The Graphical and Simplex Methods. INTRODUCTIONLinear programming (LP) is an application of matrix algebra used to solve a broad class of problems that can be represented by a system of linear equations. A linear equation is an algebraic equation whose variable quantity or quantities are in the first power only and whose graph is a straight line. LP problems are characterized by an objective function that is to be maximized or minimized, subject to a number of constraints. Both the objective function and the constraints must be formulated in terms of a linear equality or inequality. Typically; the objective function will be to maximize profits (e. The following assumptions must be satisfied to justify the use of linear programming: Linearity.
All functions, such as costs, prices, and technological require- ments, must be linear in nature. Certainty. All parameters are assumed to be known with certainty. Nonnegativity. Negative values of decision variables are unacceptable. Two approaches were commonly used to solve LP problems: Graphical method.
Simplex method. Now, however, MSExcel is much easier to use. The graphical method is limited to LP problems involving two decision variables and a limited number of constraints due to the difficulty of graphing and evaluating more than two decision variables. This restriction severely limits the use of the graphical method for real- world problems.
Irish pound - Wikipedia https://en.wikipedia.org/wiki/Irish Cymbalta official prescribing information for healthcare professionals. Includes: indications, dosage, adverse reactions, pharmacology and more. There’s more to her story but you should check it out in video form below. We also get some great anecdotes from the burgeoning computer industry of the CONSTRUCTING LINEAR PROGRAMMING PROBLEMS AND SOLVING THEM GRAPHICALLY. We will use the following Bridgeway Company case to introduce the graphical method and.
The graphical method is presented first here, however, because it is simple and easy to understand and it is a very good learning tool. The computer- based simplex method is much more powerful than the graphical method and provides the optimal solution to LP problems containing thousands of decision variables and constraints. It uses an iterative algorithm to solve for the optimal solution. Moreover, the simplex method provides information on slack variables (unused resources) and shadow prices (opportunity costs) that is useful in performing sensitivity analysis.
Excel uses a special version of the simplex method, as will be discussed later. Bridgeway Company manufactures a printer and keyboard. The contribution margins of the printer and keyboard are $3. Two types of skilled labor are required to manufacture these products: soldering and assembling. A printer requires 2 hours of soldering and I hour of assembling.
A keyboard requires 1 hour of soldering and 1 hour of assembling. Bridgeway has 1,0. There are no constraints on the supply of raw materials. Demand for keyboards is unlimited, but at most 3. Bridgeway wishes to maximize its weekly total contribution margin. Constructing the Linear Programming Problem for Maximization of the Objective Function.
However, investors in.
Constructing the LP problem requires four steps: Step 1. Define the decision variables.
Step 2. Define the objective function. Determine the constraints. Declare sign restrictions. STEP 1: DEFINE THE DECISION VARIABLES. In any LP problem, the decision variables should completely describe the decisions to be made. Bridgeway must decide how many printers and keyboards should be manufactured each week. With this in mind, the decision variables are defined as follows: X = Number of printers to produce weekly.
The Money Market Hedge: How It Works
Y = Number of keyboards to produce weekly. STEP 2: DEFINE THE OBJECTIVE FUNCTION.
The objective function represents the goal that management is trying to achieve. The goal in Bridgeway's case is to maximize (max) total contribution margin.
For each printer that is sold, $3. For each keyboard that is sold, $2. Thus, the total contribution margin for Bridgeway can be expressed by the following objective function equation: max Z = $3. X + $2. 0Ywhere the variable Z denotes the objective function value of any LP problem. In the Bridgeway case, Z equals the total contribution margin that will be realized when an optimal mix of products X (printer) and Y (keyboard) is manufactured and sold. STEP 3: DETERMINE THE CONSTRAINTS. A constraint is simply some limitation under which the enterprise must operate, such as limited production time or raw materials.
In Bridgeway's case, the objective function grows larger as X and Y increase. In other words, if Bridgeway were free to choose any values for X and Y, the company could make an arbitrarily large contribution margin by choosing X and Y to be very large.
The values of X and Y, however, are restricted by the following three constraints: Constraint 1. Each week, no more than 1,0.
Thus, constraint l may be expressed by: 2. X + Y . The inequality sign means that the total soldering time for both products X and Y cannot exceed the 1,0. Constraint 2. Each week, no more than 8.
Thus, constraint 2 may be expressed by: X + Y . Because of limited demand, at most 3. This constraint can be expressed as follows: X . To complete the formulation of an LP problem, the following question must be answered for each decision variable: Can the decision variable assume only nonnegative values, or is it allowed to assume both positive and negative values? In most LP problems, positive values are assumed.
For example, in the Bridgeway case, production cannot be less than zero units. Therefore, the sign restrictions are: X > = 0 Y > = 0. These four steps and the formulation of the LP problem for Bridgeway are summarized in Exhibit 1. This LP problem provides the necessary data to develop a graphical solution. Graphical Solution to the Maximization Linear Programming Problem. The following are two of the most basic concepts associated with LP: Feasible region. Optimal solution.
The graphical solution involves two steps: Step 1. Graphically determine the feasible region. Search for the optimal solution. STEP 1: GRAPHICALLY DETERMINE THE FEASIBLE REGION. The feasi- ble region represents the set of all feasible solutions to an LP problem. In the Bridgeway case, the feasible region is the set of all points (X, Y) satisfying the constraints in Exhibit 1.
For a point (X, Y) to be in the feasible region, (X, Y) must satisfy all the above inequalities. A graph containing these constraint equations is shown in Exhibit 1. Note that the only points satisfying the nonnegativity constraints are the points in the first quadrant of the X, Y plane. This is indicated by the arrows pointing to the right from the y- axis and upward from the x- axis.
Thus, any point that is outside the first quadrant cannot be in the feasible region. In plotting equation 2. X + Y < = 1,0. How much of product X could be produced if all resources were allocated to it? In this equation, a total of 1,0.
If all 1,0. 00 hours are allocated to product X, 5. On the other hand, how much of product Y could be produced if all resources were allocated to it? If all 1,0. 00 soldering hours are allocated to produce Y, then 1,0. Thus, the line on the graph expressing the soldering time constraint equation extends from the 5.
A on the x- axis to the 1,0. B on the y- axis. The equation associated with the assembling capacity constraint has been plotted on the graph in a similar manner. If 8. 00 assembling hours are allocated to product X, then 8. If, on the other hand, 8. Y, then 8. 00 keyboards can be produced.
This analysis results in line CD. Since equation X < 3. X, the line expressing the equation on the graph does not touch the y- axis at all. It extends from the 3. E on the x- axis and runs parallel to the y- axis, thereby signifying that regardless of the number of units of X produced, no more than 3.
X can ever be sold. Exhibit 1. 6- 2 shows that the set of points in the quadrant that satisfies all constraints is bounded by the five- sided polygon HDGFE. Any point on this polygon or in its interior is in the feasible region.
Any other point fails to satisfy at least one of the inequalities and thus falls outside the feasible region. STEP 2: SEARCH FOR THE OPTIMAL SOLUTION.
Having identified the feasible region for the Bridgeway case, we now search for the optimal solution, which will be the point in the feasible region that maximizes the objective function. In Bridgeway's case, this is: max Z = $3. X + $2. 0YTo find the optimal solution, we graph lines so that all points on a particular line have the same Z- value. In a maximization problem, such lines are called isoprofit lines; in a minimization problem, they are called isocost lines. The parallel lines are created by assigning various values to Z in the objective function to provide either higher profits or lower costs.
A graph showing the isoprofit lines for Bridgeway Company appears in Exhibit 1. The isoprofit lines are broken to differentiate them from the lines that form the feasible region. To draw an isoprofit line, any Z- value is chosen, then the x- and y- intercepts are calculated. For example, a contribution margin value of $6,0. X + $2. 0(0) $6,0.
YX = 2. 00 Y = 3. Since all isoprofit lines are of the form $3.
X + $2. 0Y = contribution margin, they all have the same slope. Consequently, once an isoprofit line is drawn, all other isoprofit lines can be found by moving parallel to the initial line. Another isoprofit line is found by selecting a contribution margin of $9,0. X + $2. 0(0) $9,0. YX = 3. 00 Y = 4.
Isoprofit lines move in a northeast direction; that is, upward and to the right. After a while, the isoprofit lines will no longer intersect the feasible region. The isoprofit line intersecting the last vertex of the feasible region defines the largest Z- value of any point in the feasible region and indicates the optimal solution to the LP problem. In Exhibit 1. 6- 3, the isoprofit line passing through point G is the last isoprofit line to intersect the feasible region. Thus, point G is the point in the feasible region with the largest Z- value and is therefore the optimal solution to the Bridgeway problem. Note that point G is located at the intersection of lines 2.
X + Y = 1,0. 00 and X + Y = 8. Solving these two equations simultaneously results in: X = 2. Y = 6. 00. The optimal value of Z (i. X and Y into the objective function.
Thus, the optimal value of Z is: max Z = $3.
Client List - Nationwide Barcode. This is a partial list of our clients who have purchased EAN and UPC Barcodes from Nationwide Barcode. Click here for Nationwide Barcode Client References# Twelve. Linenrus#1 Gluten Free.
Ling Ho. 1 Ancient Won Entertainment. Lingua Service. 1 God Productions. Linguini Sisters, Inc.
Love Entertainment. Linkedus Music. 1 Soul Entetainment LLCLinkstar. Day Pill. Linkxalot.
Feet Of Meat. Linora Corporation. Watches. Lionel B. All Natural. Lionheart Medical. Clean. Lions Roar Ent.
Entertainment, LLCLiplidz, LLC1. Entertainment. Lips Speak Magazine. Faces Coffee LLCLiquid Innovators LLC1.
Museums. Liquid Marin. Words Productions. Liquid Noise Records. Company Limited. Liquid Zeolite Company, Inc.
Records & Publishing. Liquidation Direct LLC1. St Earthborn Environmental Technologies, Lp. Liquidation Solutions.
Records LLC. Liquilock. Store. Liquimedlock Inc. Ne 1. St Avenue LLCLisa Larue Music. LLCLisa’s Pet Supplies. Zodiacs, Inc. Lisi International. Street Kitchen. Lisskiss. Cards. Listen/Hear Collective.
Tequila. Litenz. 12. Combustion, LLCLithia Mineral Water Inc. Xu Recordings. Litmocracy. Records. Little Bean Creations. Creations. Little Birdie Soap. Irthmark. Little Devil Apparel.
Ontario Inc. Little Friend. Alberta Ltd. Little Frog Publishing. Smoke. Little Giant Products, Inc. Network. Little Goodall. Bags. Little Green Mouse. Hair Care Inc. Little Green Mouse Limited.
Acandles. Little Guy Productions. Hf – One Human Family, LLCLittle Halo LLC1- Source Engineering Pte Ltd. Little Johnny’s Bbq. St Phoenix Inc. Little Kids Instruments. Step Pest Control Services Pte Ltd.
Little Light Ministries. Up Records. Little Miss Fairytale.
Cute Quilts. Little Miss Higgins Music. Dudes And A Zombie Productions. Little Miss Muffet Stamps. Tone Runts. Little Naturals.
Troublemakers. Little Ones, LLC. Turnt Up. Little Rig Speakers. Shekels Bread. Little Smarty Pantz. Vision Productions.
Little Swan Lake Winery. Products. Little Trendy Baby. Vintage, LLCLittle Works, USA2. Ontario Ltd. Littlebits Electronics Inc. Glorify God Publishing Company. Littlegreen Trading Ltd.
Hour Distribution. Litttle Friend. 24 Kt Wines LLCLiv Foods, LLC2. Marketing, Inc. Live How You Live. Companies. Live Image Music Group. Kt Wines LLCLive Sound Purification, Inc.
Ligne LLCLive Well 3. Th Night Productions.
Livelife Pilates Productions. Eight Muzik. Livelloo International, Inc. Bars & A Band. Lively Entertainment. Bars & A Band, Inc. Liverpoolair. com. Entertainment. Livetraxs.
Bhip. Livi. 2Bhip Inc. Livia’ Global. 2Brothers. Sisters Foods. Livin Dreamz Diaryz. Brothers. 2Sisters, LLCLiving Intentions. Cole Ent/Money Train Muzik Inc.
Living Real Entertainment. D Industries Inc. Living Tree Soaps, LLC. Good. 2B Bakery & Eatery.
Livingud. 2Guard. Lizaire Group. 2Heavy Enterprises. Lizbeth, Inc. 2K Entertainment. Lj’s Gourmet Caramel Popcorn. M Designs. Ljs Media Ltd. Mex And Maiselph. Ljx Inc. 2Nd Story Software.
Lll Transport. 2Ndtimearoundbarrels. Lloyd Thrasher. 2S Products. Lltc Mfg. 2. Troublemakers. Lm Wireless. 2Welve Ink.
Lni Industries Inc. Yaface, Nyaface, Onyaface, 4.
Yaface Productions. Loafers Baking Company. Mile Sauce And Salsa,LLCLobeth Group Incorporated. Minute Pregnancy Test.
Lobitos Creek Ranch. Paws, Inc. Local: A Quarterly Of People And Places, Inc. R Brand LLCLocean Labs LLC3 Stacks Entertainment.
Loft 3. 12 Inc. 3. Knox Records. Loftman Products. Morning Songs. Logan Galloway.
St Century Music Group. Logan Park Records. Degrees. Logic Technology Development LLC3.
Chutelibre. Logico LLC Dba Tektel. Gym. Logicraft Corporation. Stmall Inc. Logix Books Ltd. Enterprises Inc. Loglady Records. Music. Logsdon Farmhouse Ales. Sound And Vision. Loin. 36. 5 Inc. Lol Gourmet Snacks.
Hubs. Lol Lidz, LLC3- D Auto Parts Ltd. Lola Rose Soy Candles. D Innovations, LLCLolllimobile. D Solutions, Inc. Lolly Syrin. 3Dmd Limited. Lololucia+Hazard. Dream Records. Lololucia+Hazard.
E Engineering & Manufacturing Solutions. Lombardo Enterprises. Girls Holistic. Lombardo Music. Gjuice. Lomeli Music. Q LLCLon Lane’s Inspired Occasions. R Entertainment Group.
Lon Media Est. 3Rd Dimension Entertainment. London Town Press. Rpro Of Imago Green, LLCLondono Art Studio. Trinity. Londono Art Studio, Inc. Vnet Inc. Lone Star Production.
L’s Creek LLCLone Star Ranch & Creamery. On The Floor, LLCLone Star Smokeworks, LLC4 Paws Wash & Groom LLCLone Wolf Music Records.
The Love Of Toys! Lonely Heart Stringband. Technology Systems. Lonesome Locomotive. Forever. Long Beach Island Products.
Science. Long Chen Intercontinental Trading Co., Ltd. Dd Records. Long Miles, LLC4. Th Street Bedding. Longevity Global Inc. Karma. Longevity Inc. M Catering, LLC Dba Fathead Spices.
Longevity Premier Nutraceuticals Inc. N3 Labs, LLCLonghorn Marketing Group. Ok. Longislandbreeze. Pos. Longo Imports.
Rs Fitness Finds. Lonna Dodds. 4Th Down Magazine. Lonnie Dangerous. Trinity. Lookinglass. Wat. Lookrelaxmd Products And Services LLC5 Generaciones. Loominary. 5 Star Buys. Loop Paint. 5 Star Buys, LLCLoop Recycled Paint.
Star Distributing. Loopo, Inc. 5 Star- G Entertainment. Loopsy Lulu. 5 Starr Knives & Cutlery. Loose Leaf Tea Company. Stron Inc. Loot Lady Toys Inc. Strong Inc. Lop Ltd. Gramz Ent. L’Or D’Afrique.
Tidings. L’Or D’Afrique 1. Organic Argan Oil Conditioner. Promo. Lord & Gadpaille Corporation. Zone. Lord Cathouse. Media Group. Lorenzo Morgan Jr. Publishing, LLC5.
Recordings. Loretti. M Industry. Lorica West Corp. Mil. Lorilee’s Gourmet Brittle.
Star Lifequests, Inc. Lorne & Frankie. Waters Global Ventures. Los Angeles Pop Art, LLC5.
Waters Global Ventures LLCLos Arma. De La Sierra. 6 Line Records. Los Conios. 6 Mile Cellars.
Los Doss. 6 Sigma. Los Ilimitados De Michoacan. Pursuit. Los Panes Bakery Inc. Production Company.
Los Potrillos Food Market. X Products. Losing September. Led. Loss Control Svcs. Solutions. Lost Colony Music. Waves, Inc. Lost Empire. Enterprises LLCLost K I.
D Bracelets. 8 Beaks. Lost Mine Silverworks. Legged Recordz. Lost Moon, LLC8. Inc. orporated. Lost N Scanned. Productions. Lost Star Music Group.
Deals. Lost Tribe Designs. Tantawan Co., Ltd.
Lothill Solutions, LLC8. West Inc. Lotos Film. Media, LLCLotus Blooming Herbs LLC9. Records. Lotus Energy Drink Inc. Studios. Lotus Healing Hub/Biomat Central.
Foods Inc. Lotus Light Enterprises. Ether LLCLotus Light Enterprises Inc. A & A Limted Corp. Lou Marzeles. A & P Record. Loudboy Records. A Band Called Company. Loudedge, Inc. A Barr Simpson Co.
Loudedge, Inc. Dba Green Banana Greetings. A Book On One Page. Loudersoft M+M+MA Brief View Of The Hudson. Loudlab. A Cajun Life, LLCLouis Packaging, Inc. A Case Of The Blues.
Louise’s Country Closet. A Click Away Remotes. Louise’s Foods. A Crowded Coop. Louisiana Led Inc. A Fetish Vendetta. Louylou. A Few Failed Abortions. Lovadee. A Girls Night Out Limo LLCLove & Fleece, LLCA Hero En Route LLCLove & Joy Foods.
A Jillian Vance Design. Love Cupcakes, LLCA Line Furniture LLCLove Entertainment Inc.
A Lion Named Roar. Love Grace LLC. A Little Dog Company. Love Ink, LLCA Lyttle Goodness. Love Lifestyle, LLCA Million Smiles, Inc. Love Mason. A Modern Style. Love My Tapes. A Perfect Tan Bikini. Love Rocks Me. A Perfect Tan Bikini.
Love Your Neighborhood. A Plus Products. Love Your Pets LLCA Plus Spec Inc. Loves Creations. A Real Nut Company. Lovethebelt. com. A Safe Image. Lovett Lab.
A Sound Of Thunder, LLCLovies By Linda. A Star Furniture. Lovina Bar. A Steal Of A Deal!
Low Life Inc. A T Guys. Low Voltage And More!
A Vision Of. Lower Shannon Farms Limited. A Workmens Tool. Lowest Price. A& A International Group Inc. Lowlander Highlanders.
A& A Jewelry Supply. Lowly, The Tree Ghost.
A& F Global Investment Inc. Lowman Log Works.
A& G Food Distributors,LLCLoyatly Publishing/Distribution. A& J International Merchadise. Lp Music Corp. A& J Sleepwear. Lp Outdoors LLCA& J Sleepwear, LLCLpav. A& M Group LLCLpav LLCA& N Performance Laboratory. Lpb, LLCA& N Performance Laboratory, LLCLpe Assets, LLCA.
Solutions LLCLps Technologies, LLCA. Lyric. Lqi LLC Dba Navimaze.
A. Morejon Multiple Services. Lrg Ent/Luxile Ent. A. B. E Productions. Lrpm Ltd. A. B. E. Lrs Apparel, Ltd. A. D. Ls Mayer. A. D. U. F. Inc. orporated.
Lsl LLCA. Fritz & Co. Aberkrom. Lstreetz. A. I. S. Music Group.
Ltg Inc. A. J.’s Mix Inc. Ltw. A. K. A. Marvelicious. Lu Ruello/Pentultima. A. Q. H. Arts. Lubber Inc. A/C Multi- Media.
Lubotex Inc. A. Ltd. Lunas Productions. Abmedicalsupplies. Lunch Money Gang. Abolano Records. Lunenburg County Shipwrights Inc. LLCAcajou Gourmet. Luxzir, LLCAcala Farms, LLCLxmb Productions, LLCAcapsule Toys And Gifts LLCLyceum Recordz.
Acapulco’s. Lydell Hopson Jr. Acb Scitech, Inc. Lyfetyme X Hookmastaz. Accent Custom Components. Lymelight Music Publishing LLCAccent Custom Components. Lyn Milum. Accent Services. Lyndhurst Technologies.
Accents Et Cetera Gift Baskets Inc. Lynns Chocolate Chip Cookies.
Access Asia Industries. Lynnsong/Lynn Livigni. Access Consciousness. Lynwood Brooks. Access Fixtures. Lynx Sportswear. Access Nutraceuticals.
Lyracist – I Do It. Access Overhead Doors. Lyriqs Da Lyraciss. Accessible Products Co.
Lytham Body. Accessories & More Inc. Lyve. Henagan Music Cd. Accurate Manufacturing. M. A. Mattos Enterprises, Inc. Accuratime. M. Klemme Techology Corp.
Ace Laser, Inc. M. L. C. Productions Inc.
Achilles Wheel Records. M. R. Weatherstone/Art. Focus. Acid Lab Records. M. S. T. Entertainment. Acid Tactical. M. Y. Music And Entertainment Group.
Aclet Electronics Ltd. M2. 00. 0Acm Digital, Inc. M3 Trading Company LLCAcm Global LLCMa$$ Records. Acme Apothecary, Ltd. Maan Health Company. Acmemagick. Mabeda LLCAcomp S.
R. O. Mabels Five And Dime. Acorn Aspirations. Mac Media. Acorn Quilt & Gift Company. Mach 1 Arsenal. Acro Innovations, LLCMack CAcropolisrpm Inc.
Mackanan LLCAcross Cards. Mackenzie Scott. Acs Herbal Tea Co. Maclevins/Sisterfields. Acsa Productions Inc. Macneil Automotive Products Limited. Act/Coleman, Inc.
Macnet Design LLCAction Alphabet. Macnichol, Inc. Action Camp.
Macsimum Ventures LLCAction Cross S. L. U. Macys. Action Products And Sales, LLCMad Capital Records.
Action Wobble Inc. Mad Cats And Beehives. Actium Marketing Inc.