Problem B.1, HR7E Solve the following LP graphically R. Saltzman

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1 Problem B.1, HR7E Solve the following LP graphically R. Saltzman Maximize 4X + 6Y = Z subject to: (1) X + 2Y <= 8 (2) 5X + 4Y <= 2 (3) X >= (4) Y >= Note: There is a typograhpical error in the book regarding the last 2 nonnegativity constraints. Y X Y Z <-- optimal solution 7 and optimal Z (2) FR (1) X

2 Problem B.2, HR7E Solve the following LP graphically R. Saltzman Maximize X + 1Y = Z subject to: (1) 4X + 3Y <= 36 (2) 2X + 4Y <= 4 (3) Y >= 3 (4) X >= (5) Y >= Y X Y Z <-- optimal solution FR 4 3 (3) 2 1 (1) (2) X

3 Problem B.4, HR7E R. Saltzman Maximize 3X1 + 1X2 = Z subject to: (1) 3X1 + X2 <= 3 (2) X1 + X2 <= 2 (3) X1 <= 1 (4) X2 >= 5 (5) X1 - X2 <= a) Solve the problem graphically: X2 3 (3) X1 X2 Z (1) <-- optimal solution <-- optimal solution (5) 1 FR 75 (2) 5 (4) X1 b) Is there more than 1 optimal solution? Yes. Also, all the points between (5, 15) and (75, 75) have the same optimal value of 3.

4 Problem B.6, HR7E Ed Solver Dog Food Co. R. Saltzman a) Formulation: Let C = # of chicken-flavored biscuits per package Let L = # of liver-flavored biscuits per package Mininize.2*C +.1L = Z subject to: (1) C + L >= 4 (Nutrient A requirement) (2) 4C + 2L >= 6 (Nutrient B requirement) (3) L <= 15 (4) C >= (5) L >= b) Solve the problem graphically: L 4 C L Z <-- optimal solution & cost (3) 1 5 (2) (1) FR C

5 Problem B.8, HR7E Optimal Mix of Bathtubs R. Saltzman a) Formulation: Let A = # of model A bathtubs Let B = # of model B bathtubs Maxinize 9A + 7B = Z Total Profit subject to: (1) 125A + 1B <= 25 Steel Availability (2) 2A + 3B <= 6 Zinc Availability (3) A >= (4) B >= b) Solve the problem graphically: B A B Z <-- optimal solution (1) FR (2) A

6 Problem B.9, HR7E Mattresses & Box Springs R. Saltzman a) Formulation: Let M = # of mattresses to produce Let B = # of box springs to produce Mininize 2M + 24B = Z Total Cost subject to: (1) M + B >= 3 Minimum production requirement (2) 1M + 2B >= 4 Stitching machine requirement (3) M >= (4) B >= b) Solve the problem graphically: B M B Z <-- optimal solution (1) FR (2) M

7 Problem B.1, HR7E Making Computers R. Saltzman a) Formulation: Let A = # of Alpha 4 minicomputers to produce Let B = # of Beta 5 minicomputers to produce Maxinize 12A + 18B = Z Total Cost subject to: (1) 2A + 25B = 8 Full employment (2) A >= 1 Minimum Alpha 4 production (3) B >= 15 Minimum Beta 5 production b) Solve the problem graphically: B A B Z <-- optimal solution (2) 3 2 (3) FR 1 (1) A

8 Problem B.16, HR7E Busing Students R. Saltzman Superintendent must assign students living in 5 geographic sectors to 3 schools. 1. Different numbers of students live in each sector 2. Each high school has a capacity of 9 students 3. Some students must be bused - distances are shown in the table 4. Students living in a sector where there is a school walk ( bus miles) Goal: Find assignment that minimizes the total # of student miles traveling by bus to school. Let Xij = Number of students from sector I bused to school in sector j Data Mileage Supply From \ To School-in-Sector B School-in-Sector C School-in-Sector E (Students) Sector A Sector B Sector C Sector D Sector E (Fake) F 2 Demand \ 27 Allocations Optimal Solution (found using Solver) From \ To School-in-Sector B School-in-Sector C School-in-Sector E Row Total Sector A Sector B 5 5 Sector C 1 1 Sector D 8 8 Sector E 4 4 (Fake) F 2 2 Col. Total \ 27 Total Cost 54

9 Problem B.18, HR7E Restaurant Scheduling R. Saltzman * Open 24 hours a day * Servers work 8 hour shifts, reporting for duty at beginning of one of 6 time periods: Period Time # of Servers Required i = 1 3 am - 7 am 3 i = 2 7 am - 11 am 12 i = 3 11 am - 3 pm 16 i = 4 3 pm - 7 pm 9 i = 5 7 pm - 11 pm 11 i = 6 11 pm - 3 am 4 Goal: Find minimum # of servers required to cover the schedule. Let Xi = # of servers who begin work at start of period i, i = 1, 2, 3, 4, 5, 6. X1 X2 X3 X4 X5 X6 Σ No. of Servers <-- optimal solution Cost of Server (via Solver) Period Time X1 X2 X3 X4 X5 X6 LHS RHS 1 3 am - 7 am >= am - 11 am >= am - 3 pm >= pm - 7 pm >= pm - 11 pm >= pm - 3 am >= 4 That is: Minimize X1 + X2 + X3 + X4 + X5 + X6 = Z subject to: X1 + X6 >= 3 X1 + X2 >= 12 X2 + X3 >= 16 X3 + X4 >= 9 X4 + X5 >= 11 X5 + X6 >= 4 All Xj >=, for j = 1, 2, 3, 4, 5, 6

10 Problem B.19, HR7E Birdhouse Builder R. Saltzman a) Formulation: Let W = # of Wren Birdhouses to build Let B = # of Bluebird Birdhouses to build Maxinize 6W + 15B = Z Total Profit subject to: (1) 4W + 2B <= 6 Labor availability (2) 4W + 12B <= 12 Lumber availability (3) W >= (4) B >= b) Solve the problem graphically: B W B Z <-- optimal solution (2) (1) FR W

11 Problem B.25, HR7E Advertising Agency R. Saltzman a) Formulation: Let T = # of TV spots to run Let S = # of Sunday newspaper ads to run Maxinize 35T + 2S = Z Total Exposure (in 1's) subject to: (1) 3T + 125S <= 1 Advertising Budget (2) T >= 5 Minimum # of TV spots (3) T <= 25 Maximum # of TV spots (4) S >= 1 Minimum # of Sunday ads b) Solve the problem graphically: S T S Z <-- optimal solution (1) (2) (3) 3 FR 2 1 (4) T

12 Problem B.26, HR7E Factories & Warehouses R. Saltzman Unit Shipping Costs & Capacities To Warehouse Production From A B C Capability Factory 1 $ 6 $ 5 $ 3 6 Factory 2 $ 8 $ 1 $ 8 8 Factory 3 $ 11 $ 14 $ 18 1 Capacity a) Write the objective function and constraints: Objective: Minimize 6X1A + 5X1B + 3X1C + 8X2A + 1X2B + 8X2C + 11X3A + 14X3B + 18X3C Constraints: X1A + X1B + X1C = 6 X2A + X2B + X2C = 8 X3A + X3B + X3C = 1 X1A + X2A + X3A = 7 X1B + X2B + X3B = 12 X1C + X2C + X3C = 5 Plus 9 nonnegativity constraints: all variables (cells) must be at least.

13 Problem C.1, HR7E Transportation Problem R. Saltzman To From Los Angeles Calgary Panama City Supply Mexico City $ 6 $ 18 $ 8 1 Detroit $ 17 $ 13 $ 19 6 Ottawa $ 2 $ 1 $ 24 4 Demand a) Find an initial solution using the northwest-corner method: To From Los Angeles Calgary Panama City Supply Mexico City Detroit Ottawa 4 4 Demand b) Find an initial solution using the lowest-cost method: To From Los Angeles Calgary Panama City Supply Mexico City Detroit Ottawa 4 4 Demand c) The total cost of the northwest-corner solution = $ 3,12 The total cost of the lowest-cost solution = $ 2,

Introduction to Operations Research

Introduction to Operations Research Unit 1: Linear Programming Terminology and formulations LP through an example Terminology Additional Example 1 Additional example 2 A shop can make two types of sweets