MT 2351 Chapter 2 An Introduction to Linear Programming
MT 2352 Courtesy of NPR: The Mathematician Who Solved Major Problems &segNum=14& George Dantzig
MT 2353 General Form of an LP Model
MT 2354 General Form of an LP Model where the cs, as and bs are constants determined from the problem and the xs are the decision variables
MT 2355 Components of Linear Programming An objective Decision variables Constraints Parameters
MT 2356 Assumptions of the LP Model Divisibility - basic units of xs are divisible Proportionality - as and cs are strictly proportional to the xs Additivity - each term in the objective function and constraints contains only one variable Deterministic - all cs, as and bs are known and measured without error Non-Negativity (caveat)
MT 2357 Sherwood Furniture Company Recently, Sherwood Furniture Company has been interested in developing a new line of stereo speaker cabinets. In the coming month, Sherwood expects to have excess capacity in its Assembly and Finishing departments and would like to experiment with two new models. One model is the Standard, a large, high-quality cabinet in a traditional design that can be sold in virtually unlimited quantities to several manufacturers of audio equipment. The other model is the Custom, a small, inexpensive cabinet in a novel design that a single buyer will purchase on an exclusive basis. Under the tentative terms of this agreement, the buyer will purchase as many Customs as Sherwood produces, up to 32 units. The Standard requires 4 hours in the Assembly Department and 8 hours in the Finishing Department, and each unit contributes $20 to profit. The Custom requires 3 hours in Assembly and 2 hours in Finishing, and each unit contributes $10 to profit. Current plans call for 120 hours to be available next month in Assembly and 160 hours in Finishing for cabinet production, and Sherwood desires to allocate this capacity in the most economical way.
MT 2358 Sherwood Furniture Company – Linear Equations
MT 2359 Sherwood Furniture Company – Graph Solution
MT Sherwood Furniture Company – Graph Solution Constraint 1
MT Sherwood Furniture Company – Graph Solution Constraint 1
MT Sherwood Furniture Company – Graph Solution Constraint 2
MT Sherwood Furniture Company – Graph Solution Constraint 1 & 2
MT Sherwood Furniture Company – Graph Solution Constraint 3
MT Sherwood Furniture Company – Graph Solution Constraint 1, 2 & 3
MT Sherwood Furniture Company – Graph Solution
MT Sherwood Furniture Company – Graph Solution
MT Sherwood Furniture Company – Solve Linear Equations
MT Sherwood Furniture Company – Solve Linear Equations
MT Sherwood Furniture Company – Solve Linear Equations
MT Sherwood Furniture Company – Graph Solution Optimal Point (15, 20)
MT Sherwood Furniture Company – Slack Calculation
MT Sherwood Furniture Company - Slack Variables Max 20x x 2 + 0S 1 + 0S 2 + 0S 3 s.t. 4x 1 + 3x 2 + 1S 1 = 120 8x 1 + 2x 2 + 1S 2 = 160 x 2 + 1S 3 = 32 x 1, x 2, S 1,S 2,S 3 >= 0
MT Sherwood Furniture Company – Graph Solution 2 3 1
MT Sherwood Furniture Company – Slack Calculation Point 1 Point 1
MT Sherwood Furniture Company – Graph Solution 2 3 1
MT Sherwood Furniture Company – Slack Calculation Point 2 Point 2
MT Sherwood Furniture Company – Graph Solution 2 3 1
MT Sherwood Furniture Company – Slack Calculation Point 3 Point 3
MT Sherwood Furniture Company – Slack Calculation Points 1, 2 & 3 Point 1Point 2Point 3
MT Sherwood Furniture Company – Slack Variables For each constraint the difference between the RHS and LHS (RHS-LHS). It is the amount of resource left over. Constraint 1; S 1 = 0 hrs. Constraint 2; S 2 = 0 hrs. Constraint 3; S 3 = 12 Custom
MT 23532
MT 23533
MT 23534
MT 23535
MT 23536
MT 23537
MT Pet Food Company A pet food company wants to find the optimal mix of ingredients, which will minimize the cost of a batch of food, subject to constraints on nutritional content. There are two ingredients, P1 and P2. P1 costs $5/lb. and P2 costs $8/lb. A batch of food must contain no more than 400 lbs. of P1 and must contain at least 200 lbs. of P2. A batch must contain a total of at least 500 lbs. What is the optimal (minimal cost) mix for a single batch of food?
MT Pet Food Company – Linear Equations
MT Pet Food Company – Graph Solution
MT Pet Food Company – Graph Solution Constraint 1
MT Pet Food Company – Graph Solution Constraint 1
MT Pet Food Company – Graph Solution Constraint 2
MT Pet Food Company – Graph Solution Constraint 1 & 2
MT Pet Food Company – Graph Solution Constraint 3
MT Pet Food Company – Graph Solution Constraint 1, 2 & 3
MT Pet Food Company – Solve Linear Equations
MT Pet Food Company – Graph Solution
MT Pet Food Company – Solve Linear Equations
MT Pet Food Company – Solve Linear Equations
MT Pet Food Company – Graph Solution Optimal Point (300, 200)
MT Pet Food Company – Slack/ Surplus Calculation
MT Pet Food Co. – Linear Equations Slack/ Surplus Variables Min 5P 1 + 8P 2 + 0S 1 + 0S 2 + 0S 3 s.t. 1P 1 + 1S 1 = 400 1P 2 - 1S 2 = 200 1P 1 + 1P 2 - 1S 3 = 500 P 1, P 2, S 1,S 2,S 3 >= 0
MT Pet Food Co. – Slack Variables For each constraint the difference between the RHS and LHS (RHS-LHS). It is the amount of resource left over. Constraint 1; S 1 = 100 lbs.
MT Pet Food Co. – Surplus Variables For each constraint the difference between the LHS and RHS (LHS-RHS). It is the amount bt which a minimum requirement is exceeded. Constraint 2; S 2 = 0 lbs. Constraint 3; S 3 = 0 lbs.
MT 23556
MT 23557
MT 23558
MT 23559
MT 23560
MT Special Cases Alternate Optimal Solutions No Feasible Solution Unbounded Solutions
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions A B
MT Alternate Optimal Solutions
MT Alternate Optimal Solutions
MT 23574
MT 23575
MT 23576
MT 23577
MT 23578
MT Special Cases Alternate Optimal Solutions No Feasible Solution Unbounded Solutions
MT No Feasible Solution
MT No Feasible Solution
MT No Feasible Solution
MT 23583
MT 23584
MT 23585
MT Special Cases Alternate Optimal Solutions No Feasible Solution Unbounded Solutions
MT Unbounded Solutions
MT Unbounded Solutions
MT Unbounded Solutions
MT Unbounded Solutions
MT 23591
MT 23592
MT 23593
MT 23594
MT 23595