Suppose all supplies and demands are integers in a transportation problem. Then the optimal solution to the LP-relaxation of the transportation problem is always integral. True False
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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.A company manufacture two different products. For the coming week 120 hours of labor are available for manufacturing the two products work-hours can be allocated for production or either products l . In addition ,since both products generate a good profit , management is interested in using All 120 hours during the week . Each unit produced of product A requires 3 hours of labor and each unit of product B requires 2.5 hours . (a) Define an equation which states that total work -hours used for producing x units of product A and y units or product B equal 120 . (b) how many units of product A can be produced if 30 units of Product B are produced? (c) if management decides to produce one product only ,what is the maximum quantity which can be produced or product A ? The maximum of product B?A firm operates with the production function Q = K2L Q is the number of units of output per day when the firm rents K units of capital and employs L workers each day. What are the contingent and unconditional input demand curves?
- A Swiss watchmaker wants to create a production plan for the next 4 months.Projected orders for the company’s products are listed in the table. Over the 4 month period,watches may be produced in one month and stored in inventory to meet some later month’sdemand. Because of seasonal factors, the cost of production is not constant, as shown in thetable. The cost of holding a watch in inventory for 1 month is $4. The maximum numberof watches that can be held in inventory is 250. The inventory level at the beginning of theplanning horizon is 200 watches; the inventory level at the end of the planning horizon is to be100. Formulate a linear program (do not solve) to determine the optimal amount to produce ineach month so that demand is met while minimizing the total cost of production and inventory.Shortages are not permitted. Clearly define all decision variables and constraintsIn "Blingblong Takes the World", a famous online role-playing game, the titular character's total damage is computed by taking the product of the critical rate x and critical damage y under the constraint x + % maximum damage to an opponent if c = 100%. = c. Find the critical rate and critical damage needed to inflict theQ.1/ A retail store stocks two types of shirts A and B. These are packed in attractive cardboard boxes. During a week the store can sell a maximum of 400 shirts of type A and a maximum of 300 shirts of type B. The storage capacity, however, is limited to a maximum of 600 of both types combined. Type A shirt fetches a profit of Rs. 2/- per unit and type B a profit of Rs. 5/- per unit. How many of each type the store should stock per week to maximize the total profit? Formulate a mathematical model of the problem.
- The demand function for a good is given by the equation P = a − bQ while the totalcost function is TC = dQ 2+ eQ + f, where a, b, d, e and f are positive constants.(a) Derive the equation for profit.(b) Derive an expression for the value of Q for which profit is maximised.Consider a rectangle in the xy-plane, with corners at (0, 0), (a, 0), (0, b), and (a, b). If (a, b) lies on the graph of the equation y = 30 - x, find a and b such that the area of the rectangle is maximized. What economic interpretations can be given to your answer if the equation y = 30 - x represents a demand curve and y is the price corresponding to the demand x?Suppose that a patient receives a daily dose of 50mg/L of a certain drug such that 42% of it is eliminated from the body each day. If on a certain Monday, the concentration of the drug measured in their body (shortly after the daily dose) is 55 mg/L, write down the Discrete-Time Dynamical System describing the dynamics of the concentration xt of the drug in the body (in mg/L, t days after that Monday). Then write down the general solution to this DTDS.
- (a) A firm makes products A and B and has a total production capacity of 9 tonnes per day, Aand B requiring the same production capacity. The firm has a permanent contract to supplyat least 2 tonnes of A and 3 tonnes of B per day to another company. Each tonne of Arequires 20 machine- hour production time and each tonne of B requires 50 machine-hourproduction time, the daily maximum possible production time available is 360 hours. Profitper unit of product A is 80 and that of B is 120. Formulate a linear programming modelfor the problem.(b) Let xij be the amount shipped from source i to destination j in a 5X5 transportation problem, and let Cij be the corresponding transportation cost per unit. The amounts of supply at sources 1,2,3,4 and 5 are 210, 150,250,190 and 150 units respectively and the demands at destinations 1,2,3,4 and 5 are 205, 220,180,180 and 165 units respectively. The cost of shipping from source 1 to destinations 1,2,3,4 and 5 are 29, 28, 31, 27 and 32 respectively. The cost of shipping from source 2 to destinations 1,2,3,4 and 5 are 26, 27, 30, 37 and 31 respectively. The cost of shipping from source 3 to destinations 1,2,3,4 and 5 are 32, 38, 29, 27 and 35 respectively. The cost of shipping from source 4 to destinations 1,2,3,4 and 5 are 33, 39, 27, 29 and 33 respectively. The cost of shipping from source 5 to destinations 1,2,3,4 and 5 are 35,31, 34, 23 and 35 respectively. (i)Find the associated optimal cost. (ii)Determine each non basic variable.The demand function for a good is given by the equation P = a − bQ while the totalcost function is TC = dQ2 + eQ + f, where a, b, d, e and f are positive constants.(a) Derive the equation for profit.(b) Derive an expression for the value of Q for which profit is maximised.