Formulate the linear programming model for this situation.
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The poultry farmer decided to make his own chicken scratch by combining alfalfa and corn in rail car quantities. A rail car of corn costs $400 and a rail car of alfalfa costs $200. The farmer's chickens have a minimum daily requirement of vitamin K (500 milligrams) and iron (400 milligrams), but it doesn't matter whether those elements come from corn, alfalfa, or some other grain. A unit of corn contains 150 milligrams of vitamin K and 75 milligrams of iron. A unit of alfalfa contains 250 milligrams of vitamin K and 50 milligrams of iron. Formulate the linear programming model for this situation.
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- Lemingtons is trying to determine how many Jean Hudson dresses to order for the spring season. Demand for the dresses is assumed to follow a normal distribution with mean 400 and standard deviation 100. The contract between Jean Hudson and Lemingtons works as follows. At the beginning of the season, Lemingtons reserves x units of capacity. Lemingtons must take delivery for at least 0.8x dresses and can, if desired, take delivery on up to x dresses. Each dress sells for 160 and Hudson charges 50 per dress. If Lemingtons does not take delivery on all x dresses, it owes Hudson a 5 penalty for each unit of reserved capacity that is unused. For example, if Lemingtons orders 450 dresses and demand is for 400 dresses, Lemingtons will receive 400 dresses and owe Jean 400(50) + 50(5). How many units of capacity should Lemingtons reserve to maximize its expected profit?The Pigskin Company produces footballs. Pigskin must decide how many footballs to produce each month. The company has decided to use a six-month planning horizon. The forecasted monthly demands for the next six months are 10,000, 15,000, 30,000, 35,000, 25,000, and 10,000. Pigskin wants to meet these demands on time, knowing that it currently has 5000 footballs in inventory and that it can use a given months production to help meet the demand for that month. (For simplicity, we assume that production occurs during the month, and demand occurs at the end of the month.) During each month there is enough production capacity to produce up to 30,000 footballs, and there is enough storage capacity to store up to 10,000 footballs at the end of the month, after demand has occurred. The forecasted production costs per football for the next six months are 12.50, 12.55, 12.70, 12.80, 12.85, and 12.95, respectively. The holding cost incurred per football held in inventory at the end of any month is 5% of the production cost for that month. (This cost includes the cost of storage and also the cost of money tied up in inventory.) The selling price for footballs is not considered relevant to the production decision because Pigskin will satisfy all customer demand exactly when it occursat whatever the selling price is. Therefore. Pigskin wants to determine the production schedule that minimizes the total production and holding costs. Can you guess the results of a sensitivity analysis on the initial inventory in the Pigskin model? See if your guess is correct by using SolverTable and allowing the initial inventory to vary from 0 to 10,000 in increments of 1000. Keep track of the values in the decision variable cells and the objective cell.The Pigskin Company produces footballs. Pigskin must decide how many footballs to produce each month. The company has decided to use a six-month planning horizon. The forecasted monthly demands for the next six months are 10,000, 15,000, 30,000, 35,000, 25,000, and 10,000. Pigskin wants to meet these demands on time, knowing that it currently has 5000 footballs in inventory and that it can use a given months production to help meet the demand for that month. (For simplicity, we assume that production occurs during the month, and demand occurs at the end of the month.) During each month there is enough production capacity to produce up to 30,000 footballs, and there is enough storage capacity to store up to 10,000 footballs at the end of the month, after demand has occurred. The forecasted production costs per football for the next six months are 12.50, 12.55, 12.70, 12.80, 12.85, and 12.95, respectively. The holding cost incurred per football held in inventory at the end of any month is 5% of the production cost for that month. (This cost includes the cost of storage and also the cost of money tied up in inventory.) The selling price for footballs is not considered relevant to the production decision because Pigskin will satisfy all customer demand exactly when it occursat whatever the selling price is. Therefore. Pigskin wants to determine the production schedule that minimizes the total production and holding costs. As indicated by the algebraic formulation of the Pigskin model, there is no real need to calculate inventory on hand after production and constrain it to be greater than or equal to demand. An alternative is to calculate ending inventory directly and constrain it to be nonnegative. Modify the current spreadsheet model to do this. (Delete rows 16 and 17, and calculate ending inventory appropriately. Then add an explicit non-negativity constraint on ending inventory.)
- The Pigskin Company produces footballs. Pigskin must decide how many footballs to produce each month. The company has decided to use a six-month planning horizon. The forecasted monthly demands for the next six months are 10,000, 15,000, 30,000, 35,000, 25,000, and 10,000. Pigskin wants to meet these demands on time, knowing that it currently has 5000 footballs in inventory and that it can use a given months production to help meet the demand for that month. (For simplicity, we assume that production occurs during the month, and demand occurs at the end of the month.) During each month there is enough production capacity to produce up to 30,000 footballs, and there is enough storage capacity to store up to 10,000 footballs at the end of the month, after demand has occurred. The forecasted production costs per football for the next six months are 12.50, 12.55, 12.70, 12.80, 12.85, and 12.95, respectively. The holding cost incurred per football held in inventory at the end of any month is 5% of the production cost for that month. (This cost includes the cost of storage and also the cost of money tied up in inventory.) The selling price for footballs is not considered relevant to the production decision because Pigskin will satisfy all customer demand exactly when it occursat whatever the selling price is. Therefore. Pigskin wants to determine the production schedule that minimizes the total production and holding costs. Modify the Pigskin model so that there are eight months in the planning horizon. You can make up reasonable values for any extra required data. Dont forget to modify range names. Then modify the model again so that there are only four months in the planning horizon. Do either of these modifications change the optima] production quantity in month 1?VITRAN Bus Services purchases diesel fuel from Domino Gas Supply. In addition to fuel cost, Domino Gas Supply charges VITRAN Bus Services $250 per order to cover the expenses of delivering and transferring the fuel to VITRAN Bus Services storage tanks. The lead time for new shipment from Domino Gas Supply is 10 days, the cost of holding a gallon of fuel in the storage tanks is $0.04 per month, or $0.48 per year and the annual usage of fuel is 150,000 gallons. VITRAN Bus Services buses operate 300 days a year. What is the optimal order quantity for VITRAN Bus Services? How frequently should VITRAN Bus Services order to replenish the gasoline supply?You own a hamburger franchise and are planning to shut down operations for the day, but you are left with 12 buns, 15 defrosted beef patties, and 10 opened cheese slices. Rather than throw them out, you decide to use them to make burgers that you will sell at a discount. Plain burgers each require 1 beef patty and 1 bun, double cheeseburgers each require 2 beef patties, 1 bun, and 2 slices of cheese, while regular cheeseburgers each require 1 beef patty, 1 bun, and 1 slice of cheese. How many of each should you make?
- Carlo and Anita make mailboxes and toys in their craft shop near Lincoln. Each mailbox requires 1 hour of work from Carlo and 1 hourfrom Anita. Each toy requires 1 hour of work from Carlo and 4 hoursfrom Anita. Carlo cannot work more than 6 hours per week and Anita cannot work more than 12 hours per week. If each mailbox sells for $12 and each toy sells for $23, then how many of each should they make to maximize their revenue? What is their maximum revenue? Carlo and Anita should make, x mailboxes and y toys. Their maximum revenue is:Chip Bilton sells sweatshirts at State U football games.He is equally likely to sell 200 or 400 sweatshirts at eachgame. Each time Chip places an order, he pays $500 plus$5 for each sweatshirt he orders. Each sweatshirt sells for$8. A holding cost of $2 per shirt (because of the opportunitycost for capital tied up in sweatshirts as well as storagecosts) is assessed against each shirt left at the end of agame. Chip can store at most 400 shirts after each game.Assuming that the number of shirts ordered by Chip must be a multiple of 100, determine an ordering policy that maximizes expected profits earned during the first three games of theseason. Assume that any leftover sweatshirts have a value of $6.Chip Bilton sells sweatshirts at State U football games. He is equally likely to sell 200 or400 sweatshirts at each game. Each time Chip places an order, he pays $500 plus $5 for each sweatshirt he orders. Each sweatshirt sells for $8. A holding cost of $2 per shirt (because of the opportunity cost for capital tied up in sweatshirts as well as storage costs) is assessed against each shirt left at the end of a game. Chip can store at most 400 shirts after each game. Assuming that the number of shirts ordered by Chip must be a multiple of 100, determine an ordering policy that maximizes expected profits earned during the first three games of the season. Assume that any leftover sweatshirts have a value of $6.
- (Need both parts a and b) During the next four months, a customer requires, respectively, 500, 650, 1000, and 700 units of a commodity, and no backlogging is allowed (that is, the customer’s requirements must be met on time). Production costs are $50, $80, $40, and $70 per unit during these months. The storage cost from one month to the next is $20 per unit (assessed on ending inventory). It is estimated that each unit on hand at the end of month 4 can be sold for $60. Assume there is no beginning inventory. A. What is the objective function in this problem? B. What are the constraints in this problem? Write algebraic expressions for eachA poultry farmer in Lufyanyama has obtained a loan from the Bank to boost his poultry business. He provides you with data to help him optimize the sales. The data is that Old hens can be bought for K20 each but young one cost K50 each. The old hens lay 30 eggs per week, and young ones 50 eggs per week, each egg being worth 30ngwee. A hen cost K10 per week to feed. If a person has only K800 to spend on hens, how many of each kind should he buy to get a profit of more than K600 per week assuming that he cannot house more than 200 hens? a) Formulate the problem as a linear programming model b) Using the Big M – method, how many hens should he buy of each kind to maximize the profit per week? c) Identify the binding and non-binding constraints and justify your choiceBag Company produces leather jackets and handbags. A jacket requires of leather and a handbag only . The labor requirements for the two products are 12 and 5 hours respectively. The current weekly supplies of leather and labor are limited to and 1,850 hours. The company sells the jackets for $350 and the handbags for $120. The objective is to determine the production schedule that maximizes the net revenue. In this document, write the mathematical formulation of both the primal and the dual problems. In the Excel file used in problem 1, solve the primal problem and generate a Solver sensitivity report in a new worksheet. In this document, use the Solver sensitivity report to answer the following two questions: What is the maximum purchase price the company should pay for additional leather? What is the maximum amount the company should pay for additional labor?