A company produces two products. The relevant data is shown below:Product 1Suppose thatDemand rate (units/year)Production rate (units/year)Production cost ($/unit)Inventory carrying rate (%/unit/year)2,0004,0002025Product 212,00016,0001525the setup cost for each product is same and $400.the total number of setups used for producing Product 1 and Product 2 is limited by 5.Using Lagrange multiplier technique, determine the optimum order quantity for each product and theminimum total cost.

To find the optimum order quantity for each product and the minimum total cost, we can use the…

Answered: A company produces two products. The…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter4: Linear Programming Models

Section: Chapter Questions

Problem 73P

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Assume the demand for a companys drug Wozac during the current year is 50,000, and assume demand will grow at 5% a year. If the company builds a plant that can produce x units of Wozac per year, it will cost 16x. Each unit of Wozac is sold for 3. Each unit of Wozac produced incurs a variable production cost of 0.20. It costs 0.40 per year to operate a unit of capacity. Determine how large a Wozac plant the company should build to maximize its expected profit over the next 10 years.
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?
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?
Based on Zangwill (1992). Murray Manufacturingruns a day shift and a night shift. Regardless of thenumber of units produced, the only production costduring a shift is a setup cost. It costs $8000 to run theday shift and $4500 to run the night shift. Demand forthe next two days is as follows: day 1, 2000; night 1,3000; day 2, 2000; night 2, 3000. It costs $1 per unitto hold a unit in inventory for a shift.a. Determine a production schedule that minimizesthe sum of setup and inventory costs. All demandmust be met on time. (Note: Not all shifts have tobe run.)b. After listening to a seminar on the virtues of theJapanese theory of production, Murray has cut thesetup cost of its day shift to $1000 per shift and thesetup cost of its night shift to $3500 per shift. Nowdetermine a production schedule that minimizes thesum of setup and inventory costs. All demand mustbe met on time. Show that the decrease in setupcosts has actually raised the average inventorylevel. Is this reasonable?
(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 each
A company produces three different products; x,y, and z. For the production of 1 unit of product x, 1 unit of A and 1 unit of B are used as input. 1 unit of A and 2 units of B are used to produce 1 unit of product y. To produce 1 unit of product z, only 1 unit of A is used. The company holds 40 units of A and 20 units of B periodically in total. By the way, the company can not produce product y more than the twice of product z. On the other hand, the sale price of 1 unit of product x is 10 TL, product y is 15 TL and product z is 12 TL. Furthermore, the cost of 1 unit of product x is 8 TL, product y is 9 TL and product z is 7 TL. According to above information, what should be the company’s optimal product mix to maximize its profit? Construct the problem as a Linear programming model. Solve the above problem by using Simplex Method.
A company produces three types of items. A singlemachine is used to produce the three items on a cyclicalbasis. The company has the policy that every item isproduced once during each cycle, and it wants to determinethe number of production cycles per year that will minimizethe sum of holding and setup costs (no shortages areallowed). The following data are given:Pi number of units of product i that could be producedper year if the machine were entirely devoted toproducing product iDi annual demand for product iKi cost of setting up production for product ihi cost of holding one unit of product i in inventoryfor one yeara Suppose there are N cycles per year. Assuming thatduring each cycle, a fraction N1of all demand for eachproduct is met, determine the annual holding cost andthe annual setup cost.b Let qi* be the number of units of product i producedduring each cycle. Determine the optimal value of N(call it N*) and qi*. c Let EROQi be the optimal production run size forproduct i if…
Wallace Company makes and sells a single product. Each unit of product requires two hours of labor with a wage rate of $8 per hour. The company has budgeted to sell 8,000 units and to produce 10,000 units during the current month. The product requires three pounds of material for each unit produced. Budgeted direct labor costs for the current month would be: $192,000 $128,000 $80,000 $64,000 $160,000

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