1. Use the minimum-entry and u-v methods to solve the following balanced minimum-cost trans- portation problem: M₁ M₂ M3 M4 Supplies W₁ 2 2 6 6 10 W₂ 4 5 4 3 30 W3 1 6 9 6 20 Demands 8 10 20 22 Use Bland's rule to choose entering and leaving variables where variables/cells are ordered from top to bottom and left to right.
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- 23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes 1/2 of the food and 1/3 of the manufactured goods. Manufacturing consumes 1/2 of the food and 2/3 of the manufactured goods. Assuming the economy is closed and in equilibrium, find the relative outputs of the farming and manufacturing industries.2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.7 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 3 Food C 1 1 1Suppose that you have two products, (Product 1 and 2), which can be produced in one of two plants (Plant 1 and 2). The corresponding unit processing times of each product on each plant, and their unit costs are given. Each plant has a minimum working hour limit, as shown in the table below.Decision variables for the problem are defined as x: number of product 1 produced, and y:number of product 2 produced. Formulated the problem as an LP minimizing total cost and solve it by graphical method. Explicitly show the isocost lines, the intersection of the costraints and the optimal solution on the graph. Then, according to your graph, answer the following questions.(a) What is the shadow price of the Constraint 1(Plant 1 Constraint)? Compute and show on graph.(b) What is the shadow price of the Constraint 2(Plant2 Constraint)? Compute and show on graph.(c) What is the reduced cost of x? (How much should the objective function coefficient of x reduced from 6 for it to become nonzero in the…
- A factory manufactures three products. These products are processed through 3 different conduction stages. The time required to manufacture each of these 3 products and the daily capacity of the stages are given by the following data Time per unit (minutes) stage Product 1 Product 2 Product 3 Stage capacity min/day 1 1 1 430 2 3 2 460 3 1 4 420 Use simplex method to determine the daily number of units to be manufactured of each product given that the profit per unit for product 1, 2 and 3 are 3, 2 and 5 respectively. Assume that all amounts produced are consumed by the market. Fully interpret your solution.Thank you7. Use Lagrange multipliers to give an alternate solution. Find two positive numbers whose product is 100 and whose sum is a minimum.
- Use the minimum-entry and u-v methods to solve the following minimum-cost transportationproblem. Assuming there is no cost associated with the artificial warehouse or market.Use Bland’s rule to choose entering and leaving variables where variables/cells are ordered from topto bottom and left to right.Suppose the following graphical representation of columns in optimization model and RHS vector (b). Which pair of column vectors represents the optimal solution? (Suppose that all costs are equal) Suppose minimization model. Select one: Ο α-β a-y Ο α-δ O α-ε B-Y O B-8 O y-8 Y-E O unbounded criteria function O infeasible solution Clear my choice4. A medium-sized bakery has just opened in Slovakia. A loaf of bread is currently selling for 14 koruna (the Slovakian currency) over and above the cost of intermediate goods (flour, yeast, and so on). Assuming that labor is the only variable factor of pro- duction, the following table gives the production function for the bread. WORKERS LOAVES OF BREAD 1 15 30 3 42 4 52 60 6. 66 7 70 a. Suppose the current wage rate in Slovakia is 119 koruna per hour. How many workers will the bakery employ? b. Suppose the economy of Slovakia begins to grow, incomes rise, and the price of a loaf of bread is pushed up to 20 koruna. Assuming no increase in the price of labor, how many workers will the bakery hire? c. An increase in the demand for labor pushes up wages to 125 koruna per hour. What impact will this increase in cost have on employment and output in the bakery at the 20-koruna price of bread? d. If all firms behaved like our bakery, would the allocation of resources in Slovakia be…