OPSM 301 Operations Management Koç University OPSM 301 Operations Management Class 25: Applied LP continued Zeynep Aksin zaksin@ku.edu.tr

A Transportation Problem: Tropicsun Ocala 4 Orlando 5 Leesburg 6 Processing Plants Mt. Dora 1 Eustis 2 Clermont 3 Groves Distances (in miles) Supply Capacity 21 225,000 600,000 200,000 275,000 400,000 300,000 50 40 35 30 22 55 20 25

Defining the Decision Variables Xij = # of bushels shipped from node i to node j Specifically, the nine decision variables are: X14 = # of bushels shipped from Mt. Dora (node 1) to Ocala (node 4) X15 = # of bushels shipped from Mt. Dora (node 1) to Orlando (node 5) X16 = # of bushels shipped from Mt. Dora (node 1) to Leesburg (node 6) X24 = # of bushels shipped from Eustis (node 2) to Ocala (node 4) X25 = # of bushels shipped from Eustis (node 2) to Orlando (node 5) X26 = # of bushels shipped from Eustis (node 2) to Leesburg (node 6) X34 = # of bushels shipped from Clermont (node 3) to Ocala (node 4) X35 = # of bushels shipped from Clermont (node 3) to Orlando (node 5) X36 = # of bushels shipped from Clermont (node 3) to Leesburg (node 6)

Defining the Objective Function Minimize the total number of bushel-miles. MIN: 21X14 + 50X15 + 40X16 + 35X24 + 30X25 + 22X26 + 55X34 + 20X35 + 25X36

Defining the Constraints Capacity constraints X14 + X24 + X34 <= 200,000 } Ocala X15 + X25 + X35 <= 600,000 } Orlando X16 + X26 + X36 <= 225,000 } Leesburg Supply constraints X14 + X15 + X16 = 275,000 } Mt. Dora X24 + X25 + X26 = 400,000 } Eustis X34 + X35 + X36 = 300,000 } Clermont Nonnegativity conditions Xij >= 0 for all i and j

Implementing the Model

An Employee Scheduling Problem: Air-Express Day of Week Workers Needed Sunday 18 Monday 27 Tuesday 22 Wednesday 26 Thursday 25 Friday 21 Saturday 19 Shift Days Off Wage 1 Sun & Mon $680 2 Mon & Tue $705 3 Tue & Wed $705 4 Wed & Thr $705 5 Thr & Fri $705 6 Fri & Sat $680 7 Sat & Sun $655

Defining the Decision Variables X1 = the number of workers assigned to shift 1 X2 = the number of workers assigned to shift 2 X3 = the number of workers assigned to shift 3 X4 = the number of workers assigned to shift 4 X5 = the number of workers assigned to shift 5 X6 = the number of workers assigned to shift 6 X7 = the number of workers assigned to shift 7

Defining the Objective Function Minimize the total wage expense. MIN: 680X1 +705X2 +705X3 +705X4 +705X5 +680X6 +655X7

Defining the Constraints Workers required each day 0X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 + 0X7 >= 18 } Sunday 0X1 + 0X2 + 1X3 + 1X4 + 1X5 + 1X6 + 1X7 >= 27 } Monday 1X1 + 0X2 + 0X3 + 1X4 + 1X5 + 1X6 + 1X7 >= 22 }Tuesday 1X1 + 1X2 + 0X3 + 0X4 + 1X5 + 1X6 + 1X7 >= 26 } Weds. 1X1 + 1X2 + 1X3 + 0X4 + 0X5 + 1X6 + 1X7 >= 25 } Thurs. 1X1 + 1X2 + 1X3 + 1X4 + 0X5 + 0X6 + 1X7 >= 21 } Friday 1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 0X6 + 0X7 >= 19 } Saturday Nonnegativity conditions Xi >= 0 for all i

Implementing the Model

An Investment Problem: Retirement Planning Services, Inc. A client wishes to invest $750,000 in the following bonds. Years to Company Return Maturity Rating Acme Chemical 8.65% 11 1-Excellent DynaStar 9.50% 10 3-Good Eagle Vision 10.00% 6 4-Fair Micro Modeling 8.75% 10 1-Excellent OptiPro 9.25% 7 3-Good Sabre Systems 9.00% 13 2-Very Good

Investment Restrictions No more than 25% can be invested in any single company. At least 50% should be invested in long-term bonds (maturing in 10+ years). No more than 35% can be invested in DynaStar, Eagle Vision, and OptiPro.

Defining the Decision Variables X1 = amount of money to invest in Acme Chemical X2 = amount of money to invest in DynaStar X3 = amount of money to invest in Eagle Vision X4 = amount of money to invest in MicroModeling X5 = amount of money to invest in OptiPro X6 = amount of money to invest in Sabre Systems

Defining the Objective Function Maximize the total annual investment return. MAX: .0865X1 + .095X2 + .10X3 + .0875X4 + .0925X5 + .09X6

Defining the Constraints Total amount is invested X1 + X2 + X3 + X4 + X5 + X6 = 750,000 No more than 25% in any one investment Xi <= 187,500, for all i 50% long term investment restriction. X1 + X2 + X4 + X6 >= 375,000 35% Restriction on DynaStar, Eagle Vision, and OptiPro. X2 + X3 + X5 <= 262,500 Nonnegativity conditions Xi >= 0 for all i

Implementing the Model

Product Mix Decisions: Kristen Cookies offers 2 products Sale Price of Chocolate Chip Cookies: $5.00/dozen Cost of Materials: $2.50/dozen Sale Price of Oatmeal Raisin Cookies: $5.50/dozen Cost of Materials: $2.40/dozen Maximum weekly demand of Chocolate Chip Cookies: 100 dozen Oatmeal Raisin Cookies: 50 dozen Total weekly operating expense $270 ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................

Product Mix Decisions Total time available in week: 20 hrs

Margin per dozen Chocolate Chip cookies = $2.50 Product Mix Decisions Margin per dozen Chocolate Chip cookies = $2.50 Margin per dozen Oatmeal Raisin cookies = $3.10 Margin per oven minute from Chocolate Chip cookies = $2.50 / 10 = $ 0.250 Margin per oven minute from Oatmeal Raisin cookies = $3.10 / 15 = $ 0.207 ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................

LP for Optimal Product Mix Selection xcc: Dozens of chocolate chip cookies sold. xor: Dozens of oatmeal raisin cookies sold. Max 2.5 xcc + 3.1 xor subject to 8 xcc + 5 xor < 1200 10 xcc + 15 xor < 1200 4 xcc + 4 xor < 1200 xcc < 100 xor < 50 Technology Constraints Market Constraints

over the next four months: Anadolu Truck A. Ş. operates a small facility that assembles street cleaning trucks. The company has the following firm orders for its product and production capacity over the next four months: February March April May Demand 200 350 400 Capacity 300 450 Many of the parts used in the assembly of the trucks are imported, and the company estimates that product costs will change (due to the fluctuations in the exchange rates) over the next four months as follows: February March April May Cost per Truck (TL) 15 Billion 14 Billion 16 Billion 17 Billion The cost of holding one completed truck in the inventory for one month is estimated to be 1 Billion TL. Write a Linear Program to minimize the total cost of the company.

Decision Variables: XFF: Quantity of trucks produced and sold in February XFM: Produced in Feb. sold in March XFA: Produced in Feb. sold in April XFY: Produced in Feb. sold in May XMM: March/March XMA: March/April XMY: March/May XAA: April/April XAY: April/May XYY: May/May Objective Function: Minimize Total Cost = ($1*XFM+$2*XFA+$3*XFY+$15(XFF+XFM+XXF+XFY)+ $1*XMA+$2*XMY+$14(XMM+ XMA+XMY) + $1*XAY+$16 (XAA+XAY)+$17*XYY )

All variables should be ≥ 0 Constraints: XFF+XFM+XFA+XFY ≤ 300 XFF ≥ 200 XMM+XMA+XMY ≤ 400 XFM+XMM ≥ 350 XAA+XAY ≤ 450 XFA+XMA+XAA ≥ 400 XYY ≤ 450 XFY+XMY+XAY+XYY ≥ 350 All variables should be ≥ 0

Better Products, Inc., manufactures three products on two machines. In a typical week, 40 hours are available on each machine. The profit contribution and production time in hours per unit are as follows: Category Product 1 Product 2 Product 3 Profit/unit $30 $50 $20 Machine 1 time/unit 0.5 2.0 0.75 Machine 2 time/unit 1.0   Two operators are required for machine 1; thus, 2 hours of labor must be scheduled for each hour of machine 1 time. Only one operator is required for machine 2 time. A maximum of 100 labor-hours is available for assignment to the machines during the coming week. Other production requirements are that product 1 cannot account for more than 50% of the units produced and that product 3 must account for at least 20% of the units produced. Formulate a linear programming model that can be used to determine the number of units of each product to produce to maximize the total profit contribution

Unit Production Requirements: X1: Number of Product1 to be produced. X2: Number of Product2 to be produced. X3: Number of Product3 to be produced. Objective; maximize profits: (30. X1+50. X2+20. X3) Simply unit profits x number of units to be produced Subject to: Machine working hours constraint (machines have limited capacity) 0,5X1 + 2X2 + 0,75X3 ≤ 40 for machine 1 X1 + X2 + 0,5X3 ≤ 40 for machine 2 Labor Hours Constraint: (we have limited labor hours=100 to be allocated over machines) [0,5X1 + 2X2 + 0,75X3] + 2 . [X1 + X2 + 0,5X3] ≤ 100 labor time req. for machine1 + 2x labor time req. for machine2 ≤ Total labor hrs available Unit Production Requirements: And, X1, X2, X3 are non-negative