Here is the new homework question. Please review the General Homework instructions due to the instructor being so strict. Were there any podcasts or Homework document notes that you needed me to upload? Thanks!

Also- I will have the graded LP1 back tomorrow or Saturday for you to review in case there are any adjustments that need to be made to this assignment since it says to review notes that she makes on the other assignment during grading. I will upload it once I have it.

General Homework Instructions BADM 3963 Read and follow the instructions on each homework assignment. They may vary from assignment to assignment. Your name must be the first thing typed in every Word document and every Excel worksheet. Work submitted without your name will not be graded. Consider your work to be professional presentations. They should be well-organized and a positive reflection of your ability to communicate. Make sure that your work is easy to read and follow o In Excel use a separate worksheet for each problem - with labeled tabs. o In Word use white space, headings, indentations to make the report more readable When reporting answers: write complete sentences, use correct grammar and spell check your work. If you are asked to report answers: \"Reported\" means answers should tell a complete story. Here is an example: o Inadequate, unacceptable answer: 5 o Acceptable answer: The NPV for project D is $5 million. When a Word document is used to report results from Excel, any significant tables or graphs should be copied into the Word document from Excel. o The entire spreadsheet should NOT be copied into Word. o The Excel file will also be submitted. Learn to re-size Excel tables and graphs so the work fits on a single page in Word. o At this stage in your career, you are responsible for learning how to use the software appropriately. When submitting two files (one Word and one Excel) you must attach both before you hit the Submit button in Blackboard. o Never submit more than two files - one Word and one Excel would be the max. BADM 3963 Linear Programming Formulation homework #2 Summer 2017 Write the LP formulations for the following problems in a Word document. Submit Word document in BB Assignments - review comments & notes from LP hmwk #1 before submitting this assignment. Due 7/10, 9pm 1. Management at the Riverside Diner is trying to decide how to allocate a monthly advertising budget of $1800. They are considering newspaper advertising and radio advertising. Aside from the total budget, the following requirements must be met: at least 25% of the budget must be spent on each type of advertising the amount of money spend on newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. The index for newspaper advertising is estimated at 50 and the index for radio advertising is 80. Write the linear programming formulation for Riverside to use to determine how much of its budget that should be allocated to each type of advertising in order to maximize total audience exposure. 2. Madisen Furniture of Overland Park, Kansas produces two types of desks: executive desks and secretary desks at two plants. Plant 1 operates on a double shift of 80 hours per week. Plant currently operates 50 hours per week. The table below shows production time and standard costs at each plant for each type of desk. Executive desks are sold for a price of $350 and secretary desks for $275. The company has been experiencing cost overruns lately and has set a weekly budget constraint on production costs. The weekly budget for total production of executive desks is $2000, while the budget for secretary desks is $2200. Management would like to determine the number of each type of desk that should be produced at each plant in order to maximize PROFIT. Item Plant Executive desk Secretary desk Executive desk Secretary desk 1 1 2 2 Production time (hours/desk) 7 4 6 5 Standard cost ($/desk) 250 200 260 180 Write the linear programming formulation for this problem. 3. Mr. Barry Fisher, head dietician for Seattle Grace Hospital, is responsible for planning and managing patient diets. Mr. Fisher is currently planning a diet for patients who are restricted to a diet consisting of two food sources. The nutritional requirements that must be met daily are at least: 1,000 units of nutrient A, 2,000 units of nutrient B, and 1,500 units of nutrient C. Each ounce of food source #1 contains 100 units of A, 400 units of B, and 200 units of C. Each ounce of food source #2 contains 200 units of A, 240 units of B, and 200 units of C. Food source #1 costs $6.05 per ounce and food source #2 costs $8.32 per ounce. Write the linear programming formulation that can be used to determine the minimal cost combination of the food sources that will meet all nutrition requirements. 4. The Landsmere Shipping Company runs a cargo jet between London, England and New rk, New Jersey. To keep operational expenses in line, the jet will not depart until all decks are loaded with cargo. The aircraft has three decks: lower, middle, and upper. The jet cannot carry more than 120 total tons of cargo for each leg of the trip. No more than 50 tons of cargo should be carried on the lower deck. For balance purposes, the middle deck must carry one-third of the load of the lower deck and the upper deck must carry two-fifths of the load of the lower deck. However, no more than 75 tons of cargo should be loaded on the middle and upper decks combined. The profit from the shipping is $18 per ton for cargo on the lower deck $21 for cargo on the middle deck, and $32 for cargo on the upper desk. Write the linear programming formulation to determine the best allocation of cargo on the three decks. 5. The KK Cosmetics Company is introducing three new products for the fall season, which the marketing department currently is calling Mad, Mud, and Mod. These three products are made from three ingredients, code-named A, B, and C. The milligrams of each ingredient required to make one ounce of each final product are shown in the following table. Product Mad Mud Mod Ingredient (milligrams per ounce of product) Ing A Ing B Ing C 4 7 8 3 9 7 6 3 12 Each milligram of Ingredient A costs $0.15, each milligram of ingredient B costs $0.23, and each milligram of ingredient C costs $0.55. Each ounce Mad will be sold for $18, each ounce of Mud will be sold for $25, each ounce of Mod will be sold for $39 The firm currently has 400, 800, and 1,000 milligrams respectively, on hand of the ingredients A, B, and C. Write the linear programming formulation to determine the optimal mix of Mad, Mud, and Mod that KK should produce to maximize profit. 6. Standard Motors, Inc., sells standard automobiles and station wagons. The company makes $3000 profit on each standard auto that it sells and $4000 on each station wagon. The manufacturer cannot supply more than 300 standard autos and 200 station wagons per month. Dealer preparation time requires 2 hours for each standard auto and 3 hours for each station wagon. The company has 900 hours of shop time available each month for new car preparation. Write the linear programming formulation to determine how many standard autos and station wagons should be ordered to maximize profit. Linear Programming Formulation The most critical step in LP is the formulation - just like any analysis, if you don't have the problem correctly stated then you will be solving the wrong problem. At this step one of the biggest challenges is to not try to solve the problem, we just want the mathematical formulation. Start by reading the problem completely and carefully. 1. Decision variables Determine what the question is, what do you need to decide. Is it how many of each product to produce, how many ounces (lbs, gms, cups....) of each ingredient to put in a mixture, how many people to assign to a shift??? Elements that you will NOT likely have to decide: profit or cost per item - it's a given(or the information will be given for you to calculate it) TOTAL profit or cost will be an output of the solution (again NOT a decision to be made) resource usage - hours per item, hours available - those will be given... nutrients required nutrients delivered by the ingredients What you must decide will be written out as the decision variables. Decision variables represent the choices that must be made. This is the first formal part of the LP formulation. Decision variable definitions must be specific and explicit. It's not good enough to say: A = the amount of ingredient A to put in mix. If A has a cost per ounce, a contribution per ounce, etc.... Then the definition must say A = the number of ounces of ingredient A to put in the mix. If you are doing a product mix problem to determine weekly production, the definition should say: B = the number of product B to produce per week Note: you do not have to use X, Y or X1, X2 as variable names - use letters that help you remember what's going on in the problem. See Flair Furniture example in text pg 252- the problem is about tables and chairs so we use T = number of tables to be produced per week C = number of chairs to be produced per week Once you have the decision variables defined, it may be helpful to organize the data in a table - put the decision variables across the top (column for each) and organize known parameters in table - see Table 7.2 pg 252. If you look at Table 7.2 and then across to pg 253 where the complete problem is stated, you should be able to see how the numbers in the table have been \"peeled off\" to write the mathematical functions for the objective function and the constraints. 2. Objective function Once you have defined variables, you must write a mathematical function that can be used to calculate the objective. LP problems will only have one objective function. This mathematical function must use the decision variables that you have defined in step 1. In most of our problems, the objective will be to maximize profit or minimize cost. Remember again, that we are not trying to solve the problem at this stage. It might help some of you to think of it this way: I know you could figure profit if we said we are selling apples that give a profit of $1.00 and oranges that give a profit of $1.50 and we sold 3 apples and 4 oranges (Total profit = 1.00*3 + 1.50*4 = $900) That is Profit = profit per apple * number of apples + profit per orange * number of oranges. For the LP formulation, we don't know the numbers of fruits yet so we use variables to hold the place A = number of apples, R= number of oranges (don't like to use O as a variable name, looks too much like the number 0) Profit = 1.00*A + 1.50*R In the LP formulation the objective would be Maximize Profit = 1.00*A + 1.50*R 3. Constraints Constraints are written similarly to the objective function - they are linear functions written in terms of the decision variables (and you must use the decision variables that you have defined.) Each constraint will have a right hand side (rhs), that is the limit or requirement stated in the problem. (rhs is an established item in LP terminology) Your text gives the general idea of a resource constraint as: Amount used = Amount required Contract requirement for a certain product Number of item A's produced >= Contract for number of A's Demand limit for a certain product Number of item B's produced = 0 HL >= 0 ********If you are asked to write an LP formulation, this is what should be presented: AS = number of Aqua-Spa model tubs to produce per week HL = number of Hydro-Lux model tubs to produce per week Maximize profit = 250AS + 300HL Subject to: Pumps 1AS + 1HL = 0 HL >= 0 2. The Elixer Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics in different proportions. One gram of ingredient 1 contributes 3 units, and ingredient 2 contributes 1 unit of antibiotic A. The drug requires 6 units. At least 4 units of antibiotic B are required, ingredient 1 contributes 1 unit per gram and ingredient 2 contributes 1 unit per gram. At least 12 units of antibiotic C are required; ingredient 1 contributes 2 units per gram and ingredient 2 contributes 6 units per gram. The cost for a gram of ingredient 1 is $80 and the cost for a gram of ingredient 2 is $50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at the minimum cost. Formulate as a linear programming model 1. What do we need to decide? \"to determine the number of grams of each ingredient that must go into the drug\" X1 = number of grams of ingredient 1 to put in the drug X2 = the number of grams of ingredient 2 to put in the drug Note: It is VITAL that you put the units in the definition (grams). Don't be confused by the problem, read it carefully - we are told the units of antibiotic A per gram of ingredient, units of antibiotic B per gram of ingredient, units of antibiotic C per gram of ingredient and cost per gram of ingredient... all of this should lead you to define the variables as GRAMS of ingredient Organize data in a table Cost per gram Vit A per gram Vit B per gram Vit C per gram Ing 1 X1 80 3 1 2 Ing 2 X2 50 1 1 6 Needed 6 4 12 2. Write the objective function - again the problem helps us out here \"at the minimum cost.\" Minimize cost = 80X1 + 50X2 3. Write the constraints We have requirements for 3 different antibiotics: A, B, and C - we need three constraints A B C 3X1 + 1X2 >= 6 1X1 + 1X2 >= 4 2X1 + 6X2 >= 12 X1 >= 0 X2 >= 0 Note: we must have at least xx units of each antibiotic, at least means greater than or equal to, >= One thing that you can do to check your formulation (especially your variable definition) is check the units in the formulas... For this objective function we have 80 $ $ * X 1gram 50 * X 2 gram gram gram The grams will cancel out and we are left with $ + $ - that checks out - what we expect for cost is $$$$$$ So the LP formulation is: X1 = number of grams of ingredient 1 to put in the drug X2 = the number of grams of ingredient 2 to put in the drug Minimize cost = 80X1 + 50X2 Subject to: 3X1 + 1X2 >= 6 1X1 + 1X2 >= 4 2X1 + 6X2 >= 12 X1 >= 0 X2 >= 0 3. One type of constraint that always seems to be a problem is actually too simple for words. Go back to the Blue Ridge hot tub example. What if the marketing department has signed a contract that requires us to make at least 10 Aqua-Spa models for a certain client. That requirement means another constraint has to be added. The constraint simply has to ensure that we make at least 10 Aqua-Spas Mathematically: AS >= 10 Simple, right??? BADM 3963 Summer 2017 Linear Programming Solution Be able to identify the following on a graphically displayed LP problem: o constraints, feasible region, and objective function Use the graphical method of \"pushing the objective\" to identify the optimal corner point(s) Use Excel Solver to solve a variety of LP problems o Interpret Solver output - report the optimal solution and objective attainment Read textbook Ch 7, section 7.4 - focus on IsoProfit Line method omit Corner Point Solution method), section 7.5 (omit QM, just look at Excel), sections 7.6, 7.7 Read Supplemental notes on LP graphical solutions Watch podcasts on solving LPs in Excel 7/17/17 Homework LP Solution homework will be posted in BB Assignments folder BADM 3963 Summer 2017 Linear Programming Solution Be able to identify the following on a graphically displayed LP problem: o constraints, feasible region, and objective function Use the graphical method of \"pushing the objective\" to identify the optimal corner point(s) Use Excel Solver to solve a variety of LP problems o Interpret Solver output - report the optimal solution and objective attainment Read textbook Ch 7, section 7.4 - focus on IsoProfit Line method omit Corner Point Solution method), section 7.5 (omit QM, just look at Excel), sections 7.6, 7.7 Read Supplemental notes on LP graphical solutions Watch podcasts on solving LPs in Excel 7/17/17 Homework LP Solution homework will be posted in BB Assignments folder Amber Henson BADM 3963 Linear Programming Formulation homework #1 Summer 2017 Write the LP formulations for the following problems in a Word document. Submit Word document in BB Assignments Due 7/6, 9pm 1.Teethree, Inc. is a small manufacturer of sports equipment whose management has decided to begin producing golf bags. Two models will be produced - a standard bag and a deluxe bag. The bags require work in four departments as indicated in the table below. Each standard bag contributes $15 profit and each deluxe bag contributes $12 profit. The amount of time available in each department for production of golf bags is as follows: Cutting, 630 hours; Sewing, 600 hours; Finishing, 708 hours; and Packaging, 135 hours. Department Cutting Sewing Finishing Packaging Standard Bag requirement per bag 0.7 hour 0.5 hour 1 hour 0.10 hour Deluxe bag requirement per bag 1 hour 0.8 hour 0.67 hour 0.25 hour Write the linear programming formulation to help Teethree determine the best mix of the two bags to produce. Variables Let; X1 = Number of standard bags to be produced X2 = Number of deluxe bags to be produced Maximize profit = 15X1 + 12X2 Subject to; 0.70X1 + X2 = 0 (non-negativity constraint) 2. Epsum Chemical Co. produces two products that are sold as raw materials to companies that manufacture cosmetics. Based on an analysis of current inventory levels and potential demand for the coming month, Epsum's management specified that the combined production for products A and B must total at least 325 gallons. Separately, a major customer's order for 105 gallons of product A must be met. Product A requires 2 hours of processing time per gallon, and product B requires 1 hour of processing time per gallon. For the coming month, 500 hours of processing time are available. Production costs are $2 per gallon for product A and $3 per gallon for product B. Write the linear programming formulation to help Epsum determine how many gallons of each product should be produced for a minimum cost. Variables Let; X1 = Number of gallons of product A to be produced X2 = Number of gallons of product B to be produced Minimize cost = 2X1 + 3X2 Subject to; X1 + X2 >= 325 (Combined number of gallons to produce) X1 >= 105 (number of gallons of product A to produce) 2X1 + X2 = 0 (non-negativity constraint) 3. The Checotah Chamber of Commerce sponsors public service programs and is planning next year's advertising mix to promote the programs. Television, radio, and online ads are being proposed. The following table shows estimates of audience reach, costs, and individual upper limits on each type of ad. Type of ad Cost per ad Audience reached Upper limit on per ad ads of this type Television $2,000 100,000 10 Radio $300 18,000 20 Online $600 40,000 10 The budget for ads is limited to $18,500. To ensure a balanced used of the three media, radio ads must not exceed 50% of the total number of ads and television ads must account for at least 10% of the total number of ads. Write the linear programming formulation to help the Chamber determine the best mix of ads that will maximize the total audience reached. Variables Let; X1 = number of television ads to be used for the best mix X2 = number of radio ads to be used for the best mix X3 = number of online ads to be used for the best mix Maximize total audience = 100,000X1 + 18,000X2 + 40,000X3 Subject to; X1 = 0.1*(X1 + X2 + X3) (minimum number of radio ads allowed) 4. Mad Manufacturing makes three components that are sold to makers of air conditioners. The components are processed on two machines: a shaper and a grinder. Times required for each component on each machine are as follows: Component Shaper (minutes) Grinder (minutes) A 6 4 B 4 5 C 4 2 The shaper is available for 120 hours and the grinder for 110 hours. No more than 200 units of component C can be sold, but up to 1,000 units of each of the other components can be sold. The company has orders for 600 units of component A that must be met. The profit contributions for components A, B, and C are $8, $6, and $9, respectively. Write the linear programming formulation for Mad Manufacturing to determine the best production mix of the three components. Variables Let; X1 = number of units of component A to be processed X2 = number of units of component B to be processed X3 = number of units of component C to be processed Maximize profit = 8X1 + 6X2 + 9X3 Subject to; 6X1 + 4X2 + 4X3 = 600 (component A units demanded) BADM 3963 Summer 2017 Learning Objectives Linear Programming Formulation Describe/Define/Explain the components, terms and assumptions of Linear programming o Decision variable, constraint, objective function, non-negativity constraint, optimal solution, feasible region, binding constraint, non-binding constraint Be able to formulate simple LP models Read textbook Ch 7, sections 7.1,7.2, 7.3 pg 270 Holiday Meal Turkey Ranch - study the formulation Ch 8 - study all formulations, don't worry about solutions yet Read Supplemental LP formulation notes 7/6/17 LP hmwk #1 Will be posted in BB Assignments folder 7/10/17 LP hmwk #2 Will be posted in BB Assignments folder EXAM 2 Must be completed by 7/13/17, 9pm Linear Programming Formulation The most critical step in LP is the formulation - just like any analysis, if you don't have the problem correctly stated then you will be solving the wrong problem. At this step one of the biggest challenges is to not try to solve the problem, we just want the mathematical formulation. Start by reading the problem completely and carefully. 1. Decision variables Determine what the question is, what do you need to decide. Is it how many of each product to produce, how many ounces (lbs, gms, cups....) of each ingredient to put in a mixture, how many people to assign to a shift??? Elements that you will NOT likely have to decide: profit or cost per item - it's a given(or the information will be given for you to calculate it) TOTAL profit or cost will be an output of the solution (again NOT a decision to be made) resource usage - hours per item, hours available - those will be given... nutrients required nutrients delivered by the ingredients What you must decide will be written out as the decision variables. Decision variables represent the choices that must be made. This is the first formal part of the LP formulation. Decision variable definitions must be specific and explicit. It's not good enough to say: A = the amount of ingredient A to put in mix. If A has a cost per ounce, a contribution per ounce, etc.... Then the definition must say A = the number of ounces of ingredient A to put in the mix. If you are doing a product mix problem to determine weekly production, the definition should say: B = the number of product B to produce per week Note: you do not have to use X, Y or X1, X2 as variable names - use letters that help you remember what's going on in the problem. See Flair Furniture example in text pg 252- the problem is about tables and chairs so we use T = number of tables to be produced per week C = number of chairs to be produced per week Once you have the decision variables defined, it may be helpful to organize the data in a table - put the decision variables across the top (column for each) and organize known parameters in table - see Table 7.2 pg 252. If you look at Table 7.2 and then across to pg 253 where the complete problem is stated, you should be able to see how the numbers in the table have been \"peeled off\" to write the mathematical functions for the objective function and the constraints. 2. Objective function Once you have defined variables, you must write a mathematical function that can be used to calculate the objective. LP problems will only have one objective function. This mathematical function must use the decision variables that you have defined in step 1. In most of our problems, the objective will be to maximize profit or minimize cost. Remember again, that we are not trying to solve the problem at this stage. It might help some of you to think of it this way: I know you could figure profit if we said we are selling apples that give a profit of $1.00 and oranges that give a profit of $1.50 and we sold 3 apples and 4 oranges (Total profit = 1.00*3 + 1.50*4 = $900) That is Profit = profit per apple * number of apples + profit per orange * number of oranges. For the LP formulation, we don't know the numbers of fruits yet so we use variables to hold the place A = number of apples, R= number of oranges (don't like to use O as a variable name, looks too much like the number 0) Profit = 1.00*A + 1.50*R In the LP formulation the objective would be Maximize Profit = 1.00*A + 1.50*R 3. Constraints Constraints are written similarly to the objective function - they are linear functions written in terms of the decision variables (and you must use the decision variables that you have defined.) Each constraint will have a right hand side (rhs), that is the limit or requirement stated in the problem. (rhs is an established item in LP terminology) Your text gives the general idea of a resource constraint as: Amount used = Amount required Contract requirement for a certain product Number of item A's produced >= Contract for number of A's Demand limit for a certain product Number of item B's produced = 0 HL >= 0 ********If you are asked to write an LP formulation, this is what should be presented: AS = number of Aqua-Spa model tubs to produce per week HL = number of Hydro-Lux model tubs to produce per week Maximize profit = 250AS + 300HL Subject to: Pumps 1AS + 1HL = 0 HL >= 0 2. The Elixer Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics in different proportions. One gram of ingredient 1 contributes 3 units, and ingredient 2 contributes 1 unit of antibiotic A. The drug requires 6 units. At least 4 units of antibiotic B are required, ingredient 1 contributes 1 unit per gram and ingredient 2 contributes 1 unit per gram. At least 12 units of antibiotic C are required; ingredient 1 contributes 2 units per gram and ingredient 2 contributes 6 units per gram. The cost for a gram of ingredient 1 is $80 and the cost for a gram of ingredient 2 is $50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at the minimum cost. Formulate as a linear programming model 1. What do we need to decide? \"to determine the number of grams of each ingredient that must go into the drug\" X1 = number of grams of ingredient 1 to put in the drug X2 = the number of grams of ingredient 2 to put in the drug Note: It is VITAL that you put the units in the definition (grams). Don't be confused by the problem, read it carefully - we are told the units of antibiotic A per gram of ingredient, units of antibiotic B per gram of ingredient, units of antibiotic C per gram of ingredient and cost per gram of ingredient... all of this should lead you to define the variables as GRAMS of ingredient Organize data in a table Cost per gram Vit A per gram Vit B per gram Vit C per gram Ing 1 X1 80 3 1 2 Ing 2 X2 50 1 1 6 Needed 6 4 12 2. Write the objective function - again the problem helps us out here \"at the minimum cost.\" Minimize cost = 80X1 + 50X2 3. Write the constraints We have requirements for 3 different antibiotics: A, B, and C - we need three constraints A B C 3X1 + 1X2 >= 6 1X1 + 1X2 >= 4 2X1 + 6X2 >= 12 X1 >= 0 X2 >= 0 Note: we must have at least xx units of each antibiotic, at least means greater than or equal to, >= One thing that you can do to check your formulation (especially your variable definition) is check the units in the formulas... For this objective function we have 80 $ $ * X 1gram 50 * X 2 gram gram gram The grams will cancel out and we are left with $ + $ - that checks out - what we expect for cost is $$$$$$ So the LP formulation is: X1 = number of grams of ingredient 1 to put in the drug X2 = the number of grams of ingredient 2 to put in the drug Minimize cost = 80X1 + 50X2 Subject to: 3X1 + 1X2 >= 6 1X1 + 1X2 >= 4 2X1 + 6X2 >= 12 X1 >= 0 X2 >= 0 3. One type of constraint that always seems to be a problem is actually too simple for words. Go back to the Blue Ridge hot tub example. What if the marketing department has signed a contract that requires us to make at least 10 Aqua-Spa models for a certain client. That requirement means another constraint has to be added. The constraint simply has to ensure that we make at least 10 Aqua-Spas Mathematically: AS >= 10 Simple, right??? An Example LP Problem Blue Ridge Hot Tubs produces two models of hot tubs: Aquas & Hydros. The table below indicates the resources used by each model and the profit contributions. Pumps Labor Tubing Unit Profit Aqua Hydro 1 9 hours 12 feet $350 1 6 hours 16 feet $300 There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available. Management wants to determine the optimum number of each model to product in order to maximize total profit. Formulation: A = number of Aqua Spas to produce H = number of Hydros to produce Maximize Profit = 350A + 300H St 1A + 1H = 450(amount available for radio advertising in a month) N >= 900(amount available for newspaper advertising in a month) R >= 0 N >= 0 2. Madisen Furniture of Overland Park, Kansas produces two types of desks: executive desks and secretary desks at two plants. Plant 1 operates on a double shift of 80 hours per week. Plant currently operates 50 hours per week. The table below shows production time and standard costs at each plant for each type of desk. Executive desks are sold for a price of $350 and secretary desks for $275. The company has been experiencing cost overruns lately and has set a weekly budget constraint on production costs. The weekly budget for total production of executive desks is $2000, while the budget for secretary desks is $2200. Management would like to determine the number of each type of desk that should be produced at each plant in order to maximize PROFIT. Item Plant Executive desk Secretary desk Executive desk Secretary desk 1 1 2 2 Production time (hours/desk) 7 4 6 5 Standard cost ($/desk) 250 200 260 180 Write the linear programming formulation for this problem. LP formulation; Decision variables Let; E1 = number of executive desks produced weekly in Plant 1 S1 = number of secretary desks produced weekly in Plant 1 E2 = number of executive desks produced weekly in Plant 2 S2 = number of executive desks produced weekly in Plant 2 Maximize profit = 100E1 + 75S1 + 90E2 + 95S2 Subject to; 7E1 + 4S1 = 0(non-negativity constraint) 3. Mr. Barry Fisher, head dietician for Seattle Grace Hospital, is responsible for planning and managing patient diets. Mr. Fisher is currently planning a diet for patients who are restricted to a diet consisting of two food sources. The nutritional requirements that must be met daily are at least: 1,000 units of nutrient A, 2,000 units of nutrient B, and 1,500 units of nutrient C. Each ounce of food source #1 contains 100 units of A, 400 units of B, and 200 units of C. Each ounce of food source #2 contains 200 units of A, 240 units of B, and 200 units of C. Food source #1 costs $6.05 per ounce and food source #2 costs $8.32 per ounce. Write the linear programming formulation that can be used to determine the minimal cost combination of the food sources that will meet all nutrition requirements. LP formulation; Decision variables S1 = number of ounces of food source #1 in the combination in a day S2 = number of ounces of food source #2 in the combination in a day Minimize cost = 6.05S1 + 8.32S2 Subject to; 100S1 + 200S2 >=1000(nutrient A available daily) 400S1 + 240S2 >= 2000(nutrient B available daily) 200S1 + 200S2 >= 1500(nutrient C available daily) S1 >= 0 S2 >= 0 4. The Landsmere Shipping Company runs a cargo jet between London, England and New rk, New Jersey. To keep operational expenses in line, the jet will not depart until all decks are loaded with cargo. The aircraft has three decks: lower, middle, and upper. The jet cannot carry more than 120 total tons of cargo for each leg of the trip. No more than 50 tons of cargo should be carried on the lower deck. For balance purposes, the middle deck must carry one-third of the load of the lower deck and the upper deck must carry two-fifths of the load of the lower deck. However, no more than 75 tons of cargo should be loaded on the middle and upper decks combined. The profit from the shipping is $18 per ton for cargo on the lower deck $21 for cargo on the middle deck, and $32 for cargo on the upper desk. Write the linear programming formulation to determine the best allocation of cargo on the three decks. LP formulation; Decision variables Let; L = tons of cargo allocated to the lower deck for each leg of the trip M = tons of cargo allocated to the middle deck for each leg of the trip U = tons of cargo allocated to the upper deck for each leg of the trip Maximize profit = 18L + 21M + 32U Subject to; L + M + U = 0 U >= 0 M >= 0 5. The KK Cosmetics Company is introducing three new products for the fall season, which the marketing department currently is calling Mad, Mud, and Mod. These three products are made from three ingredients, code-named A, B, and C. The milligrams of each ingredient required to make one ounce of each final product are shown in the following table. Product Mad Mud Mod Ingredient (milligrams per ounce of product) Ing A Ing B Ing C 4 7 8 3 9 7 6 3 12 Each milligram of Ingredient A costs $0.15, each milligram of ingredient B costs $0.23, and each milligram of ingredient C costs $0.55. Each ounce Mad will be sold for $18, each ounce of Mud will be sold for $25, each ounce of Mod will be sold for $39 The firm currently has 400, 800, and 1,000 milligrams respectively, on hand of the ingredients A, B, and C. Write the linear programming formulation to determine the optimal mix of Mad, Mud, and Mod that KK should produce to maximize profit. Profit = sales - costs LP formulation; Decision variables Let; MA = number of ounces of Mad to produce MU = number of ounces of Mud to produce MO = number of ounces of Mod to produce Maximize profit = 11.39MA + 18.63MU + 30.81MO Subject to; 4MA + 3MU + 6MO = 0 MU >= 0 MO >= 0 6. Standard Motors, Inc., sells standard automobiles and station wagons. The company makes $3000 profit on each standard auto that it sells and $4000 on each station wagon. The manufacturer cannot supply more than 300 standard autos and 200 station wagons per month. Dealer preparation time requires 2 hours for each standard auto and 3 hours for each station wagon. The company has 900 hours of shop time available each month for new car preparation. Write the linear programming formulation to determine how many standard autos and station wagons should be ordered to maximize profit. LP formulation; Decision variables Let; SA = number of standard automobiles to order per month SW = number of station wagons to order per month Maximize profit = 3000SA + 4000SW Subject to; SA = 0 SW >= 0 Amber Henson BADM 3963 Linear Programming Formulation homework #2 Summer 2017 Write the LP formulations for the following problems in a Word document. Submit Word document in BB Assignments - review comments & notes from LP hmwk #1 before submitting this assignment. Due 7/10, 9pm 1. Management at the Riverside Diner is trying to decide how to allocate a monthly advertising budget of $1800. They are considering newspaper advertising and radio advertising. Aside from the total budget, the following requirements must be met: at least 25% of the budget must be spent on each type of advertising the amount of money spend on newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. The index for newspaper advertising is estimated at 50 and the index for radio advertising is 80. Write the linear programming formulation for Riverside to use to determine how much of its budget that should be allocated to each type of advertising in order to maximize total audience exposure. LP formulation; Decision variables Let; N = amount of dollars spent in newspaper advertising in a month R = amount of dollars spent in radio advertising in a month Maximize total audience exposure = 0.5N + 0.8R Subject to; R + N = 450(amount available for radio advertising in a month) N >= 900(amount available for newspaper advertising in a month) R >= 0 N >= 0 First comment; So is it wrong to use 0.5 and 0.8? Second comment; The 450 is 25% of $1,800 = $450 900 is twice 450. Must one use ratios instead of calculating the exact values because I preferred using R>= 450 instead of R>= 0.25*(R+N). 2. Madisen Furniture of Overland Park, Kansas produces two types of desks: executive desks and secretary desks at two plants. Plant 1 operates on a double shift of 80 hours per week. Plant currently operates 50 hours per week. The table below shows production time and standard costs at each plant for each type of desk. Executive desks are sold for a price of $350 and secretary desks for $275. The company has been experiencing cost overruns lately and has set a weekly budget constraint on production costs. The weekly budget for total production of executive desks is $2000, while the budget for secretary desks is $2200. Management would like to determine the number of each type of desk that should be produced at each plant in order to maximize PROFIT. Item Plant Executive desk Secretary desk Executive desk Secretary desk 1 1 2 2 Production time (hours/desk) 7 4 6 5 Standard cost ($/desk) 250 200 260 180 Write the linear programming formulation for this problem. LP formulation; Decision variables Let; E1 = number of executive desks produced weekly in Plant 1 S1 = number of secretary desks produced weekly in Plant 1 E2 = number of executive desks produced weekly in Plant 2 S2 = number of executive desks produced weekly in Plant 2 Maximize profit = 100E1 + 75S1 + 90E2 + 95S2 Subject to; 7E1 + 4S1 = 0(non-negativity constraint) 3. Mr. Barry Fisher, head dietician for Seattle Grace Hospital, is responsible for planning and managing patient diets. Mr. Fisher is currently planning a diet for patients who are restricted to a diet consisting of two food sources. The nutritional requirements that must be met daily are at least: 1,000 units of nutrient A, 2,000 units of nutrient B, and 1,500 units of nutrient C. Each ounce of food source #1 contains 100 units of A, 400 units of B, and 200 units of C. Each ounce of food source #2 contains 200 units of A, 240 units of B, and 200 units of C. Food source #1 costs $6.05 per ounce and food source #2 costs $8.32 per ounce. Write the linear programming formulation that can be used to determine the minimal cost combination of the food sources that will meet all nutrition requirements. LP formulation; Decision variables S1 = number of ounces of food source #1 in the combination in a day S2 = number of ounces of food source #2 in the combination in a day Minimize cost = 6.05S1 + 8.32S2 Subject to; 100S1 + 200S2 >=1000(nutrient A available daily) 400S1 + 240S2 >= 2000(nutrient B available daily) 200S1 + 200S2 >= 1500(nutrient C available daily) S1 >= 0 S2 >= 0 4. The Landsmere Shipping Company runs a cargo jet between London, England and New rk, New Jersey. To keep operational expenses in line, the jet will not depart until all decks are loaded with cargo. The aircraft has three decks: lower, middle, and upper. The jet cannot carry more than 120 total tons of cargo for each leg of the trip. No more than 50 tons of cargo should be carried on the lower deck. For balance purposes, the middle deck must carry one-third of the load of the lower deck and the upper deck must carry two-fifths of the load of the lower deck. However, no more than 75 tons of cargo should be loaded on the middle and upper decks combined. The profit from the shipping is $18 per ton for cargo on the lower deck $21 for cargo on the middle deck, and $32 for cargo on the upper desk. Write the linear programming formulation to determine the best allocation of cargo on the three decks. LP formulation; Decision variables Let; L = tons of cargo allocated to the lower deck for each leg of the trip M = tons of cargo allocated to the middle deck for each leg of the trip U = tons of cargo allocated to the upper deck for each leg of the trip Maximize profit = 18L + 21M + 32U Subject to; L + M + U = 0 U >= 0 M >= 0 Is it a must to use ratio constraints when you can calculate because I opted for calculating one third of tons in lower deck that is 1/3*50 = 50/3. This applies also to upper deck maximum which is 2/5*50 = 20. Must one use M = 1/3*L instead of M= 0 MU >= 0 MO >= 0 6. Standard Motors, Inc., sells standard automobiles and station wagons. The company makes $3000 profit on each standard auto that it sells and $4000 on each station wagon. The manufacturer cannot supply more than 300 standard autos and 200 station wagons per month. Dealer preparation time requires 2 hours for each standard auto and 3 hours for each station wagon. The company has 900 hours of shop time available each month for new car preparation. Write the linear programming formulation to determine how many standard autos and station wagons should be ordered to maximize profit. LP formulation; Decision variables Let; SA = number of standard automobiles to order per month SW = number of station wagons to order per month Maximize profit = 3000SA + 4000SW Subject to; SA = 0 SW >= 0 Kindly specify the ratio constraint. Amber Henson BADM 3963 Linear Programming Formulation homework #2 Summer 2017 Write the LP formulations for the following problems in a Word document. Submit Word document in BB Assignments - review comments & notes from LP hmwk #1 before submitting this assignment. Due 7/10, 9pm 1. Management at the Riverside Diner is trying to decide how to allocate a monthly advertising budget of $1800. They are considering newspaper advertising and radio advertising. Aside from the total budget, the following requirements must be met: at least 25% of the budget must be spent on each type of advertising the amount of money spend on newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. The index for newspaper advertising is estimated at 50 and the index for radio advertising is 80. Write the linear programming formulation for Riverside to use to determine how much of its budget that should be allocated to each type of advertising in order to maximize total audience exposure. LP formulation; Decision variables Let; N = amount of dollars spent in newspaper advertising in a month R = amount of dollars spent in radio advertising in a month Maximize total audience exposure = 0.5N + 0.8R Subject to; R + N = 450(amount available for radio advertising in a month) N >= 900(amount available for newspaper advertising in a month) R >= 0 N >= 0 2. Madisen Furniture of Overland Park, Kansas produces two types of desks: executive desks and secretary desks at two plants. Plant 1 operates on a double shift of 80 hours per week. Plant currently operates 50 hours per week. The table below shows production time and standard costs at each plant for each type of desk. Executive desks are sold for a price of $350 and secretary desks for $275. The company has been experiencing cost overruns lately and has set a weekly budget constraint on production costs. The weekly budget for total production of executive desks is $2000, while the budget for secretary desks is $2200. Management would like to determine the number of each type of desk that should be produced at each plant in order to maximize PROFIT. Item Plant Executive desk Secretary desk Executive desk Secretary desk 1 1 2 2 Production time (hours/desk) 7 4 6 5 Standard cost ($/desk) 250 200 260 180 Write the linear programming formulation for this problem. LP formulation; Decision variables Let; E1 = number of executive desks produced weekly in Plant 1 S1 = number of secretary desks produced weekly in Plant 1 E2 = number of executive desks produced weekly in Plant 2 S2 = number of executive desks produced weekly in Plant 2 Maximize profit = 100E1 + 75S1 + 90E2 + 95S2 Subject to; 7E1 + 4S1 = 0(non-negativity constraint) 3. Mr. Barry Fisher, head dietician for Seattle Grace Hospital, is responsible for planning and managing patient diets. Mr. Fisher is currently planning a diet for patients who are restricted to a diet consisting of two food sources. The nutritional requirements that must be met daily are at least: 1,000 units of nutrient A, 2,000 units of nutrient B, and 1,500 units of nutrient C. Each ounce of food source #1 contains 100 units of A, 400 units of B, and 200 units of C. Each ounce of food source #2 contains 200 units of A, 240 units of B, and 200 units of C. Food source #1 costs $6.05 per ounce and food source #2 costs $8.32 per ounce. Write the linear programming formulation that can be used to determine the minimal cost combination of the food sources that will meet all nutrition requirements. LP formulation; Decision variables S1 = number of ounces of food source #1 in the combination in a day S2 = number of ounces of food source #2 in the combination in a day Minimize cost = 6.05S1 + 8.32S2 Subject to; 100S1 + 200S2 >=1000(nutrient A available daily) 400S1 + 240S2 >= 2000(nutrient B available daily) 200S1 + 200S2 >= 1500(nutrient C available daily) S1 >= 0 S2 >= 0 4. The Landsmere Shipping Company runs a cargo jet between London, England and New rk, New Jersey. To keep operational expenses in line, the jet will not depart until all decks are loaded with cargo. The aircraft has three decks: lower, middle, and upper. The jet cannot carry more than 120 total tons of cargo for each leg of the trip. No more than 50 tons of cargo should be carried on the lower deck. For balance purposes, the middle deck must carry one-third of the load of the lower deck and the upper deck must carry two-fifths of the load of the lower deck. However, no more than 75 tons of cargo should be loaded on the middle and upper decks combined. The profit from the shipping is $18 per ton for cargo on the lower deck $21 for cargo on the middle deck, and $32 for cargo on the upper desk. Write the linear programming formulation to determine the best allocation of cargo on the three decks. LP formulation; Decision variables Let; L = tons of cargo allocated to the lower deck for each leg of the trip M = tons of cargo allocated to the middle deck for each leg of the trip U = tons of cargo allocated to the upper deck for each leg of the trip Maximize profit = 18L + 21M + 32U Subject to; L + M + U = 0 U >= 0 M >= 0 5. The KK Cosmetics Company is introducing three new products for the fall season, which the marketing department currently is calling Mad, Mud, and Mod. These three products are made from three ingredients, code-named A, B, and C. The milligrams of each ingredient required to make one ounce of each final product are shown in the following table. Product Mad Mud Mod Ingredient (milligrams per ounce of product) Ing A Ing B Ing C 4 7 8 3 9 7 6 3 12 Each milligram of Ingredient A costs $0.15, each milligram of ingredient B costs $0.23, and each milligram of ingredient C costs $0.55. Each ounce Mad will be sold for $18, each ounce of Mud will be sold for $25, each ounce of Mod will be sold for $39 The firm currently has 400, 800, and 1,000 milligrams respectively, on hand of the ingredients A, B, and C. Write the linear programming formulation to determine the optimal mix of Mad, Mud, and Mod that KK should produce to maximize profit. Profit = sales - costs LP formulation; Decision variables Let; MA = number of ounces of Mad to produce MU = number of ounces of Mud to produce MO = number of ounces of Mod to produce Maximize profit = 11.39MA + 18.63MU + 30.81MO Subject to; 4MA + 3MU + 6MO = 0 MU >= 0 MO >= 0 6. Standard Motors, Inc., sells standard automobiles and station wagons. The company makes $3000 profit on each standard auto that it sells and $4000 on each station wagon. The manufacturer cannot supply more than 300 standard autos and 200 station wagons per month. Dealer preparation time requires 2 hours for each standard auto and 3 hours for each station wagon. The company has 900 hours of shop time available each month for new car preparation. Write the linear programming formulation to determine how many standard autos and station wagons should be ordered to maximize profit. LP formulation; Decision variables Let; SA = number of standard automobiles to order per month SW = number of station wagons to order per month Maximize profit = 3000SA + 4000SW Subject to; SA = 0 SW >= 0 Amber Henson BADM 3963 Linear Programming Formulation homework #2 Summer 2017 Write the LP formulations for the following problems in a Word document. Submit Word document in BB Assignments - review comments & notes from LP hmwk #1 before submitting this assignment. Due 7/10, 9pm 1. Management at the Riverside Diner is trying to decide how to allocate a monthly advertising budget of $1800. They are considering newspaper advertising and radio advertising. Aside from the total budget, the following requirements must be met: at least 25% of the budget must be spent on each type of advertising the amount of money spend on newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. The index for newspaper advertising is estimated at 50 and the index for radio advertising is 80. Write the linear programming formulation for Riverside to use to determine how much of its budget that should be allocated to each type of advertising in order to maximize total audience exposure. LP formulation; Decision variables Let; N = amount of dollars spent in newspaper advertising in a month R = amount of dollars spent in radio advertising in a month Maximize total audience exposure = 0.5N + 0.8R Subject to; R + N = 450(amount available for radio advertising in a month) N >= 900(amount available for newspaper advertising in a month) R >= 0 N >= 0 First comment; So is it wrong to use 0.5 and 0.8? Second comment; The 450 is 25% of $1,800 = $450 900 is twice 450