Exam 19: Linear Programming

The property manager of a city government issues chairs, desks, and other office furniture to city buildings from a centralized distribution centre. Like most government agencies, it operates to minimize its costs of operations. In this distribution centre, there are two types of standard office chairs, Model A and Model B. Model A is considerably heavier than Model B, and costs $20 per chair to transport to any city building; each model B costs $14 to transport. The distribution centre has on hand 400 chairs-200 each of A and B. The requirements for shipments to each of the city's buildings are as follows: Building 1 needs at least 100 of A Building 2 needs at least 150 of B. Building 3 needs at least 100 chairs, but they can be of either type, mixed. Building 4 needs 40 chairs, but at least as many B as A. Write out the objective function and the constraints for this problem. (Hint: there are eight variables-chairs for building 1 cannot be used to satisfy the demands for another building).

A company has the following inputs and cost structure. Product Ingredient A 4 2 8 Ingredient B 2 3 3 Ingredient C 4 5 3 Ingredient D 7 5 7 Cost per unit 0.25 0.21 0.69 They must also use the following minimum quantities. A 100, B 120, C 140, D 160 The is also a maximum of Z of 200 units. c) Calculate how many of each product (X, Y, Z) should be produced. d) How much is the minimized total cost?

What are corner points? What is their relevance to solving linear programming problems?

For the two constraints given below, which point is in the feasible region of this minimization problem? (1) 14x + 6y > 42 (2) x - y > 3

In sensitivity analysis, a zero shadow price (or dual value) for a resource ordinarily means that

One requirement of a linear programming problem is that the objective function must be expressed as a linear equation.

What combination of a and b will yield the optimum for this problem? Maximize $6a + $15b, subject to (1) 4a + 2b < 12 and (2) 5a + 2b < 20 and (3) a, b ≥ 0.

The requirements of linear programming problems include an objective function, the presence of constraints, objective and constraints expressed in linear equalities or inequalities, and ________.

A linear programming problem contains a restriction that reads "the quantity of S must be no less than one-fourth as large as T and U combined." Formulate this as a constraint ready for use in problem solving software.

A company has the following usage of inputs, constraints on inputs and profit margins to make their three products. Product Wiring Drilling Assembly Profit X202 0.6 0.41 0.8 14 Y303 0.1 0.57 0.47 9.75 Z404 0.35 0.3 0.26 12.25 There are maximum amounts of the following: 1625 hours for wiring, 1750 hours for drilling and 1175 hours for Assembly. Make sure that the units produced are integers. Solve for the above problem. The company must make at least 700 X202, 600 Y303 and 500 Z404. a) How many units of each product will be made? b) What is the maximum profit? c) Which input has the least amount of slack?

A feedlot is trying to decide on the lowest cost mix that will still provide adequate nutrition for its cattle. Suppose that the numbers in the chart represent the number of grams of ingredient per 100 grams of feed and that the cost of Feed X is $5/100 grams, Feed Y is $3/100 grams, and Feed X is $8/100 grams. Each cow will need 50 grams of A per day, 20 grams of B, 30 grams of C, and 10 grams of D. If the feedlot can get no more than 200 grams per day per cow of any of the feed types determine the constraints governing the problem. Ingredient X Y Z 10 15 5 30 10 20 40 0 20 0 20 30

A linear programming problem contains a restriction that reads "the quantity of S must be no more than one-fourth as large as T and U combined." Formulate this as a constraint ready for use in problem solving software.

Consider the following constraints from a two-variable linear program. (1) X ≥ 1 (2) Y ≥ 1 (3) X + Y ≤ 9 If these are the only constraints, which of the following points (X, Y) cannot be the optimal solution?

What are the requirements of all linear programming problems?

What is the feasible region in a linear programming problem?

Which of the following represents valid constraints in linear programming?

A synonym for shadow price is ________.

Suppose that a constraint is given by X + Y≤10. If another constraint is given to be 3X + 2Y≥15 determine the corners of the feasible solution. If the profit from X is 5 and the profit from Y is 10, determine the maximum profit.

Lost Maples Winery makes three varieties of contemporary Mission Hill Country wines: Austin Formation (a fine red), Ste. Genevieve (a table white), and Los Alamos (a hearty pink Zinfandel). The raw materials, labour, and contribution per case of each of these wines is summarized below. Grapes Variety A bushels Grapes Variety B bushels Sugar KGS Labour (man- hours) Contrib. per case Austin Formation 5 0 2 3 \ 24 Ste. Genevieve 0 4 0 1 \ 28 Los Alamos 2 2 2 4 \ 20 The winery has 2900 bushels of Variety A grapes, 2080 bushels of Variety B grapes, 900 pounds of sugar, and 1260 man-hours of labour available during the next week. The firm operates to achieve maximum contribution. Answer the following questions. a. For maximum contribution, how much of each wine should be produced? b. How much contribution will be made by selling the output? c. Is there any sugar left over? If so, how much?

Sensitivity analysis helps to