow and how much additional profit would she make? What factor constrains her from borrowing even more money than this amount (indicated in your answer to the previous question)?
C. When Ayu Ting-ting looked at the solution in (A), she realized that it would be physically difficult for her to prepare all the hot dogs and barbecue sandwiches indicated in this solution. She believes she can hire a friend of hers to help her for $100 per game. Based on the results in (A) and (B), is this something you think she could reasonably do and should do?
D. Ayu Ting-ting seems to be basing her analysis on the assumption that everything will go as she plans. What are some of the uncertain factors in the model that could go wrong and adversely affect Ayu Ting-ting’s analysis? Given these uncertainties and the results in (A), (B), and (C), what do you recommend that Ayu Ting-ting do?
Cases 3; Si Raja Jalanan Shipping and Transport Company
Luna Maya is the manager of the Java-area office of the “Si Raja Jalanan” Shipping and Transport Company. She is in the process of negotiating a new shipping contract with Polychem, a company that manufactures chemicals for industrial use. Polychem wants “Si Raja Jalanan” to pick up and transport waste products from its six plants to three waste disposal sites. Luna maya is very concerned about this prop osed arrangement. The chemical wastes that will be hauled can be hazardous to humans and the environment if they leak. In addition, a number of towns and communities in the region where the plants are located prohibit hazardous materials from being shipped through their municipal limits. Thus, not only will the shipments have to be handled carefully and transported at reduced speeds, they will also have to traverse circuitous routes in many cases.
Luna maya has estimated the cost of shipping a barrel of waste from each of the six plants to each of the three waste disposal sites (in US $) as shown in the following table:
|Waste Disposal Site|
|Plant||Bantar Gebang||Tandes||Sunan Kuning|
The plants generate the following amounts of waste products each week:
|Plant||Waste per Week (bbl)|
The three waste disposal sites at bantar Gebang, Tandes, and Sunan Kuning can accommodate a maximum of 65, 80, and 105 barrels per week, respectively. In addition to shipping directly from each of the six plants to one of the three waste disposal sites, Luna maya is also considering using each of the plants and waste disposal sites as intermediate shipping points. Trucks would be able to drop a load at a plant or disposal site to be picked up and carried on to the final destination by another truck, and vice versa. Si Raja Jalanan would not incur any handling costs because Polychem has agreed to take care of all local handling of the waste materials at the plants and the waste disposal sites. In other words, the only cost “Si Raja Jalanan” incurs is the actual transportation cost. So Luna maya wants to be able to consider the possibility that it may be cheaper to drop and pick up loads at intermediate points rather than ship them directly. Luna mayaestimates the shipping costs per barrel between each of the six plants to be as follows:
The estimated shipping cost per barrel (US $) between each of the three waste disposal sites is as follows:
|Waste Disposal Site|
|Waste Disposal Site||Bantar Gebang||Tandes||Sunan Kuning|
Luna maya wants to determine the shipping routes that will minimize Si Raja Jalanan’s total cost in order to develop a contract proposal to submit to Polychem for waste disposal. She particularly wants to know if it would be cheaper to ship directly from the plants to the waste sites or if she should drop and pick up some loads at the various plants and waste sites. Develop a model to assist Luna mayaand solve the model to determine the optimal routes.
Tujuan: maksimalkan keuntungan
190X + 240Y
- Batasan cetakan:18X + 15Y < 3600
- Batasan pembakaran: 0.27X + 0.58Y < 105
- Batasan pola: 0.16X + 0.2Y < 40
- Batasan jumlah: 32.8X + 20Y < 6000
X dibuat 60 batch dan Y dibuat 152 batch
D. pemakaian sumberdaya
- batasan cetakan terpakai 56 jam
- batasan pembakaran terpakai 104.36 jam
- batasan waktu untuk pola terpakai 40 jam.
- Batasan jumlah bahan terpakai 5008 pon
F. jika X = 0, maka
Jika x =0 maka maksimum laba terletak pada Y= 181
Analisis sensitifitas untuk model normal:
Analisis sensitifitas untuk model x=0
X = potongan pizza
Y = hotdog
Batasan ruang oven maksimal
= 3 feet x 4 feet x16
= 27.648 inchi persegi
X membutuhkan 14 inchi
Y membutuhkan 16 inchi
Z membutuhkan 25 inchi
14*X + 16*Y + 25*Z < 27.648
Modal $1500, dikurangi $600 u sewa oven menjadi $900
Biaya X = 0.75
Biaya Y = 0.45
Biaya Z = 0.9
0.75*X + 0.45*Y + 0.9*Z < 900
X = 1.5-0.75 = 0.75
Y = 1.5-0.45 = 1.05
Z = 2.25-0.9 = 1.35
MAX: 0.75*X + 1.05*Y + 1.35*Z
X = Y+Z
14*X + 16*Y + 25*Z < 27648;
0.75*X + 0.45*Y + 0.9*Z < 900;
Saran: menyewa stan karna keuntungan lebih dari $1000
B. meminjam uang untuk modal
Pinjaman bisa dilakukan hingga modal mencapai $1.131,05
Artinya meminjam sebesar = $1.131,05 – $900
= $231,05 Keuntungan yang didapat menjadi:
Faktor penghambat adalah kapasitas oven. Modal yang lebih besar dari $1131.05 tidak akan bertambah keuntungan karena keterbatasan oven.
C. Bayar sewa $100 artinya menjadi tambahan biaya, mempengaruhi batasan modal menjadi:
0.75*X + 0.45*Y + 0.9*Z < 800
Pada kondisi (A) keuntungan bersih even pertama akan menjadi $165 karena $1000 akan digunakan untuk menyewa stan.
sedangkan pada kondisi (B) keuntungan bersih even pertama $490.
Sedangkan pada game kedua sampai kelima akan mendapat tambahan keuntungan $600 karena tidak perlu menyewa oven. Sehingga bayaran teman $100 pergame masih sangat bisa dilakukan.
D. ketidak pastian terletak pada realisasi X Y Z yang terjual.
Bantar Gebang (65)
Dipenuhi langsung dari semarang, rinciannya
Dari semarang (pabrik) = 29
Didatangkan dari malang = 36
Bandung = 64 (produksi bandung 26, didatangkan dari Tegal 38)
Sunan Kuning (105)
Gudang Bantar Gebang = 27