# Joint pricing and production management: a geometric programming approach with consideration of cubic production cost function

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## Abstract

Coordination and harmony between different departments of a company can be an important factor in achieving competitive advantage if the company corrects alignment between strategies of different departments. This paper presents an integrated decision model based on recent advances of geometric programming technique. The demand of a product considers as a power function of factors such as product's price, marketing expenditures, and consumer service expenditures. Furthermore, production cost considers as a cubic power function of outputs. The model will be solved by recent advances in convex optimization tools. Finally, the solution procedure is illustrated by numerical example.

## Keywords

Geometric programming Production management Cubic cost production function Jointness## Introduction

Based on the literature in this area, the GP method is an excellent method to solve a typical algebraic nonlinear optimization (Duffin et al. 1967; Boyd et al. 2007). One of the notable properties of GP is that new solution methods can solve even large-scale GPs extremely efficiently and reliably (Boyd et al. 2007).

Literature review summary on application of geometric programming

Author (s) | DD | DV | Power functional relations | Reliability | |||||
---|---|---|---|---|---|---|---|---|---|

Production (purchase) cost | Inventory holding cost | Demand | |||||||

Function | Fixe | Function | Fixe | Function | Fixe | ||||

Cheng (1991) | 0 | | \( C = BD^{ - \chi } r^{\kappa } \) | * | | | |||

Lee (1993) | 1 | | \( C = BQ^{ - \sigma } \) | \( h = i_{h} C \) | \( D = kp^{ - \alpha } \) | ||||

Lee and Kim (1993) | 0 | | \( C = BD^{ - \chi } \) | \( h = i_{h} C \) | \( D = kp^{ - \alpha } M^{\beta } \) | | |||

Hariri et al. (1995) | 2N-1 | | * | * | | ||||

Lee et al. (1996) | 0 | | \( C = BD^{ - \chi } \) | * | \( D = kp^{ - \alpha } \) | | |||

3 | |||||||||

Kim and Lee (1998) | 2 | | * | \( h = i_{h} C \) | \( D = kp^{ - \alpha } \) | ||||

4 | |||||||||

Chen (2000) | 1 | | \( P = Br^{ - \vartheta } p^{\alpha } \) | * | | | |||

Jung and Klein (2001) | 0 | | \( C = BD^{ - \chi } \) | \( h = i_{h} C \) | * | ||||

1 | | * | |||||||

Abuo-El-Ata et al. (2003) | 4N-1 | | * | * | random | ||||

Sadjadi et al. (2005) | 1 | | \( C = BQ^{ - \sigma } \) | \( h = i_{h} C \) | \( D = kp^{ - \alpha } M^{\beta } \) | ||||

Jung and Klein (2005) | 0 | | \( C = BD^{ - \chi } \) | \( h = i_{h} C \) | * | ||||

1 | | \( C = BQ^{ - \sigma } \) | \( h = i_{h} C \) | * | |||||

1 | | \( C = BQ^{ - \sigma } D^{ - \chi } \) | \( h = i_{h} C \) | * | |||||

Liu (2006) | 1 | | \( C = BQ^{ - \sigma } \) | * | \( D = kp^{ - \alpha } \) | ||||

Mandal and Roy (2006) | 3N-1 | | Fuzzy | Hybrid | * | * | |||

4N-1 | | Fuzzy | Hybrid | * | * | ||||

Jung and Klein (2006) | 1 | | \( C = BD^{ - \chi } \) | * | * | ||||

1 | | * | * | ||||||

1 | | \( C = BQ^{ - \sigma } D^{ - \chi } \) | * | * | |||||

Leung (2007) | 0 | | * | * | * | * | |||

Islam (2008) | 1 | | \( C = BD^{ - \chi } \) | \( h = i_{h} C \) | \( D = kM^{\beta } \) | ||||

Fathian et al. (2009) | 0 | | \( C = BD^{ - \chi } \) | * | \( D = kp^{ - \alpha } M^{\beta } S^{\tau } \) | ||||

Ghazi Nezami (2009) | 4 | | \( C = BQ^{ - \sigma } \) | \( h = i_{h} C \) | \( D = kp^{ - \alpha } M^{\beta } \) | ||||

Sadjadi et al. (2012) | 6 | | \( C = BQ^{ - \sigma } q^{\varphi } r^{\vartheta } \) | \( h = i_{h} C \) | \( D = kp^{ - \alpha } \prod\limits_{i = 1}^{I} {M^{\beta } q^{\theta } } \) | * | |||

Kotb and Fergany (2011) | 3N-1 | | \( C = BD^{ - \chi } \) | * | |||||

Ghosh and Roy (2013) | 0 | | \( C = BQ^{ - \sigma } \) | \( h = i_{h} C \) | \( D = kp^{ - \alpha } M^{\beta } \) |

Most of the GP applications above are posynomial GP with zero or a few degrees of difficulty. The degree of difficulty is the difference between the number of dual variables and the number of independent linear equations; and the greater the degrees of difficulty, the more difficult would be the solution (Creese 2010). GP requires that the expressions used are posynomials (Creese 2010; Boyd et al. 2007). In this study, an integrated model with the consideration of demand, cost, marketing and services given to customer is proposed, where total variable cost is a cubic power function of output. In fact, the current analysis goes beyond the existing literature, so that first, it introduces a significant innovation to the field by considering a new form of cost function in the context of production management. Also, we have developed a model in this paper to fill this gap in the literature by using a novel methodology in the environment of GP to get optimal solutions for the signomial models based on a transformation of these models to standard posynomial GP by using the concepts behind the relations between geometric and arithmetic means, which allows us to solve problem with high degree of difficulty (Ghazi Nezami et al. 2009; Duffin et al. 1967). And also unlike the conventional approach, the standard GP model is solved by recent advances in optimization tools (Boyd and Michael 2009). The reminder of this paper is organized as follows. In the next section, the necessary notations, assumptions and the model formulation are presented. The model formulation is discussed in “Mathematical formulation”. In “Solution approach”, the signomial GP model transformed to a standard posynomial GP model. In “A numerical example”, a numerical example is solved in order to show the implementation of the algorithm and analyze our model’s parameters behavior. Finally, a conclusion is drawn in “Conclusion”.

## Model formulation

To illustrate the idea proposed in this paper, consider a company who produces a single product using a given production process. Due to growing company’s size and escalating market competition, the company directors are trying to increase their share of the market demand. To this end, they have several options to choose generally. On the one hand, they can propel the investments outward their company, so they promote their marketing and customer service capability. In other words, in order to meet better requirements of the new fierce business environment, they can make investments to increase flexibility and reliability of its production process. They can also choose a combination of the overall strategy. In this condition, the manufacturer must make several decisions simultaneously, so that it leads to increasing company profit.

### Notations and assumptions

| Annual demand |

| Production cost per unit |

| Inventory holding cost rate (%/per unit time) (0 < |

| Storage area requirement for each item |

| Available floor/storage area |

| Total budget available to the marketing methods and customer service level |

| Maintenance costs per production cycle |

| Total cost of interest and depreciation for a production process per production cycle |

| Percent of total market demand that should be covered by manufacturer |

| Total market demand |

| Total resources available |

| Resource requirements for each item |

| The goal associated to number of production cycles |

| The upper limit on the reliability of the production process |

| The lower limit on the reliability of the production process |

Decision variables | |

| Selling price per unit |

| Volume of investments in marketing method |

| Volume of investments in customer service strategy |

| Economic production quantity |

| Production process reliability, in other word percent of non-defective items in a batch |

| Setup cost (representing process flexibility) |

Assumptions | |

I | Replenishment is instantaneous |

II | No excess stock is carried, and no shortage and lost sales are allowed |

III | The production quantity is produced in batches (lots) |

IV | All batches are subject to a 100 % inspection policy and all defective items are discarded |

In addition, the following power function relations are defined for the model:

- 1.In current model, the demand per unit time is described as a decreasing power function of price per unit, and marketing expenditures in various approaches, and customer service expenditures in different scenario act as increasing power functions according to the following equation:$$ D \, (p,M_{j} ,S_{l} ) = kp^{ - \alpha } \mathop \prod \limits_{j = 1}^{J} M_{j}^{{\beta_{j} }} \prod\limits_{l = 1}^{L} {S_{l}^{{\tau_{l} }} } ,\,\,\,{\text{where}}\,\,k\,\,{\text{and}}\,\alpha > \,0\,\,{\text{and}}\,\,0 < \beta_{j} ,\tau_{l} < 1 $$(1)
In actuality, Eq. 1 indicates that when selling price increases, the demand per unit time decreases. On the other hand, when marketing and customer service expenditures increase, the demand per unit time increases. Also, requiring

*k*> 0 is an obvious condition since*D*must be nonnegative. - 2.
The total variable cost (

*C*_{2}*Q*) considered as a cubic power function of output. Generally, a cost function a function that shows the relation between the magnitude of cost and of output. The existence of such a function is postulated upon the following assumptions: (1) there is a fixed body of plant and equipment; (2) the prices of input factors such as wage rates and raw material prices remain constant; (3) no changes occur in the skill of the workers, managerial efficiency, or in the technical methods of production.

Money expenses of production depend upon the prices and quantities of the factors of production used. Since prices are assumed to remain unchanged, the shape of the cost function will be determined by the physical quantities of the factors used up at different levels of operation.

And since these quantities are functionally related to output, their relation to cost can be represented by a cost-output function. Thus, the underlying determinant of cost behavior is the pattern of change in the factor ingredients as output varies.

*S*curve; from Fig. 2, it can be seen that how the total cost curve first increases gradually and then rapidly. Hence, by considering Eq. 2 as the equation of the total variable cost curve form explained above,

*e*

_{1},

*e*

_{3}> 0 and

*e*

_{2}< 0 (Gujarati 2003).

- 3.Total cost of interest and depreciation per production cycle is inversely related to a setup cost and directly related to production process reliability according to the following equation:$$ E(C_{1} ,r) = \,dC_{1}^{ - \delta } r^{\theta } ,\,\,\,{\text{where}}\,\,d,\,\delta ,\,\theta \ge 0 $$(3)
In the equation,

*b*is a scaling constant, \( \delta \) is setup elasticity to interest and depreciation cost, and \( \theta \) is production process reliability elasticity to interest and depreciation cost.

## Mathematical formulation

*q*(

*t*) is inventory level in time \( \left( {q(t) = \left\{ \begin{gathered} rQ\,\,\,\,\,\,\,{\text{at}}\,\,t = 0 \hfill \\ 0\,\,\,\,\,\,\,\,\,\,\,{\text{at}}\,t = \,T \hfill \\ \end{gathered} \right.} \right) \) and \( T = \frac{rQ}{D} \), is the cycle length. This model is similar to those proposed by Panda and Maiti (2009).

But it is quite obvious that manufacture to increase their profits is faced with a series of restrictions including:

## Solution approach

*v*

_{ n }are positive numbers and

*k*

_{ n }are nonnegative weights which their summation must equal to one, \( \left( {\sum\limits_{n = 1}^{N} {\kappa_{n} = 1} } \right) \), in this equation if taking

*κ*

_{ n }·

*v*

_{ n }≡

*u*

_{ n }in mind, result will be as follows:

*U*

_{1}and

*U*

_{2}.

*Z*).

## A numerical example

\( k = 11 \times 10^{9} , \) | \( \alpha = 2, \) | \( \beta_{1} = 0.0086, \) | \( \beta_{2} = 0.0057, \) | \( \tau_{1} = 0.0053, \) |

\( \tau_{2} = 0.0083, \) | \( e_{1} = 0.03, \) | \( e_{2} = 0.452, \) | \( e_{3} = 2. 7 0 3, \) | \( h = 0.15, \) |

\( t = 55, \) | \( P_{M} = 5 \times 10^{3} , \) | \( \psi = 0.22, \) | \( B = 187000, \) | \( d = 50, \) |

\( H = 0.93, \) | \( L = 0.71, \) | \( \delta = 1.5, \) | \( \theta = 1, \) | \( \gamma = 0.2, \) |

\( a = 1000, \) | \( f = 10, \) | \( F = 1900, \) | \( b = 4, \) | \( R = 1200, \) |

Thus, the posynomial GP model according to the above parameters is a very difficult GP problem (degree difficulty = 14).

Intrinsically, it is complex to solve the model by solution procedures discussed in literature for a GP (Duffin et al. 1967). Hence, in order to find the optimal solution of the problem a MATLAB-based modeling system (CVX) (Boyd and Michael 2009) is used.

\( Z = \$ 1 1 , 1 0 3 , 4 3 6\,\, \) | \( Q^{ * } = 2 2 0, \) | \( p^{ * } = \$ 1 1 8 9.3\,, \) | \( M_{1}^{ * } = \$ 4 8 0 6 8 , { } \) |

\( M_{2}^{ * } = \$ 3 1 8 7 4. 7 5\,, \) | \( S_{1}^{ * } = \$ 2 9 6 4 0. 5 3\,, \) | \( S_{2}^{ * } = \$ 4 6 3 9 2. 9 6\,, \) | \( r^{ * } = 0. 8 6, \) |

\( C_{1}^{ * } = \,\$ 5. 2 9\,, \) |

### Sensitivity analysis

In this sub-section, due to high degrees of nonlinearities of the model, the behavior of the decision variables in situation optimal with regard to changes in some parameters of the model are analyzed. In other words, the study discuses some managerial insights by studying how the optimal solution would vary as the inputs values change.

#### Effects of changes in *α* on optimal solution

*α*

_{0}= 1.8, and

*n*= 0,…, 9. According to, Figs 3, 4, 5, 6, respectively, reveal that as price elasticity of demand increases, the manufacturer has to decrease its product price with proper intensity in order to maintain the acceptable market share (Fig. 3). As a result of the reduced optimal selling price, manufacturer’s profit decreases as well (Fig. 4), so that, the manufacturer has to decrease marketing and customer service expenditures in order to escape from the loss limits (Fig. 5). Also, the manufacturer in order to minimize additional cost of production (i.e., inventory holding cost) and maintenance costs need to reduce lot-size (Fig. 6) and increases reliability of process (Fig. 7). Generally, when the lot-size decreases, setup cost will increase as well (Fig. 8).

#### Effects of changes in *β* _{ j } and *τ* _{ l } on the optimal solution

This sub-section organized as follows. First, we investigate the effect of increasing in values of *β* _{ j } (*j* = 1, 2) and *τ* _{ l } (*l* = 1, 2) on the optimal solution. Second, we investigate the effect of increasing in values *β* _{ j }(j = 1, 2) and *τ* _{ l } (*l* = 1, 2), while simultaneously, *α* will also increases. Finally, we investigate a gradual increases in both *β* _{2} and *τ* _{1} while decreases *β* _{1} and *τ* _{2} simultaneously such that *β* _{ j } + *τ* _{ l } = constant.

##### Increasing both *β* _{ j } (*j* = 1, 2) and *τ* _{ l } (*l* = 1, 2)

*β*

_{ j }(

*j*= 1, 2) and

*τ*

_{ l }(

*l*= 1, 2) are examined as follows: \( \beta_{j} (\tau_{l} ) = \beta_{0j} (\tau_{0l} ) + (0.0008) \times n \) where \( \beta_{01} = 0.0066 \), \( \tau_{01} = 0.0033 \) \( \beta_{02} = 0.0037 \), \( \tau_{02} = 0.0063 \), and \( n = 0, \, 1, \ldots ,9 \). According to Figs. 9 and 10, it is seen that when

*β*

_{ j }(

*j*= 1, 2) and

*τ*

_{ l }(

*l*= 1, 2) increase simultaneously, it results in increasing price of product. Therefore, price increase results in increasing profit. This is because of the rise in marketing and customer service elasticity of demand, the marketing and customer service expenditures are increasing in response to customers’ requests as well (Fig. 11). As a result, the company that gain more profit, has to increase its product price.

##### Increasing *β* _{ j } (*j* = 1, 2), *τ* _{ l } (*l* = 1, 2) and *α*

The effect of changes in *β* _{ j } (*j* = 1, 2)*,τ* _{ l } (*l* = 1, 2) and *α* on the optimal values of decision variables are discussed in this part. The values of *β* _{ j } (*j* = 1, 2), *τ* _{ l } (*l* = 1, 2) are increased according to the equation *β* _{ j } (*τ* _{ l }) = *β* _{0j } (*τ* _{0l }) + (0.0008 × *n*), while simultaneously, \( \alpha \)is increased according to \( \alpha_{k} = \alpha_{0} + (n \times 0.02), \) for \( n = 0, \ldots ,9 \) such that \( \beta_{01} = 0.0066 \), \( \tau_{01} = 0.0033, \)

*β*

_{ j }(

*j*= 1, 2),

*τ*

_{ l }(

*l*= 1, 2) and

*α*, the optimal selling price and total profit decreases subsequently. This is because of company response to combination results of changes in

*β*

_{ j }(

*j*= 1, 2),

*τ*

_{ l }(

*l*= 1, 2) and

*α*[i.e., where effects of changes

*α*overcome on the effects of changes in

*β*

_{ j }(

*j*= 1, 2) and

*τ*

_{ l }(

*l*= 1, 2)]. Also, Fig. 14 shows that the company should not be oblivious to the effects of changes in values of marketing and customer service elasticity, and in order to minimize the total cost, the company has to decrease the marketing expenditures through marketing method 1 and also decrease the customer’s service expenditures through customer’s service type 2. On the other hand, in order to respond to the needs of different customer segments, the company has to increase the marketing expenditures through marketing method 1, and also increase the customer’s service expenditures through customer‘s service type 2.

*β*

_{ j }(

*j*= 1, 2),

*τ*

_{ l }(

*l*= 1, 2) and

*α*, the lot-size decreases and the process reliability increases. Generally, when that the effect of change in

*α*overcomes the effects of changes in

*β*

_{ j }(

*j*= 1, 2) and

*τ*

_{ l }(

*l*= 1, 2), it can be concluded that customers are highly sensitive to the price of product and company cannot just retain its customer through marketing and offering services. Therefore, as explained in the previous section, the company has to decrease its product price in order to keep its customer. On the other hand, by increasing process reliability and reducing lot-size, the company has to reduce its total cost such as maintenance costs and inventory holding cost in order to escape from the limit of losses.

##### Increasing *β* _{2} and *τ* _{1} while decreasing *β* _{1} and *τ* _{2}

*β*

_{1}and

*τ*

_{1}are calculated according to the equation

*β*

_{1}(

*τ*

_{ 1 }) =

*β*

_{01}(

*τ*

_{01}) ± (0.00033 ×

*n*), while simultaneously,

*β*

_{2}and

*τ*

_{2}are calculated according to

*β*

_{2}(

*τ*

_{2}) =

*β*

_{02}(

*τ*

_{02}) ± (0.00026 ×

*n*), where

*β*

_{01}= 0.0066, \( \tau_{01} = 0.0033, \), \( \beta_{02} = 0.0037 \), \( \tau_{02} = 0.0063 \) and \( n = 0, \ldots ,9 \). According to Figs. 17, 18, 19, as

*β*

_{ j }(

*j*= 1, 2) and

*τ*

_{ l }(

*l*= 1, 2) get closer to each other (i.e.,

*n*= 5), the product price, and total annual profit decreases. This is mainly due to the fact that when different methods of marketing and various types of customer’s service are equally important from the customer’s perspective, increasing investment on all options are useless. From this point of view, it is advisable to invest in a diverse range of marketing methods and customer service types in order to efficient use of limited budget.

#### Effect of changes in *γ* on optimal solution

The effects of changes in reliability elasticity of maintenance cost on the optimal solution are investigated by considering different values for *γ* as follow: *γ* = 0.2 + (0.05 × *n*), for \( n = 0, \ldots ,9 \).

#### Effects of changes in *b* on optimal solution

The effects of changes in *b* on the optimal solution are investigated by considering different values for *b* as follows: \( b = 4 + (1 \times n) \), for \( n = 0, \ldots ,9 \).

*b*leads to a higher optimal selling price, lower total profit, lower optimal marketing and customer service expenditures, higher process reliability, smaller optimal production lot-size, lower optimal setup costs per production cycle. The implication of these behaviors is that the company has to decrease additional cost in order to prevent of decreasing profit (Fig. 30).

## Conclusions

To cope with the issue of integrating the marketing and production strategies for a company that produces a single product, an integrated model with using the GP technique is proposed in this paper. In this paper, the production cost is considered as a cubic power function of outputs. The model is restricted with available limited storage space constraint for acceptable products, available limited budget for marketing and service quality, and necessary to achieve a minimum market share is planned. Also, in this paper the defective items are considered, so that only %*r* of products can be good to be used. According to the descriptions, the resulting GP model is very hard, with the 14 degrees of difficulty, so it is very complex to solve the model intrinsically.

To the best of our knowledge, this research is one of the primary works using the concepts behind the relations between geometric–arithmetic means and CVX modeling system for solving type of models with a high degree of difficulty, hence the literature considering this approach in production planning is still scarce. Therefore, many possible future research avenues can be defined in this context, and will be solved using the toolbox and concepts.

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