# Operations Management Linear Programming Module B - Part 2

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B-2. Issue B.23. 1. Gross Distributors bundles and circulates modern supplies. A standard shipment can be bundled in a class A holder, a class K compartment, or a class T holder. The benefit from utilizing every kind of compartment is: \$8 for every class A holder, \$6 for every class K compartment, and \$14 for every class T compartment. The measure of pressing material required by each A, K and T co

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﻿Operations Management Linear Programming Module B - Part 2

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Problem B.23 1. Net Distributors bundles and circulates modern supplies. A standard shipment can be bundled in a class A compartment, a class K holder, or a class T holder. The benefit from utilizing each kind of compartment is: \$8 for each class A holder, \$6 for each class K holder, and \$14 for each class T compartment. The measure of pressing material required by each A, K and T holder is 2, 1 and 3 lbs., separately. The measure of pressing time required by each A, K, and T holder is 2, 6, and 4 hours, individually. There is 120 lbs of pressing material accessible every week. Six packers must be utilized full time (40 hours for every week each). Decide what number of compartments to pack every week.

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Packing material (lbs.) Packing time (hrs.) Container Profit \$8 2 A 2 6 \$6 1 K 4 3 \$14 T =240 Amount accessible  120 Problem B.23

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2x A + x K + 3x T  120 (lbs.) 2x A + 6x K + 4x T = 240 (hours) x A , x K , x T  0 : Maximize: 8x A + 6x K + 14x T Problem B.23 x i = Number of class i holders to pack every week. i=A, K, T

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Linear Programming Solutions Unique Optimal Solution. Various Optimal Solutions. Infeasible (no arrangement). x + y  800 x  1000 x, y  0 Unbounded (unending arrangement). Boost 3x + 2y x + y  1000

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Computer Solutions Optimal estimations of choice factors and target work. Affectability data for target work coefficients. Affectability data for RHS (right-hand side) of limitations and shadow cost.

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Computer Solutions Enter information from detailing in Excel. 1 push for the coefficients of goal. 1 push for coefficients & RHS of every requirement. 1 last line for arrangement (choice variable) values. Select Solver from the Tools Menu.

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Computer Solutions - Solver

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Computer Solutions - Solver

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Computer Solutions - Solver Parameters

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Computer Solutions Set Target Cell: to estimation of target capacity. E3 Equal To: Max or Min By Changing Cells: = Sol'n values (choice variable qualities). B7:D7 Subject to the Constraints: Click Add to include every requirement: LHS =,  ,  RHS

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Computer Solutions - Adding Constraints Cell Reference: LHS area Select sign : <=, =, >= Constraint: RHS area

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Computer Solutions - Adding Constraints first limitation. Click Add. Rehash for second limitation.

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Computer Solutions Click Options to set up Solver for LP.

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Computer Solutions - Solver Options Check "on" Assume Linear Model and Assume Non-Negative.

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Computer Solutions Click Solve to locate the ideal arrangement.

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Computer Solutions - Solver Results

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Computer Solutions - Optimal Solution Optimal arrangement is to utilize: 0 A compartments 17.14 K holders 34.29 T holders Maximum benefit is \$583 every week. Really \$582.857… in Excel qualities are adjusted.

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Computer Solutions Optimal arrangement is to utilize: 0 class A holders. 17.14 class K compartments. 34.29 class T compartments. Most extreme benefit is \$582.857 every week. Select Answer and Sensitivity Reports and snap OK. New pages show up in Excel.

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Sensitivity Analysis Projects how much an answer will change if there are changes in factors or info information. Shadow cost (double) - Value of one extra unit of an asset.

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Computer Solution - Sensitivity Report

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Computer Solution - Sensitivity Report Microsoft Excel 8.0e Sensitivity Report Worksheet: [probb.23.xls]Sheet1 Report Created: 1/31/01 9:53:27 PM Adjustable Cells Final Reduced Objective Allowable Cell Name Value Cost Coefficient Increase Decrease \$B\$7 Sol'n values A cont. 0 - 1.142857143 8 1.142857143 1E+30 \$C\$7 Sol'n values K cont. 17.14285714 0 6 8 1E+30 \$D\$7 Sol'n values T cont 34.28571429 0 14 1E+30 1.6 Optimal arrangement: 0 class A compartments 17.14285… class K holders 34.28571… class T holders Profit = 0(8) + 17.14285(6) + 34.28571(14) = \$582.857

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Computer Solution - Sensitivity Report

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Sensitivity for Objective Coefficients the length of coefficients are in range demonstrated, then current arrangement is still ideal, however benefit may change! Current arrangement is ideal the length of: Coefficient of x An is between - boundlessness and 9.142857 Coefficient of x K is between - vastness and 14 Coefficient of x T is in the vicinity of 12.4 and interminability

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Sensitivity for Objective Coefficients If benefit for class K compartment was 12 (not 6), what is ideal arrangement?

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Sensitivity for Objective Coefficients If benefit for class K holder was 12 (not 6), what is ideal arrangement? x A =0, x K =17.14, x T =34.29 (same as before) benefit = 685.71 (more than before!)

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Sensitivity for Objective Coefficients If benefit for class K compartment was 16 (not 6), what is ideal arrangement?

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Sensitivity for Objective Coefficients If benefit for class K compartment was 16 (not 6), what is ideal arrangement? Diverse! Resolve issue to get arrangement.

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Computer Solution - Sensitivity Report

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Sensitivity for RHS values Shadow cost is change in target an incentive for every unit change in RHS the length of progress in RHS is inside range. Each extra lb. of pressing material will expand benefit by \$4.2857... for up to 60 extra lbs. Each extra hour of pressing time will expand benefit by \$0.2857... for up to 480 extra hours.

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Sensitivity for RHS values Suppose you can purchase 50 more lbs. of pressing material for \$250. Would it be a good idea for you to get it?

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Sensitivity for RHS values Suppose you can purchase 50 more lbs. of pressing material for \$250. Would it be a good idea for you to get it? NO. \$250 for 50 lbs. is \$5 per lb. Benefit increment is just \$4.2857 per lb.

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Sensitivity for RHS values How much would you pay for 50 more lbs. of pressing material?

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Sensitivity for RHS values How much would you pay for 50 more lbs. of pressing material? \$214.28 50 lbs.  \$4.2857/lb. = \$214.2857...

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Sensitivity for RHS values If change in RHS is outside range (from suitable increment or abatement), then we can not tell how the target esteem will change.

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Extensions of Linear Programming Integer programming (IP): Some or all factors are confined to whole number qualities. Permits "assuming… then" imperatives. Substantially harder to comprehend (more PC time). Nonlinear programming: Some requirements or goal are nonlinear capacities. Permits more extensive scope of circumstances to be demonstrated. Significantly harder to explain (more PC time).

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{ Integer Programming 1 on the off chance that we construct a plant in St. Louis 0 generally. 1 in the event that we assemble an industrial facility in Chicago 0 generally. We will construct one manufacturing plant in Chicago or St. Louis. x 1 + x 2  1 We will manufacture one processing plant in either Chicago or St. Louis. x 1 + x 2 = 1 If we work in Chicago, then we won't work in St. Louis. x 2  1 - x 1

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Harder Formulation Example You are making a speculation portfolio from 4 venture alternatives: stocks, land, T-charges (Treasury-bills), and money. Stocks have a yearly rate of return of 12% and a hazard measure of 5. Land has a yearly rate of return of 10% and a hazard measure of 8. T-bills have a yearly rate of return of 5% and a hazard measure of 1. Money has a yearly rate of return of 0% and a hazard measure of 0. The normal danger of the portfolio can not surpass 5. No less than 15% of the portfolio must be in real money. Define a LP to expand the yearly rate of return of the portfolio.

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Midnight - 4 am 4 am - 8 am 3 6 Another Formulation Example A business works 24 hours a day and representatives work 8 hour shifts. Movements may start at midnight, 4 am, 8 am, twelve, 4 pm or 8 pm. The quantity of workers required in every 4 hour time of the day to serve request is in the table beneath. Plan a LP to limit the quantity of representatives to fulfill the request. 8 am - twelve Noon - 4 pm 4 pm - 8 pm 8 pm - midnight 12 9 13 15