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Goal ProgrammingGoal ProgrammingGoal programming which reflects the Simon's theory of “satisficing” is widely applied techniques for modeling modern decision-making problems. The advantage of using goal programming over other techniques is with dealing with real-world decision problems is that it reflects the way manages actually make decisions.Goal programming allows decision maker to incorporate environmental, organizational, and managerial consideration into model through goal levels and priorities.
Goal Programming originally introduced by Goal Programming originally introduced by A. Charnes and W.W. Cooper and further A. Charnes and W.W. Cooper and further developed by Y.Ijiri, S. M. Lee, and others is developed by Y.Ijiri, S. M. Lee, and others is similar to linear programming concept.similar to linear programming concept.
Goal programming can be employed in Goal programming can be employed in decision problems with a single goal decision problems with a single goal (objective) and multiple sub goals, as well as (objective) and multiple sub goals, as well as in cases having multiple goals and sub goals. in cases having multiple goals and sub goals.
With in goal programming model, goals may With in goal programming model, goals may be achieved only at the expense of other be achieved only at the expense of other goals.goals.
The goal programming necessitates The goal programming necessitates the establishment of a weighting the establishment of a weighting system for the goals such that lower-system for the goals such that lower-ranked (or weighted) goals are ranked (or weighted) goals are considered only after higher-ranked considered only after higher-ranked goals have been satisfied or have goals have been satisfied or have reached the point beyond which no reached the point beyond which no further improvement is desirable. This further improvement is desirable. This weights can be ordinal or cardinal.weights can be ordinal or cardinal.
Goal programming is a form of linear Goal programming is a form of linear programming, goal programming models programming, goal programming models must be formulated under the same must be formulated under the same limitations, assumptions, and conditions limitations, assumptions, and conditions as linear programming models (linearity, as linear programming models (linearity, divisibility, determinism, etc.). divisibility, determinism, etc.).
Goal programming problems can also be Goal programming problems can also be solved by using the simplex method (in a solved by using the simplex method (in a modified form)modified form)
Goal Programming has been widely Goal Programming has been widely applied to decision problems in applied to decision problems in business organizations, government business organizations, government agencies, and nonprofit institutions. agencies, and nonprofit institutions. Examples include the following:Examples include the following:
Academic administration planningAcademic administration planning Manpower planningManpower planningAccounting analysisAccounting analysis Marketing logisticsMarketing logisticsAdvertising media schedulingAdvertising media scheduling Military strategiesMilitary strategiesBlood bank logisticsBlood bank logistics Organizational analysisOrganizational analysisCapital budgetingCapital budgeting Personnel administrationPersonnel administrationComputer resource allocationComputer resource allocationPolicy analysisPolicy analysisDecision support system planningDecision support system planning Portfolio managementPortfolio managementEconomic policy analysisEconomic policy analysis Production schedulingProduction schedulingEducational system planningEducational system planning Project managementProject managementEnergy resources planningEnergy resources planning Quality controlQuality controlEnvironmental protectionEnvironmental protection Research and developmentResearch and developmentFacilities layout and location decisionsFacilities layout and location decisions Transportation logisticsTransportation logisticsFinancial analysisFinancial analysis Urban planningUrban planningHealth care delivery planningHealth care delivery planningWater resources planningWater resources planningInventory managementInventory management
Example 13.1 Product Mix Example 13.1 Product Mix ProblemProblem
A manufacturing A manufacturing company produces three company produces three products, 1, 2, and 3. The products, 1, 2, and 3. The three products have three products have resource requirements as resource requirements as follows:follows:
Labor (hr/unit) Materials (lb/unit) Profit ($/unit)
Product 1 5 4 3
Product 2 2 6 5
Product 3 4 3 2
At present the firm has a normal production capacity of 240 hours of labor available daily and a daily supply of 400 pounds of material.
Maximize Z= 3X1+5X2+2X3
SUBJECT TO
5X1+6X2+3X3≤240
4X1+6X2+3X3≤400
X1,X2,X3≥0
This model has a single objective, profit This model has a single objective, profit maximization. Now considering the maximization. Now considering the management developed set of goals, arranged management developed set of goals, arranged in order of their importance to the firm.in order of their importance to the firm.
1.1. Because of labor relations difficulties, Because of labor relations difficulties, management desires to avoid underutilization management desires to avoid underutilization of normal production capacity (i.e., no layoffs of normal production capacity (i.e., no layoffs of workers).of workers).
2.2. Management has established a satisfactory Management has established a satisfactory profit level of $500 per day.profit level of $500 per day.
3.3. Overtime is to be minimized as much as Overtime is to be minimized as much as possible.possible.
4.4. Management wants to minimize the purchase Management wants to minimize the purchase of additional materials because of handling of additional materials because of handling and storage problems.and storage problems.The goal constraints developed are as follows:The goal constraints developed are as follows:
Labor UtilizationLabor UtilizationIn order to reflect the possibility of underutilization of In order to reflect the possibility of underutilization of labor (as well as overtime), the original linear labor (as well as overtime), the original linear programming constraint is reformulated as programming constraint is reformulated as
240ddx4x2x5 11321
The variable are referred to as deviational variablesdeviational variables. They represent the number of hours less than (underutilization) and the number of hours
exceeding (overtime) for the amount of production
determined by the values of X1,X2,X3.
dd 11and
d 1240
d240 1
In the analysis, one of the deviational variable,
must always be zero in the solution.
It is not possible to physically have both underutilization and over utilization at the same time.
A constraint in which we attempt to minimize or both is referred to as a goal constraintgoal constraint..
dd 11or
dd 21,
The objective function for underutilization The objective function for underutilization is specified as follows:is specified as follows:
Minimize Minimize
PP11 is the preemptive priority designation for this goal.
The term reflects the fact that the first priority goal of the firm is to minimize , the underutilization of labor.
The first goal is to minimize ( drive it as close to zero as possible)
dP 11Z
dp 11
d 1
d 1
The minimization of overtime the fact that management has ranked third is reflected in the objective function as follows:
Minimize
P3 designates minimization of overtime, as the third priority goal.
Z represents a multidimensional function composed of various priority factors and associated income immensurable objective criteria.
d 1
dpdp 1311Z
d 1
Profit LevelProfit Level Management’s second goal is to achieve the
satisfactory profit level of $500. This goal constraint is formulated as
Where is underachievement of the profit goal and is the overachievement of the profit goal. The goal is reflected in the objective function by minimizing
at the second priority level.
Minimize
500ddx2x5x3 22321
d 2
d 2
d 2
dpdpdp 132211Z
Purchase of MaterialsPurchase of Materials Managements final goal is that daily material
purchases in excess of 400 pounds be minimized. Formulating, the goal constraint
where where is the over utilization of normal material requirement and is the purchase of extra materials. The objective function at the fourth priority level
The last term reflects management’s desire to minimize the
Purchase of extra materials at a level of priority below those of the other three goals.
400ddx3x6x4 33321
d 3
d 3
dpdpdpdp 34132211MinimizeZ
The goal programming model for the problem can be The goal programming model for the problem can be summarized as:summarized as:
The last term reflects the managements desire to minimize the purchase of extra material at a level of priority below those of the other three goals.goal programming model can be summarized as follows:
subject to
dpdpdpdp 34132211MinimizeZ
dpdpdpdp 34132211MinimizeZ
240ddx4x2x5 11321
500ddx2x5x3 22321
400ddx3x6x4 33321
0,,,,,,,, ddddddxxx 332211321
Solution of this problem requires that the deviations from the goals specified in the objective function be minimized.
The value of the deviational variable associated with the highest preemptive priority (P1) must be first minimized to the fullest possible extent.
when no further improvement is possible or desired for this goal, the value of the deviational variable ( )
associated with the next highest priority factor, P2 is minimized, and so on.
The solution procedures is a modified simplex approach.
Z represents the sum of unattained portions of each of the goals at different priority levels.
d 1
d 2
Example 13.2 Weighted Example 13.2 Weighted GoalsGoals
A small manufacturing firm produces A small manufacturing firm produces washers and dryers. Production of either washers and dryers. Production of either product requires 1 hour of production product requires 1 hour of production time. The plant has a normal production time. The plant has a normal production capacity of 40 hours per week. A maximum capacity of 40 hours per week. A maximum of 24 washers and 30 dryers can be stored of 24 washers and 30 dryers can be stored per week. The profit margin is $80 for a per week. The profit margin is $80 for a washer and $40 for a dryer. The manager washer and $40 for a dryer. The manager has established the following goals, has established the following goals, arranged in order of their priority.arranged in order of their priority.
PP11: Avoid underutilization of normal : Avoid underutilization of normal production capacity.production capacity.
PP22: Produce as many washers and : Produce as many washers and dryers as possible. However, since the dryers as possible. However, since the profit margin for a washer is twice profit margin for a washer is twice that for a dryer, the manager has that for a dryer, the manager has twice as much desire to achieve the twice as much desire to achieve the production of washers as to achieve production of washers as to achieve the production of dryers.the production of dryers.
PP33: Minimize overtime as much as : Minimize overtime as much as possible.possible.
Production CapacityProduction Capacity
The first goal constraint reflects the production time requirements for both products..
where X1 and X2 are the respective numbers of washers and dryers produced.
The deviational variable, , reflects underutilization of the normal production capacity of 40 hours per week, while overtime.
Priority goals 1 and 3 can be reflected as
40ddxx 1121
d 1
d 1
dpdp 1311MinimizeZ
Storage constraintStorage constraint
The production goal constraints are:The production goal constraints are:
The first goal constraint represents the underachievement of the production goal for washers.
The second goal constraint is the underachievement of the production goal for dryers.
The production goals have been eliminated, because these goal levels represent absoluteabsolute maximum values (i.e., storage capacities) not to be exceeded.
30
24
dxdx
32
21
d 2
d 3
dd 32and
This type of constraint is referred to as system system constraintconstraint because deviation in the positive and/or negative direction is prohibited..
Second priority goal is reflected in the objective function as follows:
The goal programming model is formulated as:
subject to:
dpdp 1311MinimizeZ
dpd2p 3222
dpdpp 133222d 1p1MinimizeZ
0,,,,,
30
24
40
ddddxxdxdx
ddxx
321121
32
21
1121
Example 13.3 Deviational Example 13.3 Deviational Variable Goal ConstraintVariable Goal Constraint
Extending from the previous problem, the added goal that overtime not exceed 10 hours per week, if possible. The priority level of this new goal places it between the old P1 and P2 levels.
The production goal constraint:
Our new goal is that overtime be restricted to 10 hours, which is formulated as
40ddxx 1121
10ddd 441
Another way to formulate the same goal constraint in terms of decision variables is adding the allowed overtime of 10 hours to the original production requirement goal as as follows:
dpdpdd4p2MinimizeZ
becomesfunctionobjectivenewThe
50ddxx
14332p32d1p1
4421
The new second priority goal specifies that the amount of overtime in excess of 10 hours is to be minimized. This goal is not incompatible with the goal of minimizing overtime.
The new goal programming model is The new goal programming model is
Minimize Minimize
0,,
10
30
24
40
toSubject
2Z
ddxddd
dxdx
ddxx
dpdpdpdp
111
441
32
21
1121
14334211
Example 13.4 Recreational Example 13.4 Recreational Facility FundingFacility Funding
A city parks and recreational authority has A city parks and recreational authority has been given a federal grant of $600,000 to been given a federal grant of $600,000 to expand its public recreational facilities. expand its public recreational facilities. Four different types of facilities have been Four different types of facilities have been requested by city council members requested by city council members speaking for their constitutes: gymnasiums, speaking for their constitutes: gymnasiums, athletic fields, tennis courts, and swimming athletic fields, tennis courts, and swimming pools. The total demand by various pools. The total demand by various neighborhoods has been for 7 gyms, 10 neighborhoods has been for 7 gyms, 10 athletic fields, 8 tennis courts, and 12 athletic fields, 8 tennis courts, and 12 swimming pools.swimming pools.
Facility Cost ($) Required Acres
Expected Usage
(People/week)
Gymnasium 80,000 4 1500
Athletic 24,000 8 3000
Tennis Court 15,000 3 500
Swimming Pool
40,000 5 1000
Each facility costs a certain amount, requires a certain number of acres, and has an expected usage. These parameters are summarized in the following table:
The park authority has located 50 acres of land for construction (although more land could be located if necessary).The authority has established the following list of prioritized goals:
PP11:The authority must spend the total grant :The authority must spend the total grant (otherwise the amount not spent will be (otherwise the amount not spent will be returned to the federal government).returned to the federal government).
PP22: The park authority desires that the : The park authority desires that the facilities be used weekly by 20,000 or facilities be used weekly by 20,000 or more people.more people.
PP33: If more land is acquired, the additional : If more land is acquired, the additional amount should be limited to 10 acres.amount should be limited to 10 acres.
PP44: The authority would like to meet the : The authority would like to meet the demands of the city council members for demands of the city council members for new facilities. However, this priority new facilities. However, this priority should be weighted according to the should be weighted according to the number of people expected to use each number of people expected to use each facility.facility.
PP55: The park authority wants to avoid : The park authority wants to avoid securing land beyond the 50 acres securing land beyond the 50 acres presently available.presently available.
Funding ConstraintFunding ConstraintThe cost requirement for the various facilities are shown in goal constraint:
where X1,X2,X3,X4 are number of facilities of each type to be constructed.
The deviational variable is the portion of the grant not spent.
The deviational variable has been eliminated, the first priority goal is reflected in the objective function as follows:
000,600dx000,40000,15x000,24x000,80 14321
d 1
d 1
dp 11ZMinimize
Facility UseFacility Use The Expected total weekly usage for all the
facilities is formulated as
The deviational variables are the amounts of weekly underutilization or over utilization of the facilities. The priority 2 goal of minimizing under utilization is shown in objective function as
000,20ddx1000x500x3000x1500 224321
dpdp 2211ZMinimize
Land RequirementsLand Requirements The land requirements for the various facility types
are reflected in the equation as
The deviational variables represent the amount by which the number of acres used is less than 50, , and the excess above 50 acres, . The park authority desires that the amount of land in excess of 50 acres be limited, to 10 acres.
This goal is reflected in the objective function by minimization of
at the priority 3 level. This goal and the priority 5 goal are shown in objective function as
50ddx5x3x8x4 334321
d 3
d 3
10ddd 443
d 4
dpdpdpdp 35432211ZMinimize
Facility DemandFacility Demand
The demand for facilities is shown in four The demand for facilities is shown in four goal constraint.goal constraint.
12ddx
8ddx
10ddx
7ddx
884
773
662
551
dpd2pdpd6pd3pdpdpdp 3584746454432211ZMinimize
The complete goal programming model is formulated as
Subject to:
dpdp2dpdp6dp3dpdpdp 3584746454432211ZMinimize
0ddx12ddx
8ddx
10ddx
7ddx
10ddd
50ddx5x3x8x4
000,20ddx1000x500x3000x1500
000,600dx000,40x000,15x000,24x000,80
jjj
884
773
662
551
443
334321
224321
14321
,,
Example 13.5 Multiperiod Example 13.5 Multiperiod Investment ProblemInvestment Problem
A investment firm has $1,000,000 to invest in four alternatives: stocks, bonds, savings certificates, and real estate. The firm wishes to determine the mix of investments that will maximize the cash value at the end of 6 years. Investment opportunities in stocks and bonds are available at the beginning of each of the next 6 year. Each dollar invested in stocks at the beginning of each year will return $1.20 ( a profit of $0.20) 2 years later, which can be immediately reinvested in any alternative. Each dollar invested in bonds at the beginning of each year will return $1.40 3 years later, which can be reinvested immediately.
Investment opportunities in savings certificates are available only once, at the beginning of the second year. Each dollar invested in certificates at the beginning of the second year will return $1.80 4 years later. Investment opportunities in real estate are available at the beginning of the fifth and sixth years. Each dollar invested in real estate will return $1.10 a year later.The management of the firm wishes to determine the optimal mix of investments in the various alternatives that will achieve the following goals, listed in the order of their importance.P1:: In order to maximize risk, the total amount invested in stocks and bonds should be limited to 40% of the total investment.
P2: The amount invested in savings certificates should be at least 25% of the total investment.
P3: Real estate is expected to be very attractive in the future. Thus, management would like to invest at least $300,000 in real estate.
P4: The total cash value by the end of the sixth year should be maximized.
Si= amount of money invested in stocks at the beginning of year i; i=1,2,3,4,5
Bi=amount of money invested in bonds
C2= amount of money invested in saving certificates in year 2
Ri=amount of money invested in real estate
Ii= amount of money held idle and not invested during year i;
Decision VariablesDecision Variables
Year 1 1 Year 2 Year 3 Year 4 Year 5 Year 6
S1
S3 S5S2 S4
B1
B4
B2 B3
C2
R5R6
System constraintsSystem constraints
Investment opportunity constraint in first year Year 1: S1 +B1+I1=1,000,000
I1 is the amount of money not invested at the beginning of year 1.
Second year, the investment will be S2,,B2, and C2.
Investment Opportunities Amounts AvailableYear 2: S2+B2+C2+I2 =I1
Year 3: S3+B3+I3 =I2+1.2S1
Investment Opportunities Amounts Available
Year 4: S4+B4+I4 =I3+1.2S2+1.4B1
Year 5: S5+R5+I5 =I4+1.2S3+1.4B2
Year 6: R6+I6
=I5+1.2S4+1.4B3+1.8C2+1.1R5
Goal ConstraintGoal Constraint
We can formulate the four goal constraint as follows:
P1: The total amount invested in stocks and bonds, should not exceed 40% of the total investment in all the alternatives,
4
1ii
5
1ii BS
04.04.06.06.0
Minimize,arrangingRe
4.0
intconstragoalfollowingformulatecanWe
ddRCBS
d
RCBSddBS
RCBS
11
6
5ii2
4
1ii
5
1ii
1
5
1i
4
1i
6
5ii2i111
4
1ii
5
1ii
6
5ii2
4
1ii
5
1ii
PP22:: Since the amount invested in savings certificates should be at least 25% of the total investment, we should minimize from the following goal constraint:
P3: For the real estate investment, we should minimize in the following goal constraint:
d 2
025.075.025.025.0 ddRCBS 22
6
5ii2
4
1ii
5
1ii
d 3
000,300ddR 3
6
5i3i
PP4: Our goal is to maximize the total cash value by the end of the sixth year. The investments alternatives are S5,B4, and R6. By setting a cash value at an arbitrarily large number M ($500,000,00) and minimize , we will be maximizing the cash value. We can formulate as follows:
The complete goal programming model can be summarized as:
d 4
MdIR1.1B4.1S2.1 46645
0,,,,,,
000,000,5001.14.12.1
000,300
025.075.025.025.0
04.04.06.06.0
ddIRCBS
dIRBS
ddR
ddRCBS
ddRCBS
iiiiiii
46645
33
6
5ii
22
6
5ii2
4
1ii
5
1ii
11
6
5ii2
4
1ii
5
1ii
0IIRR1.1C8.1B4.1S0IIRB4.1SS0IIBB4.1SS
0IICBS
000,000,1IBS
dpdpdpdpZ
6565234
545253
434142
21222
111
44332211
2.1
2.1
2.1
tosubject
Minimize
General Goal Programming General Goal Programming ModelModel
The general goal programming model can be formulated as follows:
where Pk is the preemptive priority weight (Pk>>>Pk+1) assigned to goal k (k=0 is reserved for system constraint),
are the numerical( differential) weights assigned to the deviational variables of goal i at a given priority level k, represent the negative and positive deviations, aij is the technological coefficients of xj in goal i, and bi, is the ith goal level.
0,,
m,.......2,1i
tosubject
ZMinimize
ddx
bddxa
dwdwp
iij
i11j
n
1jij
iikiik
k
0k
m
1ik
ww ikikand
dd iiand
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