Importance
1. Assists financial analyst in selecting an investment portfolio from a variety of stock and bond investment alternative that would maximize the return of investment.
2. Helps marketing managers to determine the best way to allocate a fixed advertising budget among alternative advertising media such as radio, television, newspaper and magazines that would maximize advertising effectiveness.
3. Helps the manager of a company to select the proper warehouse where it should ship the right quantity of products to customers in order to minimize the total transportation costs.
Properties
1. They deal with maximizing or minimizing some quantity.
2. There are restrictions or constraints that limit the degree to which the objective can be pursued.
Linear Programming Model
1. Objective. Linear programming algorithms require that a single goal or objective be specified. There are two general types of objectives: maximization and minimization. A maximization problem involves profits, revenues, efficiency or rate of return. Conversely, a minimization problem involves cost in time, distance traveled, or scrap. The profit cost per unit of output or input is summarized by the objective function.
2. Decision variables. These represent choices available to the decision maker in terms of amounts of either inputs or outputs.
3. Constraints are limitations that restrict the alternative available to decision making. There are three types of constraints: less than or equal to, greater than or equal to, and simply equal. A less that or equal to constraint implies an upper limit on the amount of some scare resource available for use. A greater than or equal to constraint specifies a lower bound that must be achieved in the final solution. The equal constraint is more restrictive in the sense that it specifies exactly what a decision variable should equal a linear programming model can consist of one or more constraints.
3. Parameters. A linear programming model consists of a mathematical statement of the objective and a mathematical statement of each constraint. These statements consist of symbol (X1, X2) that represent the decision variables and numerical value called parameters. The parameters are fixed value: the model is solved given these values.
Assumption on Linear Programming Models
1. Linearity: The impact of decision variables is linear in constraints and the objective function.
2. Divisibility: non-integer value of decision variables are acceptable.
3. certainty: values of parameters are known and constant
4. Non-negativity . Negative value of decision variables are unacceptable.
Formulating a Linear Programming
1. Identify the decision variables
2. write out the objective function
3. Formulate each of the constraints
Let the problem guide you in formulating a linear program.
A. If the first words of a problem are "A manager wants to maximize profits" this would lead to maximize (profits)"
B. " A department has 1000 pounds of raw material available to prepare its products" (raw material less than or equal to 1000 pounds)
C. A supervisor would like to know much of each two products to make (X1 = quantity pf product 1, X2 = quantity of product 2)
Source: Quantitative Techniques in Business by Rex Book Store
1. MarMar Inc, handles radio and television promotional jobs and placements for a wide range of clients. The agency's objective is to maximize the total audience exposure for its clients products. Research indicates that each radio spot results in 1000 exposures while each television spot contributes an independent 3000 exposures. A radio spot costs P5000 and a television spot costs P20,000. Clients provide Ad-Board with a maximum monthly budget of P1,000,000. Contracts with radio networks require a minimum of 100 spots per month. The agency employs account executives to place the spots. past experience indicates that each radio spot takes 20 hours of executive effort and each television spot uses 40 hours. The agency's account executives are available for 4000 hours per month.
2. An appliance manufacturer produces two models of electric fans: stand fans and desk fans. Both models acquire fabrications and assembly work. Each stand fan uses 4 hours of fabrication and two hours of assembly and each desk fan uses two hours of fabrication and six hours of assembly. There are 600 fabrication hours available per week and 480 hours of assembly. Each stand fan contributes P400 to profit and each desk fan contributes P300 to profit. How many stand fans and desk fans must the manufacturer produce in order to maximize the profit?
3. Moore's Meat packing company produces a hotdog mixture in 1000-pound batches. The mixture contains three ingredients- chicken, beef, and cereal. The cost pound of each of these ingredients is as follows:
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ingredient cost/lb
chicken $3
beef 5
cereal 2
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Each batch must contain the following a) at least 200 pounds of chicken, b) at least 400 pounds of beef and c) no more than 300 pounds of cereal. The company wants to know the optimal mixture of ingredients that maximizes cost. Formulate a linear programming model for this problem.
4. The Southern Sporting Goods Company makes basketballs and footballs. Each product is produced from two resources - rubber and leather. The resource requirements for each product and total resources available are as follows.
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Product Resource Requirements per Unit
Rubber (lb) Leather (ft^2)
Basketball 3 4
Football 2 5
Total resources 500 800
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Each basketball produced results in a profit of $12, and each football earns $16 in profit. Formulate a linear programming model to determine the number of basketballs and footballs to produce in order to maximize profit.
5. A hospital dietitian prepares breakfast menus every morning for the hospital patients. Part of the dietitian's responsibility is to make sure that minimum daily requirements for vitamins A and B are met. At the same time, the cost of the menus must be kept as low as possible. The main breakfast staples providing vitamins A and B are eggs, bacon, and cereal. The vitamin requirements and vitamin contributions for each staple follow.
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Vitamin Contributions
Vitamin mg/egg mg/bacon strip mg/cereal cup minimum daily requirements
A 2 4 1 16
B 3 2 1 12
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An egg costs $0.04, a bacon strip costs $0.03, and a cup of cereal costs $0.02. The dietitian wants to know how much of each staple to serve per order to meet the minimum daily vitamin requirements while minimizing total cost. Formulate a linear programming model for this problem.
6. The Pyrotec Company produces three electrical products - clocks, radios, and toasters. These products have the following resource requirements.
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Resource Requirements
product cost/unit labor hours/unit
clock $7 2
radio 10 3
toaster 5 2
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The manufacturer has a daily production budget of $2,000 and a maximum of 660 hours of labor. maximum daily customer demand is for 200 clocks, 300 radios, and 150 toasters. Clock sell for $15, radios for $20 and toasters for $12. The company desires to know the optimal product mix that will maximize profit. Formulate a linear programming model for this problem.
7. Fred Friendly owns an automobile dealership in Tampa, Florida. Fred is presently attempting to determine how many cars to order from the factory in Detroit. Fred has a budget of $210,00 to purchase new cars. He stocks three different styles - the Eagle (a full-size car), the Hawk (a medium-size car), and the Sparrow (a compact). The costs to Fred of the various models are as follows.
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Car Cost
Eagle $8,000
Hawk 5,000
sparrow 4,500
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Fred sells Eagles for $11,000, Hawks for $8,500, and sparrows for $7,000. Based on past sales, Fred knows that the maximum demand for eagles is 80 cars; for Hawks, 175 cars; and for Sparrows, 250 cars. Fred has 8,000 square feet of space available on his lot to store new cars. An Eagle takes up 50 square feet, a Hawk requires 35 square feet, and a Sparrow requires 25 square feet. Fred wants to know how many cars of each style he should order from the distributor in order to maximize his total profit. Formulate a linear programming model for this problem.
