DM545 Linear and Integer Programming Lecture 2 The Simplex Method Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark
Introduction Solving LP Problems Outline Preliminaries 1. Introduction Diet Problem 2. Solving LP Problems Fourier-Motzkin method 3. Preliminaries Fundamental Theorem of LP Gaussian Elimination 2
Introduction Solving LP Problems Outline Preliminaries 1. Introduction Diet Problem 2. Solving LP Problems Fourier-Motzkin method 3. Preliminaries Fundamental Theorem of LP Gaussian Elimination 3
Introduction Solving LP Problems Outline Preliminaries 1. Introduction Diet Problem 2. Solving LP Problems Fourier-Motzkin method 3. Preliminaries Fundamental Theorem of LP Gaussian Elimination 4
Introduction Solving LP Problems The Diet Problem (Blending Problems) Preliminaries • Select a set of foods that will satisfy a set of daily nutritional requirement at minimum cost. • Motivated in the 1930s and 1940s by US army. • Formulated as a linear programming problem by George Stigler • First linear programming problem • (programming intended as planning not computer code) min cost/weight subject to nutrition requirements: eat enough but not too much of Vitamin A eat enough but not too much of Sodium eat enough but not too much of Calories ... 5
Introduction Solving LP Problems The Diet Problem Preliminaries Suppose there are: • 3 foods available, corn, milk, and bread, and • there are restrictions on the number of calories (between 2000 and 2250) and the amount of Vitamin A (between 5,000 and 50,000) Food Cost per serving Vitamin A Calories Corn $0.18 107 72 2% Milk $0.23 500 121 Wheat Bread $0.05 0 65 6
Introduction Solving LP Problems The Mathematical Model Preliminaries Parameters (given data) F = set of foods N = set of nutrients a ij = amount of nutrient j in food i , ∀ i ∈ F , ∀ j ∈ N c i = cost per serving of food i , ∀ i ∈ F F mini = minimum number of required servings of food i , ∀ i ∈ F F maxi = maximum allowable number of servings of food i , ∀ i ∈ F N minj = minimum required level of nutrient j , ∀ j ∈ N N maxj = maximum allowable level of nutrient j , ∀ j ∈ N Decision Variables number of servings of food i to purchase/consume, ∀ i ∈ F x i = 7
Introduction Solving LP Problems The Mathematical Model Preliminaries Objective Function: Minimize the total cost of the food � Minimize c i x i i ∈ F Constraint Set 1: For each nutrient j ∈ N , at least meet the minimum required level � a ij x i ≥ N minj , ∀ j ∈ N i ∈ F Constraint Set 2: For each nutrient j ∈ N , do not exceed the maximum allowable level. � a ij x i ≤ N maxj , ∀ j ∈ N i ∈ F Constraint Set 3: For each food i ∈ F , select at least the minimum required number of servings x i ≥ F mini , ∀ i ∈ F Constraint Set 4: For each food i ∈ F , do not exceed the maximum allowable number of servings. x i ≤ F maxi , ∀ i ∈ F 8
Introduction Solving LP Problems The Mathematical Model Preliminaries system of equalities and inequalities � min c i x i i ∈ F � a ij x i ≥ N minj , ∀ j ∈ N i ∈ F � a ij x i ≤ N maxj , ∀ j ∈ N i ∈ F x i ≥ F mini , ∀ i ∈ F x i ≤ F maxi , ∀ i ∈ F 9
Introduction Solving LP Problems Mathematical Model Preliminaries Graphical Representation: Machines/Materials A and B x 2 Products 1 and 2 max 6 x 1 + 8 x 2 5 x 1 + 10 x 2 ≤ 60 4 x 1 + 4 x 2 ≤ 40 x 1 ≥ 0 x 2 ≥ 0 5 x 1 + 10 x 2 ≤ 60 x 1 4 x 1 + 4 x 2 ≤ 40 6 x 1 + 8 x 2 = 16 10
Introduction Solving LP Problems In Matrix Form Preliminaries max c 1 x 1 + c 2 x 2 + c 3 x 3 + . . . + c n x n = z s.t. a 11 x 1 + a 12 x 2 + a 13 x 3 + . . . + a 1 n x n ≤ b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + . . . + a 2 n x n ≤ b 2 . . . a m 1 x 1 + a m 2 x 2 + a m 3 x 3 + . . . + a mn x n ≤ b m x 1 , x 2 , . . . , x n ≥ 0 c 1 a 11 a 12 . . . a 1 n x 1 b 1 x 2 b 2 c 2 a 21 a 22 . . . a 2 n c = , A = , x = , b = . . . . . . ... . . . . . . . . . . . . c n a 31 a 32 . . . a mn x n b m z = c T x max A x ≤ b x ≥ 0 11
Introduction Solving LP Problems Linear Programming Preliminaries Abstract mathematical model: Parameters, Decision Variables, Objective, Constraints The Syntax of a Linear Programming Problem max / min c T · x c ∈ R n objective func. s.t. A · x � b A ∈ R m × n , b ∈ R m constraints x ∈ R n , 0 ∈ R n x ≥ 0 Essential features: continuity, linearity (proportionality and additivity), certainty of parameters • Any vector x ∈ R n satisfying all constraints is a feasible solution. • Each x ∗ ∈ R n that gives the best possible value for c T x among all feasible x is an optimal solution or optimum • The value c T x ∗ is the optimum value 12
Introduction Solving LP Problems Preliminaries • The linear programming model consisted of 9 equations in 77 variables • Stigler, guessed an optimal solution using a heuristic method • In 1947, the National Bureau of Standards used the newly developed simplex method to solve Stigler’s model. It took 9 clerks using hand-operated desk calculators 120 man days to solve for the optimal solution • The original instance: http://www.gams.com/modlib/libhtml/diet.htm 13
Introduction Solving LP Problems AMPL Model Preliminaries ✞ ☎ # diet.mod set NUTR; set FOOD; param cost {FOOD} > 0; param f_min {FOOD} >= 0; param f_max { i in FOOD} >= f_min[i]; param n_min { NUTR } >= 0; param n_max {j in NUTR } >= n_min[j]; param amt {NUTR,FOOD} >= 0; var Buy { i in FOOD} >= f_min[i], <= f_max[i] minimize total_cost: sum { i in FOOD } cost [i] ∗ Buy[i]; subject to diet { j in NUTR }: n_min[j] <= sum {i in FOOD} amt[i,j] ∗ Buy[i] <= n_max[j]; ✝ ✆ 14
Introduction Solving LP Problems AMPL Model Preliminaries ✞ ☎ ✞ ☎ # diet.dat param amt (tr): data; A C B1 B2 := BEEF 60 20 10 15 set NUTR := A B1 B2 C ; CHK 8 0 20 20 set FOOD := BEEF CHK FISH HAM MCH FISH 8 10 15 10 MTL SPG TUR; HAM 40 40 35 10 MCH 15 35 15 15 param: cost f_min f_max := MTL 70 30 15 15 BEEF 3.19 0 100 SPG 25 50 25 15 CHK 2.59 0 100 TUR 60 20 15 10 ; ✝ ✆ FISH 2.29 0 100 HAM 2.89 0 100 MCH 1.89 0 100 MTL 1.99 0 100 SPG 1.99 0 100 TUR 2.49 0 100 ; param: n_min n_max := A 700 10000 C 700 10000 B1 700 10000 B2 700 10000 ; # % ✝ ✆ 15
Introduction Python Script Solving LP Problems Preliminaries Data ✞ ☎ ✞ ☎ from gurobipy import ∗ # Nutrition values for the foods nutritionValues = { categories, minNutrition, maxNutrition = ( ’hamburger’ , ’calories’ ): 410, multidict({ ( ’hamburger’ , ’protein’ ): 24, ’calories’ : [1800, 2200], ( ’hamburger’ , ’fat’ ): 26, ’protein’ : [91, GRB.INFINITY], ( ’hamburger’ , ’sodium’ ): 730, ’fat’ : [0, 65], ( ’chicken’ , ’calories’ ): 420, ’sodium’ : [0, 1779] }) ( ’chicken’ , ’protein’ ): 32, ( ’chicken’ , ’fat’ ): 10, foods, cost = multidict({ ( ’chicken’ , ’sodium’ ): 1190, ’hamburger’ : 2.49, ( ’hot dog’ , ’calories’ ): 560, ’chicken’ : 2.89, ( ’hot dog’ , ’protein’ ): 20, ’hot dog’ : 1.50, ( ’hot dog’ , ’fat’ ): 32, ’fries’ : 1.89, ( ’hot dog’ , ’sodium’ ): 1800, ’macaroni’ : 2.09, ( ’fries’ , ’calories’ ): 380, ’pizza’ : 1.99, ( ’fries’ , ’protein’ ): 4, ’salad’ : 2.49, ( ’fries’ , ’fat’ ): 19, ’milk’ : 0.89, ( ’fries’ , ’sodium’ ): 270, ’ice cream’ : 1.59 }) ( ’macaroni’ , ’calories’ ): 320, ✝ ✆ ( ’macaroni’ , ’protein’ ): 12, ( ’macaroni’ , ’fat’ ): 10, ( ’macaroni’ , ’sodium’ ): 930, ( ’pizza’ , ’calories’ ): 320, ( ’pizza’ , ’protein’ ): 15, ( ’pizza’ , ’fat’ ): 12, ( ’pizza’ , ’sodium’ ): 820, ( ’salad’ , ’calories’ ): 320, ( ’salad’ , ’protein’ ): 31, 16 ( ’salad’ , ’fat’ ): 12,
Introduction Solving LP Problems Preliminaries ✞ ☎ # Model diet.py m = Model( "diet" ) # Create decision variables for the foods to buy buy = {} for f in foods: buy[f] = m.addVar(obj=cost[f], name=f) # The objective is to minimize the costs m.modelSense = GRB.MINIMIZE # Update model to integrate new variables m.update() # Nutrition constraints for c in categories: m.addConstr( quicksum(nutritionValues[f,c] ∗ buy[f] for f in foods) <= maxNutrition[c], name=c+ ’max’ ) m.addConstr( quicksum(nutritionValues[f,c] ∗ buy[f] for f in foods) >= minNutrition[c], name=c+ ’min’ ) # Solve m.optimize() ✝ ✆ 17
Introduction Solving LP Problems Outline Preliminaries 1. Introduction Diet Problem 2. Solving LP Problems Fourier-Motzkin method 3. Preliminaries Fundamental Theorem of LP Gaussian Elimination 17
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