OPSM 301 Operations Management

Class 13&14:

Linear Programming using Excel

Koç University

Zeynep Aksinzaksin@ku.edu.tr

Example: Giapetto's Woodcarving

Two types of toys are manufactured: soldiers and trains Soldiers:

– Sells for $27– Uses raw materials worth $10– Each soldier increases variable labor and overhead costs by $14

Trains:– Sells for $21– Uses raw materials worth $9– Each train increases variable labor and overhead costs by $10

Giapetto's Woodcarving

Manufacture requires skilled labor of two types– Carpentry– Finishing

Resource requirements by product– Soldier: 1 hour of carpentry and 2 hours of finishing– Train: 1 hour of carpentry and 1 hour of finishing

Total resources available– Unlimited raw materials– 80 hours of carpentry– 100 hours of finishing labor

Giapetto's Woodcarving

Demand – for trains is unlimited– At most 40 soldiers can be sold each week

Objective is to maximize weekly profit

Formulate as a linear program (LP)

Towards the Mathematical Model:

Define (decision variables)– x1 : number of soldiers produced each week– x2 : number of trains produced each week

Objective function:– maximize weekly profit = weekly profit from soldiers + weekly

profit from trains

Constraints: – each week, no more than 100 hours of finishing time may be used– each week, no more than 80 hours of carpentry time may be used– each week, the number of soldiers produced should not exceed 40

because of limited demand

The Linear Programming Model:

max 3x1 + 2x2

subject to2x1 + x2 100 (finishing hours)

x1 + x2 80 (carpentry hours)

x1 40 (demand for soldiers)

x1 0 (nonnegativity constraint)

x2 0 (nonnegativity constraint)

Giapetto's Woodcarving: The LP Model

max 3x1 + 2x2

subject to2x1 + x2 100 (finishing hours)

x1 + x2 80 (carpentry hours)

x1 40 (demand for soldiers)

x1 0 (nonnegativity constraint)

x2 0 (nonnegativity constraint)

Where– x1 : number of soldiers produced each week– x2 : number of trains produced each week

The Excel Model

                                                    

    soldiers trainsTotal(objective)    

  changing cells 20 60     

  profit 3 2 180                

    soldiers trains used capacity  

  finishing 2 1 100 100   carpenter 1 1 80 80   demand     20 40                                                                               

Filled in by Excel Solver

The optimal solution for Giapetto is to produce 20 soldiers and 60 trains per week, resulting in an optimal profit of $180. (The maximum possible profit attainable is $180, which can be achieved by producing 20 soldiers and 60 trains)

Reading the variable information

C10 min/un

B15 min/un

C5 min/un

A15 min/un.

A10 min/un.

B15 min/un.

D10 min/un

D5 min/un

P QSales price:90 $/unitMax demand:100 units/week

Sales price: 100 $/unitMax Demand:50 units/week

RM120$/un

RM220$/un

RM320$/un

Products P and Q are produced using the given process routing. 4 machines are used:A,B,C,D. (available for 2400 min/week)The price and raw material costs are given.

Problem:Formulate an LP to find the product mix that maximizes weekly profit.

i.e. How many of each product should we produce given the capacity and demand constraints?What is the bottleneck of this process?

Purchase Part 5$/un

Source: Paul Jensen

Example 1:Product Mix Problem

10

LP Formulation Decision variables:

– P:Amount of product P to produce per week– Q:Amount of product Q to produce per week

Objective Function: Maximize Profit– Max 45P+60 Q

Constraints: Machine hours used should be less than or equal to 2400 minutes:

– A: 15 P + 10 Q <= 2400– B: 15 P + 30 Q <= 2400– C: 15 P + 5 Q <= 2400– D: 10 P + 5 Q <= 2400

Production should not exceed demand:– P<=100– Q<=50

Non-negativity– P>=0,– Q>=0 11

Solver Solution

                   P Q          changing cells 100 30        

 objective Coefficients 45 60        

  Profit 6300                        

 Constraint Coefficients: P Q L.H.S. Value   R.H.S.  

  Machine A 15 10 1800<= 2400   Machine B 15 30 2400<= 2400   Machine C 15 5 1650<= 2400   Machine D 10 5 1150<= 2400   Demand P 1 0 100<= 100   Demand Q 0 1 30<= 50                                              

12

The Huntz Company purchases cucumbers and makes two kinds of pickles: sweet and dill. The company policy is that at least 30%, but no more than 60%, of the pickles be sweet. The demand for pickles is

SWEET:5000 jars + additional 3 jars for each $1 spent on advertisingDILL:4000 jars + additional 5 jars for each $1 spent on advertising

Sweet and dill pickles are advertised separately.  The production costs are:   SWEET:0.60 $/jar DILL:0.85 $/jar

and the selling prices are:   SWEET:1.45 $/jar DILL:1.75 $/jar

Huntz has $16,000 to spend on producing and advertising pickles. Formulate an appropriate Linear Program.

Example Problem

Xs: Number of Sweet pickle jars produced.Xd: Number of Dill pickle jars produced.As: Amount of advertisement done for Sweet Pickles in dollarsAd: Amount of advertisement done for Dill Pickles in dollars Objective; maximize profits: (Revenue - Cost) max[(1,45* Xs + 1,75* Xd)-( 0,6* Xs + 0,85* Xd + As + Ad)]Subject to:

Demand Constraints:Xs ≤ 5,000 + 3AsXd ≤ 4,000 + 5AdBudget Constraint:0,6* Xs + 0,85* Xd + As + Ad ≤ 16,000Ratio Constraint:Xs / (Xs + Xd) ≥ 0,3 0,7 Xs – 0,3 Xd ≥ 0Xs / (Xs + Xd) ≤ 0,6 0,4 Xs – 0,6 Xd ≤ 0

Non-negativity Constraints:Xs, Xd, As, Ad ≥0

Solution: LP Formulation