OPSM 301 Operations Management
Class 25:Applied LP continued
Koç University
Zeynep [email protected]
A Transportation Problem:Tropicsun
Distances (in miles)CapacitySupply
275,000
400,000
300,000 225,000
600,000
200,000Mt. Dora
1
Eustis
2
Clermont
3
Groves
Ocala
4
Orlando
5
Leesburg
6
Processing Plants
21
50
40
3530
22
55
25
20
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 constraintsX14 + X24 + X34 <= 200,000 } Ocala
X15 + X25 + X35 <= 600,000 } Orlando
X16 + X26 + X36 <= 225,000 } Leesburg
Supply constraintsX14 + X15 + X16 = 275,000 } Mt. Dora
X24 + X25 + X26 = 400,000 } Eustis
X34 + X35 + X36 = 300,000 } Clermont
Nonnegativity conditionsXij >= 0 for all i and j
Implementing the Model
An Employee Scheduling Problem:Air-Express
Day of Week Workers NeededSunday 18
Monday 27
Tuesday 22
Wednesday 26
Thursday 25
Friday 21
Saturday 19
Shift Days Off Wage1 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 day0X1 + 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 conditionsXi >= 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 toCompany 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 investedX1 + X2 + X3 + X4 + X5 + X6 = 750,000
No more than 25% in any one investmentXi <= 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 conditionsXi >= 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
Maximum weekly demand of
Oatmeal Raisin Cookies: 50 dozen
Total weekly operating expense $270
Product Mix Decisions
ProcessingTimes
Mix &Dish
Load Oven& Set
Bake Cool Pack Receive Total
ChocolateChip
8 mins.(6+2/tray)
1 9 5 2/tray 1 26
OatmealRaisin
5 mins.(3+2/tray)
1 14 2 2/tray 1 25
Resource You RM+Oven Oven RM RM
Total time available in week: 20 hrs
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
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 200 350 400 350
Capacity 300 400 450 450
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 MarchXFA: Produced in Feb. sold in April XFY: Produced in Feb. sold in May XMM: March/MarchXMA: 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 )
Constraints:XFF+XFM+XFA+XFY ≤ 300
XFF ≥ 200 XMM+XMA+XMY ≤ 400
XFM+XMM ≥ 350XAA+XAY ≤ 450
XFA+XMA+XAA ≥ 400XYY ≤ 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:
CategoryProduct 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 1.0 0.5
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
0,5 321
1 XXX
X
0,2 321
3 XXX
X
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 producedSubject 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
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