Optimum procurement plan for assembled computers on anniversary sale: the case of small businesses.

Erinjeri, Jinson J. ; Mesak, Hani I. ; Ker, Jun-Ing 等

ABSTRACT

The present article introduces, formulates and solves an anniversary sale problem for the first time in the literature. The problem is formulated as an integer linear program to select the best combination of suppliers from whom to purchase several types of components to assemble units of a computer system. Factors which need to be accounted for when selecting vendors include varying purchase prices, minimum purchase requirements, shipping considerations, demand requirements and operations circumstances unique to small businesses. The proposed approach when applied to the situation of a local computer store located in the Northeastern Louisiana region of the U.S. is found to be beneficial in (1) determining the least cost budget for the anniversary sale, (2) identifying the optimum procurement plan, and (3) enhancing the competitive edge of the store through the reduction of the selling price of assembled computers on sale. The approach implemented in this article to solve the anniversary sale problem promotes the use of simple operational research tools supplemented by managerial judgement when it comes to confronting problems related to small businesses.

INTRODUCTION

Computer prices throughout the world are declining and there is stiff competition with the advent of online shopping in the US. Reputable companies like Dell, Gateway, Compaq, and HP are selling computers online directly to customers at low prices especially on special deals (Mainelli, 2001). In fact it has been estimated that in the year 2002, about 31% of the 35 million desktop PCs sold in the United States were white boxes cobbled together by small vendors whereas the remaining were sold by branded companies (Ray, 2003). All branded companies outsource the various parts, assemble them and sell the final assembled product to the customers. Local privately--owned computer stores do the same assembly but not at a large scale, as the companies mentioned above. The average retail-selling price of desktop computers has fallen since January 2001 (Baker, 2002). The advent of online shopping and e-transaction has further dropped the prices of computer systems. Also, the shift in purchaser choice to notebook (laptop) computers creates a fierce competitive environment in which local stores selling desktop computers operate. Therefore, local computer stores will have to cut down costs in order to compete with market prices and each dollar saved becomes crucial with such stiff competition. The problem faced by the manager of a privately-owned local store is how to order the various parts from a pool of potential vendors with the least possible cost. Selecting the most appropriate suppliers is considered an important strategic management decision that may impact all areas of a small business enterprise. Purchases from suppliers account for a large percentage of total costs for many firms (Jayaraman, Srivastava & Benton, 1999). Also, vendor selection is influenced by a variety of factors. For example, Dickson (1966) mentions 23 different criteria considered in vendor selection. Among these the price, availability, delivery, buyer-vendor relationship, and quality objectives of the buyer are particularly important factors in deciding how much to order from potential vendors.

The present article addresses the problem of selecting vendors for assembling computers on an anniversary sale related to a small business establishment. The importance of studying problems facing small enterprises is evident from the fact that they constitute a critical component of the U.S. economy. The U.S. Small Business Administration generally defines small businesses as those firms that employ fewer than 500 employees. In 1998 they represented more than 99% of all employers and employed 52% of all private workers. During the five-year period from 1992 to 1996, small businesses created about 11.8 million new jobs while large firms (those with 500 or more employees) actually lost about 640,000 jobs. The U.S. economy witnessed about a 6.5% rise in small businesses during the 1988-1994 period. According to the Federal Bureau of Labor Statistics, small firms are expected to generate nearly 60% of new jobs between 1995 and 2005 (Ahire, 2001).

ANNIVERSARY SALES

Numerous retailers seek to increase sales revenues by having anniversary sales. An anniversary sale is a sales promotion celebrating a store's anniversary. Retailers such as IKEA, JC Penney, Nordstrom, and Viking in the U.S. use their anniversaries to gain customer attention. Typically, retailers expect increased sales revenue as a result of an anniversary sale.

Anniversary sales are commonly used by retail establishments to promote new merchandize while attracting customers by discounting older merchandize. Nordstrom is one of the many retailers that hope to gain customer attention for new merchandize, as well as sell out of season clearance items. Nordstrom sells heavily discounted items, such as footwear when it has annual anniversary sale (Burton, 2000). It relies on this annual occurrence to boost revenues (Brown, 2000).

Anniversary sales also aid retailers in attracting consumer attention during non-peak times of the year. Pillowtex Corporation implemented an annual anniversary sale for its Cannon brand of bedding, towels, and blankets. Because the month of February was an "uneventful" period for the company, the vice president of advertising designed an ad campaign to attract attention for the anniversary sale. His campaign goal was for consumers to associate the month of February with the Cannon Anniversary Sale (Wolf, 1999).

In honor of its 85th anniversary, Frigidaire will showcase its new logo, as well as its Electrolux line by conducting an anniversary sale. Frigidaire's campaign goal differs from that Pillowtex. Frigidaire is not only seeking consumer attention, but is also searching for dealerships to carry its products. The sale is planned for September through November of 2003. The anniversary sale will be set in motion by a gala in New York where the company will honor its 10 longest standing dealers (Greenburg, 2003).

Consumers will not only be able to shop cheaper as a result of anniversary sales, but also will be able to fly cheaper. In the past, Southwest airlines have enhanced its competitiveness by conducting anniversary sales (Shifrin, 1997). Sales promotions provide airlines with a way to stimulate customers to fly during non-peak times by offering them incentives in the form of deeply discounted airline tickets. When American Trans Air observed its 25th anniversary, it offered tickets for as low as $30 (Aviation Week and Space Technology, 2002).

Celebrating an anniversary sale offers companies a variety of benefits. Although numerous retailers and airlines are conducting anniversary sales, they are not necessarily having this type of sales promotion for the same reasons. One retailer may promote new lines of products, while an airline may encourage travelers to fly. No matter the reason for a sales promotion, the bottom line is that it offers a way for companies to boost revenues. However, retailers and companies in the airline industry are not alone in the arena of anniversary sales; computer stores also have anniversary sales.

To purchase the various parts for assembling a computer on an anniversary sale, it is important for the store manager to source parts cost effectively. Also, it is important to note that all the vendors have varying prices of components, shipping costs and the manager of that small enterprise needs to take into account constraints that are unique to this line of business to secure the required components optimally for the special sale.

LITERATURE HIGHLIGHTS

The problem of selecting vendors or suppliers for buying components for assembling computers for an anniversary sale in small business establishments has not been addressed before in the literature to the best knowledge of the authors. As a matter of fact, the authors could not find any article related to the broad subject of "anniversary sales" published in any academic journal! The authors of this article, however, have benefited a great deal from ideas related to studies published on the "vendor selection problem" in formulating their "anniversary sale problem". In this regard, we review below some notable vendor selection studies that are relevant to the scope of our study for the purpose of highlighting the main aspects of similarity as well as discrepancy between their modeling efforts and ours. We show in this article that while the vendor selection problem is basically unifunctional, the anniversary sale problem is multifunctional in nature. To keep the cost of purchasing component parts to a minimum, the firm must decide which vendors to do business with, and what order of quantities to place with each vendor. Weber and Current (1993) refer to this pair of decisions as the "vendor selection problem". Degraeve, Labro and Roodhooft (2000) and Current and Weber (1994) provide excellent reviews of mathematical programming approaches to analyze such decisions.

Jayaraman, Srivastava and Benton (1999) developed a mixed integer-programming model where a buyer may purchase the same product(s) from more than one supplier. If the volume is large enough, demand requirements are split among several suppliers. The model developed by those authors makes allocations based on demand, taking into account quality requirements, restrictions in storage and production capacity and production lead-time. While our formulated problem shares the same objective of minimizing purchase costs, the model is developed using pure integer programming with assumptions and aims different from the model developed above.

Rosenthal, Zydiak and Chaudhry (1995) have studied a variety of bundling scenarios, where selection of a supplier is based on demand requirements and the type of bundling such as pure bundling (buy one get one free), mixed bundling (per unit discounts) and the generalization of these two types of bundling scenarios. A study by Van Buer, Enrique and Zydiak (1997) reflects the cost effective way of purchasing from multiple vendors as the pricing of each item in the bundle is dependent on the pricing of other items in the bundle. The main aim of the articles cited above is to select the best vendors with regards to cost effectiveness and other related considerations. The focus of our article is not on how to purchase different items in a bundle. Rather, the quantity to be assembled for an anniversary sale is forecasted and the purchase orders of different components are placed in a cost effective way considering a unique set of constraints governing the operations of business. The number of components to be purchased is not as voluminous as in the previous articles to warrant consideration of price discounts.

More recently, Degraeve and Roodhooft (2000) developed a multi-period, multi-item, multi-vendor mathematical optimization program leading to the minimization of total costs associated with the purchasing strategy subject to eight types of constraints. Our model, in contrast, is much different as it serves different purposes. It is a single period as the anniversary sale lasts for only one month in a given year, and the ordering of various components takes place almost at the same time. For an operations research model to be implementable by small businesses, it is often recommended that the model be as simple as possible and in the mean time captures the basic aspects of the real situation (Ahire, 2001).

In the next section, the anniversary sale problem facing a small business that selects computer assembly as its main line of business is highlighted and formulated as an integer programming problem. In the third section, the approach presented in the second section is applied for the situation of a computer store manager who aims at securing different components required to assemble certain number of forecasted computers to be sold on the special sale. The detailed structure of the operationalized mathematical model is relegated to an appendix. The fourth section summarizes and concludes the paper.

FORMULATION OF AN ANNIVERSARY SALE PROBLEM

The store manager has to secure for its anniversary sale the required components to assemble a certain quantity of desktop computers from a number of potential vendors. The vendors were short listed to five for this study. Choosing vendors was based on a review of the prices offered for the various components and the present ordering policy at the local store.

The manager of the computer store is faced with the task of purchasing hardware and software components (a total of eighteen components) from five different vendors in the most effective way. The payment and shipping options of various vendors are shown in Table 1. Table 1 show that transactions with most vendors can be made through the cash on delivery (COD) method of payment, except for vendors 4 and 5. Transactions with vendors 4 and 5 can be made through credit card and therefore it does not include these additional charges. Despite the associated charges, the method of payment preferred by the store is COD. The manager of the local store had a maximum available fund for credit card transactions of approximately $1,000.00.

Table 1 also indicates that, with the exception of vendor 5, all vendors ship the parts by ground United Postal Service (UPS). There are no shipping costs associated with vendor 5 because one of the employees resides in the vicinity of the location of such vendor, and he picks up the parts as needed. Shipping charges from various vendors may include an additional charge of about $8.00 per order if the method of payment is cash on delivery (COD). The cost incurred due to shipping is considered proportional to the weight of the goods. However, it is important to note that shipping rates vary between various zones. The anniversary sale lasts for a period of one month and therefore it is the management policy of the local store to ship the key items from vendors who are located in the same or near by zones as demarcated by UPS. The key items are those hardware components, which have limited warranty and are susceptible to damage. The main reason for this is the relatively long time taken to process the replacement of a failed component from the vendor. The key hardware components include Hard Drive, CPU and Cases. Dictated basically by shipping cost considerations, it is the company policy to secure the majority of the requirements related to these components from vendors in zones 7 and 8 only. Table 1 also shows the vendors and their related zones within their respective States.

The quality aspect of the components is a default factor in the modern day as all the parts are under warranty for a period of one year. This warranty is applicable to customers who buy a computer on a special sale. In addition, there is ninety days warranty on labor too. In this regard, there is a certain amount of a particular branded component (300 Watts Case) that has to be purchased from a certain vendor (Vendor (1)), regardless of price to maintain an ongoing good buyer-vendor relationship. Purchases from vendor 5 include only 17" Monitors and Windows XP operating Systems. Also, there are some vendors from whom a minimum purchase has to be made from each; otherwise additional charges have to be paid.

The total budget to be determined for the anniversary sale includes advertising and shipping costs. In addition, COD charges need to be added depending on the purchases made from vendors. Because the additional costs mentioned above had never exceeded 8% of anniversary sale budgets in the past, minimization of the purchase cost of components appears to be an appropriate objective function in this case. The related constraints for the above situation are enumerated below:

1 Constraints to meet the demand requirements of each component.

2 Constraints to satisfy the minimum purchase requirements for certain vendors.

3 Constraints to purchase certain branded parts from a specific vendor and to restrict purchases of certain items from vendors due to their unavailability.

4 Constraints to meet the zone criteria of the shipment policy.

5 A constraint to satisfy the budget available for purchases on credit.

The approach followed to solve the problem highlighted above starts with forecasting the number of desktop computers anticipated to be sold on the store's anniversary sale (an integer number). Second, the number of each component to be purchased from different vendors to assemble the forecasted number of computers is then determined. Finally, a mathematical program is formulated to determine the optimum number of components to purchase from each vendor, and hence the related budget for the event. For the proposed approach to be implementable by concerned small businesses, an access to small computers is required. The OR techniques would be simple and the related software should be made as user friendly as possible (Dickson, 1966).

MODEL STRUCTURE

The cost minimization problem is structured using the integer variables Xij's. The decision variable Xij represents the number of ith component to be purchased from the jth vendor. The cost coefficient Cij (sales tax included) represents the price associated with the ith component purchased from the jth vendor. Based on the objective function and the associated constraints mentioned earlier, the pure integer program takes the mathematical structure shown below.

Minimize [[SIGMA].sub.i] [[SIGMA].sub.j] [C.sub.ij] [X.sub.ij]

Subject to:

1 [[SIGMA].sub.j] [X.sub.ij] = [d.sub.i] where [d.sub.i] is the demand requirement for component i.

2 [[SIGMA].sub.i] [[SIGMA].sub.j] [C.sub.ij] [X.sub.ij] [greater than or equal to] [m.sub.j] for some j, where [m.sub.j] represents the minimum amount of $ purchases to be made from vendor j.

3 [X.sub.ij] = [n.sub.ij] where [n.sub.ij] represents a predetermined number of [i.sup.th] components to be purchased from a certain j vendor.

4 [[summation].sub.j=1-3] [X.sub.2j] [greater than or equal to] [k.sub.1] (Zone constraint for the Case component)

5 [[summation].sub.j=1-3] [X.sub.2j] [greater than or equal to] [k.sub.2] (Zone constraint for the Processor component)

6 [[summation].sub.j=1-3] [X.sub.2j] [greater than or equal to] [k.sub.3] (Zone constraint for the Hard Drive component) where [k.sub.i] is the smallest integer [greater than or equal to] [d.sub.j]/2; i = 1,2,3

7 [[summation].sub.i=0-18] [C.sub.i4] [X.sub.i4] + [[summation].subi=0-18] [C.sub.i5] [X.sub.i5] [less than or equal to] # 1,000 where $1,000 is the maximum available limit of purchases on credit.

The total budget of the anniversary sale includes, in addition to the least purchasing cost of components, the shipping, advertising and cash on delivery charges. Accordingly, it is given by

Total budget = [summation over (i)] [summation over (j)]Cij xij * + sc + ac + cod, where

[X.sup.*.sub.ij] = Optimum value of the decision variable [X.sub.ij];

sc = Shipping cost related to the optimum purchasing plan;

ac = Advertising cost for the anniversary sales; and

cod = cash on delivery charges related to the optimum purchasing plan.

An application of the above highlighted approach is illustrated next.

IMPLEMENTATION

In this section, the approach highlighted above is applied to the case of the computer store for the purpose of determining its 2003 anniversary sale budget. Exponential smoothing is first applied to forecast the number of desktop computers to be sold on this occasion. The model developed in the previous section is operationalized afterwards to determine the least purchasing cost of related required components. The addition of shipping, marketing and financial costs to that cost brings about the needed budget.

Forecasting the Number of Computers on Sale

Forecasting the number of computers to be sold is important as purchasing the required different components from vendors has to be made before the computers are put on sale. In this article, an extrapolation method of forecasting using historical data is employed to predict the number of computers to be sold on special sale. Exponential smoothing is appropriate for small businesses operating in simple and relatively stable environments. It is one of the most popular and cost effective of the extrapolation methods of forecasting (Armstrong & Brodie, 1999). Double Exponential smoothing was applied to weigh the more recent data heavily and smooth out erratic fluctuations. It is employed when there is an upward trend and no clear seasonal pattern in the data (Lapin & Whisler, 2002). The number of computers sold on anniversary sales since the establishment of the local store in 1993 is depicted in Table 2.

The mathematical representation of the double exponential smoothing forecasting procedure is given by the following expressions (see Lapin and Whisler 2002 for more details):

[F.sub.t+1] = [T.sub.t] + [b.sub.t]

[T.sub.t] = [alpha] [Y.sub.t] + (1- [alpha]) ([T.sub.t-1] + [b.sub.t-1])

[b.sub.t] = [gamma] ([T.sub.t] - [T.sub.t-1]) + (1- [gamma])[b.sub.t-1]

where

t is the current time period; [Y.sub.t] is the current actual value; [T.sub.t] is the smoothed value for period t; [F.sub.t-1] is the forecast for period t + 1; [b.sub.t] is the smoothed trend line slope; and [alpha] and [gamma] smoothing parameters

To initialize the procedure, we set [T.sub.2] = [Y.sub.1] and [b.sub.2] = [Y.sub.2] - [Y.sub.1]. Using MINITAB software package, the values of the smoothing parameters [alpha] = 0.894 and [gamma] = 0.175 were determined in such a way to minimize the Mean Square. Deviations (MSD) given by

MSD = [SIGMA][([Y.sub.t] - [F.sub.t]).sup.2]/n - m,

where n denotes the number of observations and m is the number of smoothing parameters. The actual and forecasted number of computers is shown in Figure 1.

[FIGURE 1 OMITTED]

The forecasting procedure brings about a value of 9.15 which is rounded down to 9 computers for the year 2003 anniversary sale to produce a conservative forecast.

Model Solution

The number of main components for each desktop computer assembled to be sold during the one-month 2003-anniversary sale is 18. These components together with their purchasing prices after taxes in U.S. dollars from five vendors are depicted in Table 3.

We observe from Table 3 that some of the elements of the cost matrix Cij shown above are not provided. This is attributed to the fact that certain branded components have to be either purchased from certain vendors or some components are unavailable with some vendors. The detailed pure integer-programming model shown in the Appendix has been solved using LINDO (Linear Interactive Discrete Optimizer) software. The package accommodates up to 2000 variables in total including a maximum of 200 integers and a maximum of 4000 constraints. An IBM--compatible personal computer that runs at 1.8 GHz. (Pentium 4 processor) with a 256 MB SDRAM shared memory was used to obtain the optimal solution in less than 2 seconds of CPU time after a total of 333 iterations. The optimal solution obtained is shown in Table 4.

Tables 3 and 4 indicate that not all the nine items of a certain component need to be purchased from the vendor that offers them at the least price. The related minimum purchasing cost for the anniversary sale is found to be $ 5605.46. Table 4 shows that the local store will continue to purchase components from each of the five potential vendors.

The dollar share of purchases, however, varies significantly among vendors with vendor 5 having the least share (3.57%) and vendor 3 having the maximum share (46.74%). The total budget related to the 2003 store anniversary sale is estimated at $6079.64. It includes a shipping cost, an advertising cost and cash on delivery charges estimated, based on past experience, at $250, $200 and $24 (3 x 8 = 24), respectively.

SUMMARY AND CONCLUSIONS

This paper introduces an approach to solve an anniversary sale problem for the first time in the literature. The problem is formulated as an integer linear program to select the best combination of suppliers from whom to purchase several forecasted types of components to assemble a computer system on the basis of a defined criterion, given different constraints. The approach when applied to the situation of a local computer store was found to be beneficial in (1) determining the least cost budget for the anniversary sale, (2) identifying the optimum procurement plan, and (3) enhancing the competitive edge of the store through the reduction of the selling price of assembled computers on sale. Our approach promotes the use of simple operations research tools supplemented by managerial judgment when it comes to solving problems related to small businesses.

The model developed for small business establishments runs very fast on the personal computer using commercially available software packages like LINDO. For the problem formulated in the Appendix incorporating 90 integer decision variables and 48 constraints, less than 2 seconds of CPU time were needed to arrive at the optimal solution. The approach introduced in this paper is realistic and generalizable in the sense that it can handle any type and number of assembled computer systems together with any number of available vendors while being adaptable to other small firms sharing this line of business with differing operating conditions. The modeling effort developed in this paper is exploratory, revealing possibilities for extensions in the future. One straightforward extension is introducing transportation costs explicitly in the model. This could be done simply by augmenting the objective function with a transportation cost term representing the different delivery options and their costs (Lapin & Whisler, 2002). Another extension is to treat some of the constraints as regular (system) constraints and treat others (e.g. purchases on credit) as goal constraints, and use goal programming to solve the problem. A goal program is a multiple objective model that allows a solution to satisfy some but not necessarily every goal of the decision maker (Ignizio, 1985). An additional challenging extension would be to consider cash on delivery (COD) charges explicitly in the objective function of the mathematical model through the incorporation of zero-one variables in the modeling effort (Lee, Moore & Taylor, 1990). In addition, in forecasting the number of assembled computers to be put on sale, the forecasting mechanism could be made more sophisticated through incorporating advertising in its mathematical structure (Stewart & Kamins, 2003).

Small businesses are started and made successful by entrepreneurs who share unique traits, such as the need for achievement, a willingness to take risks, a passion for business, and self confidence (Johnson, 1990). However, as business complexity grows, the need to switch to more formal planning and decision making based on systematic analysis of multifaceted business information also grows (Longnecker, Moore & Petty, 2000). Operational research tools are perfectly suited for helping such analyses, with associated payoffs that use small firms' scarce technical and managerial resources effectively and efficiently (Mehra & Satish, 1990; Alpar & Reeves, 1990). In this regard, this article serves as an eye opener to local stores to apply mathematical tools in day-to-day business and also for researchers to focus on small local stores' operations. The application of such models can boost profits to the local store and also benefit the customers with low price products.

This model is not limited to computer stores but can be applied to other businesses, which include assembled products such as mountain bikes/ race bikes and small scale electronic manufacturing units. Finally, assuming that the users of our model might be intimidated by the idea of computer-generated purchasing plan especially if they were inexperienced in using computers, the external user interface must be easy to use and control while still maintaining full model functionality. It is thus recommended that every effort should be made to make the operating system user friendly, compact and in the mean time cost effective (Brown & Mesak, 1992).

APPENDIX

Detailed Pure Integer Program

MIN 35 X11 + 33 X12 + 31 X13 + 28 X14 + 62 X21 + 58 X22 + 59 X23 + 59 X31 + 54.5 X32 + 60 X33 + 29 X41 + 28 X42 + 28 X43 + 29 X44 + 103 X51 + 100 X52 + 96 X53 + 92 X54 + 10 X61 + 9.5 X62 + 9 X63 + 6.99 X64 + 58 X71 + 56 X72 + 56 X73 + 43.88 X74 + 23 X81 + 23 X82 + 23 X83 + 19.88 X84 + 36 X91 + 35 X92 + 35 X93 + 33.88 X94 + 31 X101 + 28 X102 + 27 X103 + 17.88 X104 + 128 X111 + 125 X112 + 122 X113 + 117 X114 + 99.99 X115 + 17 X121 + 5.5 X122 + 6 X123 + 4.99 X124 + 10 X131 + 9.5 X132 + 8 X133 + 3.99 X134 + 6 X141 + 7 X142 + 6 X143 + 5 X144 + 6 X154 + 5 X161 + 6 X162 + 5 X163 + 2.99 X164 + 4 X171 + 5 X172 + 5 X173 + 2.99 X174 + 101 X181 + 85 X182 + 83 X183 + 94.75 X185

Subject to:

1. Constraints to meet the demand requirements of each component as forecasted:

1) X11 + X12 + X13 + X14 + X15 = 9 2) X21 + X22 + X23 + X24 + X25 = 9 3) X31 + X32 + X33 + X34 + X35 = 9 4) X41 + X42 + X43 + X44 + X45 = 9 5) X51 + X52 + X53 + X54 + X55 = 9 6) X61 + X62 + X63 + X64 + X65 = 9 7) X71 + X72 + X73 + X74 + X75 = 9 8) X81 + X82 + X83 + X84 + X85 = 9 9) X91 + X92 + X93 + X94 + X95 = 9 10) X101 + X102 + X103 + X104 + X105 = 9 11) X111 + X112 + X113 + X114 + X115 = 9 12) X121 + X122 + X123 + X124 + X125 = 9 13) X131 + X132 + X133 + X134 + X135 = 9 14) X141 + X142 + X143 + X144 + X145 = 9 15) X151 + X152 + X153 + X154 + X155 = 9 16) X161 + X162 + X163 + X164 + X165 = 9 17) X171 + X172 + X173 + X174 + X175 = 9 18) X181 + X182 + X183 + X184 + X185 = 9

2. Constraints to restrict the minimum purchases from vendors 3 and 4:

19) 31 X13 + 59 X23 + 60 X33 + 28 X43 + 96 X53 + 9 X63 + 56 X73 + 23 X83 + 35 X93 + 27 X103 + 122 X113 + 6 X123 + 8 X133 + 6 X143 + 5 X163 + 5 X173 + 83 X183 >= 300

20) 28 X14 + 29 X44 + 92 X54 + 6.99 X64 + 43.88 X74 + 19.88 X84 + 33.88 X94 + 17.88 X104 + 117 X114 + 4.99 X124 + 3.99 X134 + 5 X144 + 6 X154 + 2.99 X164 + 2.99 X174 >= 100 96

3. Constraints to purchase certain branded parts from a specific vendor (vendor 1) and to restrict purchases of certain items from vendors due to their unavailability:

21) X15 = 0 22) X24 = 0 23) X25 = 0 24) X34 = 0 25) X35 = 0 26) X44 = 0 27) X45 = 0 28) X55 = 0 29) X65 = 0 30) X75 = 0 31) X85 = 0 32) X95 = 0 33) X105 = 0 34) X125 = 0 35) X135 = 0 36) X145 = 0 37) X151 = 0 38) X152 = 0 39) X153 = 0 40) X155 = 0 41) X165 = 0 42) X175 = 0 43) X184 = 0 44) X11 = 4

4. Constraints to meet the zone criteria of the shipment policy:

45) X11 + X12 + X13 >= 5 46) X21 + X22 + X23 >= 5 47) X51 + X52 + X53 >= 5

5. A constraint to satisfy the budget available for purchases on credit:

48) 28 X14 + 29 X44 + 92 X54 + 6.99 X64 + 43.88 X74 + 19.88 X84 + 33.88 X94 + 17.88 X104 + 117 X114 + 4.99 X124 + 3.99 X134 + 5X144 + 6 X154 + 2.99 X164 + 2.99 X174 + 99.99 X115 + 94.75 X185 <= 1000 END

REFERENCES

Ahire, Sanjay L. (2001). Role of management science techniques in operations improvement in small firms. Production and Inventory Management Journal, 42 (2), 14-21.

Alpar, Paul & Sharon Reeves (1990). Predictors of MS/OR applications in small businesses. Interfaces, 20, 2-11. Armstrong, J. Scott & Roderick J. Brodie (1999). Forecasting for Marketing, Quantitative Methods in Marketing (Second Edition). London: International Thompson Business Press.

ATA Awarded Loan Banking. Aviation Week and Space Technology, 49, 2002.

Baker, Stephen (2002, November). Slow desktop PC sales drag down, NPD Techworld.

Brown, Ann (2000, March). If the price tag fits..... Black Enterprise, 8.

Brown, K.L. & H.I. Mesak (1992). Scheduling professionals in the retail pharmacy chain. OMEGA: International Journal of Management Science, 20, 671-678.

Burton, Thomas M. (2000, September). Nordstrom's chief resigns amid shift in management. Wall Street Journal (Eastern Edition), B6.

Current, John & Charles Weber (1994). Application of facility location modeling constructs to vendor selection problems. European Journal of Operational Research, 76, 387-392.

Degraeve, Zeger, Eva Labro & Filip Roodhooft (2000). An evaluation of vendor selection models from a total cost of ownership perspective. European Journal of Operational Research, 125, 34- 59.

Degraeve, Zeger & Filip Roodhoft (2000). A mathematical programming approach for procurement using activity based costing. Journal of Business, Finance and Accounting, 27: (2), 69-98.

Dickson, G.W. (1966). An analysis of vendor selection systems and decisions. Journal of Purchasing, 2, 5-17.

Greenburg, Karl (June 30--July 7 2003). Frigidaire warms up for 85th, Brandweek, 14.

Ignizio, J.P. (1985). Introduction to Linear Goal Programming, Sage Paper Series on Quantitative Applications in the Social Sciences, Series No. 56. Beverly Hills, CA, Sage Publications.

Jayaraman, Vaidyanathan, Rajesh Srivastava & W.C. Benton (1999). Supplier selection and order quantity: A comprehensive model. J of Supply Chain Mgmt, 35, 50-58.

Johnson, B.Y. (1990). Toward a multidimensional model of entrepreneurship: The case of achievement motivation and the entrepreneur. Entrepreneurship theory and Practice, 14, 39-54.

Lapin & Whisler (2002). Quantitative Decision Making with Spreadsheet Applications (Seventh Edition). Belmont, CA: Duxbury Press.

Lee, Sang M., Lawrence J. Moore & Bernard W. Taylor, III (1990). Management Science (Third Edition). Needham Heights, MA: Allyn and Bacon.

Longenecker, J.G., C.W. Moore & J.W. Petty (2000). Small business management: An entrepreneural emphasis (Third Edition). Cincinnati, Ohio: South Western College Publication.

Mainelli, Tom (2001). It's a buyer's market: PC prices plummet, PC World.

Mehra & Satish (1990). Applying MS/OR techniques to small businesses. Interfaces, 20, 38-41.

MINITAB User's Guide for Data Analysis and Quality Tools (Release 12, 1998). Minitab Inc..

Ray, Tiernan (2003, March). The black hole of white-box workstations, ECT News Network.

Rosenthal, Edward C., James L. Zydiak & Sohail S. Chaudhry (1995). Vendor selection with bundling. Decision Sciences, 26, 35-48.

Shifrin, Carole A. (1997). Reno air comes of age. Aviation Week and Space Technology, 48-53.

Stewart, David W. & Michael A. Kamins, Marketing communications (2003). In: Weitz, Barton A. & Robin Wensley (Eds.), Handbook of Marketing. Thousand Oaks, California: Sage Publications, 282-309.

Van Buer, Michael G., Enrique R. Venta & James L. Zydiak (1997). Heuristic approaches to purchasing bundles of multiple products from multiple vendors. Journal of Business Logistics, 18, 125-140.

Weber, C.A. & J.R. Current (1993). A multi-objective analysis to vendor selection. European Journal of Operational Research, 68(2), 173-184.

Wolf, Alan E. (1999, February). P'tex bows Cannon ad campaign, Home Textiles Today, 4.

Jinson J. Erinjeri, Louisiana Tech University Hani I. Mesak, Louisiana Tech University Jun-Ing Ker, Louisiana Tech University

Table 1
Shipping and payment highlights for Vendors

 Purchases Shipping
Vendor COD on Credit Mode Zone (State)

1 Allowed Possible UPS 7 (Texas)
2 Allowed Possible UPS 7 (Texas)
3 Allowed Possible UPS 7 (Illinois)
4 Not Allowed Possible UPS 5 (California)
5 Not Allowed Possible No 8 (Louisiana)

TABLE 2
Number of Computers Sold Since Store Establishment

Year Number of Computers

1994 5
1995 7
1996 11
1997 9
1998 10
1999 9
2000 8
2001 8
2002 9

TABLE 3
Cost Matrix of Various Components from Different Vendors in Dollars

 Vendors: Vendor 1 Vendor 2 Vendor 3

 Components:

1 Case 300 Watts C11 = 35 C12 = 33 C13 = 31
2 Intel Celeron Processor C21 = 62 C22 = 58 C23 = 59
 1.7 Ghz
3 Motherboard C31 = 59 C32 = 54.50 C33 = 60
4 Memory 256MB SDRAM C41 = 29 C42 = 28 C43 = 28
5 Hard Drive 80GB C51 = 103 C52 = 100 C53 = 96
6 Floppy Disk Drive C61 = 10 C62 = 9.50 C63 = 9
7 CD-RW 48X C71 = 58 C72 = 56 C73 = 56
8 CD ROM 56X C81 = 23 C82 = 23 C83 = 23
9 DVD ROM 16X C91 = 36 C92 = 35 C93 = 35
10 Video Card 32 MB C101 = 31 C102 = 28 C103 = 27
11 Monitor 17" 0.23 C111 = 128 C112 = 125 C113 = 122
 dotpitch
12 Modem 56K C121 = 17 C122 = 5.50 C123 = 6.00
13 Ethernet card C131 =10 C132 = 9.50 C133 = 8
14 Sound card C141 = 6 C142 = 7 C143 = 6
15 3-piece Subwoofer - - -
 Speakers
16 Keyboard C161 = 5 C162 = 6 C163 = 5
17 Mouse C171 = 4 C172 = 5 C173 = 5
18 Windows XP C181 = 101 C182 = 85 C183 = 83

 Vendors: Vendor 4 Vendor 5

 Components:

1 Case 300 Watts C14 = 28 -
2 Intel Celeron Processor - -
 1.7 Ghz
3 Motherboard - -
4 Memory 256MB SDRAM - -
5 Hard Drive 80GB C54 = 92 -
6 Floppy Disk Drive C64 = 6.99 -
7 CD-RW 48X C74 = 43.88 -
8 CD ROM 56X C84 = 19.88 -
9 DVD ROM 16X C94 = 33.88 -
10 Video Card 32 MB C104 = 17.88 -
11 Monitor 17" 0.23 C114 = 117 C115 = 99.99
 dotpitch
12 Modem 56K C124 = 4.99 -
13 Ethernet card C134 = 3.99 -
14 Sound card C144 = 5 -
15 3-piece Subwoofer C154 = 6 -
 Speakers
16 Keyboard C164 = 2.99 -
17 Mouse C174 = 2.99 -
18 Windows XP - C185 = 94.75

TABLE 4
Optimum Component Selection

 Components Vendor 1 Vendor 2 Vendor 3

1 Case 300 Watts X11 = 4 X12 = 0 X13 = 5
2 Intel Celeron Processor 1.7 Ghz X21 = 0 X22 = 9 X23 = 0
3 Motherboard X31 = 0 X32 = 9 X33 = 0
4 Memory 256MB SDRAM X41 = 0 X42 = 9 X43 = 0
5 Hard Drive 80GB X51 = 0 X52 = 0 X53 = 9
6 Floppy Disk Drive X61 = 0 X62 = 0 X63 = 0
7 CD-RW 48X X71 = 0 X72 = 0 X73 = 0
8 CD ROM 56X X81 = 9 X82 = 0 X83 = 0
9 DVD ROM 16X X91 = 0 X92 = 9 X93 = 0
10 Video Card 32 MB X101 = 0 X102 = 0 X103 = 0
11 Monitor 17" 0.23 dotpitch X111 = 0 X112 = 0 X113 = 7
12 Modem 56K X121 = 0 X122 = 9 X123 = 0
13 Ethernet card X131 = 0 X132 = 0 X133 = 0
14 Sound card X141 = 2 X142 = 0 X143 = 0
15 3-piece Subwoofer Speakers X151 = 0 X152 = 0 X153 = 0
16 Keyboard X161 = 0 X162 = 0 X163 = 0
17 Mouse X171 = 0 X172 = 0 X173 = 0
18 Windows XP X181 = 0 X182 = 0 X183 = 9
 Purchase cost in U.S. dollars 359.00 1629.00 2620.00

 Components Vendor 4 Vendor 5

1 Case 300 Watts X14 = 5 X15 = 0
2 Intel Celeron Processor 1.7 Ghz X24 = 0 X25 = 0
3 Motherboard X34 = 0 X35 = 0
4 Memory 256MB SDRAM X44 = 0 X45 = 0
5 Hard Drive 80GB X54 = 0 X55 = 0
6 Floppy Disk Drive X64 = 9 X65 = 0
7 CD-RW 48X X74 = 9 X75 = 0
8 CD ROM 56X X84 = 9 X85 = 0
9 DVD ROM 16X X94 = 0 X95 = 0
10 Video Card 32 MB X104 = 9 X105 = 0
11 Monitor 17" 0.23 dotpitch X114 = 0 X115 = 2
12 Modem 56K X124 = 0 X125 = 0
13 Ethernet card X134 = 9 X135 = 0
14 Sound card X144 = 7 X145 = 0
15 3-piece Subwoofer Speakers X154 = 9 X155 = 0
16 Keyboard X164 = 9 X165 = 0
17 Mouse X174 = 9 X175 = 0
18 Windows XP X184 = 0 X185 = 0
 Purchase cost in U.S. dollars 797.48 199.98