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Unformatted text preview: IE 505 MATHEMATICAL PROGRAMMING
Lecture #1 Date: September 16th 2008 1 2 3 4 5 6 7 8 9 10 Problem Modeling Algorithm “Solution” Sensitivity/ Post Optimality
11 Linear Programming
• Linear Programming is the problem of minimizing a linear cost function subject to linear constraints 12 The Linear Programming Problem (LP)
• Components: – Decision Variables: Variables whose values are to be determined – Parameters: (Uncontrolled decision variables) Environmental factors which are not under the control of the decision maker. – Objective Function: The goal is always to maximize or minimize some linear function of the decision variables. 13 The Linear Programming Problem (LP)
• The LP problem we shall study can be formulated as: max c1 x1 ... cn xn s.t. a11x1 ... a1n xn b1 . . . am1 x1 ... amn xn bm x1 , x2 ,..., xn 0
14 15 16 The Diet Problem
History of the Diet Problem The diet problem is one of the first optimization problems to be studied back in the 1930's and 40's. For a moderately active man weighing 154 pounds, how much of each of 77 foods should be eaten on a daily basis so that the man's intake of 9 nutrients will be at least equal to the recommended dietary allowances (RDSs) suggested by the National Research Council in 1943, with the cost of the diet being minimal? The nutrient RDAs required to be met in Stigler's experiment were calories, protein, calcium, iron, vitamin A, thiamine, riboflavin, niacin, and ascorbic acid. 17 The 9 nutrients and their respective recommended daily amounts were: Calories 3,000 Calories Protein 70 grams Calcium .8 grams Iron 12 milligrams Vitamin A 5,000 IU Thiamine (Vitamin B1) 1.8 milligrams Riboflavin (Vitamin B2) 2.7 milligrams Niacin 18 milligrams Ascorbic Acid (Vitamin C) 75 milligrams 18 One of the early researchers to study this problem was George Stigler. He made an educated guess of the optimal solution to linear program using a heuristic method. Through "trial and error, mathematical insight and agility," Stigler was able to eliminate 62 of the foods from the original 77 (these foods were removed based because they lacked nutrients in comparison to the remaining 15). From the reduced list, Stigler calculated the required amounts of each of the remaining 15 foods to arrive at a costminimizing solution to his question. According to Stigler's calculations, the annual cost of his solution was $39.93 in 1939 dollars. The specific combination of foods and quantities is as follows: Stigler's 1939 Diet Food Annual Quantities Annual Cost Wheat Flour 370 lb. $13.33 Evaporated Milk 57 cans 3.84 Cabbage 111 lb. 4.11 Spinach 23 lb. 1.85 Dried Navy Beans 285 lb. 16.80 Total Annual Cost $39.93
19 One of the early researchers to study this problem was George Stigler. He made an educated guess of the optimal solution to linear program using a heuristic method. Through "trial and error, mathematical insight and agility," Stigler was able to eliminate 62 of the foods from the original 77 (these foods were removed based because they lacked nutrients in comparison to the remaining 15). From the reduced list, Stigler calculated the required amounts of each of the remaining 15 foods to arrive at a costminimizing solution to his question. According to Stigler's calculations, the annual cost of his solution was $39.93 in 1939 dollars. The specific combination of foods and quantities is as follows: Stigler's 1939 Diet Food Annual Quantities Annual Cost Wheat Flour 370 lb. $13.33 Evaporated Milk 57 cans 3.84 Cabbage 111 lb. 4.11 Spinach 23 lb. 1.85 Dried Na39.93
20 One of the early researchers to study this problem was George Stigler. He made an educated guess of the optimal solution to linear program using a heuristic method. Through "trial and error, mathematical insight and agility," Stigler was able to eliminate 62 of the foods from the original 77 (these foods were removed based because they lacked nutrients in comparison to the remaining 15). From the reduced list, Stigler calculated the required amounts of each of the remaining 15 foods to arrive at a costminimizing solution to his question. According to Stigler's calculations, the annual cost of his solution was $39.93 in 1939 dollars. The specific combination of foods and quantities is as follows: Stigler's 1939 Diet Food Annual Quantities Annual Cost Wheat Flour 370 lb. $13.33 Evaporated Milk 57 cans 3.84 Cabbage 111 lb. 4.11 Spinach 23 lb. 1.85 Dried Navy Beans 285 lb. 16.80 Total Annual Cost $39.93
21 Seven years after Stigler made his initial estimates, the development of George Dantzig's Simplex algorithm made it possible to solve the problem without relying on heuristic methods. The exact value was determined to be $39.69 (using the original 1939 data). The linear program consisted of nine equation in 77 unknowns. It took nine clerks using handoperated desk calculators 120 man days to solve for the optimal solution of $39.69. Stigler's guess ($39.93) for the optimal solution was off by only 24 cents per year. 22 23 24 25 The Diet Problem
• Find the least costly combination of the 77 foods so that 9 nutrients are satisfied • Let • Each food i gives aij units of nutrient j, i=1, …, 77 ; j=1, …, 9. • Each unit of food i costs $ci s i=1,…, 77. 26 The Diet Problem
• Let Xi = the amount of food i consumed , i = 1,…,77. • The total cost to be minimized is: c X
i i 1 77 i 27 The Product Mix Problem
• The diet should satisfy the nutrition requirement of the 9 nutrients. a
i 1 77 ij X i bj j 1,..., 9 Xj 0 j 1,..., 77
28 The Diet Problem
• The model is: min c X
j j 1 ij X j 77 j subject to (s.t) a
j 1 77 bi i 1,..., 9 j 1,..., 77
29 Xj 0 30 31 32 33 34 35 Let xi 1000s of barrel of petroleum from i used. Minimize the total cost Production requirement of each product Availability of each type of petroleum 36 Minimize the total cost Production requirement of each product Availability 37 The Product Mix Problem
• A manufacturing facility wishes to determine the best mix of products so as to maximize the total profit • The facility manufactures n different products • Each product requires a certain combination of m different resources • Each unit of product i requires aij units of resource j, i=1, …, m; j=1, …, n. • The facility has bi units of resource i available on hand, i=1, …, m. • Each unit of product j return cj units of profit, j=1,…, n. 38 The Product Mix Problem
• Let Xj = the number of units of product j the facility will produce, j = 1,…,n. • The total profit to be maximized is: c X
j 1 j n j 39 The Product Mix Problem
• The facility cannot use more resources than available. a
j 1 n ij X j bi i 1,..., m Xj 0 j 1,..., n
40 The Product Mix Problem
• The model is: max c j X j
j 1 n subject to (s.t) a
j 1 n ij X j bi i 1,..., m j 1,..., n
41 Xj 0 A related problem
• Suppose now that – the management would like to determine “fair market values” for each unit of available resources • Let yi denote the fair market value for each unit of resource i, i=1,…,m. • These values will be determined in such a way that the facility would have an incentive to rent its resources as opposed to using them in their production if an outside company is willing to pay the corresponding market value
42 A related problem
• If the company uses its resources in its production as opposed
to renting them out, it would lose b y
i 1 i m i • This can be viewed as opportunity cost • For each unit of product j the company chooses not to produce it would lose cj units 43 A related problem
• However, each unit of product j not manufactured would release aij units of resource i, i=1, …,m. • Therefore market value of resources released would be: a
i 1 m ij yi cj j • In order for the company to have an incentive to rent out its resources, we should have 44 A related problem
The management obtain the following formulation: min bi yi
i 1 m s.t.
DUAL a
i 1 m ij yi c j j 1,..., n i 1,..., m
45 yi 0 Assembly
• A manufacturing facility wishes to determine the maximum number of a product, P, that can be produced, say in a week. • • • Product P is assembled from m different parts. Each unit of product P requires ai units of part i, i=1, …, m; Each unit of part i requires bi units of production time. • The factory can work B hours each week 46 Assembly
• Let Xi = the number of units of part i produced in a week, i = 1 ,…,m. • The total number of product P produced is min i ( X i / ai ) 47 Assembly
• The facility cannot work more than B hours each week b X
i 1 i m i B Xi 0 i 1,..., m
48 Assembly
• The model is: max min i ( X i / ai ) s.t Nonlinear b X
i 1 i m i B j 1,..., m Xj 0 49 Assembly
• Linear version max y s.t y X i / ai i ,1..., m b X
i 1 i m i B j 1,..., m
50 Xj 0 ...
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This note was uploaded on 04/15/2010 for the course INDUSTRIAL ie513 taught by Professor Zeynephuygur during the Spring '10 term at Bilkent University.
 Spring '10
 zeynephuygur

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