goal programming

• Goals are prioritized in some sense, and their level of aspiration is stated.

• An optimal solution is attained when all the goals are reached as close as possible to their aspiration level, while satisfying a set of constraints.

• There are two types of goal programming models:– Nonpreemtive goal programming - no goal is pre-determined to dominate

any other goal.

– Preemtive goal programming - goals are assigned different priority levels. Level 1 goal dominates level 2 goal, and so on.

13.5 Goal Programming

• A company is considering three forms of advertising.

• Goals– Goal 1: Spend no more $25,000 on advertising.

– Goal 2: Reach at least 30,000 new potential customers.

– Goal 3: Run at least 10 television spots.

NONPREEMTIVE GOAL PROGRAMMINGAn Advertisement Example

Cost per Ad Customers

Television 3000 1000Radio 800 500Newspaper 250 200

Cost per Ad Customers

Television 3000 1000Radio 800 500Newspaper 250 200

• If these were constraints rather than goals we would have:

3000X1 + 800X2 + 250X3 25,000

1000X1 + 500X2 + 200X3 30,000

X1 10

• No feasible solution exists that satisfies all the constraints.

• When these constraints are simply goals they are to be reached as close as possible.

An Advertisement Example

• Detrimental variablesUi = the amount by which the left hand side

falls short of (under) its right had side value.

Ei = the amount by which the left side exceeds its right had side value.

• The goal equations3000X1 + 800X2 + 250X3 + U1 – E1 = 25,000

1000X1 + 500X2 + 200X3 + U2 – E2 = 30,000

X1 + U3 – E3 = 10


• The objective is to minimize the penalty of not meeting the goals, represented by the detrimental variables

E1, U2, U3.


25,000 30,000 10

• The penalties are estimated to be as follows:

– Each extra dollar spent on advertisement above $25,000 cost the company $1.

– There is a loss of $5 to the company for each customer not

being reached, below the goal of 30,000.

– Each television spot below 10 is worth 100 times each

dollar over budget.


• It is assumed that no advantage is gained by overachieving a goal.

Minimize 1E1 + 5U2 + 100U3

s.t.3000X1 + 800X2 + 250X3 + U1 – E1 = 25,0001000X1 + 500X2 + 200X3 + U2 – E2 = 30,000

X1 + U3 – E3 = 10

All variables are non-negative.

An Advertisement Example – The goal programming model

• The NECC is planning next month production of its two bicycles B2 and S10.

• Data– Both models use the same seats and tires.– 2000 seats are available; 2400 tires are available. – 1000 gear assembly are available (used only in the S10

model).– Production time per unit: 2 hours for B2; 3 hours for

S10.– Profit: $40 for each B2; 10$ for each S10.

PREEMTIVE GOAL PROGRAMMING - New England Cycle Company

– Priority 1: Fulfill a contract for 400 B2 bicycles to be delivered next month.

NECC – Prioritized Goals

Priority 4: At least 200 tires left

over at the end of the month.

At least 100 gear assemblies left over at the end of the month.

Priority 2: Produce at least 1000 total bicycles during the month.

Priority 3: Achieve at least

$100,000 profit for the month.

Use no more than 1600 labor-hours during the month.

Management wants to determine the production schedule that best meets its prioritized schedule.

New England Cycle Company Example

NECC - SOLUTION

• Decision variablesX1 = The number of B2s to be produced next month

X2 = The number of S10s to be produced next month

• Functional / nonnegativity constraints

0X,XTires2400X2X2

assembliesGear1000XSeats2000XX2

21

21

2

21

• Goal constraints Priority 1 (goal 1): Production of at least 400 B2sPriority 1 (goal 1): Production of at least 400 B2s

XX11 + U+ U11 - E - E11 = 400 = 400

Priority 2 (goal 2): Production of at least 1000 total cyclesPriority 2 (goal 2): Production of at least 1000 total cyclesXX11 + X + X22 + U+ U22 - E - E22 = 1000 = 1000

Priority 3 (goal 3) Profit of at least $100,000Priority 3 (goal 3) Profit of at least $100,000

.04X .04X11 + .10X + .10X22 + U+ U33 - E - E33 = 100 (in $1000) = 100 (in $1000)Priority 3 (goal 4) Use a maximum of 1600 labor hoursPriority 3 (goal 4) Use a maximum of 1600 labor hours

2X 2X11 + 3X + 3X22 + U+ U44 - E - E44 = 1600 = 1600

Priority 4 (goal 5) At least 200 leftover tiresPriority 4 (goal 5) At least 200 leftover tires 2X 2X11 + 2X + 2X22 + U+ U55 - E - E55 = 2200 = 2200

Priority 4 (goal 6) At least 100 leftover gear assemblyPriority 4 (goal 6) At least 100 leftover gear assemblyXX22 + U+ U66 - E - E66 = 900 = 900

NECC - SOLUTION

• Priority level objectives

Priority 1: Underachieving a production of 400 B2s:Minimize U1

Priority 2: Underachieving a total production of 1000: Minimize U2

NECC - SOLUTION


– Priority 3: Underachieving a $100,000 profit

Using more than 1600 labor-hours

Minimize 30U3 + E4

NECC - SOLUTION

Each $1,000 short of the $100,000 goal is considered 30 times as important as utilizing an extra labor-hour.


Priority 4: Using more than 2200 tires

Using more than 900 gear assemblies

Minimize E5 + 2E6

NECC - SOLUTION

Each leftover gear assembly is deemed twice as important as leftover tire.

– Solve the linear goal programming for priority 1 objective, under the set of regular constraints and goal constraint as

shown below Minimize U1

ST

NECC - The solution procedure

X1 + U1 - E1 = 400

Tires2400X2X2Gear1000XSeats2000XX2

21

2

21

X1 = 400, thus

U1 = 0, and priority 1 goal is fully achieved.

– Solve the linear goal programming for priority 2 level objective, under the set of original constraints plus the constraint X1 400 (maintain the

level of achievement of the priority 1 goal).Minimize U2

ST


Every point that satisfies X1 + X2 1000 yields U2 = 0, and therefore, priority 2 goal is fully achieved.

Tires2400X2X2Gear1000XSeats2000XX2

21

2

21

X1 400X1 + X2 + U2 - E2 = 1000

– Solve the linear goal programming for priority 3 level objective, under the set of original constraints plus the constraint X1 400 (maintain the level of achievement of the priority 1 goal), plus the constraint X1 + X2 1000 (maintain the level of achievement of the priority 2 goal).

• Every point in the range X1 = 400 and 600 X2 800 is optimal for this model; 30U3 + E4 = 1720 is the level of achievement for the priority 3 goal.


– Solve the linear goal programming for priority 4 level objective, and notice that after the previous step the feasible region is reduced to a segment of a straight line between the points (400,600) and (400,800).

• X1 =400; 600 X2 700 and E5 + 2E6 = 0


• In summary NECC should produce– 400 B2 model– Between 600 and 700 S10 model

NECC - Solution Summary

goal programming

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