Operations Research Homework #5
2007.10.22 1. Consider the following Linear Programming problem:
max 2x1+ 4x2+ 3x3+ x4 s.t. 3x1+ x2+ x3 + 4x4 ≤12
x1−3x2+ 2x3 + 3x4 ≤7 2x1+ x2+ 3x3− x4 ≤10 x1 ≥0, x2 ≥0, x3 ≥0, x4 ≥0.
The optimal solution to this problem is x1 = 0; x2 = 10.4; x3 = 0; x4 = 0.4 with the optimal value 42. You may image that this problem be to maximize the profit of 4 products subject to three resource constraints.
(a) Suppose the coefficient of a non-basic variable decreases. You have taken a variable that you did not want to use in the first place and then made it less profitable. Of course, you are still not going to use it. Let the coefficient of x1 be reduced to 1 and use the primal feasibility, dual feasibility and complementary slackness to verify the optimal solution does not change.
(b) What if you raise the coefficient of x1? Intuitively, raising it just a little bit should not matter, but raising the coefficient a lot might induce you to make x1 >0. Find the break even point of the coefficient of x1 to make x1 >0.
(c) If the coefficient of a basic variable, say x2, decreases, the contribution of the second product to the net profit is then reduced. You might tolerate a small such reduction, but a sufficient large reduction will make you want to set x2 = 0 instead of x2 = 10.4. Discuss whether you want to tolerate if the contribution of from 4 per unit to 2 per unit?
(d) If the coefficient of a basic variable, say x4, goes up, then your profit goes up and you still want to use the variable. However, if it goes high enough, it may force you to adjust your current optimal solution.
You may want to (i) arrange more resources for producing more x4 by cutting the amount of x2 that competes the same resource with x4; or (ii) find another basis that replaces x2 to make x4 even possible.
Find the entire price range of increasing for case (i) to happen and the break even price for case (ii) to happen. Which new product is used to replace x2 in the optimal production plan?
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2. A lumber company has three sources of wood and five markets where wood is demanded. The annual quantity of wood available in the three sources of supply are 15, 20, and 15 million board feet respectively. The amount that can be sold at the five markets is 11, 12, 9, 10, and 8 million board feet, respectively. The company currently transports all of the wood by train. It wishes to evaluate its transportation schedule, possibly shifting some or all of its transportation to ships. The unit cost of shipment (in $10,000 along the various routes using both methods is described in the table below. For the routes that shipping was not feasible (the ”none” entries in the cost table), substitute a large number, say 250, for it.
Supply Market 1 Market 2 Market 3 Market 4 Market 5
A 51 62 35 45 56
B 59 68 50 39 46
C 49 56 53 51 37
Cost per Unit of Rail Transport
Supply Market 1 Market 2 Market 3 Market 4 Market 5
A 48 68 48 none 54
B 66 75 55 49 57
C none 61 64 59 50
Cost per Unit of Ship Transport
The management needs to decide to what extent to continue to rely on rail transportation.
(a) How much does it cost use rail transport exclusively?
(b) For the same transit schedule in (a) if ships were used instead of trains, how much is the cost?
(c) How much does it cost to use ships exclusively?
(d) For the same transit schedule in (c) if trains were used instead of ships, how much in the cost?
(e) How much does it cost to use the cheapest available mode of trans- portation on each route?
(f) Show that the optimal cost of (e) must be the minimum of the optimal costs of (a), (b), (d), and (d).
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