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**Linear Programming We are to learn two topics today: LP formulation**

determining LP solutions Graphical approach Simplex method – will be taken up in next lecture Tutorial Questions: Ch 2, 3,6,9,12,16,20 (to p2) (to p38) (to p61)

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**LP formulation Note that Chapter 4 has presented 8 LP formulations**

Here, we pick up the following three examples for discussions Example 1 (Investment) Example 2 (Marketing) Example 3 (Transportation) (to p3) (to p17) (to p30) (to p1)

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**Example 1 (Investment) Please refer to the handouts(p112)**

Spend 5 mins to read through them Question asked was to formulate its LP problem! Solution: (to p59) (to p4)

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How to achieve it? From lecture 1, we know LP formulation involve the following steps: Step 1: define decision variables Step 2: define the objective function Step 3: state all the resource constraints Step 4: define non-negativity constraints Overall LP formulation (to p5) (to p6) (to p8) (to p15) (to p16) (to p2)

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**Step 1: define decision variables**

Question: how much to invest in four alternative choices: Let, x1 = amount invested in municipal bonds ($) x2 = amount invested in certificates of deposit ($) x3 = amount invested in treasury bills ($) x4 = amount invested in growth stock fund($) (to p4)

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**Step 2: define the objective function**

Expected profits are: 8.5% for x1 = 0.085x1, 5% for x2 = 0.05x2 6.5% for x3 = 0.065x3 13% for x4 = 0.13x4 Thus, objective function is (to p7)

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**Objective function maximize Z = $0.085x1 + 0.05x2 + 0.065 x3+ 0.130x4**

(to p4)

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**Step 3: state all the resource constraints**

Four constraints Not more than 20%for x1 X2 should not exceed total of other investments At least 30% spent for x2 and x3 X2 and x3 should be greater than x1 and x4, by ratio of at least 1.2 to 1 Note: total investment is $70,000 All constraints are: (to p9) (to p10) (to p11) (to p12) (to p13) (to p14) (to p4)

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**Not more than 20%for x1 Given that total investment is $70,000**

20% of it = $14,000 Thus this constraint could be expressed as x1 14,000 (to p8)

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**X2 should not exceed total of other investments**

Total 4 investments, x1, x2, x3 and x4 Thus, other investments would be x1, x3 and x4 This constraint is then expressed as: x2 x1 + x3 + x4 Or x2 - x1 - x3- x4 0 (to p8)

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**At least 30% spent for x2 and x3**

That is, 30% of $70,000 = $21,000 Or x2 + x3 21,000 (to p8)

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**That is x2 + x3 1.2(x1 + x4) or -1.2x1 + x2 + x3 - 1.2 x4 0**

X2 and x3 should be greater than x1 and x4, by ratio of at least 1.2 to 1 That is x2 + x3 1.2(x1 + x4) or -1.2x1 + x2 + x x4 0 (to p8)

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**total investment is $70,000 That is, x1 + x2 + x3 + x4 = 70,000**

(to p8)

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**Over constraints x1 14,000 x2 - x1 - x3- x4 0 x2 + x3 21,000**

(to p8)

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**Step 4: define non-negativity constraints**

That is, x1, x2, x3, x4 0 (to p4)

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**Overall LP formulation**

maximize Z = $0.085x x x x4 subject to x1 14,000 x2 - x1 - x3- x4 0 x2 + x3 21,000 -1.2x1 + x2 + x x4 0 x1 + x2 + x3 + x4 = 70,000 x1, x2, x3, x4 0 (at least 20% on x1) (x2 spent not more than total investment) (at least 20% on x2 and x3) (x2+x3 has at least 1.2 ratio to x1+x4 (total investment is $70,000) (to p2) Note: it is up to you if you like to arrange in this format or let it be the way in according to how constraints were stated!

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**Example 2 (Marketing) Please read the handout (118)**

Read it for 5 mins and then try to formulate it! How to start? Again we following the four steps (to p59) (to p18)

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**4 procedural steps Step 1: define decision variables**

Step 2: define the objective function Step 3: state all the resource constraints Step 4: define non-negativity constraints Overall LP formulation (to p19) (to p20) (to p22) (to p28) (to p29)

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**Step 1: define decision variables**

Question: how to achieve a max exposure to three of possible media Let x1 = number of television commercials x2 = number of radio commercials x3 = number of newspaper aids (to p18)

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**Step 2: define the objective function**

Exposure are: (refer to the attached table) 20,000 for x1 12,000 for x2 9,000 for x3 Thus the objective function is maximize Z = ,000x1 + 12,000x2 + 9,000x3 (to p21) (to p18)

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**Audience exposure rate**

(to p20)

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**Step 3: state all the resource constraints**

We have the following resources: Budget limit $100,000 Television time for four commercials Radio time for 10 commercials Newspaper space for 7 ads Resources for no more than 15 commercials and/or ads. (to p) (to p24) (to p25) (to p26) (to p27) (to p18)

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Budget limit $100,000 All associated cost to each variables has a budget of $100,000 Thus, $15,000x1 + 6,000x 2+ 4,000x3 100,000 (to p22)

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**Television time for four commercials**

That is x1 4 (max of 4 TV ads) (to p22)

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**Radio time for 10 commercials**

That is, x2 10 (max of 10 radio ads) (to p22)

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**Newspaper space for 7 ads**

That is, x3 7 (max of 7 newspaper ads) (to p22)

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**Resources for no more than 15 commercials and/or ads**

That is x1 + x2 + x3 15 (max of 15 ads can be subscribed) (to p22)

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**Step 4: define non-negativity constraints**

That is, x1, x2, x3 0 (to p18)

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**Overall LP formulation**

maximize Z = 20,000x1 + 12,000x2 + 9,000x3 subject to $15,000x1 + 6,000x 2+ 4,000x3 100,000 x1 4 x2 10 x3 7 x1 + x2 + x3 15 x1, x2, x3 0 (note it is advisable to include explanation here!) (to p2)

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**Example 3 (Transportation)**

Please read the handout (121) Read it for 5 mins and then try to formulate it! How to start? Again we following the four steps (to p60) (to p31)

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**4 procedural steps Step 1: define decision variables**

Step 2: define the objective function Step 3: state all the resource constraints Step 4: define non-negativity constraints Overall LP formulation (to p32) (to p33) (to p34) (to p36) (to p37)

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**Step 1: define decision variables**

We are to determine all combination of shipping from warehouse i to store j, i=1,2,3;j=A,B,C That is, xij, where i is warehouse, j is store or X1A= no of TV shipped from warehouse I to A store X1B = no of TV shipped from warehouse 1 to B store X1C = no of TV shipped from warehouse 1 to C store X2A = no of TV shipped from warehouse 2 to A store X2B = no of TV shipped from warehouse 2 to B store X2C = no of TV shipped from warehouse 2 to C store X3A = no of TV shipped from warehouse 3 to A store X3B = no of TV shipped from warehouse 3to B store X3C = no of TV shipped from warehouse 3 to C store OR Xij = no of TV shipped from warehouse i to j store, i=1,2,3;j=A,B,C (to p31)

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**Step 2: define the objective function**

Here, we attempt to min the shipping cost, i.e. Cost Minimize Z = $16x1A + 18x1B + 11x1C + 14x2A + 12x2B + 13x2C + 13x3A + 15x3B + 17x3C (to p31)

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**Step 3: state all the resource constraints**

Our resources are: Warehouse supply of televisions sets: Retail store demand for television sets: 1- Cincinnati A - New York 150 2- Atlanta B - Dallas 250 3- Pittsburgh C - Detroit 200 total total 600 So, the resource constraints are: x1A + x1B+ x1 300 x2A+ x2B + x2C 200 x3A+ x3B + x3C 200 x1A + x2A + x3A = 150 x1B + x2B + x3B = 250 x1C + x2C + x3C = 200 How to get these? Note, how this Problem really looks Like? (to p31) (to p35)

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**Transportation Problem**

Total supply 300 200 200 700 150 250 200 Total demand 600 Note: total supply is greater than total demand. Thus, we set demand constraint to = and supply to <= (to p34)

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**Step 4: define non-negativity constraints**

That is, xij 0, i=1,2,3; j=A,B,C (to p31)

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**Overall LP formulation**

Minimize Z = $16x1A + 18x1B + 11x1C + 14x2A + 12x2B + 13x2C + 13x3A + 15x3B + 17x3C Subject to x1A + x1B+ x1C 300 x2A+ x2B + x2C 200 x3A+ x3B + x3C 200 x1A + x2A + x3A = 150 x1B + x2B + x3B = 250 x1C + x2C + x3C = 200 xij 0, i=1,2,3; j=A,B,C (to p2)

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**LP - Graphical approach**

Here, we now try to solve an LP formulation using a graphical approach, that is Plot all equations onto a graph and determine Z values of decision variables that satisfying all resource constraints How it works? Types of LP solutions (to p39) (to p53)

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**LP - Graphical approach**

Consider a simple LP problem: maximize Z=$40x1 + 50x2 subject to 1x1 + 2x2 40 ………(e1) 4x2 + 3x2 120 ……..(e2) x1 ……...(e3) x2 ………(e4) Objective function Resource Constraints Solution steps of graphical approach (to p40)

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**Solution steps of graphical approach**

Step 1: take one resource constraint and treat it as an equation and draw them onto a graph Step 2: determine the region that satisfying the inequality equation of step 1 Step 3: repeat steps 1-2 for all resource constraints Step 4: super imposing objective function to determine the solution set for decision variables (to p41)

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**Step 1: Consider 1x1 + 2x2 40 ……… (1) And teating it as**

How to draw them ? (to p42)

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**1x1 + 2x2 = 40 How to plot? Consider any points on x and y axis**

in turn, then we have : x1=0, x2=20 and x2=0, x1=40 Step2: Where is 1x1 + 2x2 40 ? (to p43)

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**Determine 1x1 + 2x2 40 This region We take a point Is known**

as feasible region! We take a point at (0,0) to check which side satisfying this constraint! (0,0) At (0,0), 1*0 + 0*0 40 ? Yes, then all points on the side of (0,0) are feasible region Repeat these steps for equation e2 (to p44)

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Constraint e2 (to p45) When combine this to the one we obtained before, then we have ….

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**Constraints e1 to e2 Interception point (x1,x2)=(24,8) (0,20) (30,0)**

(to p46) When add e3 and e4, we have ……

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Common region of e1 to e4 Now, we proceed to Step 4, that is impose Objective function Z onto this graph ………. How to plot it? (to p47)

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maximize Z=$40x1 + 50x2 To plot this equation Z, we can simply assign an aribitrary number for Z, Say, Z = 800 Then =$40x1 + 50x2 When x2 = 0, then x1= 20 ….. A plotting point (20,0) When x1 = 0, then x2 = 16 ….a plotting point (0,16) When plotting it on the graph, we would have …. (to p48) Where is the optimal solution?

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Optimal solution The optimal solution is at the corner point of the feasible region! To obtain the optimal point, we use a ruler move toward the right hand side and the last corner point Solution (to p49) For max problem if min then it is on the left hand side

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**Optimal point This is the solution (x1,x2)=(24,8)**

(to p50) Let check to see if it is the optimal point/solution ….

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Checking! Optimal solution X1=0 X2-0 Z=0 More exercise! …….. (to p51)

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**Another example Consider Minimize Z = $6x1 + 3x2 subject to**

2x1 + 4x2 16 …………. (e1) 4x1 + 3x2 24 …………...(e2) x1 …………..(e3) x2 …………..(e4) (to p52) Feasible region for constraints e1 to e4 ………

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**Resource constraints Feasible region Imposing function Z And also**

x1 0, x2 0 2x1 + 4x2 16 4x1 + 3x2 24 Imposing inequalities Note, Min Z is at the most left Hand side of the Z function Imposing function Z (to p38)

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**Types of LP solutions Two types of LP solution:**

The solution – the unique solution Multiple solutions Question: Do all LP problems have a solution? (to p54) (to p55) (to p56)

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**The solution – unique solution**

It is referred to ….there is only one optimal solution Z has satisfying all LP formulation …. i.e. only one (corner) point from the feasible region (to p53)

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Multiple solutions It refers to that we have more than one set of solutions satisfying to the LP formulation. That is, the optimal function Z lies onto one of resource constraints that forms the feasible region …… and that all points (x1,x2) on that region are optimal solutions to the LP problem For Max Z, all points between here are optimal solutions (to p53)

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**LP solutions LP solution does not exit if our LP problems are:**

Infeasible problems Unbound problems (to p57) (to p58) (to p1)

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An Infeasible Problem Every possible solution violates at least one constraint: maximize Z = 5x1 + 3x2 subject to 4x1 + 2x2 8 x1 4 x2 6 x1, x2 0 Here, we cannot find a region that has satisfying all constraints, i.e. no feasible region! (to p56)

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An Unbounded Problem Value of objective function increases indefinetly: maximize Z = 4x1 + 2x2 subject to x1 4 x2 2 x1, x2 0 Note: there is no bounded feasible region, thus it cannot find Max Z. But it has solution if Z is a Min problem! (Why?) (to p56)

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(to p3) (to p17)

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(to p30)

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4-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Linear Programming: Modeling Examples Chapter 4.

4-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Linear Programming: Modeling Examples Chapter 4.

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