Multi-objective marketing planning inventory model: A geometric programming approach

Applied Mathematics and Computation 205 (2008) 238–246

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Multi-objective marketing planning inventory model: A geometricprogramming approach

Sahidul IslamDepartment of Mathematics, Bengal College of Engineering and Technology, Durgapur-12, West Bengal, India

a r t i c l e i n f o

Keywords:Max–min methodLeast square methodGlobal criterion methodHybrid methodDemandSet-up costMulti-objectiveInventory

0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.07.037

E-mail address: [email protected]

a b s t r a c t

In this paper, we have formulated a multi-objective marketing planning inventory modelunder the limitations of space capacity and the total allowable shortage cost constraints.Here, we have discussed multi-objective geometric programming problem based on globalcriterion method. The different objective functions are combined to a single objective func-tion by global criterion method. Geometric programming technique is used to find the opti-mal solutions for different preferences on objective functions. Finally, the solutionprocedure is illustrated by a numerical example.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

In most of the inventory problems, the unit price of an item is considered as independent in nature. Actually, it relates tothe demand of that item. When the demand of an item is high, it is produced in large numbers. Fixed costs of production arespread over a large number of items. Hence the unit cost of the item decreases, i.e., the unit price of an item inversely relatesto the demand of that item. Cheng [4] formulated the EOQ problem with this idea and solved through GP method. Duffinet al. [5] discussed the basic theories on GP with engineering application in their books. Another famous book on GP andits application appeared in 1976 [2]. They described GP with positive or zero degree of difficulty.

But there may be some problems on GP with negative degree of difficulty. Sinha et al. [11] proposed it theoretically. Usinggeometric programming technique Abou-El-Ata and Kotb [1] studied the multi-item EOQ inventory model with varyingholding cost under two restrictions and solved by geometric programming technique. Sahidul et al. [9,10] discussed modifiedgeometric programming problem and also developed a inventory model solved by modified geometric programmingtechnique.

Similarly the marketing cost which includes the advertisement and promotion cost directly affect the demand of an item.The manufacturing companies increase the advertisement cost and give some advantages (like promotion and incentives) totheir sales representatives according to their performances. Lee and Kim [7] studied the marketing planning problem con-sidering such idea to solve by GP method.

The Society becomes more complex and as the competitive environment develops, businesspersons are finding that theyrequire deeming multiple objectives. Almost every imperative real world problem involves more than one objective. In suchcases, decision makers evaluate best possible approximate solution alternatives according to multiple criteria. Over the lasttwo decades tremendous amount of research effort has been expanded on multi-objective decision making leading to thepublication of many interesting results in the literature [3,6,8].

In this paper, we have formulated a multi-objective marketing planning inventory model under the limitations of spacecapacity and the total allowable shortage cost constraints and solved by global criterion method. The different objective

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S. Islam / Applied Mathematics and Computation 205 (2008) 238–246 239

functions are combined to a single objective function by global criterion method. Geometric programming technique is usedto derive the optimal solutions for different preferences on objective functions and is illustrated by a numerical example.

2. Mathematical model

2.1. Mathematical formulation

The following basic notations and assumptions are used in the proposed model.

NotationsS shortage amount (a decision variable)h inventory carrying cost per item per unit timeD demand rate, units per unit time (a decision variable)Q order quantity, i.e., number of units ordered per order (a decision variable)M marketing expenditure (a decision variable)R amount of stock at time t = 0c0 unit production costc2 shortage cost per itemc3 set-up cost per orderw1 storage space area per unit itemw total available storage areaS1 total allowable shortage cost

Assumptions:

(i) Production is instantaneous,(ii) Demand is uniform,

(iii) The demand rate is directly proportional to the marketing expenditure, i.e., D = aMb, a > 0, b > 0,(iv) The unit production cost is inversely proportional to demand rate, i.e., c0 = cD�d, where c > 0 is a scaling constant and

d > 1 is called price elasticity.

Let the amount of stock is R at time t = 0. In the interval (0, T = t1 + t2), the inventory level gradually decreases to meetdemand. By this process the inventory level reaches zero level at time t1 and then shortages are allowed to occur in the inter-val (t1,T), the cycle then repeats itself. The differential equation for the instantaneous inventory q(t) at time t in (0,T) is givenby

dqðtÞdt¼ �D for 0 6 t 6 T ð2:1:1Þ

with the conditions q(0) = R,q(T) = �S and q(t1) = 0.Solution of the differential equation is given by

qðtÞ ¼ R� Dt for 0 6 t 6 t1

¼ Dðt1 � tÞ for t1 6 t 6 T

So, Dt1 = R, S = Dt2, Q = DT.Holding cost = hc0

R t10 qðtÞdt ¼ hc0ðQ�SÞ2

2Q T.For each period a fixed amount of shortage is allowed and there is a penalty cost c2 per item of unsatisfied demand per

unit time.Shortage cost = c2

R Tt1ð�qðtÞÞdt ¼ c2S2

2Q T.

Production cost = c0Q = ca�dM�bdQ.Advertisement cost = MQ.The total inventory related cost = set up cost + holding cost + shortage cost

¼ hc0ðQ � SÞ2

2QT þ c3 þ c2

S2

2QT:

The total average inventory related cost, i.e., T C1(M,Q,S)

¼ hðQ � SÞ2

2Qca�dM�bd þ c3

aMb

Qþ c2

S2

2Q;

¼ 12

hca�dM�bdQ � hca�dM�bdSþ hca�dM�bdS2

2Qþ c3

aMb

Qþ c2

S2

2Q: ð2:1:2Þ

240 S. Islam / Applied Mathematics and Computation 205 (2008) 238–246

And total additional cost = marketing cost + production cost = MQ + c0 Q.

So; total average additional cost i:e:; TC2ðMÞ ¼ aM1þb þ ca1�dMbð1�dÞ ð2:1:3Þ

Note: When shortages are not allowed, i.e., when c2 ?1 then

TC01ðM;QÞ ¼ 12

hca�dM�bdQ þ c3aMb

Q; ð2:1:4Þ

TC01ðMÞ ¼ aM1þb þ ca1�dMbð1�dÞ: ð2:1:5Þ

The manufacturing organization produces some items and stocks these items in a warehouse. The manufacturing companiesor organizations use huge advertisement for their products in order to increase the level of demand. Still they have somelimitations regarding total space capacity, total allowable shortage cost, etc. In this phenomenon the organization is inter-ested in minimizing the inventory related cost (including setup cost and shortage cost) and additional cost (includes mar-keting cost and production cost) simultaneously.

So, the problem is to minimize total average inventory costs and also to minimize total average of additional cost underthe limitations of space capacity, total allowable shortage cost.

Hence the multi-objective inventory model (MOIM) is as follows:

Min TC1ðM;Q ; SÞ ¼ 12


2Qþ c3

aMb

Qþ c2

S2

2Q

Min TC2ðMÞ ¼ aM1þb þ ca1�dMbð1�dÞ

subject to w1ðQ � SÞ 6 w;

c2S2

2Q6 S1;

M;Q ; S > 0:

ð2:1:6Þ

The above problem can be written as a multi-objective geometric programming problem (MOGPP)



2Qþ c3

aMb

Qþ c2

S2

2Q

Min TC2ðMÞ ¼ aM1þb þ ca1�dMbð1�dÞ

subject tow1

wðQ � SÞ 6 1;

c2S2

2S1Q6 1;

M;Q ; S > 0:

ð2:1:7Þ

3. Mathematical analysis

3.1. Multi-objective geometric programming problem (MOGPP)

A multi-objective geometric programming problem can be stated as

Find x ¼ ðx1; x2; . . . . . . :; xnÞT so as to

Minimize f 1ðxÞ ¼XT0

1

i¼1

c01i

Yn

r¼1

xa0

1irr

Minimize f 2ðxÞ ¼XT0

2

i¼1

c02i

Yn

r¼1

xa0

2irr

. . . . . . . . . : . . . . . . . . . : . . . . . . . . . : . . . . . . . . . : . . . . . . . . . :

Minimize f kðxÞ ¼XT0

k

i¼1

c0ki

Yn

r¼1

xa0

kirr

subject to gpðxÞ ¼XTp

s¼1

cps

Yn

r¼1

xapsrr 6 1; p ¼ 1;2 . . . ;m

x > 0;

ð3:1:1Þ

where c0jið> 0Þ; cksð> 0Þ; a0

jir; ajir are all real numbers for j ¼ 1;2; . . . ; k; i ¼ 1;2; . . . ; T0j ; k ¼ 1;2; . . . ;m; s ¼ 1;2; . . . ; Tk.


Let X be a set of constraints of (3.1.1) such that

X ¼ fx 2 RnjgiðxÞ 6 bj; j ¼ 1;2; . . . ;m; and x ¼ ðx1; x2; . . . ; xnÞT with xi P 0 for i ¼ 1;2; . . . ::;ng

The multi-objective optimization problem is convex if all the objective functions and the feasible region are convex. Somebasic on Pareto optimal solutions are introduced below.

Pareto optimality

In single objective optimization problems, the main focus is on the decision variable space whereas in the multi-objec-tive framework, we are often more interested in the objective space. In multi-objective programming problems, multipleobjectives are usually non-commensurable and cannot be combined into a single objective. In MOGPP, the objectives aresimultaneously optimized. But due to an intrinsic conflicting nature among the objectives it is not possible to find a singlesolution that would be optimal for all the objectives simultaneously. Consequently, the aim in solving MOGPP is to find acompromise or satisfying solution of the DM. There is no natural ordering in the objective space because it is only partiallyordered.

However, some of the objective vectors can be extracted for examination. Such vectors are those where none of the com-ponents can be improved without deterioration to at least one of the other components. This definition is usually calledPareto-optimality, which is laid, by French–Italian economist and sociologist Vilfredo Pareto.

Definition 3.1.1. Let x* be the optimal solution of the following problem:

Minimize f rðxÞr ¼ 1;2; . . . ; k;

subject to x 2 X:

The point x* is known as ideal solution and rth objective function value at x*, i.e., fr(x*) is known as ideal objective value.

Definition 3.1.2. A solution x* is said to be a Complete optimal solution to the MOGPP (3.1.1) if there exists x* 2 X such thatfr(x*) 6 fr(x) (r = 1,2, . . .,k) for all x 2 X.

In general, the objective functions of the MOGPP conflict with each other, a complete optimal solution does not alwaysexist and so Pareto (or non-dominated) optimality concept is introduced.

Definition 3.1.3. A decision vector x* 2 X is Pareto optimal solution if there does not exist another decision vector x 2 X suchthat fr (x) 6 fr(x*) for all r = 1,2, . . .,k and fj(x) – fj(x*) for at least one j, j 2 {1,2, . . .,k}.

Definition 3.1.4. A decision vector x* 2 X is weakly Pareto optimal solution if there does not exist another decision vector x 2 Xsuch that fr(x) < fr(x*) for all r = 1,2, . . . ,k.

Definition 3.1.5. A solution x* 2 X is said to be a locally Pareto optimal solutionto the MOGPP if and only if there exists an r < 0such that x* is Pareto optimal in X \ N(x*,r), i.e., there does not exist another x* 2 X \ N(x*,r) such that fi(x) 6 fi(x*).

Now, we introduce Global criterion method, which have been used in this paper to achieve at least local Pareto optimalsolutions.

3.2. Global criterion method

In this method, the distance between some reference point and the feasible objective region is minimized. The decisionmaker has to select the reference point and the metric for measuring the distances. In this way, the multiple objective func-tions are transferred into a single objective function. We suppose that the weighting coefficients wr are real numbers suchthat wr P 0 "r = 1,2, . . . ,k and

Pkr¼1wr ¼ 1. The weighted Lp-problem for minimizing distances is stated as

Minimize Lpðf ðxÞÞ ¼Xk

r¼1

wr jfrðxÞ � frðx�Þjp !1

p

subject to x 2 X for 1 6 p <1:

ð3:2:1Þ

3.3. Hybrid method

Following Changkong and Haimes [3], the hybrid problem combining Lp and e-constraint method is as follows:


Minimize Lpðf ðxÞÞ ¼Xk

r¼1

wr jfrðxÞ � frðx�Þjp !1

p

subject to f r 6 wr; x 2 X for 1 6 p <1;

where wr P 0 8r ¼ 1;2; . . . ; k;Xk

r¼1

wr ¼ 1;

and erð6 frðx�ÞÞ is a real number for all r ¼ 1;2; . . . ; k:

ð3:3:1Þ

For p ¼ 1; Minimize L1ðf ðxÞÞ ¼Xk

r¼1

wr frðxÞ � frðx�Þj j

subject to x 2 X:

ð3:3:2Þ

The objective function L1(f(x)) is the sum of the weighted deviations, which is to be minimized and the problem (3.3.2) isknown as weighted sum problem:

For p ¼ 2; Minimize L2ðf ðxÞÞ ¼Xk

r¼1

wrjfrðxÞ � frðx�Þj2 !1

2

subject to x 2 X:

ð3:3:3Þ

When p becomes larger, the minimization of the deviation becomes more and more important. The problem (3.3.3) is knownas least square problem.

Finally, when p ?1, the only thing that matters is the weighted relative deviation of one objective function, i.e.,

Minimize L1ðf ðxÞÞ ¼ Maxr¼1;2;...:;k

Xk

r¼1

wr jfrðxÞ � frðx�Þj !

subject to x 2 X:

ð3:3:4Þ

The above method is called ‘min–max’ method or Tchebycheff method. Problem (3.3.4) is non-differentiable like its un-weighted counterpart. Correspondingly, it can be solved in a differentiable form as long as the objective and the constraintfunctions are differentiable and fr(x*) is known globally. In this case, the problem (3.3.4) becomes

Minimize k

subject to wrðfrðxÞ � frðx�ÞÞ 6 k for all r ¼ 1;2; . . . ; k;

x 2 X; k 2 R:

ð3:3:5Þ

Theorem 3.3.1. The solution of weighted Lp-problem (when 1 6 p <1) is Pareto optimal solution if all the weighting coefficientsare positive.

Theorem 3.3.2. The solution of weighted Tchebycheff problem (L1) is weakly Pareto optimal if all the weighting coefficients arepositive.

Theorem 3.3.3. Weighted Tchebycheff problem has at least one Pareto optimal solution.

Theorem 3.3.4. Let a decision vector x* 2 X be given, solve the problem

MinimizeXk

r¼1

frðxÞ

subject to f rðxÞ 6 frðx�Þ for all r ¼ 1;2; . . . ; k;

x P 0:

ð3:3:6Þ

Let ø(x*) be the optimal objective value. The decision vector x* 2 X is Pareto optimal if and only if it is a solution of (3.3.6) so that/ðx�Þ ¼

Pkr¼1frðx�Þ.

Proof. The proofs of the four theorems are followed by Miettinen [8]. h

When fr(x) and (r = 1,2,, . . . ,k) and gj(x) (j = 1,2, . . . ,k) are polynomial and/signomial functions, the problems (3.3.2), (3.3.3)and (3.3.5) may be reduced to a single objective geometric programming problem.


3.4. Global criterion method to solve the MOGPP (2.1.7)

The multi-objective inventory model (MOIM) may be solved by several techniques. Some of those are Fuzzy geometricprogramming technique and global criterion method. Here global criterion is used to find the compromise solution of prob-lem (2.1.7). In this method the objective functions are combined to a single objective function.

Let wr P 0, r = 1,2 are the normalized weights (i.e. w1 + w2 = 1) corresponding to the objective functions TC1(M,Q,S)and TC2(M). TC1 and TC2 are the ideal objective values of TC1(M,Q,S) and TC2(M), respectively (Deductions are shown inAppendix A). TC1 and TC2 are obtained objective functions for TC1(M,Q,S) and TC2(M), respectively, without takingconstraint by using geometric programming (GP) method. The weighted Lp-problem according to Miettinen [8] is asfollows:

Minimize UpðM;Q ; SÞ ¼ ðw1jTC1ðM;Q ; SÞ � TC1jp þw2jTC2ðMÞ � TC2jpÞ1p ð3:4:1Þ

subject to same constraints as in (2.1.7), for 1 6 p�1.Case 1. The weighted sum problem, i.e., for p = 1 in (3.4.1) is given as

Minimize U1ðM;Q ; SÞ ¼ w1ðTC1ðM;Q ; SÞ � TC1Þ þw2ðTC2ðMÞ � TC2Þ

subject to the same constraints as in (2.1.7).Since w1, w2, TC1 and TC2 are independent of the decision variables, so it is enough to solve the following problem:

Minimize V1ðM;Q ; SÞ ¼ w1TC1ðM;Q ; SÞ þw2TC2ðMÞsubject to the same constraints as in ð2:1:7Þ;where U1ðM;Q ; SÞ ¼ V1ðM;Q ; SÞ � ðw1TC1 þw2TC2Þ:

ð3:4:2Þ

The above problem (3.4.2) is a signomial geometric programming problem with DD = 6 and can be easily solved by GPmethod.

Case 2. The least square problem, i.e., for p = 2 in (3.4.1) is given as

Minimize U2ðM;Q ; SÞ ¼ ½w1ðTC1ðM;Q ; SÞ � TC1Þ2 þw2ðTC2ðMÞ � TC2Þ2�12 ð3:4:3Þ

subject to the same constraints as in (2.1.7).To obtain the standard form of GP problem of the above weighted quadratic problem we introduce two new variables y1

and y2 which are the upper bounds of TC1(M,Q,S) � TC1 and TC2(M) � TC2, respectively (i.e TC1(M,Q,S)�TC1 6 y1 andTC2(M) � TC2 6 y2), we may rewrite the above problem (3.4.3) as follows:

Minimize V2ðD;Q ;YÞ ¼ w1y21 þw2y2

2

subject toTC1ðM;Q ; SÞ

TC1� y1

TC16 1;

TC2ðMÞTC2

� y2

TC26 1;

w1

wðQ � SÞ 6 1;

c2S2

2S1Q6 1;

M;Q ; S; y1; y2 > 0;

ð3:4:4Þ

where U2ðM;Q ; SÞ ¼ ðV2ðM;Q ; SÞÞ12.

The problem (3.4.4) is also signomial GP problem with DD = 8 and can be easily solved by GP method.Case 3. The Tchebycheff problem (i.e., for p ?1) is given as

Minimize U3ðM;Q ; SÞ ¼Maximize ðw1jTC1ðM;Q ; SÞ � TC1j;w2jTC2ðMÞ � TC2jÞ ð3:4:5Þ

subject to the same constraints as in (2.1.7).We introduce a new variable k, which is maximum value between

w1ðTC1ðM;Q ; SÞ � TC1Þ and w2ðTC2ðMÞ � TC2Þ; i:e:; w1ðTC1ðM;Q ; SÞ � TC1Þ 6 k and w2ðTC2ðMÞ � TC2Þ 6 k:


Then the above problem (3.4.5) is reduces to the problem as follows:

Table 1Equal p

p

121

Table 2More p

p

121

Table 3More p

p

121

Minimize k

subject toTC1ðM;Q ; SÞ

TC1� k

w1TC16 1;

TC2ðMÞTC2

� kw2TC2

6 1;

w1

wðQ � SÞ 6 1;

c2S2

2S1Q6 1;

M;Q ; S > 0:

ð3:4:6Þ

The above problem (3.4.6) is again a signomial GP problem with DD = 8 and can be easily solved by GP technique.

4. Numerical examples

A multinational manufacturing company produces soft drinks. It is given that the inventory holding cost (h) of the drinksis 25% of the unit production cost, the set-up cost (c3) is $130 and the shortage cost (c2) is $10. The annual demand of thedrinks is varies directly with the marketing expenditure, i.e., D = 10M1.5 and the unit production cost is varies inversely withthe demand, i.e., c0 = 5.05M�1.8. The storage area per unit item is 3 m2. The total available storage area and total allowableshortage cost are w = 225 m2 and S1 = $.085.

The ideal value (as computed in Appendix A) of TC1(M,Q,S) is TC1 = $123.1274 and that of TC2(M) is TC2 = $122.2257. Thecompany decides to know the optimal values of the inventory related cost (TC1(M,Q,S)), additional cost (TC2(M)), marketingcost M*, order quantity Q*, and shortage amount S*.

The above problem can be formulated as multi-objective geometric programming problem. The optimal solutions of thisMOGPP by global criterion method for p = 1, p = 2 and p ?1 are given in Tables 1–3 for different preference values (i.e., dif-ferent weights) of the objective functions.

Table 1 gives different optimal solutions when the DM supplies equal preferences to the inventory related cost objectivefunction TC1(M,Q,S) and additional cost objective function TC2(M). Here, TC�1ðM

�;Q �; S�Þ give minimum value when p ?1where as TC�2ðM

�Þ give minimum value when p = 1.The above Table 2 shows different optimal solutions when the DM supplies more preference to the inventory related cost

objective function TC1 (M,Q,S) than the additional cost objective function TC2(M). Here, TC�1ðM�;Q �; S�Þ give minimum value

when p ?1 whereas TC�2ðM�Þ give minimum value when p = 1.

references to the two objective functions, i.e., for (w1, w2) = (0.5, 0.5)

M* Q* S* TC�1ðM� ;Q�; S�Þ TC�2ðM

�Þ

0.9300382 39.51303 0.5809763 128.0386 123.03280.9196067 37.23608 0.5836205 127.6902 123.63510.8458588 32.01543 0.5797730 127.5276 125.2465

reference to the inventory related cost function, i.e., (w1, w2) = (0.6, 0.4)

M* Q* S* TC�1ðM�;Q�; S�Þ TC�2ðM

�Þ

0.9306990 38.83385 0.5819986 127.8774 123.22920.9123236 36.57991 0.5838419 127.6460 123.79970.8458587 32.01543 0.5797731 127.5276 125.2465

reference to the additional cost objective function, i.e., (w1, w2) = (0.4, 0.6)

M* Q* S* TC�1ðM�;Q�; S�Þ TC�2ðM

�Þ

0.8966431 38.19715 0.5651637 128.8857 122.51620.9293127 38.46130 0.5824761 127.8178 123.32770.8961108 35.32355 0.5837589 127.5862 124.1366


Table 3 shows different optimal solutions when the DM supplies more preference to the additional cost objective functionTC2(M) than the inventory related cost objective function TC1(M,Q,S). Here, TC�1ðM

�;Q �; S�Þ give minimum value when p ?1whereas TC�2ðM

�Þ give minimum value when p = 1.

5. Conclusion

Here, we have discussed multi-objective geometric programming problem based on global criterion method. We havealso formulated the multi-objective inventory optimizations model of economic production and marketing planning prob-lem under two constraints. The different objective functions are combined to a single objective function by global criterionmethod. GP technique is used to derive the optimal solutions for different preferences on objective functions. In Tables 1–3we have shown the optimal solution’s of my problem for different preference values of the objective functions. This tech-nique can be applied to solve the different decision making problems (like in inventory and other areas).

Acknowledgements

The author wish to acknowledge the helpful comments and suggestions of the referee’s. I would like to express my sin-cere gratitude to Dr. Tapan Kumar Roy, Assistant Professor, Department of Mathematics, Bengal Engineering and ScienceUniversity, Shibpur, Howrah-711103.

Appendix A

Working rule for finding the ideal objective values TC01 and TC02,



2Qþ c3

aMb

Qþ c2

S2

2Q

subject to M,Q,S > 0.The above problem is a primal geometric programming problem with DD = 1. The corresponding dual problem is

Maximize dw ¼ hca�d

2w1

� �w1 hca�d

w2

� �w2 hca�d

2w3

� �w3 c3aw4

� �w4 c2

2w5

� �w5

subject to the normality and orthogonality conditions:

w1 �w2 þw3 þw4 þw5 ¼ 1;� bdw1 þ bdw2 � bdw3 þ bw4 ¼ 0;w1 �w3 �w4 �w5 ¼ 0;�w2 þ 2w3 þ 2w5 ¼ 0;w1;w2;w3;w4;w5 > 0:

Solving the dual variables in terms of w3 we get

w1 ¼ w3 þ2d� 1

2d; w2 ¼ 2w3 þ

d� 1d

; w4 ¼12; w5 ¼

d� 12d

:

Since the dual variables are always positive so d > 1. Substituting the values of the above dual variables into the dual objec-tive function and then differentiating with respect to w3 equal to zero, we get w�3 ¼

ðd�1Þ22d .

The values of the other dual variables are

x�1 ¼d2;x�2 ¼ d� 1; x4 ¼

12; x�5 ¼

d� 12d

where d > 1:

Substituting the values of the dual decision variables in the dual objective function of the dual problem we getdw� ¼ ðhca�d

2w�1Þw�1 ðhca�d

w�2Þw�2 ðhca�d

2w�3Þw�3 ðc3a

w�4Þw�4 ð c2

2w�5Þw�5 .

Following Duffin et al. [5] we get the optimum value of the primal objective function as TC�1 ¼ dw�.The decision variables of the primal problem can be obtains from the following primal–dual relationships:

hca�dM�bdQ2w�1

¼ dw�;

c3aMb

Qw�4¼ dw�;

c2S2

2Qw�5¼ dw�:


Solving the above system of non-linear equations, we get

M� ¼ hcc3

2ad�1w�1w�4ðdw�Þ2

! 1bðd�1Þ

;

Q � ¼ c3aw�4dw�

hcc3


! 1d�1

;

S� ¼ 2w�5dw�c3ac2w�4dw�

hcc3


! 1d�1

0@

1A

0:5

:

The ideal objective value TC1 is defined as TC1 ¼ TC�1ðM�;Q �; S�Þ.

In a similar way, we can find the optimal value of the additional cost objective function TC2(M).

Minimize TC2ðMÞ ¼ aM1þb þ ca1�dMbð1�dÞ

subject to M > 0.The above problem is a primal geometric programming problem with DD = 0. The corresponding dual problem is

Maximize dx ¼ ax1

� �x1 ca1�d

x2

� �x2

subject to the normality and orthogonality conditions

x1 þx2 ¼ 1;ðbþ 1Þx1 þ bð1� dÞx2 ¼ 0;x1;x2 > 0:

Solving the system of linear equations of the above dual problem, we get the dual decision variables are

x�1 ¼bðd� 1Þbdþ 1

; x�2 ¼bþ 1bdþ 1

; d > 1:

Substituting the values of the dual decision variables in the dual objective function of the dual problem we getdx� ¼ ð a

x�1Þx�1 ðca1�d

x�2Þx�2 . Following Duffin et al. (1967) we get the optimum value of the primal objective function as TC�2 ¼ dw�.

The decision variables of the primal problem can be obtain from the following primal–dual relationships:

aM1þb

x�1¼ dx�:

Solution of the above equation, we get M� ¼ ðx�1dx�

a Þ1

1þb.The ideal objective value TC2 is defined as TC2 ¼ TC�2ðMÞ:

References

[1] M.O. Abou-El-Ata, K.A.M. Kotb, Multi-item EOQ inventory model with varying holding cost under two restrictions: a geometric programming approach,Production Planning & Control 8 (5) (1997) 608–611.

[2] G.S. Beightler, D.T. Phillips, Applied Geometric Programming, Wiley, New York, 1976.[3] V. Changkong, Y.Y. Haimes, Multi-objective Decision Making, North- Holland Publishing, New York, Amsterdam, 1983.[4] T.C.E. Cheng, An economic production quantity model with demand-dependent unit cost, European Journal of Operational Research 39 (1989)

174–179.[5] R.J. Duffin, E.L. Peterson, C. Zener, Geometric Programming Theory and Applications, Wiley, New York, 1966.[6] M. Ehrogott, Multi-criterion Optimization, Springer, Heidelberg, Berlin, 2005.[7] W.J. Lee, D.S. Kim, Optimal and heuristic decision strategies for integrated production and marketing planning, Decision Sciences 24 (1993) 1203–1213.[8] K.M. Miettinen, Non-linear Multi-objective Optimization, Kluwer’s Academic Publishing, 1999.[9] Sahidul Islam, T.K. Roy, An economic production quantity model with flexibility and reliability consideration and demand dependent unit cost with

negative degree of difficulty: geometric programming approach, Tamsui Oxford Journal of Management Science, Taiwan 20 (1) (2004) 01–17.[10] Sahidul Islam, T.K. Roy, Modified geometric programming problem and its applications, Journal of Applied Mathematics and Computing, Korea 17

(1–2) (2005) 121–144.[11] S.B. Sinha, A. Biswas, M.P. Biswal, Geometric programming problems with negative degree of difficulty, European Journal of Operational Research 28

(1987) 101–103.

Multi-objective marketing planning inventory model: A geometric programming approach

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