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Chapter 2, PP 33, Section 2.1: A Simple Maximization Problem (Reference: An introduction to Management Science by Anderson, Sweeney and Williams)Par, Inc., is a small manufacturer of golf equipment and suppliers whose management has decided to move into the market for medium and high priced golf bags. Pars distributor is enthusiastic about the new product line and has agreed to buy all the golf bags Par produces over the next three months. After a thorough investigation of the steps involved in manufacturing a gold bag, management determined that each golf bag produced will require the following operations:1. Cutting and dyeing the material2. Sewing3. Finishing (inserting umbrella holder, club separators, etc.)4. Inspection and packagingThe director of manufacturing analyzed each of the operations and concluded that if the company produces a medium priced standard medium, each bag will require 7/10 hour in the cutting and dyeing department, hour in the sewing department, 1 hour in the finishing department, and 1/10 hour in the inspection and packaging department. The more expensive deluxe model will require 1 hour for cutting and dyeing, 5/6 hour for sewing, 2/3 hour for finishing, and hour for inspection and packaging.Pars production is constrained by a limited number of hours available in each department. After studying departmental workload projections, the director of manufacturing estimates that 630 hours for cutting and dyeing, 600 hours for sewing, 708 hours for finishing, and 135 hours for inspection and packaging will be available for the production of golf bags during the next three months. The accounting department analyzed the production data, assigned all relevant variable costs, and arrived at prices for both bags that will result in a profit contribution of $10 for every standard bag and $9 for every deluxe bag produced. Formulate a mathematical model of the Par, Inc., problem that can be used to determine the number of standard bags and the number of deluxe bags to produce in order to maximize total profit contribution.Formulation of Simple Maximization Linear Programming ProblemQ.1) A company has three operational departments (weaving, processing and packing) with capacity to produce three different types of clothes namely suitings, shirtings and woolens yielding a profit of Rs. 2, Rs. 4 and Rs. 3 per meter respectively. One meter suitings require 3 minutes in weaving, 2 minutes in processing and 1 minute in packing. Similarly, one meter of shirting requires 4 minutes in weaving, 1 minute in processing and 3 minutes in packing and one meter of woolens requires 3 minutes in each department. In a week, total run time of each department is 60, 40 and 80 hours for weaving, processing and packing departments respectively. Formulate the linear programming problem to find the product mix to maximize the profit.

Q2) RMC, Inc. is a small firm that produces a variety of chemical products. In a particular production process, three raw materials are blended (mixed together) to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of 2/3 ton of material 1, ton of material 2 and 3/5 of material 3. A ton of solvent base is a mixture of ton of material 1, 1/5 ton of material 2, and 3/10 ton of material 3. After deducting relevant costs, the profit contribution is $40 for every ton of fuel additive produced and $30 for every ton of solvent base produced. RMCs production is constrained by a limited availability of the three raw materials. For the current production period, RMC has available the following quantities of each raw material:Raw MaterialAmount Available for Production

Material 120 tons

Material 25 tons

Material 321 tons

Formulate the linear programming problem to maximize the total profit contribution.

Q3) A small manufacturer employs 5 skilled men and 10 semi-skilled men for making a product in two qualities: a deluxe model and an ordinary model. The production of a deluxe model requires 2-hour work by a skilled man and 2-hour work by a semi-skilled man. The ordinary model requires 1-hour work by a skilled man and 3-hour work by a semi-skilled man. According to worker unions rules, no man can work more than 8 hours per day. The profit of the deluxe model is Rs. 1000 per unit and that of the ordinary model is Rs. 800 per unit. Formulate a linear programming model for this manufacturing situation to determine the production volume of each model such that the total profit is maximized. Q4) A firm manufactures three products A, B and C. Their profits per unit are Rs. 300, Rs. 200 and Rs. 400, respectively. The firm has two machines and the required processing time in minutes on each machine for each product is given in the following table:Product

ABC

Machine1435

2224

Machines 1 and 2 have 2000 and 2500 machine-minute, respectively. The upper limits for the production volumes of the product A, B and C are 100 units, 200 units and 50 units, respectively. But, the firm must produce a minimum of 50 units of the product A. Develop a LP model for this manufacturing situation to determine the production volume of each product such that the total profit is maximized.

Q5) Johnson & Johnson has two products Deluxe Stayfree and Deluxe carefree. To produce one unit of deluxe Stayfree, 2 units of material A, 4 units of material B and 2 units of material C are required. To produce one unit of Deluxe Carefree, 3 units of Material A and 2 units of material B and 1 units of material C are required. Not more than 16 units of material A can be used and at least 16 units of material B must be used and the use of material C in total should be equal to 16. The profit contribution per unit of Deluxe Stayfree and Deluxe Carefree are Rs. 6 and Rs. 8 respectively. Formulate as a linear programming problem.Q6) Consider a milk producer wants to feed his cows adequately but at the same time with minimum cost. Suppose the adequate feed menu consists of a minimum of 50 units of protein, 60 units of calcium and 40 units of carbohydrates. Now, the job of the milk producer is to find out the quantities of two feeds 1 and 2 that he can buy at Rs. 150 and Rs. 200 per unit respectively subject to the minimum requirements of all the three nutrients given. The following table gives the nutrients availability per unit of the feeds concerned and the minimum requirements. NutrientsFeed 1Feed 2Minimum Requirements

Protein2250

Calcium3260

Carbohydrate2440

Formulate a suitable linear programming problem and obtain its optimum solution by graphical method. Q7) A firm produces two products A and B. Both of the products have to be processed on two machines G and H. One unit of product requires four minutes of processing on machine G and three minutes of processing on machine H. One unit of product B requires three minutes of processing on machine G and four minutes of processing on machine H. The firm only has 1650 minutes of processing time available per week on machine G and 2250 minutes of processing time available on machine H. The profit contributions per unit of A and B are expected to be Rs. 45 and Rs. 55, respectively. Formulate the firms problem as an LPP, find the optimal resource allocation and product mix. Q9) A small furniture manufacturer produces tables and chairs. Each product must go through three stages of the manufacturing process: assembly, finishing, and inspection. Each table requires 3 hours of assembly, 2 hours of finishing and 1 hour of inspection. Each chair requires 2 hrs of assembly, 2 hrs of finishing and 1 hr of inspection. The profit per table is $ 120 while the selling price per chair is $ 80. Currently, each week there are 200 hours of assembly time available, 180 hours of finishing time and 40 hours of inspection time. Formulate the problem as an LPP, and find the optimal production schedule.

Q10) The Versatech Corporation has decided to produce three new products. Five branch plants now have excess product capacity. The unit manufacturing cost of the first product would be $31, $29, $32, $28 and $29 in plants 1, 2, 3, 4 and 5, respectively. The unit manufacturing cost of the second product would be $45, $41, $46, $42 and $43 in plants 1, 2, 3, 4, and 5, respectively. The unit manufacturing cost of the third product would be $38, $35 and $40 in plants 1, 2 and 3, respectively, whereas plants 4 and 5 do not have the capability for producing this product. Sales forecasts indicate that 600, 1000 and 8000 units of products 1, 2 and 3, respectively, should be produced per day. Plants 1, 2, 3, 4, and 5 have the capacity to produce 400, 600, 400, 600 and 1000 units daily, respectively, regarding of the product or combination of products involved. Assume that any plant having the capability and capacity to produce them can produce any combination of the products in any quantity. Management wishes to know how to allocate the new products to the plants to minimize total manufacturing cost. Formulate and solve the problem.

A Simple Minimization ProblemM & D Chemicals produces two products that are sold as raw materials to companies manufacturing bath soaps and laundry detergents. Based on an analysis of current inventory levels and potential demand for the coming month, M&Ds management specified that the combined production for products A and B must total at least 350 gallons. Separately, a major customers order for 125 gallons of product A must also be satisfied. 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, 600 hours of processing time are available. M&Ds objective is to satisfy these requirements at a minimum total production cost. Production costs are $2 per gallon for product A and $3 per gallon for product B. Find the minimum cost production schedule.

Graphical Solution problem of Linear Programming Problem:Q1) A company makes two kinds of leather belts. Belts A is a high quality belt, and belt B is of lower quality. The respective profits are Rs. 4.00 and Rs. 3.00 per belt. Each belt of type A requires twice as much time as a belt of type B, and if all belts were of type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 belts per day (Both A and B combined). Belt A requires a fancy buckle and only 400 per day are available. There are only 700 buckles a day available for belt B. Determine the optimal product mix. Q2) Use graphical method to solve the LPP:

subject to the constraints

.Q3) Use graphical method to solve the LPP:

subject to the constraints

.

Q4) Use graphical method to solve the LPP:

subject to the constraints

.

Try yourself Q21, Q22, Q24 and Q28. MarketingAn advertising Company wishers to plan an advertising campaign in three different mediatelevision, radio, and magazines. The purpose of the advertising is to reach as many potential customers as possible. Results of a market study are given below.

Television

Prime day(rs.)Prime time (Rs.)Radio(Rs.)Magazines(Rs.)

Cost of an advertising unit40,00075,00030,00015,000

Number of potential customers Reached per unit4,00,0009,00,0005,00,0004,00,000

Number of women customers reached per unit3,00,0004,00,0002,00,0001,00,000

The company does not want to spend more than Rs. 8,00,000 on advertising. It further requires thati) at least 2 million exposures take place among womenii) advertising on television be limited to Rs. 5,00,000iii) at least 3 advertising units be bought on prime day and two units during prime time; andiv) the number of advertising units on radio and magazine should each be between 5 and 10Formulate the linear programming model in order to maximize total number of potential customers reached.

Problem related to solving a Linear Programming problem with Simplex Method.High Tech Industries imports electronic components that are used to assemble two different models of personal computers. One model is called the Deskpro, and the other model is called the Portable. HighTechs management is currently interested in developing a weekly production schedule for both products. The Deskpro generates a profit contribution of $50 per unit, and the Portable generates a profit contribution of $40 per unit. For next weeks production, a maximum of 150 hours of assembly time can be made available. Each unit of Deskpro requires 3 hours of assembly time, and each unit of the Portable requires 5 hours of assembly time. In addition, HighTech currently has only 20 Portable display components in inventory; thus, no more than 20 units of the Portable may be assembled. Finally, only 300 square feet of warehouse space can be made available for new production. Assembly of each Deskpro requires 8 square feet of warehouse space; similarly, each Portable requires 5 square feet.

Problem related to solving a Linear Programming problem with Two Phase Simplex Method.Q1) G. J. Breveries Ltd. have two bottling plants, one located at G and the other at J. Each plant produces three drinks - whisky, beer and brandy names A, B and C respectively. The number of bottles produced per day is as follows:DrinkPlant

GJ

Whisky 15001500

Beer 30001000

Brandy 20005000

A market survey indicated that during the month of July, there will be a demand of 20,000 bottles of whisky, 40,000 bottles of beer and 44,000 bottles of brandy. The operating costs per day of plants at G and J are 600 and 400 monetary units. For how many days each plant be run in July so as to minimize the production cost, while still meeting the market demand? Solve by two-phase Simplex Method?Q2) Air Force is experimenting with three types of bombs P, Q and R in which three kinds of explosive, viz., A, B and C will be used. Taking the various factors into consideration, it has been decided to use at most 600 kg of explosive A, at least 480 kg of explosive B and exactly 540 of explosive C. Bomb P requires 3, 2, 2 kg. of A, B and C respectively. Bomb Q requires 1, 4, 3 kg of A, B and C respectively. Bomb R requires 6, 2, 3 kg of A, B, and C respectively. Now bomb P will give the equivalent of a 2 ton explosion, bomb Q will give a 3 ton explosion and Bomb R will give a 4 ton explosion. Under what production schedule can be Air Force make the biggest bang? Solve by Two-Phase Simplex Method.

CASE STUDY

TJs Inc, makes three nut mixes for sale to grocery chains located in the Southeast. The three mixes, referred to as the Regular Mix, the Deluxe Mix, and the Holiday Mix, are made by mixing different percentages of five types of nuts. In preparation for the fall season, TJs has just purchased the following shipments of nuts at the prices shown:Type of NutShipment Amount (pounds)Cost per Shipment ($)

Almond60007500

Brazil75007125

Filbert75006750

Pecan60007200

Walnut75007875

The Regular Mix consists of 15% almonds, 25% Brazil nuts, 25% filberts, 10% pecans, and 25% walnuts. The Deluxe Mix consists of 20% of each type of nut, and the Holiday Mix consists of 25% almonds, 15% Brazil nuts, 15% filberts, 25% pecans, and 20% walnuts.TJs accountant analyzed the cost of packaging materials, sales price per pound, and so forth, and determined that the profit contribution per pound is $1.65 for the Regular Mix, $2.00 for the Deluxe Mix, and $2.25 for the Holiday Mix. These figures do not include the cost of specific types of nuts in the different mixes because that cost can vary greatly in the commodity markets. Customer orders already received are summarized here:Types of MixOrders (pounds)

Regular10000

Deluxe3000

Holiday5000

Because demand is running high, it is expected that TJs will receive many more orders than can be satisfied. TJs is committed to using the available nuts to maximize profit over the fall seasons; nuts not used will be given to a local charity. Even if it is not profitable to do so, TJs president indicated that the orders already received must be satisfied. Management ReportPerform an analysis of TJs product-mix problem, and prepare a report for TJs president that summarizes your findings. Be sure to include information and analysis on the following:1. the cost per pound of the nuts included in the Regular, Deluxe and Holiday mixes.2. The optimal product mix and the total profit contribution3. Recommendations regarding how the total profit contribution can be increased if additional quantities of nuts can be purchased. 4. A recommendation as to whether TJs should purchase an additional 1000 pounds of almonds for $1000 from a supplier who overbought5. Recommendations on how profit contribution could be increased (if at all) if TJs does not satisfy all existing orders.